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Modeling

We applied modeling and quantitative methods in all aspects of our project, from micro scope to a macro scope, from mechanics to applications.

First, we designed a model to predict how the bi-regulation switch - arabinose and IPTG may influence the concentration of effective dCas9-sgRNA complex in a bacterial cell. Second, we designed a model to explain how our system can slow down bacteria's replication. Third, we designed a model to explain how our system can control plasmid copy number. Fourth, we modeled the elongation of our cell. Fifth, we applied a model to explain how our bacteria can improve the productivity of some specific bioproducts. Sixth, we designed a model to explain how our system, coupled with a quorum sensing system - can automatcally regulate the population. Finally, corresponding to our future plan, we designed a model to discuss our system's potential of reducing the intracellular gene expression noise.

Our team emphasizes the importance of quantitative methods and mathematical description of our project.

regulatory part

Introduction

In this section, we apply a deterministic model to predict the behavior of a regulatory part in our project.

Originally, we did not include the regulation of sgRNA. In this case, the sgRNA is constantly transcripted (By a J23119 promoter, see Project Description) However, we found some works reporting that dCas9 itself can change the cell's growth rate, thus we have to explain that our system works in our exprected ways. Aditionally, we know that the promoters, even if regulated by an inducer or an inhibitor, may perfrom a "leakage expression", hence we have to apply some other ways to gain a better control of the system. Both of these motivated us to design a second switch for sgRNA.

Our regulatory part includes an IPTG-activated pLac-promoted sgRNA and an Arabinose-activated pBAD-promoted dCas9. The mechanics are described in Pic 1. The formation of the effective dCas9-sgRNA complex depends on both IPTG and Arabinose inducers.

Mono-regulation

We started from the simpler case: when there is only one regulated variable, e.g. dCas9 regulated by Arabinose. Since the expression of dCas9 is promoted by pBAD promoter and regulated by Arabinose, it is reasonable to describe this process by Hill's equation

v = \frac{v_{\max }\cdot[\mathrm{Ara}]^n}{[\mathrm{Ara}]^n + K^n}

$v$ $n$ $K$ $v = v_{\max }/2$ .

Bi-regulation

We introduced the regulation of expresion of sgRNA by changing the promoter of sgRNA from J23119 to T7. This T7 promoter's activation is further regulated by a pTAC promoter. Therefore, we can further get another Hill's function for this promoter. Besides the transcription of sgRNA and dCas9 mRNA, the translation of dCas9, the binding of dCas9 and sgRNA and the degradation of all the mentioned matters are to be considered. Here, we describe this system by an ordinary differential equation system:

\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}m &amp;= \frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}} - \gamma_m m \\ \frac{\mathrm{d}}{\mathrm{d}t}c &amp;= \beta_c m - \gamma_c c + k_-b - k_+cr\\ \frac{\mathrm{d}}{\mathrm{d}t}r &amp;= \frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} - \gamma_r r + k_- b - k_+ c r\\ \frac{\mathrm{d}}{\mathrm{d}t}b &amp;= -k_- b + k_+ c r - \gamma_b b \end{align}

$m$ $c$ $r$ $b$ represents the concentration of dCas9-sgRNA concentration. The parameters are determined from both others' works and our experiments and listed as follows

denotion	explanation	value
$n_m$	Hill's coefficient of pBAD promoter
$K_m$	Half-activate Arabinose concentration of pBAD promoter
$\alpha_m$	Maximum dCas9 mRNA expression rate	0.0011 nM·min^-1
$n_r$	Hill's coefficient of pTac promoter
$K_r$	Half-activate Arabinose concentration of pTac promoter
$\alpha_r$	Maximum sgRNA expression rate	0.0011 nM·min^-1
$\beta_c$	dCas9 protein synthesis rate from dCas9 mRNA	0.0057 protein·transcript^-1·min^-1
$\gamma_c$	dCas9 degradation rate	5.6408×10^-4min^-1
$k_+$	Rate of dimerization of dCas9 and sgRNA
$k_-$	Rate of dissociation of dCas9-sgRNA complex
$\gamma_r$	sgRNA degradation rate
$\gamma_b$	dCas9-sgRNA degradation rate	5.6408×10^-4min^-1

To study the steady state of the system, we solve the equation when all particles' concentration do not vary with time (e.g. the left-hand side of the ODE systems equal to zero)

\begin{align} \frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}} - \gamma_m m &amp; = 0 \\ \beta_c m - \gamma_c c + k_-b - k_+cr &amp; = 0\\ \frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} - \gamma_r r + k_- b - k_+ c r &amp; = 0\\ -k_- b + k_+ c r - \gamma_b b &amp; = 0 \end{align}

$b$ ).

from the system above, we can yield

\frac{k_+ \gamma_b^2}{\gamma_c\gamma_r}b^2 - (k_- + \gamma_b + \frac{k_+ \gamma_b \alpha_r^*}{\gamma_c\gamma_r} + \frac{k_+ \gamma_b \alpha_r^*\beta_c}{\gamma_c\gamma_r\gamma_m})b +\frac{k_+ \alpha_m^* \alpha_r^*\beta_c}{\gamma_c\gamma_r\gamma_m} = 0

where

\alpha_r^* = \frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} \\ \alpha_m^* = \frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}}

$\gamma_m \rightarrow +\infty$ , the smaller solution is 0 and the bigger one is a positive real number

\frac{\gamma_c \gamma_r k_- + \gamma_c \gamma_b \gamma_r + k_+ \gamma_b \alpha_r}{k_+ \gamma_b^2}

$\gamma_m$ is extremely big, mRNA degrades so frequently that there is almost no dCas9 mRNA, hence no dCas9, and subsequentially no dCas9-sgRNA complex, so the zero solution is the one corresponding to the real biological situation)

Thus, we get the final solution

b = \frac{B - \sqrt{B^2 - 4AC}}{2A}

where

\begin{align} A &amp;= \frac{k_+ \gamma_b^2}{\gamma_c\gamma_r}b^2 \\ B &amp;= k_- + \gamma_b + \frac{k_+ \gamma_b}{\gamma_c\gamma_r}\frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} + \frac{k_+ \gamma_b \beta_c}{\gamma_c\gamma_r\gamma_m}\frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} \\ C &amp;= \frac{k_+ \beta_c}{\gamma_c\gamma_r\gamma_m}\frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}}\frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}} \end{align}

This equation gives the relation between the concentration of dCas9-sgRNA and the amount of inducer we add. Applying the given parameters, we can demonstrate this relationship by a 3D surface plot. Instinctively , the expression output follows the AND logic: the concentration of dCas9-sgRNA is only considerable when the concentration of both IPTG and Arabinose are high enough.

The relationship between the expression of dCas9 and the cell's growth rate

In this section we show that our system delays the bacteria's replication. Specifically, the average time of a bacteria's cell cycle depends linearly on the concentration of the dCas9-sgRNA molecule copy number. This is deduced from a coarse-grained model including the process of dCas9 and DnaA binding to the OriC site and the cell's replication. Here we assume that

The bingding process of both binding of dCas9 and DnaA are reversible
Genome DNA replication can start only when the OriC is bound to DnaA

A Markov chain on continuous time is built, with the hypothesis that the replication time is subjected to a Poisson distribution. A cell has three possible states at a certain time

OriC is bound to a dCas9 molecule
OriC is naked
The bacteria start replication

$k_1$ $k_2$ $\kappa$ ). A infinitesimal transition matrix is written to describe the transition among these states:

{\bf A} = \begin{pmatrix} -k_1-\kappa &amp; k_1 &amp; \kappa \\ k_2 &amp; -k_2 &amp; 0 \\ 0 &amp; 0 &amp; 0\end{pmatrix}

where

$k_1$ $\rightarrow$ $c$ ):

k_1 = k_1^*c

$k_2$ $\rightarrow$ state 1).
$\kappa$ $\rightarrow$ state 3)

$(1,0, 0)$ $F$ of the waiting time before replication. To do this, we only have to calculate

{\bf F}(t) = \begin{pmatrix} 1 &amp; 0 &amp; 0 \end{pmatrix} \times {\rm e}^{{\bf A}t}

${\bf F}(t)$ $F(t)$ we want.

$F(t)$ :

F(t) = 1 - \frac{\lambda_2(\lambda_1 + k_2){\rm e}^{\lambda_1t} - \lambda_1(\lambda_2 + k_2){\rm e}^{\lambda_2t}}{k_2(\lambda_2 - \lambda_1)}

where

\begin{align*} \lambda_1 = \frac{-\kappa -k_1 - k_2 - \sqrt{(\kappa + k_1 + k_2)^2 - 4\kappa k_2}}2\\ \lambda_2 = \frac{-\kappa -k_1 - k_2 + \sqrt{(\kappa + k_1 + k_2)^2 - 4\kappa k_2}}2 \end{align*}

The average waiting time is:

{\mathbb E}(T) = \int_0^{+\infty}tF\ '(t){\rm d}t = \frac{k_1 + k_2}{k_2\kappa} = \frac{k_1^*c + k_2}{k_2\kappa}

linearly related to the concentration of dCas9-sgRNA.

Plasmid Copy number Hacking

Introduction

In this section we discuss how our system can control the plasmid copy number (see Project Description). Typically the plasmids replicate themselves during the cell cycle. After a cell's division, the plasmids are equivalently distributed to both of the cells. Here we discuss a cell line's behavior over time.

plasmid_cell_line

the simulation process

In an unbiased cell division, the plasmids in a cell are distributed equivalently to both of the children cells. Therefore, while tracing a cell line, we see that a cell lose half of its plasmids after each division. It is reasonable to assume this process as a random process in which each plasmid has 1/2 probability to "disappear" (actually entering its sibling cell), and 1/2 probability to remain in the cell line we are interested in.

$k_N$ $k_C$ , the frequency that a plasmid and dCas9 decouple.

Another factor taken into account is the process that the replication of cells are also regulated by the intracellular environment to ensure that the plasmid number do not increase uncontrolled. Hence we assume that during a cell's cycle, the growth of the cell's plasmid number is a logistic process. Precisely, the frequency that plasmid replicate itself is

K(n_N, n_C) = \kappa\cdot n_N\cdot\left(1 - \frac{n_N + n_C}{n_\max}\right)

$n_N$ $n_C$ $n_\max$ $\kappa$ is a constant replicate rate.

\frac{\mathrm{d}}{\mathrm{d}t}p_{N, C} = k_C\cdot p_{N+1, C-1} + k_N\cdot p_{N-1, C+1} + K(n_{N-1}, n_C)\cdot p_{N-1, C}-(k_C + k_N + K(n_N, n_C))\cdot p_{N, C}

$T$ $\lambda = T^{-1}$ $nT\ (n\in \mathbb{N^+})$ , the cell's plasmid number undergo a rappid change

\begin{cases} n_N \rightarrow n_N'\\ n_C \rightarrow n_C' \end{cases}

where

n_N' \sim \mathcal{B}(n_N, \frac{1}{2}), \ \ \ n_C' \sim \mathcal{B}(n_C, \frac{1}{2})

$k_C$ $k_C$ $k_C$ increase, the ratio of dCas9 bound to sgRNA tend to be greater and the cell line run out of the plasmid in less generations. (see pic plasmid_time_series, plasmid_clearance_generation.png)

Gene Expression Noise Control

In this section we model to illustrate how our system can control the expression noise in a cell. Gene expression noise is explained as fluctuation of "very low copy numbers of many components" leading to "large amounts of cell-cell variation observed in isogenic populations". This noise can be either intrinsic or extrinsic. Extrinsic noise include most of the environmental factors like the nutrient and antibiotics, gene expression regulation by inhibitor or enhancer, and also the gene copy number in the cell.

In a fast growing cell, the copy number of OriC can be more than ten. In these cases, genes near the OriC may express more than genes far from OriC. Ting Lu deduced from Helmstetter-Copper model a relation between the gene's relative location to OriC and gene's copy number:

g_{q}(\lambda)=g_{0, q} \exp \left(\lambda\left[\left(1-x_{q}\right) C+D\right]\right)

$g_q$ $\lambda$ $x_q$ $k$ $T$ $kT$ .

$k$ , the frequency of replication fork formation.

Hypothesis

Each cell contains only one set of genome, which may contain multiple replication forks so that each gene's copy number in the genome may be different. The cell divide immediately after genome replication. (Our experiment results show that some cells actually contains more than one set of genome. These cells are very long and in each cell genomes are distant from each other. We treat this kind of cell as a chain of multiple cells)
The cell containing the studied genome has been replicating exponentially for several generations in a stable environment, thus the replication fork's distribution on the genome is steady (i.e. sampled from a fixed distribution)
The replication fork only forms at OriC and the formations of replication forks are independent with each other.
$k$ $T$ are constant.
$k$ . This process is a fast grade 1 process.

Analysis of gene copy number and its fluctuation

$T$ $T$ $T$ Yule-Furry process $k$ $m$ $t\ (0\leq t \leq T)$ is:

p_m(t) = \mathrm{Pr}[N(t) = m] = \mathrm{e}^{-k t}(1 - \mathrm{e}^{-k t})^{m - 1}

$n_0=1$ )

$t$ $m$ $T - t$ $T - t$ $(T - t)/T$ $T$ $(T - t)/T$ $t/T$ $m$ $t$ .

noise_ctrl_mech1

$x=t/T$ $0\leq x \leq 1$ $x = 0$ $x = 1$ corresponding to OriC), then we get

p_m(x) = \mathrm{e}^{-k T x}(1 - \mathrm{e}^{-k T x})^{m - 1}

This equation determines the copy number distribution of all the genes at all the sites. We can further deduce

\mathbb{E}[N(x)] = \mathrm{e}^{k Tx}

and

\mathbb{Var}[N(x)] = \mathrm{e}^{2k Tx} - \mathrm{e}^{k Tx}

$x$ $\mathbb{E}[N(x)]$ $\mathbb{Var}[N(x)]$ $\mathbb{Var}[N(0)]=0$ $\mathbb{E}[N(0)] = 1$ corresponds to the model hypothesis that the copy number of Ter is invariantly 1.

Variance itself is not sufficient enough to indicate the intensity of fluctuation because the average expression is also increasing. We apply CV value, the ratio of standard deviation to the average, to more precisely describe the intensity of fluctuation:

\mathrm{CV}[N(x)] = \frac{\sqrt{\mathbb{Var}N(x)}}{\mathbb{E}N(x)} = \sqrt{1-\mathrm{e}^{-k T x}}

$\mathrm{CV}[N(0)]=0$ $\mathrm{CV}[N(x)]$ $x$ , indicating that the copy number of genes near OriC in sequence tend to vary more than that of genes far from OriC in sequence.

Analysis of dCas9 binding process

k = \frac{K_m\cdot k_0}{K_m+[\mathrm{dCas9}]}

Productivity

In this section we provide a model to explain why our system is able to increase the cells' productivity to multiple types of products like GFP and indigo. It is somehow contrasting our intuition that a decline in microbiomes' growth rate may increase the microbial productivity. Therefore, it is necessary to carefully test this issue in a quantitive method.

The undesired productivity of microbiomes is thwarting this technology from being more widely applicable. Therefore, it is always concerned how to improve the productivity of microbiomes, or how to enforce these wild or engineered cells to turn more of the substrates we feed them into the bio-products we want.

A non-negligible factor thwarding the bicrobiomes from producing our bioproduct is the growth of the bicrobiomes themselve. All the cell products we want to yield from the microbiomes are products of the microbiomes' own metabolism forming the microbiomes' own biomass. Therefore, their production is under strict regulation of the microbiomes. In most engineered microbiomes, the genes we introduce to the microbiomes are foreign genes unrelated to the cells' growth and normal metabolism, and the production of these genes are undesired for the microbiomes themselves. The engineered microbiomes, stressed by these products, will activate all the possible pathways to change its gene expression pattern and to reallocate its nutrients and enzymes to produce more "necessities" for themselves. This is particularly true for bicrobiomes under steady growth rate, whose nutrients are mainly used for growth.

Actually, many studies concerns the relationship between microbiomes' growth and the way they allocate their resources. Terence Hwa et.al studied E. coli's metabolism under different nutrient and antibiotic condition by dividing the cells' proteins into different sectors. Particularly in this work, a sector defined as "Unnecessary Expression"(corresponding to the bio-products we want) is found to be negatively related to the growth rate. The authors described this relation semi-quantitively as

\lambda(\phi_U) = \lambda(\phi_U = 0)\left(1 - \frac{\phi_U}{\phi_c}\right)

$\phi_U$ $\lambda$ $\phi_U$ . Ting Lu et. al built a coarse-grained whole-cell model including the productions and functions of proteins of different sectors and further validated Hwa's model. In this model, the regulation performed by ppGpp is included and this provides an explanation of the phenomenological result in Hwa's work. ppGpp is an important gene expression regulation molecule responding to various types of environmental stress and cell's abnormal pnysiology. One of its function is to down-regulate the expression of ribosomal RNA, protein and expression affiliated proteins while up-regulate the expression of some enzymes relating to the cell's core metabolism, adapting the cell from a fast-growing state to a slow-growing state. Some optimization models not explicitly including ppGpp also discovers a similarity between cells' optimal resource allocation strategies and the ppGpp regulation strategies.

All the works mentioned above regard the growth rate as the result or equivalence of the microbiomes' protein accumulation. In our system, however, the growth rate is hacked. Because of this, the causal relationship between the cell physiology (growth rate) and the nutrient reallocation is different from these works. A better explanation of the mechanics of our system is that the hacked growth rate freed the cells from producing too much growth-necessary proteins and enabled them to produce the bio-product we want. We emphasize that our model is still a phenominological model in which the resource allocation regulation is finished in a "black box", not explicitly related to the regulation of any single pathway.

$\alpha$ $1 - \alpha$ $\alpha=0$ $\alpha=1$ corresponds to a "growth-only" state. This process is written as an ODE system

\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}n &amp;= -\frac{k_n n}{n+K_n}m\\ \frac{\mathrm{d}}{\mathrm{d}t}m &amp;= \frac{k_m n}{n+K_n}m\alpha\\ \frac{\mathrm{d}}{\mathrm{d}t}p &amp;= \frac{k_p n}{n+K_n}m(1-\alpha)\\ \end{align}

$n$ $m$ $p$ $k_n$ $k_m$ $k_p$ $K_n$ $\alpha$ is the above-mentioned allocation ratio.

$\alpha$ , when the nutrient is sufficient, the second ODE can be re-written as

\frac{\mathrm{d}}{\mathrm{d}t}m = k_m\alpha\cdot m

$\lambda = k_m\alpha$ .

\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}n &amp;= -\frac{k_n n}{n+K_n}m\\ \frac{\mathrm{d}}{\mathrm{d}t}m &amp;= \frac{n\lambda}{n+K_n}m\\ \frac{\mathrm{d}}{\mathrm{d}t}p &amp;= \frac{k_p n}{n+K_n}\left(1-\frac{\lambda}{k_m}\right)m\\ \end{align}

the equation explicitly includes the growth rate.

The ODE is difficult to solve but we can analyze the steady states about it.

$m(0) = m_0, n(0) = n_0, p(0) = p_0$ , noticing that

\frac{1}{k_n}\frac{\mathrm{d}}{\mathrm{d}t}n + \frac{1}{\lambda}\frac{\mathrm{d}}{\mathrm{d}t}m = \frac{1}{k_n}\frac{\mathrm{d}}{\mathrm{d}t}n + \frac{k_m}{k_p(k_m-\lambda)}\frac{\mathrm{d}}{\mathrm{d}t}p = 0

$\displaystyle \frac{n}{k_n} + \frac{m}{\lambda}$ $\displaystyle \frac{n}{k_n} + \frac{k_m p}{k_p(k_m-\lambda)}$ remain constant in the whole dynamic process. Provided the initial condition, we can deduce

\begin{cases} \displaystyle \frac{n}{k_n} + \frac{m}{\lambda} = \frac{n_0}{k_n} + \frac{m_0}{\lambda} \\ \displaystyle \frac{n}{k_n} + \frac{k_m p}{k_p(k_m-\lambda)} = \frac{n_0}{k_n} + \frac{k_m p_0}{k_p(k_m-\lambda)} \end{cases}

$n^{\mathrm{ss}}, m^{\mathrm{ss}}$ $p^{\mathrm{ss}}$ , the above-mentioned equation still holds:

\begin{cases} \displaystyle \frac{n^{\mathrm{ss}}}{k_n} + \frac{m^{\mathrm{ss}}}{\lambda} = \frac{n_0}{k_n} + \frac{m_0}{\lambda} \\ \displaystyle \frac{n^{\mathrm{ss}}}{k_n} + \frac{k_m p^{\mathrm{ss}}}{k_p(k_m-\lambda)} = \frac{n_0}{k_n} + \frac{k_m p_0}{k_p(k_m-\lambda)} \end{cases}

$n^{\mathrm{ss}}=0$ . Then we can solve

\begin{cases} \displaystyle m^{\mathrm{ss}} = m_0 + \frac{\lambda n_0}{k_n} \\ \displaystyle p^{\mathrm{ss}} = p_0 + \frac{n_0k_p}{k_n}(1 - \frac{\lambda}{k_m}) \end{cases}

$p^{\mathrm{ss}}$ is negatively and linearly related to the growth rate. This result is in consistent to the empirical equation provided by Terence Hwa.

Quorum Sensing

In this section we model to illustrate how our system works coupling with a quorum sensing system. We want out system to realize the population's auto-regulation - the cells stop growing fast when they sense a lot of other cells crowding around it. We realize this by applying LuxI-LuxR system, in which a kind of small molecule call AHL. This system was firstly discovered in V. fischeri as a means of intercellular communication. The cells produce exceeding amount of AHL which can move either into or out of the cells. When the population is large or dense in a region, the local AHL concentration increases and the cells sense this high concentration and respond to this by up- or down-regulating some genes' expression. In our system, the sensing of AHL results in an increment of dCas9, and consequently a down regulation in growth or replication rate (see design for the design and experiments)

A Simulation of Donor-Receptor Experiment

Our first series of experiments involve the testing of our fore-mentioned logic. We separately introduced the AHL producing parts and the AHL sensing parts to two strains of E. coli. The former is called "donor" and the latter is called "receptor". The receptor cells are evenly coated onto the solid medium and the donor cells were dropped at the center of the medium. It is expected that the donors AHL, the AHL difusses around and inhibits the growth of the receptors around it (see design)

$(0, 0)$ $(0, 0)$ . A partial differential equation (PDE) can be derived from these hypothethes:

\frac{\partial u}{\partial t} = D(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})u + C\mathrm{e}^{-\frac{x^2 + y^2}{\sigma^2}}

$x, y$ $\sigma$ $C$ $D$ [=2.0μM/(L·cm²)] denotes the diffusion constant. The randomly distributed cells stop growing when the local concentration reach 10^-6μM/L.

Here we numerically solve the equation with finite difference method. We draw an animation visualize this dynamic process. Furthermore, we randomly place 400 receptors on the plane to show their colony formation. It can be seen that the receptors distant from the center grow into colonies, while most of the receptors nearby the center grow into relatively small colonies or cannot grow into colonies.

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+<div  id='write'  class = 'is-mac'><h1><a name="modeling" class="md-header-anchor"></a><span>Modeling</span></h1><p><span>We applied modeling and quantitative methods in all aspects of our project, from micro scope to a macro scope, from mechanics to applications. </span></p><p><span>First, we designed a model to predict how the bi-regulation switch - arabinose and IPTG may influence the concentration of effective dCas9-sgRNA complex in a bacterial cell. Second, we designed a model to explain how our system can slow down bacteria&#39;s replication. Third, we designed a model to explain how our system can control plasmid copy number. Fourth, we modeled the  elongation of our cell. Fifth, we applied a model to explain how our bacteria can improve the productivity of some specific bioproducts. Sixth, we designed a model to explain how our system, coupled with a quorum sensing system - can automatcally regulate the population. Finally, corresponding to our future plan, we designed a model to discuss our system&#39;s potential of reducing the intracellular gene expression noise.</span></p><p><span>Our team emphasizes the importance of quantitative methods and mathematical description of our project.</span></p><h2><a name="regulatory-part" class="md-header-anchor"></a><span>regulatory part</span></h2><h3><a name="introduction" class="md-header-anchor"></a><span>Introduction</span></h3><p><span>In this section, we apply a deterministic model to predict the behavior of a regulatory part in our project. </span></p><p><span>Originally, we did not include the regulation of sgRNA. In this case, the sgRNA is constantly transcripted (By a J23119 promoter, see </span><a href='www.baidu.com'><span>Project Description</span></a><span>) However, we found some works reporting that dCas9 itself can change the cell&#39;s growth rate, thus we have to explain that our system works in our exprected ways. Aditionally, we know that the promoters, even if regulated by an inducer or an inhibitor, may perfrom a &quot;leakage expression&quot;, hence we have to apply some other ways to gain a better control of the system. Both of these motivated us to design a second switch for sgRNA.</span></p><p><span>Our regulatory part includes an IPTG-activated pLac-promoted sgRNA and an Arabinose-activated pBAD-promoted dCas9. The mechanics are described in Pic 1. The formation of the effective dCas9-sgRNA complex depends on both IPTG and Arabinose inducers. </span></p><h3><a name="mono-regulation" class="md-header-anchor"></a><span>Mono-regulation</span></h3><p><span>We started from the simpler case: when there is only one regulated variable, e.g. dCas9 regulated by Arabinose. Since the expression of dCas9 is promoted by pBAD promoter and regulated by Arabinose, it is reasonable to describe this process by Hill&#39;s equation</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n12" cid="n12" mdtype="math_block">
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transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2-MJMATHI-76" x="0" y="0"></use><use xlink:href="#E2-MJMAIN-3D" x="762" y="0"></use><g transform="translate(1540,0)"><g transform="translate(397,0)"><rect stroke="none" width="5490" height="60" x="0" y="220"></rect><g transform="translate(72,715)"><use xlink:href="#E2-MJMATHI-76" x="0" y="0"></use><g transform="translate(485,-150)"><use transform="scale(0.707)" xlink:href="#E2-MJMAIN-6D"></use><use transform="scale(0.707)" xlink:href="#E2-MJMAIN-61" x="833" y="0"></use><use transform="scale(0.707)" xlink:href="#E2-MJMAIN-78" x="1333" y="0"></use></g><use xlink:href="#E2-MJMAIN-22C5" x="2123" y="0"></use><use xlink:href="#E2-MJMAIN-5B" x="2623" y="0"></use><g transform="translate(2901,0)"><use xlink:href="#E2-MJMAIN-41" x="0" y="0"></use><use xlink:href="#E2-MJMAIN-72" x="750" y="0"></use><use xlink:href="#E2-MJMAIN-61" x="1142" y="0"></use></g><g transform="translate(4543,0)"><use xlink:href="#E2-MJMAIN-5D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2-MJMATHI-6E" x="393" y="513"></use></g></g><g transform="translate(60,-716)"><use xlink:href="#E2-MJMAIN-5B" x="0" y="0"></use><g transform="translate(278,0)"><use xlink:href="#E2-MJMAIN-41" x="0" y="0"></use><use xlink:href="#E2-MJMAIN-72" x="750" y="0"></use><use xlink:href="#E2-MJMAIN-61" x="1142" y="0"></use></g><g transform="translate(1920,0)"><use xlink:href="#E2-MJMAIN-5D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2-MJMATHI-6E" x="393" y="408"></use></g><use xlink:href="#E2-MJMAIN-2B" x="2944" y="0"></use><g transform="translate(3944,0)"><use xlink:href="#E2-MJMATHI-4B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2-MJMATHI-6E" x="1274" y="408"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">v = \frac{v_{\max }\cdot[\mathrm{Ara}]^n}{[\mathrm{Ara}]^n + K^n}</script></div></div><p><span>in which </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.126ex" height="1.41ex" viewBox="0 -504.6 485 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E36-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E36-MJMATHI-76" x="0" y="0"></use></g></svg></span><script type="math/tex">v</script><span> is the producing rate of dCas9 mRNA, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.394ex" height="1.41ex" viewBox="0 -504.6 600 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E132-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E132-MJMATHI-6E" x="0" y="0"></use></g></svg></span><script type="math/tex">n</script><span> is the Hill&#39;s coefficient of pBAD promoter indicating the sharpness of induction rate and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.065ex" height="1.877ex" viewBox="0 -755.9 889 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E38-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-4B" x="0" y="0"></use></g></svg></span><script type="math/tex">K</script><span> is the coefficient indicating the IPTG concentration at which </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.961ex" height="2.577ex" 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428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z"></path><path stroke-width="0" id="E39-MJMAIN-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path stroke-width="0" id="E39-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E39-MJMATHI-76" x="0" y="0"></use><use xlink:href="#E39-MJMAIN-3D" x="762" y="0"></use><g transform="translate(1818,0)"><use xlink:href="#E39-MJMATHI-76" x="0" y="0"></use><g transform="translate(485,-150)"><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-6D"></use><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-61" x="833" y="0"></use><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-78" x="1333" y="0"></use></g></g><use xlink:href="#E39-MJMAIN-2F" x="3719" y="0"></use><use xlink:href="#E39-MJMAIN-32" x="4219" y="0"></use></g></svg></span><script type="math/tex">v = v_{\max }/2</script><span>.</span></p><h3><a name="bi-regulation" class="md-header-anchor"></a><span>Bi-regulation</span></h3><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvtgctybj31930qnach.jpg' alt='' referrerPolicy='no-referrer' /></p><p><span>We introduced the regulation of expresion of sgRNA by changing the promoter of sgRNA from J23119 to T7. This T7 promoter&#39;s activation is further regulated by a pTAC promoter. Therefore, we can further get another Hill&#39;s function for this promoter. Besides the transcription of sgRNA and dCas9 mRNA, the translation of dCas9, the binding of dCas9 and sgRNA and the degradation of all the mentioned matters are to be considered. Here, we describe this system by an ordinary differential equation system:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n17" cid="n17" mdtype="math_block">
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transform="translate(278,288)"><use transform="scale(0.707)" xlink:href="#E3-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E3-MJMATHI-72" x="848" y="-213"></use></g></g><use xlink:href="#E3-MJMAIN-2B" x="4147" y="0"></use><g transform="translate(5147,0)"><use xlink:href="#E3-MJMATHI-4B" x="0" y="0"></use><g transform="translate(901,456)"><use transform="scale(0.707)" xlink:href="#E3-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E3-MJMATHI-72" x="848" y="-213"></use></g><use transform="scale(0.707)" xlink:href="#E3-MJMATHI-72" x="1200" y="-211"></use></g></g></g></g><use xlink:href="#E3-MJMAIN-2212" x="8785" y="0"></use><g transform="translate(9785,0)"><use xlink:href="#E3-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3-MJMATHI-72" x="732" y="-213"></use></g><use xlink:href="#E3-MJMATHI-72" x="10722" y="0"></use><use xlink:href="#E3-MJMAIN-2B" x="11395" y="0"></use><g transform="translate(12395,0)"><use xlink:href="#E3-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3-MJMAIN-2212" x="736" y="-213"></use></g><use xlink:href="#E3-MJMATHI-62" x="13566" y="0"></use><use xlink:href="#E3-MJMAIN-2212" x="14218" y="0"></use><g transform="translate(15218,0)"><use xlink:href="#E3-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3-MJMAIN-2B" x="736" y="-213"></use></g><use xlink:href="#E3-MJMATHI-63" x="16389" y="0"></use><use xlink:href="#E3-MJMATHI-72" x="16822" y="0"></use></g><g transform="translate(0,-4023)"><use xlink:href="#E3-MJMAIN-3D" x="277" y="0"></use><use xlink:href="#E3-MJMAIN-2212" x="1333" y="0"></use><g transform="translate(2111,0)"><use xlink:href="#E3-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3-MJMAIN-2212" x="736" y="-213"></use></g><use xlink:href="#E3-MJMATHI-62" x="3282" y="0"></use><use xlink:href="#E3-MJMAIN-2B" x="3933" y="0"></use><g transform="translate(4934,0)"><use xlink:href="#E3-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3-MJMAIN-2B" x="736" y="-213"></use></g><use xlink:href="#E3-MJMATHI-63" x="6105" y="0"></use><use xlink:href="#E3-MJMATHI-72" x="6538" y="0"></use><use xlink:href="#E3-MJMAIN-2212" x="7211" y="0"></use><g transform="translate(8211,0)"><use xlink:href="#E3-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3-MJMATHI-62" x="732" y="-213"></use></g><use xlink:href="#E3-MJMATHI-62" x="9133" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">\begin{align}
+\frac{\mathrm{d}}{\mathrm{d}t}m &= \frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}} - \gamma_m m \\
+\frac{\mathrm{d}}{\mathrm{d}t}c &= \beta_c m - \gamma_c c + k_-b - k_+cr\\
+\frac{\mathrm{d}}{\mathrm{d}t}r &= \frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} - \gamma_r r + k_- b - k_+ c r\\
+\frac{\mathrm{d}}{\mathrm{d}t}b &= -k_- b + k_+ c r - \gamma_b b
+\end{align}</script></div></div><p><span>Where </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.039ex" height="1.41ex" viewBox="0 -504.6 878 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E133-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E133-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex">m</script><span> represents the concentration of dCas9 mRNA, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.006ex" height="1.41ex" viewBox="0 -504.6 433 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E64-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E64-MJMATHI-63" x="0" y="0"></use></g></svg></span><script type="math/tex">c</script><span> represents the concentration of dCas9, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.047ex" height="1.41ex" viewBox="0 -504.6 451 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E42-MJMATHI-72" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E42-MJMATHI-72" x="0" y="0"></use></g></svg></span><script type="math/tex">r</script><span> represents the concentration of sgRNA and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.996ex" height="1.994ex" viewBox="0 -755.9 429 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E56-MJMATHI-62" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E56-MJMATHI-62" x="0" y="0"></use></g></svg></span><script type="math/tex">b</script><span> represents the concentration of dCas9-sgRNA concentration. The parameters are determined from both others&#39; works and our experiments and listed as follows</span></p><figure><table><thead><tr><th><span>denotion</span></th><th><span>explanation</span></th><th><span>value</span></th><th><span>origin</span></th></tr></thead><tbody><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.068ex" height="1.76ex" viewBox="0 -504.6 1320.8 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E44-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E44-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E44-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E44-MJMATHI-6D" x="848" y="-213"></use></g></svg></span><script type="math/tex">n_m</script></td><td><span>Hill&#39;s coefficient of pBAD promoter</span></td><td>&nbsp;</td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.646ex" height="2.344ex" viewBox="0 -755.9 1569.8 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E45-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path stroke-width="0" id="E45-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E45-MJMATHI-4B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E45-MJMATHI-6D" x="1200" y="-213"></use></g></svg></span><script type="math/tex">K_m</script></td><td><span>Half-activate Arabinose concentration of pBAD promoter</span></td><td>&nbsp;</td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.161ex" height="1.76ex" viewBox="0 -504.6 1360.8 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E46-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="0" id="E46-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E46-MJMATHI-3B1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E46-MJMATHI-6D" x="905" y="-213"></use></g></svg></span><script type="math/tex">\alpha_m</script></td><td><span>Maximum dCas9 mRNA expression rate</span></td><td><span>0.0011 nM·min</span><sup><span>-1</span></sup></td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.366ex" height="1.76ex" viewBox="0 -504.6 1018.9 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E47-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E47-MJMATHI-72" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E47-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E47-MJMATHI-72" x="848" y="-213"></use></g></svg></span><script type="math/tex">n_r</script></td><td><span>Hill&#39;s coefficient of pTac promoter</span></td><td>&nbsp;</td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.459ex" height="1.76ex" viewBox="0 -504.6 1058.9 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E49-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="0" id="E49-MJMATHI-72" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E49-MJMATHI-3B1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E49-MJMATHI-72" x="905" y="-213"></use></g></svg></span><script type="math/tex">\alpha_r</script></td><td><span>Maximum sgRNA expression rate</span></td><td><span>0.0011 nM·min</span><sup><span>-1</span></sup></td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.258ex" height="2.461ex" viewBox="0 -806.1 972.2 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E50-MJMATHI-3B2" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path stroke-width="0" id="E50-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E50-MJMATHI-3B2" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E50-MJMATHI-63" x="800" y="-213"></use></g></svg></span><script type="math/tex">\beta_c</script></td><td><span>dCas9 protein synthesis rate from dCas9 mRNA</span></td><td><span>0.0057 protein·transcript</span><sup><span>-1</span></sup><span>·min</span><sup><span>-1</span></sup></td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.146ex" height="1.877ex" viewBox="0 -504.6 924.2 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E51-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path stroke-width="0" id="E51-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E51-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E51-MJMATHI-63" x="732" y="-213"></use></g></svg></span><script type="math/tex">\gamma_c</script></td><td><span>dCas9 degradation rate</span></td><td><span>5.6408×10</span><sup><span>-4</span></sup><span>min</span><sup><span>-1</span></sup></td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.72ex" height="2.461ex" viewBox="0 -755.9 1171.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E53-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E53-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E53-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E53-MJMAIN-2212" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_-</script></td><td><span>Rate of dissociation of dCas9-sgRNA complex</span></td><td>&nbsp;</td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.176ex" height="1.877ex" viewBox="0 -504.6 936.9 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E54-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path stroke-width="0" id="E54-MJMATHI-72" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E54-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E54-MJMATHI-72" x="732" y="-213"></use></g></svg></span><script type="math/tex">\gamma_r</script></td><td><span>sgRNA degradation rate</span></td><td>&nbsp;</td><td>&nbsp;</td></tr><tr><td><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.14ex" height="1.877ex" viewBox="0 -504.6 921.3 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E55-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path stroke-width="0" id="E55-MJMATHI-62" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E55-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E55-MJMATHI-62" x="732" y="-213"></use></g></svg></span><script type="math/tex">\gamma_b</script></td><td><span>dCas9-sgRNA degradation rate</span></td><td><span>5.6408×10</span><sup><span>-4</span></sup><span>min</span><sup><span>-1</span></sup></td><td>&nbsp;</td></tr></tbody></table></figure><p><span>To study the steady state of the system, we solve the equation when all particles&#39; concentration do not vary with time (e.g. the left-hand side of the ODE systems equal to zero)</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n86" cid="n86" mdtype="math_block">
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transform="translate(0,2700)"><use xlink:href="#E4-MJMAIN-3D" x="277" y="0"></use><use xlink:href="#E4-MJMAIN-30" x="1333" y="0"></use></g><g transform="translate(0,615)"><use xlink:href="#E4-MJMAIN-3D" x="277" y="0"></use><use xlink:href="#E4-MJMAIN-30" x="1333" y="0"></use></g><g transform="translate(0,-1366)"><use xlink:href="#E4-MJMAIN-3D" x="277" y="0"></use><use xlink:href="#E4-MJMAIN-30" x="1333" y="0"></use></g><g transform="translate(0,-3451)"><use xlink:href="#E4-MJMAIN-3D" x="277" y="0"></use><use xlink:href="#E4-MJMAIN-30" x="1333" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\begin{align}
+\frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}} - \gamma_m m & = 0 \\
+\beta_c m - \gamma_c c + k_-b - k_+cr & = 0\\
+\frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} - \gamma_r r + k_- b - k_+ c r & = 0\\
+-k_- b + k_+ c r - \gamma_b b & = 0
+\end{align}</script></div></div><p><span>Since the effective molecule in our system is the dCas9-sgRNA complex, we are interested in the concentration of dCas9-sgRNA complex (e.g. the value of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.996ex" height="1.994ex" viewBox="0 -755.9 429 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E56-MJMATHI-62" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E56-MJMATHI-62" x="0" y="0"></use></g></svg></span><script type="math/tex">b</script><span>).</span></p><p><span>from the system above, we can yield</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n89" cid="n89" mdtype="math_block">
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transform="scale(0.707)" xlink:href="#E5-MJMATHI-6D" x="905" y="-211"></use></g><g transform="translate(2531,0)"><use xlink:href="#E5-MJMATHI-3B1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E5-MJMAIN-2217" x="905" y="452"></use><use transform="scale(0.707)" xlink:href="#E5-MJMATHI-72" x="905" y="-211"></use></g><g transform="translate(3625,0)"><use xlink:href="#E5-MJMATHI-3B2" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E5-MJMATHI-63" x="800" y="-213"></use></g></g><g transform="translate(808,-686)"><use xlink:href="#E5-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E5-MJMATHI-63" x="732" y="-213"></use><g transform="translate(924,0)"><use xlink:href="#E5-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E5-MJMATHI-72" x="732" y="-213"></use></g><g transform="translate(1861,0)"><use xlink:href="#E5-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E5-MJMATHI-6D" x="732" y="-213"></use></g></g></g></g><use xlink:href="#E5-MJMAIN-3D" x="26129" y="0"></use><use xlink:href="#E5-MJMAIN-30" x="27184" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-4">\frac{k_+ \gamma_b^2}{\gamma_c\gamma_r}b^2 - (k_- + \gamma_b + \frac{k_+ \gamma_b \alpha_r^*}{\gamma_c\gamma_r} + \frac{k_+ \gamma_b \alpha_r^*\beta_c}{\gamma_c\gamma_r\gamma_m})b +\frac{k_+ \alpha_m^* \alpha_r^*\beta_c}{\gamma_c\gamma_r\gamma_m} = 0</script></div></div><p><span>where</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n91" cid="n91" mdtype="math_block">
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xlink:href="#E6-MJMAIN-5D" x="0" y="0"></use><g transform="translate(278,362)"><use transform="scale(0.707)" xlink:href="#E6-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E6-MJMATHI-6D" x="848" y="-213"></use></g></g><g transform="translate(3231,0)"><use xlink:href="#E6-MJMATHI-3B1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E6-MJMATHI-6D" x="905" y="-213"></use></g></g><g transform="translate(60,-735)"><use xlink:href="#E6-MJMAIN-5B" x="0" y="0"></use><g transform="translate(278,0)"><use xlink:href="#E6-MJMAIN-41" x="0" y="0"></use><use xlink:href="#E6-MJMAIN-72" x="750" y="0"></use><use xlink:href="#E6-MJMAIN-61" x="1142" y="0"></use></g><g transform="translate(1920,0)"><use xlink:href="#E6-MJMAIN-5D" x="0" y="0"></use><g transform="translate(278,288)"><use transform="scale(0.707)" xlink:href="#E6-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E6-MJMATHI-6D" x="848" y="-213"></use></g></g><use xlink:href="#E6-MJMAIN-2B" x="3454" y="0"></use><g transform="translate(4454,0)"><use xlink:href="#E6-MJMATHI-4B" x="0" y="0"></use><g transform="translate(901,456)"><use transform="scale(0.707)" xlink:href="#E6-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E6-MJMATHI-6D" x="848" y="-213"></use></g><use transform="scale(0.707)" xlink:href="#E6-MJMATHI-6D" x="1200" y="-211"></use></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">\alpha_r^* = \frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} \\
+\alpha_m^* = \frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}}</script></div></div><p><span>This quadratic equation has two possitive solutions. However,only the smaller solution is reasonable (consider when </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.62ex" height="2.227ex" viewBox="0 -655.4 4572.4 958.9" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E57-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path stroke-width="0" id="E57-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E57-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E57-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E57-MJMAIN-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E57-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E57-MJMATHI-6D" x="732" y="-213"></use><use xlink:href="#E57-MJMAIN-2192" x="1516" y="0"></use><use xlink:href="#E57-MJMAIN-2B" x="2794" y="0"></use><use xlink:href="#E57-MJMAIN-221E" x="3572" y="0"></use></g></svg></span><script type="math/tex">\gamma_m \rightarrow +\infty</script><span>, the smaller solution is 0 and the bigger one is a positive real number</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n93" cid="n93" mdtype="math_block">
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39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="0" id="E7-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="11530" height="60" x="0" y="220"></rect><g transform="translate(60,681)"><use xlink:href="#E7-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-63" x="732" y="-213"></use><g transform="translate(924,0)"><use xlink:href="#E7-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-72" x="732" y="-213"></use></g><g transform="translate(1861,0)"><use xlink:href="#E7-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMAIN-2212" x="736" y="-213"></use></g><use xlink:href="#E7-MJMAIN-2B" x="3254" y="0"></use><g transform="translate(4254,0)"><use xlink:href="#E7-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-63" x="732" y="-213"></use></g><g transform="translate(5178,0)"><use xlink:href="#E7-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-62" x="732" y="-213"></use></g><g transform="translate(6100,0)"><use xlink:href="#E7-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-72" x="732" y="-213"></use></g><use xlink:href="#E7-MJMAIN-2B" x="7259" y="0"></use><g transform="translate(8259,0)"><use xlink:href="#E7-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMAIN-2B" x="736" y="-213"></use></g><g transform="translate(9430,0)"><use xlink:href="#E7-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-62" x="732" y="-213"></use></g><g transform="translate(10352,0)"><use xlink:href="#E7-MJMATHI-3B1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-72" x="905" y="-213"></use></g></g><g transform="translate(4677,-782)"><use xlink:href="#E7-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMAIN-2B" x="736" y="-213"></use><g transform="translate(1171,0)"><use xlink:href="#E7-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E7-MJMAIN-32" x="778" y="487"></use><use transform="scale(0.707)" xlink:href="#E7-MJMATHI-62" x="732" y="-462"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">\frac{\gamma_c \gamma_r k_- + \gamma_c \gamma_b \gamma_r + k_+ \gamma_b \alpha_r}{k_+ \gamma_b^2}</script></div></div><p><span>biologicaly, when </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.877ex" height="1.877ex" viewBox="0 -504.6 1238.8 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E58-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path stroke-width="0" id="E58-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E58-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E58-MJMATHI-6D" x="732" y="-213"></use></g></svg></span><script type="math/tex">\gamma_m</script><span> is extremely big, mRNA degrades so frequently that there is almost no dCas9 mRNA, hence no dCas9, and subsequentially no dCas9-sgRNA complex, so the zero solution is the one corresponding to the real biological situation)</span></p><p><span>Thus, we get the final solution</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n96" cid="n96" mdtype="math_block">
+		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.791ex" height="5.612ex" viewBox="0 -1660.6 9382 2416.4" role="img" focusable="false" style="vertical-align: -1.756ex; max-width: 100%;"><defs><path stroke-width="0" id="E8-MJMATHI-62" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path stroke-width="0" id="E8-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E8-MJMATHI-42" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path stroke-width="0" id="E8-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E8-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E8-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="0" id="E8-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E8-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path stroke-width="0" id="E8-MJMAIN-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E8-MJMATHI-62" x="0" y="0"></use><use xlink:href="#E8-MJMAIN-3D" x="706" y="0"></use><g transform="translate(1484,0)"><g transform="translate(397,0)"><rect stroke="none" width="7379" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#E8-MJMATHI-42" x="0" y="0"></use><use xlink:href="#E8-MJMAIN-2212" x="981" y="0"></use><g transform="translate(1981,0)"><use xlink:href="#E8-MJMAIN-221A" x="0" y="61"></use><rect stroke="none" width="4444" height="60" x="833" y="801"></rect><g transform="translate(833,0)"><use xlink:href="#E8-MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E8-MJMAIN-32" x="1073" y="408"></use><use xlink:href="#E8-MJMAIN-2212" x="1434" y="0"></use><use xlink:href="#E8-MJMAIN-34" x="2434" y="0"></use><use xlink:href="#E8-MJMATHI-41" x="2934" y="0"></use><use xlink:href="#E8-MJMATHI-43" x="3684" y="0"></use></g></g></g><g transform="translate(3064,-686)"><use xlink:href="#E8-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E8-MJMATHI-41" x="500" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-7">b = \frac{B - \sqrt{B^2 - 4AC}}{2A}</script></div></div><p><span>where</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n98" cid="n98" mdtype="math_block">
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xlink:href="#E9-MJMAIN-41" x="0" y="0"></use><use xlink:href="#E9-MJMAIN-72" x="750" y="0"></use><use xlink:href="#E9-MJMAIN-61" x="1142" y="0"></use></g><g transform="translate(1920,0)"><use xlink:href="#E9-MJMAIN-5D" x="0" y="0"></use><g transform="translate(278,362)"><use transform="scale(0.707)" xlink:href="#E9-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E9-MJMATHI-6D" x="848" y="-213"></use></g></g><g transform="translate(3231,0)"><use xlink:href="#E9-MJMATHI-3B1" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E9-MJMATHI-6D" x="905" y="-213"></use></g></g><g transform="translate(60,-735)"><use xlink:href="#E9-MJMAIN-5B" x="0" y="0"></use><g transform="translate(278,0)"><use xlink:href="#E9-MJMAIN-41" x="0" y="0"></use><use xlink:href="#E9-MJMAIN-72" x="750" y="0"></use><use xlink:href="#E9-MJMAIN-61" x="1142" y="0"></use></g><g transform="translate(1920,0)"><use xlink:href="#E9-MJMAIN-5D" x="0" y="0"></use><g transform="translate(278,288)"><use transform="scale(0.707)" xlink:href="#E9-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E9-MJMATHI-6D" x="848" y="-213"></use></g></g><use xlink:href="#E9-MJMAIN-2B" x="3454" y="0"></use><g transform="translate(4454,0)"><use xlink:href="#E9-MJMATHI-4B" x="0" y="0"></use><g transform="translate(901,456)"><use transform="scale(0.707)" xlink:href="#E9-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E9-MJMATHI-6D" x="848" y="-213"></use></g><use transform="scale(0.707)" xlink:href="#E9-MJMATHI-6D" x="1200" y="-211"></use></g></g></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-8">\begin{align}
+A &= \frac{k_+ \gamma_b^2}{\gamma_c\gamma_r}b^2 \\
+B &= k_- + \gamma_b + \frac{k_+ \gamma_b}{\gamma_c\gamma_r}\frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} + \frac{k_+ \gamma_b \beta_c}{\gamma_c\gamma_r\gamma_m}\frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} \\
+C &= \frac{k_+ \beta_c}{\gamma_c\gamma_r\gamma_m}\frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}}\frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}}
+\end{align}</script></div></div><p><span>This equation gives the relation between the concentration of dCas9-sgRNA and the amount of inducer we add. Applying the given parameters, we can demonstrate this relationship by a 3D surface plot. Instinctively , the expression output follows the AND logic: the concentration of dCas9-sgRNA is only considerable when the concentration of both IPTG and Arabinose are high enough.</span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvtiluvej30fk0bodgr.jpg' alt='' referrerPolicy='no-referrer' /></p><h2><a name="the-relationship-between-the-expression-of-dcas9-and-the-cell%27s-growth-rate" class="md-header-anchor"></a><span>The relationship between the expression of dCas9 and the cell&#39;s growth rate</span></h2><p><span>In this section we show that our system delays the bacteria&#39;s replication. Specifically, the average time of a bacteria&#39;s cell cycle depends linearly on the concentration of the dCas9-sgRNA molecule copy number. This is deduced from a coarse-grained model including the process of dCas9 and DnaA binding to the OriC site and the cell&#39;s replication. Here we assume that </span></p><ul><li><span>The bingding process of both binding of dCas9 and DnaA are reversible</span></li><li><span>Genome DNA replication can start only when the OriC is bound to DnaA</span></li></ul><p><span>A Markov chain on continuous time is built, with the hypothesis that the replication time is subjected to a Poisson distribution. A cell has three possible states at a certain time</span></p><ol start='' ><li><span>OriC is bound to a dCas9 molecule</span></li><li><span>OriC is naked</span></li><li><span>The bacteria start replication</span></li></ol><p><span>A cell can transform from state 1 to state 2 (</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.263ex" height="2.227ex" viewBox="0 -755.9 974.6 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E62-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E62-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E62-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E62-MJMAIN-31" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_1</script><span>), from state 2 to state 1(</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.263ex" height="2.227ex" viewBox="0 -755.9 974.6 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E65-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E65-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E65-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E65-MJMAIN-32" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_2</script><span>), and from state 2 to state 3 (</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.338ex" height="1.41ex" viewBox="0 -504.6 576 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E79-MJMATHI-3BA" d="M83 -11Q70 -11 62 -4T51 8T49 17Q49 30 96 217T147 414Q160 442 193 442Q205 441 213 435T223 422T225 412Q225 401 208 337L192 270Q193 269 208 277T235 292Q252 304 306 349T396 412T467 431Q489 431 500 420T512 391Q512 366 494 347T449 327Q430 327 418 338T405 368Q405 370 407 380L397 375Q368 360 315 315L253 266L240 257H245Q262 257 300 251T366 230Q422 203 422 150Q422 140 417 114T411 67Q411 26 437 26Q484 26 513 137Q516 149 519 151T535 153Q554 153 554 144Q554 121 527 64T457 -7Q447 -10 431 -10Q386 -10 360 17T333 90Q333 108 336 122T339 146Q339 170 320 186T271 209T222 218T185 221H180L155 122Q129 22 126 16Q113 -11 83 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E79-MJMATHI-3BA" x="0" y="0"></use></g></svg></span><script type="math/tex">\kappa</script><span>). A infinitesimal transition matrix is written to describe the transition among these states:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n117" cid="n117" mdtype="math_block">
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144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E10-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E10-MJMATHI-3BA" d="M83 -11Q70 -11 62 -4T51 8T49 17Q49 30 96 217T147 414Q160 442 193 442Q205 441 213 435T223 422T225 412Q225 401 208 337L192 270Q193 269 208 277T235 292Q252 304 306 349T396 412T467 431Q489 431 500 420T512 391Q512 366 494 347T449 327Q430 327 418 338T405 368Q405 370 407 380L397 375Q368 360 315 315L253 266L240 257H245Q262 257 300 251T366 230Q422 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xlink:href="#E10-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E10-MJMAIN-31" x="736" y="-213"></use></g><use xlink:href="#E10-MJMAIN-2212" x="1974" y="0"></use><use xlink:href="#E10-MJMATHI-3BA" x="2974" y="0"></use></g><g transform="translate(1288,-50)"><use xlink:href="#E10-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E10-MJMAIN-32" x="736" y="-213"></use></g><use xlink:href="#E10-MJMAIN-30" x="1525" y="-1450"></use></g><g transform="translate(4536,0)"><g transform="translate(389,1350)"><use xlink:href="#E10-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E10-MJMAIN-31" x="736" y="-213"></use></g><g transform="translate(0,-50)"><use xlink:href="#E10-MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#E10-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E10-MJMAIN-32" x="736" y="-213"></use></g></g><use xlink:href="#E10-MJMAIN-30" x="626" y="-1450"></use></g><g transform="translate(7288,0)"><use xlink:href="#E10-MJMATHI-3BA" x="0" y="1350"></use><use xlink:href="#E10-MJMAIN-30" x="38" y="-50"></use><use xlink:href="#E10-MJMAIN-30" x="38" y="-1450"></use></g></g><g transform="translate(9073,2150)"><use xlink:href="#E10-MJSZ4-239E" x="0" y="-1154"></use><g transform="translate(0,-2036.0048309178746) scale(1,0.45410628019323673)"><use xlink:href="#E10-MJSZ4-239F"></use></g><use xlink:href="#E10-MJSZ4-23A0" x="0" y="-3156"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-9">{\bf A} = \begin{pmatrix}
+ -k_1-\kappa & k_1 & \kappa \\
+ k_2 & -k_2 & 0 \\
+& 0 & 0\end{pmatrix}</script></div></div><p><span>where</span></p><ul><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.263ex" height="2.227ex" viewBox="0 -755.9 974.6 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E62-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E62-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E62-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E62-MJMAIN-31" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_1</script><span> represents the frequency that an exposed OriC is bound by a dCas9-sgRNA molecule (state 1 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.323ex" height="1.644ex" viewBox="0 -605.1 1000 707.6" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E68-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E68-MJMAIN-2192" x="0" y="0"></use></g></svg></span><script type="math/tex">\rightarrow</script><span> state 2). This value is proportion to the concentration of dCas9-sgRNA (denoted as </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.006ex" height="1.41ex" viewBox="0 -504.6 433 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E64-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E64-MJMATHI-63" x="0" y="0"></use></g></svg></span><script type="math/tex">c</script><span>):</span></li></ul><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n121" cid="n121" mdtype="math_block">
+		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-10-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.63ex" height="2.694ex" viewBox="0 -755.9 3715.7 1160" role="img" focusable="false" style="vertical-align: -0.938ex; max-width: 100%;"><defs><path stroke-width="0" id="E11-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E11-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E11-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E11-MJMAIN-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path><path stroke-width="0" id="E11-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E11-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E11-MJMAIN-31" x="736" y="-213"></use><use xlink:href="#E11-MJMAIN-3D" x="1252" y="0"></use><g transform="translate(2308,0)"><use xlink:href="#E11-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E11-MJMAIN-2217" x="736" y="452"></use><use transform="scale(0.707)" xlink:href="#E11-MJMAIN-31" x="736" y="-434"></use></g><use xlink:href="#E11-MJMATHI-63" x="3282" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-10">k_1 = k_1^*c</script></div></div><ul><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.263ex" height="2.227ex" viewBox="0 -755.9 974.6 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E65-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E65-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E65-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E65-MJMAIN-32" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_2</script><span> represents the frequency that a dCas9-sgRNA molecule decouple with OriC (state 2 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.323ex" height="1.644ex" viewBox="0 -605.1 1000 707.6" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E68-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E68-MJMAIN-2192" x="0" y="0"></use></g></svg></span><script type="math/tex">\rightarrow</script><span> state 1).</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.338ex" height="1.41ex" viewBox="0 -504.6 576 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E79-MJMATHI-3BA" d="M83 -11Q70 -11 62 -4T51 8T49 17Q49 30 96 217T147 414Q160 442 193 442Q205 441 213 435T223 422T225 412Q225 401 208 337L192 270Q193 269 208 277T235 292Q252 304 306 349T396 412T467 431Q489 431 500 420T512 391Q512 366 494 347T449 327Q430 327 418 338T405 368Q405 370 407 380L397 375Q368 360 315 315L253 266L240 257H245Q262 257 300 251T366 230Q422 203 422 150Q422 140 417 114T411 67Q411 26 437 26Q484 26 513 137Q516 149 519 151T535 153Q554 153 554 144Q554 121 527 64T457 -7Q447 -10 431 -10Q386 -10 360 17T333 90Q333 108 336 122T339 146Q339 170 320 186T271 209T222 218T185 221H180L155 122Q129 22 126 16Q113 -11 83 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E79-MJMATHI-3BA" x="0" y="0"></use></g></svg></span><script type="math/tex">\kappa</script><span> represents the frequency that a exposed OriC is bound by replication initiators and the replication starts (state 1 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.323ex" height="1.644ex" viewBox="0 -605.1 1000 707.6" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E68-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E68-MJMAIN-2192" x="0" y="0"></use></g></svg></span><script type="math/tex">\rightarrow</script><span> state 3)</span></li></ul><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvtlteo3j31f30tltbd.jpg' alt='' referrerPolicy='no-referrer' /></p><p><span>uppose initially, the bacterium&#39;s OriC not bound with dCas9-sgRNA (i.e. the initial state distribution is </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.356ex" height="2.577ex" viewBox="0 -806.1 3167.3 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E69-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E69-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E69-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E69-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E69-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E69-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E69-MJMAIN-31" x="389" y="0"></use><use xlink:href="#E69-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E69-MJMAIN-30" x="1333" y="0"></use><use xlink:href="#E69-MJMAIN-2C" x="1833" y="0"></use><use xlink:href="#E69-MJMAIN-30" x="2278" y="0"></use><use xlink:href="#E69-MJMAIN-29" x="2778" y="0"></use></g></svg></span><script type="math/tex">(1,0, 0)</script><span>). We calculate the distribution function </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.74ex" height="1.877ex" viewBox="0 -755.9 749 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E70-MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E70-MJMATHI-46" x="0" y="0"></use></g></svg></span><script type="math/tex">F</script><span> of the waiting time before replication. To do this, we only have to calculate</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n130" cid="n130" mdtype="math_block">
+		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.224ex" height="3.044ex" viewBox="0 -1007.2 10429.8 1310.7" role="img" focusable="false" style="vertical-align: -0.705ex; max-width: 100%;"><defs><path stroke-width="0" id="E12-MJMAINB-46" d="M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 612T441 617T387 618H368H304V371H333Q389 373 411 390T437 468V488H499V192H437V212Q436 244 430 263T408 292T378 305T333 309H304V62H439V0H425Z"></path><path stroke-width="0" id="E12-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 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250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path stroke-width="0" id="E12-MJMAIN-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path stroke-width="0" id="E12-MJMAINB-41" d="M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMAINB-46" x="0" y="0"></use><use xlink:href="#E12-MJMAIN-28" x="724" y="0"></use><use xlink:href="#E12-MJMATHI-74" x="1113" y="0"></use><use xlink:href="#E12-MJMAIN-29" x="1474" y="0"></use><use xlink:href="#E12-MJMAIN-3D" x="2140" y="0"></use><g transform="translate(3196,0)"><use xlink:href="#E12-MJMAIN-28" x="0" y="0"></use><g transform="translate(556,0)"><use xlink:href="#E12-MJMAIN-31" x="-15" y="-50"></use><use xlink:href="#E12-MJMAIN-30" x="1485" y="-50"></use><use xlink:href="#E12-MJMAIN-30" x="2985" y="-50"></use></g><use xlink:href="#E12-MJMAIN-29" x="4208" y="0"></use></g><use xlink:href="#E12-MJMAIN-D7" x="8015" y="0"></use><g transform="translate(9016,0)"><use xlink:href="#E12-MJMAIN-65" x="0" y="0"></use><g transform="translate(444,412)"><use transform="scale(0.707)" xlink:href="#E12-MJMAINB-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E12-MJMATHI-74" x="869" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-11">{\bf F}(t) = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \times {\rm e}^{{\bf A}t}</script></div></div><p><span>thus the third element of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.327ex" height="2.577ex" viewBox="0 -806.1 1863 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E71-MJMAINB-46" d="M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 612T441 617T387 618H368H304V371H333Q389 373 411 390T437 468V488H499V192H437V212Q436 244 430 263T408 292T378 305T333 309H304V62H439V0H425Z"></path><path stroke-width="0" id="E71-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E71-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E71-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E71-MJMAINB-46" x="0" y="0"></use><use xlink:href="#E71-MJMAIN-28" x="724" y="0"></use><use xlink:href="#E71-MJMATHI-74" x="1113" y="0"></use><use xlink:href="#E71-MJMAIN-29" x="1474" y="0"></use></g></svg></span><script type="math/tex">{\bf F}(t)</script><span> will be the probability </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.385ex" height="2.577ex" viewBox="0 -806.1 1888 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E73-MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path><path stroke-width="0" id="E73-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E73-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E73-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E73-MJMATHI-46" x="0" y="0"></use><use xlink:href="#E73-MJMAIN-28" x="749" y="0"></use><use xlink:href="#E73-MJMATHI-74" x="1138" y="0"></use><use xlink:href="#E73-MJMAIN-29" x="1499" y="0"></use></g></svg></span><script type="math/tex">F(t)</script><span> we want.</span></p><p><span>It is not hard to get an explicit expression of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.385ex" height="2.577ex" viewBox="0 -806.1 1888 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E73-MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path><path stroke-width="0" id="E73-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E73-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 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xlink:href="#E13-MJMATHI-74" x="1036" y="0"></use></g></g><use xlink:href="#E13-MJMAIN-2212" x="6802" y="0"></use><g transform="translate(7802,0)"><use xlink:href="#E13-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMAIN-31" x="824" y="-213"></use></g><use xlink:href="#E13-MJMAIN-28" x="8839" y="0"></use><g transform="translate(9228,0)"><use xlink:href="#E13-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMAIN-32" x="824" y="-213"></use></g><use xlink:href="#E13-MJMAIN-2B" x="10487" y="0"></use><g transform="translate(11487,0)"><use xlink:href="#E13-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMAIN-32" x="736" y="-213"></use></g><use xlink:href="#E13-MJMAIN-29" x="12461" y="0"></use><g transform="translate(12850,0)"><use xlink:href="#E13-MJMAIN-65" x="0" y="0"></use><g transform="translate(444,362)"><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E13-MJMAIN-32" x="824" y="-213"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-74" x="1036" y="0"></use></g></g></g><g transform="translate(4727,-716)"><use xlink:href="#E13-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMAIN-32" x="736" y="-213"></use><use xlink:href="#E13-MJMAIN-28" x="974" y="0"></use><g transform="translate(1363,0)"><use xlink:href="#E13-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMAIN-32" x="824" y="-213"></use></g><use xlink:href="#E13-MJMAIN-2212" x="2622" y="0"></use><g transform="translate(3622,0)"><use xlink:href="#E13-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMAIN-31" x="824" y="-213"></use></g><use xlink:href="#E13-MJMAIN-29" x="4659" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-12">F(t) = 1 - \frac{\lambda_2(\lambda_1 + k_2){\rm e}^{\lambda_1t} - \lambda_1(\lambda_2 + k_2){\rm e}^{\lambda_2t}}{k_2(\lambda_2 - \lambda_1)}</script></div></div><p><span>where</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n135" cid="n135" mdtype="math_block">
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y="-213"></use></g><use xlink:href="#E14-MJMAIN-2B" x="3384" y="0"></use><g transform="translate(4384,0)"><use xlink:href="#E14-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E14-MJMAIN-32" x="736" y="-213"></use></g><g transform="translate(5358,0)"><use xlink:href="#E14-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E14-MJMAIN-32" x="550" y="408"></use></g><use xlink:href="#E14-MJMAIN-2212" x="6423" y="0"></use><use xlink:href="#E14-MJMAIN-34" x="7423" y="0"></use><use xlink:href="#E14-MJMATHI-3BA" x="7923" y="0"></use><g transform="translate(8499,0)"><use xlink:href="#E14-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E14-MJMAIN-32" x="736" y="-213"></use></g></g></g></g><use xlink:href="#E14-MJMAIN-32" x="8532" y="-686"></use></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-13">\begin{align*}
+\lambda_1 = \frac{-\kappa -k_1 - k_2 - \sqrt{(\kappa + k_1 + k_2)^2 - 4\kappa k_2}}2\\
+\lambda_2 = \frac{-\kappa -k_1 - k_2 + \sqrt{(\kappa + k_1 + k_2)^2 - 4\kappa k_2}}2
+\end{align*}</script></div></div><p><span>The average waiting time is:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n137" cid="n137" mdtype="math_block">
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xlink:href="#E15-MJMAIN-32" x="736" y="-213"></use><use xlink:href="#E15-MJMATHI-3BA" x="974" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-14">{\mathbb E}(T) = \int_0^{+\infty}tF\ '(t){\rm d}t = \frac{k_1 + k_2}{k_2\kappa} = \frac{k_1^*c + k_2}{k_2\kappa}</script></div></div><p><span>linearly related to the concentration of dCas9-sgRNA.</span></p><h2><a name="plasmid-copy-number-hacking" class="md-header-anchor"></a><span>Plasmid Copy number Hacking</span></h2><h3><a name="introduction" class="md-header-anchor"></a><span>Introduction</span></h3><p><span>In this section we discuss how our system can control the plasmid copy number (see </span><a href='www.baidu.com'><span>Project Description</span></a><span>). Typically the plasmids replicate themselves during the cell cycle. After a cell&#39;s division, the plasmids are equivalently distributed to both of the cells. Here we discuss a cell line&#39;s behavior over time.</span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvtq1o7kj31db0rs76c.jpg' alt='plasmid_cell_line' referrerPolicy='no-referrer' /></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvtsbw83j30rc0pwab5.jpg' alt='plasmid_dilution.png' referrerPolicy='no-referrer' /></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvtv8ej0j31cl0q4gnn.jpg' alt='the simulation process' referrerPolicy='no-referrer' /></p><p><span>In an unbiased cell division, the plasmids in a cell are distributed equivalently to both of the children cells. Therefore, while tracing a cell line, we see that a cell lose half of its plasmids after each division. It is reasonable to assume this process as a random process in which each plasmid has 1/2 probability to &quot;disappear&quot; (actually entering its sibling cell), and 1/2 probability to remain in the cell line we are interested in. </span></p><p><span>Besides splitting, the plasmids may also replicate itself and bind or unbind to dCas9, These process, together with the division &quot;1/2 disappear&quot;, make up all the plasmids&#39; behaviors we are interested in. When bound to dCas9, the plasmid cannot replicate. Here we introduce two parameters </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.901ex" height="2.227ex" viewBox="0 -755.9 1248.9 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E74-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E74-MJMATHI-4E" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E74-MJMATHI-4E" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_N</script><span>, the frequency that a plasmid bind to a dCas9 molecule, and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.69ex" height="2.344ex" viewBox="0 -755.9 1158.4 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E85-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E85-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E85-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E85-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span>, the frequency that a plasmid and dCas9 decouple.</span></p><p><span>Another factor taken into account is the process that the replication of cells are also regulated by the intracellular environment to ensure that the plasmid number do not increase uncontrolled. Hence we assume that during a cell&#39;s cycle, the growth of the cell&#39;s plasmid number is a logistic process. Precisely, the frequency that plasmid replicate itself is</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n148" cid="n148" mdtype="math_block">
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transform="translate(7308,0)"><use xlink:href="#E16-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-4E" x="848" y="-213"></use></g><use xlink:href="#E16-MJMAIN-22C5" x="8859" y="0"></use><g transform="translate(9359,0)"><use xlink:href="#E16-MJSZ3-28"></use><use xlink:href="#E16-MJMAIN-31" x="736" y="0"></use><use xlink:href="#E16-MJMAIN-2212" x="1458" y="0"></use><g transform="translate(2236,0)"><g transform="translate(342,0)"><rect stroke="none" width="3907" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#E16-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-4E" x="848" y="-213"></use><use xlink:href="#E16-MJMAIN-2B" x="1550" y="0"></use><g transform="translate(2550,0)"><use xlink:href="#E16-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-43" x="848" y="-218"></use></g></g><g transform="translate(945,-686)"><use xlink:href="#E16-MJMATHI-6E" x="0" y="0"></use><g transform="translate(600,-150)"><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-6D"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-61" x="833" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-78" x="1333" y="0"></use></g></g></g></g><use xlink:href="#E16-MJSZ3-29" x="6606" y="-1"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-15">K(n_N, n_C) = \kappa\cdot n_N\cdot\left(1 - \frac{n_N + n_C}{n_\max}\right)</script></div></div><p><span>where </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.084ex" height="1.644ex" viewBox="0 -504.6 1327.9 707.6" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E76-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 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x="2444" y="0"></use></g></g><use xlink:href="#E17-MJMAIN-2212" x="27810" y="0"></use><use xlink:href="#E17-MJMAIN-28" x="28811" y="0"></use><g transform="translate(29200,0)"><use xlink:href="#E17-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E17-MJMATHI-43" x="736" y="-218"></use></g><use xlink:href="#E17-MJMAIN-2B" x="30580" y="0"></use><g transform="translate(31581,0)"><use xlink:href="#E17-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E17-MJMATHI-4E" x="736" y="-213"></use></g><use xlink:href="#E17-MJMAIN-2B" x="33052" y="0"></use><use xlink:href="#E17-MJMATHI-4B" x="34052" y="0"></use><use xlink:href="#E17-MJMAIN-28" x="34941" y="0"></use><g transform="translate(35330,0)"><use xlink:href="#E17-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E17-MJMATHI-4E" x="848" y="-213"></use></g><use xlink:href="#E17-MJMAIN-2C" x="36658" y="0"></use><g transform="translate(37102,0)"><use xlink:href="#E17-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E17-MJMATHI-43" x="848" y="-218"></use></g><use xlink:href="#E17-MJMAIN-29" x="38340" y="0"></use><use xlink:href="#E17-MJMAIN-29" x="38729" y="0"></use><use xlink:href="#E17-MJMAIN-22C5" x="39340" y="0"></use><g transform="translate(39840,0)"><use xlink:href="#E17-MJMATHI-70" x="0" y="0"></use><g transform="translate(503,-155)"><use transform="scale(0.707)" xlink:href="#E17-MJMATHI-4E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E17-MJMAIN-2C" x="888" y="0"></use><use transform="scale(0.707)" xlink:href="#E17-MJMATHI-43" x="1166" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-16">\frac{\mathrm{d}}{\mathrm{d}t}p_{N, C} = k_C\cdot p_{N+1, C-1} + k_N\cdot p_{N-1, C+1} + K(n_{N-1}, n_C)\cdot p_{N-1, C}-(k_C + k_N + K(n_N, n_C))\cdot p_{N, C}</script></div></div><p><span>Since the cell cycle length is relatively invariant in a stable environment, we use a fixed cell length </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.635ex" height="1.877ex" viewBox="0 -755.9 704 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E108-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" 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1372T515 1343T479 1305T449 1257T429 1200Q425 1180 425 1152T423 851Q422 579 422 549T416 498Q407 459 388 424T346 364T297 318T250 284T214 264T197 254L188 251L205 242Q290 200 345 138T416 3Q421 -18 421 -48T423 -349Q423 -397 423 -472Q424 -677 428 -694Q429 -697 429 -699Q434 -722 443 -743T465 -782T491 -816T519 -845T548 -868T574 -886T595 -899T610 -908L616 -910Q618 -912 618 -928V-943Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E18-MJSZ3-7B"></use><g transform="translate(917,0)"><g transform="translate(-15,0)"><g transform="translate(0,684)"><use xlink:href="#E18-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E18-MJMATHI-4E" x="848" y="-213"></use><use xlink:href="#E18-MJMAIN-2192" x="1605" y="0"></use><g transform="translate(2883,0)"><use xlink:href="#E18-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E18-MJMAIN-2032" x="848" y="444"></use><use transform="scale(0.707)" xlink:href="#E18-MJMATHI-4E" x="848" y="-451"></use></g></g><g transform="translate(0,-635)"><use xlink:href="#E18-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E18-MJMATHI-43" x="848" y="-218"></use><use xlink:href="#E18-MJMAIN-2192" x="1515" y="0"></use><g transform="translate(2792,0)"><use xlink:href="#E18-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E18-MJMAIN-2032" x="848" y="444"></use><use transform="scale(0.707)" xlink:href="#E18-MJMATHI-43" x="848" y="-473"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-17">\begin{cases}
+n_N \rightarrow n_N'\\
+n_C \rightarrow n_C'
+\end{cases}</script></div></div><p><span>where</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n154" cid="n154" mdtype="math_block">
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xlink:href="#E19-MJMAIN-2C" x="5042" y="0"></use><g transform="translate(5320,0)"><g transform="translate(286,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E19-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E19-MJMAIN-32" x="60" y="-686"></use></g></g><use xlink:href="#E19-MJMAIN-29" x="6347" y="0"></use><use xlink:href="#E19-MJMAIN-2C" x="6736" y="0"></use><g transform="translate(7930,0)"><use xlink:href="#E19-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E19-MJMAIN-2032" x="848" y="444"></use><use transform="scale(0.707)" xlink:href="#E19-MJMATHI-43" x="848" y="-473"></use></g><use xlink:href="#E19-MJMAIN-223C" x="9445" y="0"></use><use xlink:href="#E19-MJCAL-42" x="10501" y="0"></use><use xlink:href="#E19-MJMAIN-28" x="11165" y="0"></use><g transform="translate(11554,0)"><use xlink:href="#E19-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E19-MJMATHI-43" x="848" y="-218"></use></g><use xlink:href="#E19-MJMAIN-2C" x="12792" y="0"></use><g transform="translate(13070,0)"><g transform="translate(286,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E19-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E19-MJMAIN-32" x="60" y="-686"></use></g></g><use xlink:href="#E19-MJMAIN-29" x="14096" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-18">n_N' \sim \mathcal{B}(n_N, \frac{1}{2}), \ \ \ n_C' \sim \mathcal{B}(n_C, \frac{1}{2})</script></div></div><p><span>Putting all these together, we can perform a modified Gillespie simulation over time. We are especially  interested in the parameter </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.69ex" height="2.344ex" viewBox="0 -755.9 1158.4 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E85-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E85-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E85-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E85-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span>, which varies with both the binding box&#39;s affinity and the concentration of dCas9. Therefore, we run the simulation with different </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.69ex" height="2.344ex" viewBox="0 -755.9 1158.4 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E85-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E85-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E85-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E85-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span> values while other parameters are unchanged . The simulation result shows that when </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.69ex" height="2.344ex" viewBox="0 -755.9 1158.4 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E85-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E85-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E85-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E85-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span> increase, the ratio of dCas9 bound to sgRNA tend to be greater and the cell line run out of the plasmid in less generations. (see pic </span><code>plasmid_time_series</code><span>, </span><code>plasmid_clearance_generation.png</code><span>)</span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvtydoa3j30yi0iadik.jpg' alt='image-20191008012315005' referrerPolicy='no-referrer' /></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvu1oq3nj30fk0bojrh.jpg' alt='image-20191008012315005' referrerPolicy='no-referrer' /></p><h2><a name="gene-expression-noise-control" class="md-header-anchor"></a><span>Gene Expression Noise Control</span></h2><p><span>In this section we model to illustrate how our system can control the expression noise in a cell. Gene expression noise is explained as fluctuation of &quot;very low copy numbers of many components&quot; leading to &quot;large amounts of cell-cell variation observed in isogenic populations&quot;. This noise can be either intrinsic or extrinsic. Extrinsic noise include most of the environmental factors like the nutrient and antibiotics, gene expression regulation by inhibitor or enhancer, and also the gene copy number in the cell.</span></p><p><span>In a fast growing cell, the copy number of OriC can be more than ten. In these cases, genes near the OriC may express more than genes far from OriC. Ting Lu deduced from Helmstetter-Copper model a relation between the gene&#39;s relative location to OriC and gene&#39;s copy number:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n161" cid="n161" mdtype="math_block">
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id="MathJax-Element-19">g_{q}(\lambda)=g_{0, q} \exp \left(\lambda\left[\left(1-x_{q}\right) C+D\right]\right)</script></div></div><p><span>where </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.096ex" height="1.994ex" viewBox="0 -504.6 902.3 858.4" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="0" id="E86-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 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copy number, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.994ex" viewBox="0 -755.9 583 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E126-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E126-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\lambda</script><span> is the cell&#39;s growth rate and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.316ex" height="1.994ex" viewBox="0 -504.6 997.3 858.4" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="0" id="E88-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E88-MJMATHI-71" d="M33 157Q33 258 109 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However, both Helmstetter and Cooper&#39;s model and Ting Lu&#39;s model are deterministic and unable to be applied to analyze the random factors. Daniel L. Jones et.al included gene copy number variance as a factor of noise, but the model was coarse-grained and only genes with only one replication fork are considered. To fully expose the &quot;noisy&quot; nature of intracellular gene copy number variance, we describe the genome&#39;s replication as a stochastic process. We introduce a parameter </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.21ex" height="1.994ex" viewBox="0 -755.9 521 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E99-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E99-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span> representing the frequency that of a single OriC site forming a new replication fork, and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.635ex" height="1.877ex" viewBox="0 -755.9 704 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E108-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E108-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> as the time it takes for the replication complex to replicate the whole genome. We deduce that this process is a Yule-Furry process to gene&#39;s relative location to OriC with parameter </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.845ex" height="1.994ex" viewBox="0 -755.9 1225 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E91-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E91-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E91-MJMATHI-6B" x="0" y="0"></use><use xlink:href="#E91-MJMATHI-54" x="521" y="0"></use></g></svg></span><script type="math/tex">kT</script><span>.</span></p><p><span>In our system, the genome or plasmid DNA replication is blocked by dCas9. This can prevent the genome from forming new replication forks. Specifically, our system decreases </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.21ex" height="1.994ex" viewBox="0 -755.9 521 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E99-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E99-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>, the frequency of replication fork formation.</span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvu4noh3g30fl08hk9e.gif' alt='' referrerPolicy='no-referrer' /></p><h3><a name="hypothesis" class="md-header-anchor"></a><span>Hypothesis</span></h3><ol start='' ><li><span>Each cell contains only one set of genome, which may contain multiple replication forks so that each gene&#39;s copy number in the genome may be different. The cell divide immediately after genome replication. (Our experiment results show that some cells actually contains more than one set of genome. These cells are very long and in each cell genomes are distant from each other. We treat this kind of cell as a chain of multiple cells)</span></li><li><span>The cell containing the studied genome has been replicating exponentially for several generations in a stable environment, thus the replication fork&#39;s distribution on the genome is steady (i.e. sampled from a fixed distribution)</span></li><li><span>The replication fork only forms at OriC and the formations of replication forks are independent with each other.</span></li><li><span>The frequency that of a single OriC site forming a new replication fork </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.21ex" height="1.994ex" viewBox="0 -755.9 521 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E99-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E99-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span> and the time it takes for the replication complex to replicate the whole genome </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.635ex" height="1.877ex" viewBox="0 -755.9 704 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E108-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E108-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> are constant.</span></li><li><span>Our system works by decreasing </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.21ex" height="1.994ex" viewBox="0 -755.9 521 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E99-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E99-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>. This process is a fast grade 1 process.</span></li></ol><h3><a name="analysis-of-gene-copy-number-and-its-fluctuation" class="md-header-anchor"></a><span>Analysis of gene copy number and its fluctuation</span></h3><p><span>To study the copy number of each in a given branchy genome, we firstly need to know how this genome and its replication forks are formed. Considering a newly formed gene. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.635ex" height="1.877ex" viewBox="0 -755.9 704 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E108-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E108-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> ago, the replication fork forming the current genome was just newly formed and the OriC number of this branch is 1. In the next </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.635ex" height="1.877ex" viewBox="0 -755.9 704 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E108-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E108-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> period, DNA  replication complex bind to OriC to form new replication fork, creating a new OriC for this branch. According to hypothesis 3 and 4, the copy increasing process of OriC through the time period </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.635ex" height="1.877ex" viewBox="0 -755.9 704 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E108-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E108-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> is a </span><a href='https://onlinelibrary.wiley.com/doi/pdf/10.1002/0470011815.b2a07059'><span>Yule-Furry process</span></a><span> (a stochastic counterpart to the deterministic exponential growth model) with parameter </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.21ex" height="1.994ex" viewBox="0 -755.9 521 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E99-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E99-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>. According to the theory of Yule-Furry process, the probabilty that the number of OriC is </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.039ex" height="1.41ex" viewBox="0 -504.6 878 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E133-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E133-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex">m</script><span> at time </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.056ex" height="2.577ex" viewBox="0 -806.1 5621.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E101-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E101-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E101-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E101-MJMAIN-2264" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path stroke-width="0" id="E101-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E101-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E101-MJMATHI-74" x="0" y="0"></use><use xlink:href="#E101-MJMAIN-28" x="611" y="0"></use><use xlink:href="#E101-MJMAIN-30" x="1000" y="0"></use><use xlink:href="#E101-MJMAIN-2264" x="1777" y="0"></use><use xlink:href="#E101-MJMATHI-74" x="2833" y="0"></use><use xlink:href="#E101-MJMAIN-2264" x="3472" y="0"></use><use xlink:href="#E101-MJMATHI-54" x="4528" y="0"></use><use xlink:href="#E101-MJMAIN-29" x="5232" y="0"></use></g></svg></span><script type="math/tex">t\ (0\leq t \leq T)</script><span> is:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n179" cid="n179" mdtype="math_block">
+		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="42.961ex" height="2.928ex" viewBox="-39 -956.9 18497.1 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.091ex; max-width: 100%;"><defs><path stroke-width="0" id="E21-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 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transform="scale(0.707)" xlink:href="#E21-MJMATHI-74" x="1299" y="0"></use></g></g><g transform="translate(16444,0)"><use xlink:href="#E21-MJMAIN-29" x="0" y="0"></use><g transform="translate(389,412)"><use transform="scale(0.707)" xlink:href="#E21-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-2212" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-31" x="1655" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-20">p_m(t) = \mathrm{Pr}[N(t) = m] = \mathrm{e}^{-k t}(1 - \mathrm{e}^{-k t})^{m - 1}</script></div></div><p><span>(Note: according to hypothesis 1, the initial number of OriC is 1, therefore we are using the formula under the condition </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.706ex" height="2.344ex" viewBox="0 -755.9 2887.1 1009.2" role="img" 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type="math/tex">n_0=1</script><span>)</span></p><p><span>Furthermore, the DNA replication complex is moving from OriC to Ter in a constant speed after forming the replication fork at OriC. Suppose we have known that the number of OriC at time </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.838ex" height="1.877ex" viewBox="0 -705.6 361 808.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E112-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E112-MJMATHI-74" x="0" y="0"></use></g></svg></span><script type="math/tex">t</script><span> is </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.039ex" height="1.41ex" viewBox="0 -504.6 878 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E133-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E133-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex">m</script><span>, then after </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.313ex" height="2.11ex" viewBox="0 -755.9 2287.4 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E106-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E106-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E106-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E106-MJMATHI-54" x="0" y="0"></use><use xlink:href="#E106-MJMAIN-2212" x="926" y="0"></use><use xlink:href="#E106-MJMATHI-74" x="1926" y="0"></use></g></svg></span><script type="math/tex">T - t</script><span> forms the genome we are studying. In this </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.313ex" height="2.11ex" viewBox="0 -755.9 2287.4 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E106-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E106-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E106-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E106-MJMATHI-54" x="0" y="0"></use><use xlink:href="#E106-MJMAIN-2212" x="926" y="0"></use><use xlink:href="#E106-MJMATHI-74" x="1926" y="0"></use></g></svg></span><script type="math/tex">T - t</script><span> period, all the DNA replication complex move forward a distance </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.916ex" height="2.577ex" viewBox="0 -806.1 4269.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E109-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E109-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E109-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E109-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E109-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E109-MJMAIN-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E109-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E109-MJMATHI-54" x="389" y="0"></use><use xlink:href="#E109-MJMAIN-2212" x="1315" y="0"></use><use xlink:href="#E109-MJMATHI-74" x="2315" y="0"></use><use xlink:href="#E109-MJMAIN-29" x="2676" y="0"></use><use xlink:href="#E109-MJMAIN-2F" x="3065" y="0"></use><use xlink:href="#E109-MJMATHI-54" x="3565" y="0"></use></g></svg></span><script type="math/tex">(T - t)/T</script><span> relative to the genome. Therefore, after </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.635ex" height="1.877ex" viewBox="0 -755.9 704 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E108-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E108-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> the copy number of gene at </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.916ex" height="2.577ex" viewBox="0 -806.1 4269.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E109-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E109-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E109-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E109-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E109-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E109-MJMAIN-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E109-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E109-MJMATHI-54" x="389" y="0"></use><use xlink:href="#E109-MJMAIN-2212" x="1315" y="0"></use><use xlink:href="#E109-MJMATHI-74" x="2315" y="0"></use><use xlink:href="#E109-MJMAIN-29" x="2676" y="0"></use><use xlink:href="#E109-MJMAIN-2F" x="3065" y="0"></use><use xlink:href="#E109-MJMATHI-54" x="3565" y="0"></use></g></svg></span><script type="math/tex">(T - t)/T</script><span> relative to OriC, or </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.635ex" height="2.577ex" viewBox="0 -806.1 1565 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E110-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E110-MJMAIN-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path stroke-width="0" id="E110-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E110-MJMATHI-74" x="0" y="0"></use><use xlink:href="#E110-MJMAIN-2F" x="361" y="0"></use><use xlink:href="#E110-MJMATHI-54" x="861" y="0"></use></g></svg></span><script type="math/tex">t/T</script><span> relative to Ter, equals to </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.039ex" height="1.41ex" viewBox="0 -504.6 878 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E133-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E133-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex">m</script><span>, the number of OriC at time </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.838ex" height="1.877ex" viewBox="0 -705.6 361 808.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E112-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E112-MJMATHI-74" x="0" y="0"></use></g></svg></span><script type="math/tex">t</script><span>.</span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvu79n71j31300ryq3m.jpg' alt='noise_ctrl_mech1' referrerPolicy='no-referrer' /></p><p><span>We introduce </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.061ex" height="2.577ex" viewBox="0 -806.1 3470.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E113-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E113-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E113-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E113-MJMAIN-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path stroke-width="0" id="E113-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E113-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E113-MJMAIN-3D" x="849" y="0"></use><use xlink:href="#E113-MJMATHI-74" x="1905" y="0"></use><use xlink:href="#E113-MJMAIN-2F" x="2266" y="0"></use><use xlink:href="#E113-MJMATHI-54" x="2766" y="0"></use></g></svg></span><script type="math/tex">x=t/T</script><span> as a site&#39;s relative distance to Ter in the genome (</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.846ex" height="2.227ex" viewBox="0 -755.9 4239.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E114-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E114-MJMAIN-2264" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path stroke-width="0" id="E114-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E114-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E114-MJMAIN-30" x="0" y="0"></use><use xlink:href="#E114-MJMAIN-2264" x="777" y="0"></use><use xlink:href="#E114-MJMATHI-78" x="1833" y="0"></use><use xlink:href="#E114-MJMAIN-2264" x="2683" y="0"></use><use xlink:href="#E114-MJMAIN-31" x="3739" y="0"></use></g></svg></span><script type="math/tex">0\leq x \leq 1</script><span>, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.587ex" height="1.994ex" viewBox="0 -755.9 2405.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E115-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E115-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E115-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E115-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E115-MJMAIN-3D" x="849" y="0"></use><use xlink:href="#E115-MJMAIN-30" x="1905" y="0"></use></g></svg></span><script type="math/tex">x = 0</script><span> corresponding to Ter, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.587ex" height="1.994ex" viewBox="0 -755.9 2405.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E116-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E116-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E116-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E116-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E116-MJMAIN-3D" x="849" y="0"></use><use xlink:href="#E116-MJMAIN-31" x="1905" y="0"></use></g></svg></span><script type="math/tex">x = 1</script><span> corresponding to OriC), then we get</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n184" cid="n184" mdtype="math_block">
+		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="29.731ex" height="2.928ex" viewBox="-39 -956.9 12801 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.091ex; max-width: 100%;"><defs><path stroke-width="0" id="E22-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="0" id="E22-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 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x="6661" y="0"></use><use xlink:href="#E22-MJMAIN-2212" x="7383" y="0"></use><g transform="translate(8383,0)"><use xlink:href="#E22-MJMAIN-65" x="0" y="0"></use><g transform="translate(444,412)"><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-6B" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-54" x="1299" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-78" x="2002" y="0"></use></g></g><g transform="translate(10748,0)"><use xlink:href="#E22-MJMAIN-29" x="0" y="0"></use><g transform="translate(389,412)"><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-2212" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-31" x="1655" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-21">p_m(x) = \mathrm{e}^{-k T x}(1 - \mathrm{e}^{-k T x})^{m - 1}</script></div></div><p><span>This equation determines the copy number distribution of all the genes at all the sites. We can further deduce</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n186" cid="n186" mdtype="math_block">
+		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.351ex" height="2.928ex" viewBox="0 -956.9 6609.2 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex; max-width: 100%;"><defs><path stroke-width="0" id="E23-MJAMS-45" d="M12 666Q12 675 24 683H582Q590 680 593 672V588Q593 514 591 502T575 490Q567 490 563 495T555 517Q552 556 517 590Q486 623 445 634T340 648H282Q266 636 264 620T260 492V370H277Q329 375 358 391T404 439Q420 480 420 506Q420 529 436 529Q445 529 451 521Q455 517 455 361Q455 333 455 298T456 253Q456 217 453 207T437 197Q420 196 420 217Q420 240 406 270Q377 328 284 335H260V201Q261 174 261 134Q262 73 264 61T278 38Q281 36 282 35H331Q400 35 449 50Q571 93 602 179Q605 203 622 203Q629 203 634 197T640 183Q638 181 624 95T604 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344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path stroke-width="0" id="E23-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E23-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E23-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E23-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="0" id="E23-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E23-MJMAIN-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 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Tx}</script></div></div><p><span>The nearer the gene is to OriC, the greater </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E124-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 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indicate the intensity of fluctuation because the average expression is also increasing. We apply CV value, the ratio of standard deviation to the average, to more precisely describe the  intensity of fluctuation:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n191" cid="n191" mdtype="math_block">
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y="0"></use><use transform="scale(0.707)" xlink:href="#E26-MJMATHI-6D" x="1200" y="-213"></use><use xlink:href="#E26-MJMAIN-2B" x="1792" y="0"></use><use xlink:href="#E26-MJMAIN-5B" x="2792" y="0"></use><g transform="translate(3070,0)"><use xlink:href="#E26-MJMAIN-64" x="0" y="0"></use><use xlink:href="#E26-MJMAIN-43" x="556" y="0"></use><use xlink:href="#E26-MJMAIN-61" x="1278" y="0"></use><use xlink:href="#E26-MJMAIN-73" x="1778" y="0"></use><use xlink:href="#E26-MJMAIN-39" x="2172" y="0"></use></g><use xlink:href="#E26-MJMAIN-5D" x="5742" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-25">k = \frac{K_m\cdot k_0}{K_m+[\mathrm{dCas9}]}</script></div></div><p>&nbsp;</p><h2><a name="productivity" class="md-header-anchor"></a><span>Productivity</span></h2><p><span>In this section we provide a model to explain why our system is able to increase the cells&#39; productivity to multiple types of products like GFP and indigo. It is somehow contrasting our intuition that a decline in microbiomes&#39; growth rate may increase the microbial productivity. Therefore, it is necessary to carefully test this issue in a quantitive method.</span></p><p><span>The undesired productivity of microbiomes is thwarting this technology from being more widely applicable. Therefore, it is always concerned how to improve the productivity of microbiomes, or how to enforce these wild or engineered cells to turn more of the substrates we feed them into the bio-products we want. </span></p><p><span>A non-negligible factor thwarding the bicrobiomes from producing our bioproduct is the growth of the bicrobiomes themselve. All the cell products we want to yield from the microbiomes are products of the microbiomes&#39; own metabolism forming the microbiomes&#39; own biomass. Therefore, their production is under strict regulation of the microbiomes. In most engineered microbiomes, the genes we introduce to the microbiomes are foreign genes unrelated to the cells&#39; growth and normal metabolism, and the production of these genes are undesired for the microbiomes themselves. The engineered microbiomes, stressed by these products, will activate all the possible pathways to change its gene expression pattern and to reallocate its nutrients and enzymes to produce more &quot;necessities&quot; for themselves. This is particularly true for bicrobiomes under steady growth rate, whose nutrients are mainly used for growth.</span></p><p><span>Actually, many studies concerns the relationship between microbiomes&#39; growth and the way they allocate their resources. Terence Hwa </span><em><span>et.al</span></em><span> studied </span><em><span>E. coli</span></em><span>&#39;s metabolism under different nutrient and antibiotic condition by dividing the cells&#39; proteins into different sectors. Particularly in this work, a sector defined as &quot;Unnecessary Expression&quot;(corresponding to the bio-products we want) is found to be negatively related to the growth rate. The authors described this relation semi-quantitively as</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n201" cid="n201" mdtype="math_block">
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-949 39 -947T34 -937Q88 -883 140 -813Q428 -430 428 251Q428 453 402 628T338 922T245 1146T145 1309T46 1425Q44 1427 42 1429T39 1433T36 1436L34 1438Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-3BB" x="0" y="0"></use><use xlink:href="#E27-MJMAIN-28" x="583" y="0"></use><g transform="translate(972,0)"><use xlink:href="#E27-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E27-MJMATHI-55" x="842" y="-213"></use></g><use xlink:href="#E27-MJMAIN-29" x="2210" y="0"></use><use xlink:href="#E27-MJMAIN-3D" x="2877" y="0"></use><use xlink:href="#E27-MJMATHI-3BB" x="3932" y="0"></use><use xlink:href="#E27-MJMAIN-28" x="4515" y="0"></use><g transform="translate(4904,0)"><use xlink:href="#E27-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E27-MJMATHI-55" x="842" y="-213"></use></g><use xlink:href="#E27-MJMAIN-3D" x="6421" y="0"></use><use xlink:href="#E27-MJMAIN-30" x="7476" y="0"></use><use xlink:href="#E27-MJMAIN-29" x="7976" y="0"></use><g transform="translate(8532,0)"><use xlink:href="#E27-MJSZ3-28"></use><use xlink:href="#E27-MJMAIN-31" x="736" y="0"></use><use xlink:href="#E27-MJMAIN-2212" x="1458" y="0"></use><g transform="translate(2236,0)"><g transform="translate(342,0)"><rect stroke="none" width="1358" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#E27-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E27-MJMATHI-55" x="842" y="-213"></use></g><g transform="translate(178,-686)"><use xlink:href="#E27-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E27-MJMATHI-63" x="842" y="-213"></use></g></g></g><use xlink:href="#E27-MJSZ3-29" x="4056" y="-1"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-26">\lambda(\phi_U) = \lambda(\phi_U = 0)\left(1 - \frac{\phi_U}{\phi_c}\right)</script></div></div><p><span>were </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.876ex" height="2.461ex" viewBox="0 -755.9 1238.4 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E127-MJMATHI-3D5" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 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transform="scale(0.707)" xlink:href="#E127-MJMATHI-55" x="842" y="-213"></use></g></svg></span><script type="math/tex">\phi_U</script><span> is the ratio of &quot;Unnecessary&quot; protein mass to the total protein mass of a cell, and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.994ex" viewBox="0 -755.9 583 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E126-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E126-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\lambda</script><span> is the growth rate corresponding to the ratio </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.876ex" height="2.461ex" viewBox="0 -755.9 1238.4 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E127-MJMATHI-3D5" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z"></path><path stroke-width="0" id="E127-MJMATHI-55" d="M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E127-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E127-MJMATHI-55" x="842" y="-213"></use></g></svg></span><script type="math/tex">\phi_U</script><span>. Ting Lu </span><em><span>et. al</span></em><span> built a coarse-grained whole-cell model including the productions and functions of proteins of different sectors and further validated Hwa&#39;s model. In this model, the regulation performed by ppGpp is included and this provides an explanation of the phenomenological result in Hwa&#39;s work. ppGpp is an important gene expression regulation molecule responding to various types of environmental stress and cell&#39;s abnormal pnysiology. One of its function is to down-regulate the expression of ribosomal RNA, protein and expression affiliated proteins while up-regulate the expression of some enzymes relating to the cell&#39;s core metabolism, adapting the cell from a fast-growing state to a slow-growing state. Some optimization models not explicitly including ppGpp also discovers a similarity between cells&#39; optimal resource allocation strategies and the ppGpp regulation strategies. </span></p><p><span>All the works mentioned above regard the growth rate as the result or equivalence of the microbiomes&#39; protein accumulation. In our system, however, the growth rate is hacked. Because of this, the causal relationship between the cell physiology (growth rate) and the nutrient reallocation is different from these  works. A better explanation of the mechanics of our system is that the hacked growth rate freed the cells from producing too much growth-necessary proteins and enabled them to produce the bio-product we want. We emphasize that our model is still a phenominological model in which the resource allocation regulation is finished in a &quot;black box&quot;, not explicitly related to the regulation of any single pathway.</span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvub1zpxj31fo0sfgnq.jpg' alt='' referrerPolicy='no-referrer' /></p><p><span>In our model, there is a single-source nutrient. An engineered cell uptakes the nutrient and uses it both for its own growth and for the production of the bio-product. The ratio of nutrient used for cell&#39;s own growth to the total nutrient uptaken by the microbiomes is </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E140-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E140-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span>, and the ratio of nutrient used for bio-product production to the total nutrient uptaken by the microbiomes is </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.487ex" height="2.11ex" viewBox="0 -755.9 2362.4 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E129-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E129-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E129-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E129-MJMAIN-31" x="0" y="0"></use><use xlink:href="#E129-MJMAIN-2212" x="722" y="0"></use><use xlink:href="#E129-MJMATHI-3B1" x="1722" y="0"></use></g></svg></span><script type="math/tex">1 - \alpha</script><span>. Thus, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.745ex" height="1.994ex" viewBox="0 -755.9 2473.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E130-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="0" id="E130-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E130-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E130-MJMATHI-3B1" x="0" y="0"></use><use xlink:href="#E130-MJMAIN-3D" x="917" y="0"></use><use xlink:href="#E130-MJMAIN-30" x="1973" y="0"></use></g></svg></span><script type="math/tex">\alpha=0</script><span> corresponds to a &quot;production-only&quot; state and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.745ex" height="1.994ex" viewBox="0 -755.9 2473.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E131-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="0" id="E131-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E131-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E131-MJMATHI-3B1" x="0" y="0"></use><use xlink:href="#E131-MJMAIN-3D" x="917" y="0"></use><use xlink:href="#E131-MJMAIN-31" x="1973" y="0"></use></g></svg></span><script type="math/tex">\alpha=1</script><span> corresponds to a &quot;growth-only&quot; state. This process is written as an ODE system</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n206" cid="n206" mdtype="math_block">
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y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-27">\begin{align}
+\frac{\mathrm{d}}{\mathrm{d}t}n &= -\frac{k_n n}{n+K_n}m\\
+\frac{\mathrm{d}}{\mathrm{d}t}m &= \frac{k_m n}{n+K_n}m\alpha\\
+\frac{\mathrm{d}}{\mathrm{d}t}p &= \frac{k_p n}{n+K_n}m(1-\alpha)\\
+\end{align}</script></div></div><p><span>where </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.394ex" height="1.41ex" viewBox="0 -504.6 600 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E132-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E132-MJMATHI-6E" x="0" y="0"></use></g></svg></span><script type="math/tex">n</script><span> is the mass of nutrient, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.428ex" height="2.344ex" viewBox="0 -755.9 1045.3 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E135-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E135-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E135-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E135-MJMATHI-6E" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_n</script><span> is the cell&#39;s nutrient uptake rate, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.268ex" height="2.577ex" viewBox="0 -755.9 976.7 1109.7" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="0" id="E137-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E137-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E137-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E137-MJMATHI-70" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_p</script><span> is maximum production rate when the nutrient is sufficient and all the uptaken nutrient are used for the bio-product&#39;s production, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.19ex" height="2.344ex" viewBox="0 -755.9 1373.3 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E138-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path stroke-width="0" id="E138-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E138-MJMATHI-4B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-6E" x="1200" y="-213"></use></g></svg></span><script type="math/tex">K_n</script><span> is the nutrient corresponding to half-maximum nutrient uptaking rate, and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E140-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E140-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span> is the above-mentioned allocation ratio.</span></p><p><span>Noticing that for a given </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E140-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E140-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span>, when the nutrient is sufficient, the second ODE can be re-written as </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n209" cid="n209" mdtype="math_block">
+		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-28-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.19ex" height="5.145ex" viewBox="0 -1459.5 6970.8 2215.4" role="img" focusable="false" style="vertical-align: -1.756ex; max-width: 100%;"><defs><path stroke-width="0" id="E29-MJMAIN-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path stroke-width="0" id="E29-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E29-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E29-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E29-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E29-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="0" 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y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-28">\frac{\mathrm{d}}{\mathrm{d}t}m = k_m\alpha\cdot m</script></div></div><p><span>This corresponds to an exponential growth with a growth rate </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.822ex" height="2.344ex" viewBox="0 -755.9 3798.4 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E141-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E141-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 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id="E141-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E141-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E141-MJMATHI-3BB" x="0" y="0"></use><use xlink:href="#E141-MJMAIN-3D" x="860" y="0"></use><g transform="translate(1916,0)"><use xlink:href="#E141-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E141-MJMATHI-6D" x="736" y="-213"></use></g><use xlink:href="#E141-MJMATHI-3B1" x="3158" y="0"></use></g></svg></span><script type="math/tex">\lambda = k_m\alpha</script><span>.</span></p><div contenteditable="false" 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xlink:href="#E30-MJMATHI-4B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E30-MJMATHI-6E" x="1200" y="-213"></use></g></g></g></g><g transform="translate(4889,0)"><use xlink:href="#E30-MJSZ3-28"></use><use xlink:href="#E30-MJMAIN-31" x="736" y="0"></use><use xlink:href="#E30-MJMAIN-2212" x="1458" y="0"></use><g transform="translate(2236,0)"><g transform="translate(342,0)"><rect stroke="none" width="1361" height="60" x="0" y="220"></rect><use xlink:href="#E30-MJMATHI-3BB" x="389" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E30-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E30-MJMATHI-6D" x="736" y="-213"></use></g></g></g><use xlink:href="#E30-MJSZ3-29" x="4060" y="-1"></use></g><use xlink:href="#E30-MJMATHI-6D" x="9852" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-29">\begin{align}
+\frac{\mathrm{d}}{\mathrm{d}t}n &= -\frac{k_n n}{n+K_n}m\\
+\frac{\mathrm{d}}{\mathrm{d}t}m &= \frac{n\lambda}{n+K_n}m\\
+\frac{\mathrm{d}}{\mathrm{d}t}p &= \frac{k_p n}{n+K_n}\left(1-\frac{\lambda}{k_m}\right)m\\
+\end{align}</script></div></div><p><span>the equation explicitly includes the growth rate. </span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvudu1ijj30fk0bo3yq.jpg' alt='' referrerPolicy='no-referrer' /></p><p><span>The ODE is difficult to solve but we can analyze the steady states about it.</span></p><p><span>suppose initially </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="32.625ex" height="2.577ex" viewBox="0 -806.1 14046.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E142-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 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type="math/tex">m(0) = m_0, n(0) = n_0, p(0) = p_0</script><span>, noticing that </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n216" cid="n216" mdtype="math_block">
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xlink:href="#E31-MJMAIN-2212" x="2829" y="0"></use><use xlink:href="#E31-MJMATHI-3BB" x="3829" y="0"></use><use xlink:href="#E31-MJMAIN-29" x="4412" y="0"></use></g></g></g><g transform="translate(18602,0)"><g transform="translate(120,0)"><rect stroke="none" width="1037" height="60" x="0" y="220"></rect><use xlink:href="#E31-MJMAIN-64" x="240" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E31-MJMAIN-64" x="0" y="0"></use><use xlink:href="#E31-MJMATHI-74" x="556" y="0"></use></g></g></g><use xlink:href="#E31-MJMATHI-70" x="19879" y="0"></use><use xlink:href="#E31-MJMAIN-3D" x="20660" y="0"></use><use xlink:href="#E31-MJMAIN-30" x="21716" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-30">\frac{1}{k_n}\frac{\mathrm{d}}{\mathrm{d}t}n + \frac{1}{\lambda}\frac{\mathrm{d}}{\mathrm{d}t}m = \frac{1}{k_n}\frac{\mathrm{d}}{\mathrm{d}t}n + \frac{k_m}{k_p(k_m-\lambda)}\frac{\mathrm{d}}{\mathrm{d}t}p = 0</script></div></div><p><span>thus we know that </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.978ex" height="4.912ex" viewBox="0 -1208.2 3865.7 2114.9" role="img" focusable="false" style="vertical-align: -2.106ex;"><defs><path stroke-width="0" id="E143-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E143-MJMATHI-6B" d="M121 647Q121 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xlink:href="#E143-MJMATHI-3BB" x="207" y="-686"></use></g></g></g></svg></span><script type="math/tex">\displaystyle \frac{n}{k_n} + \frac{m}{\lambda}</script><span> and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.092ex" height="5.846ex" viewBox="0 -1459.5 7789.7 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="0" id="E144-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 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xlink:href="#E144-MJMAIN-28" x="976" y="0"></use><g transform="translate(1365,0)"><use xlink:href="#E144-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E144-MJMATHI-6D" x="736" y="-213"></use></g><use xlink:href="#E144-MJMAIN-2212" x="2829" y="0"></use><use xlink:href="#E144-MJMATHI-3BB" x="3829" y="0"></use><use xlink:href="#E144-MJMAIN-29" x="4412" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\displaystyle \frac{n}{k_n} + \frac{k_m p}{k_p(k_m-\lambda)}</script><span> remain constant in the whole dynamic process. Provided the initial condition, we can deduce</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n218" cid="n218" mdtype="math_block">
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id="MathJax-Element-31">\begin{cases}
+\displaystyle \frac{n}{k_n} + \frac{m}{\lambda} = \frac{n_0}{k_n} + \frac{m_0}{\lambda} \\
+\displaystyle \frac{n}{k_n} + \frac{k_m p}{k_p(k_m-\lambda)} = \frac{n_0}{k_n} + \frac{k_m p_0}{k_p(k_m-\lambda)}
+\end{cases}</script></div></div><p><span>holds throughout the growth-and-production process. Finally, the nutrients are exhausted and the microbiomes stop both growing and producing. Denoting the concentration of all the materials at the final state as </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.518ex" height="2.344ex" viewBox="0 -755.9 3237.1 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E145-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E145-MJMAIN-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path stroke-width="0" id="E145-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E145-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E145-MJMATHI-6E" x="0" y="0"></use><g transform="translate(600,362)"><use transform="scale(0.707)" xlink:href="#E145-MJMAIN-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E145-MJMAIN-73" x="394" y="0"></use></g><use xlink:href="#E145-MJMAIN-2C" x="1257" y="0"></use><g transform="translate(1701,0)"><use xlink:href="#E145-MJMATHI-6D" x="0" y="0"></use><g transform="translate(878,362)"><use transform="scale(0.707)" xlink:href="#E145-MJMAIN-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E145-MJMAIN-73" x="394" y="0"></use></g></g></g></svg></span><script type="math/tex">n^{\mathrm{ss}}, m^{\mathrm{ss}}</script><span> and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.785ex" height="2.344ex" viewBox="-39 -755.9 1199.2 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex; margin-left: -0.091ex;"><defs><path stroke-width="0" id="E148-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="0" id="E148-MJMAIN-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E148-MJMATHI-70" x="0" y="0"></use><g transform="translate(503,362)"><use transform="scale(0.707)" xlink:href="#E148-MJMAIN-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E148-MJMAIN-73" x="394" y="0"></use></g></g></svg></span><script type="math/tex">p^{\mathrm{ss}}</script><span>, the above-mentioned equation still holds:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n220" cid="n220" mdtype="math_block">
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xlink:href="#E33-MJMAIN-28" x="976" y="0"></use><g transform="translate(1365,0)"><use xlink:href="#E33-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E33-MJMATHI-6D" x="736" y="-213"></use></g><use xlink:href="#E33-MJMAIN-2212" x="2829" y="0"></use><use xlink:href="#E33-MJMATHI-3BB" x="3829" y="0"></use><use xlink:href="#E33-MJMAIN-29" x="4412" y="0"></use></g></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-32">\begin{cases}
+\displaystyle \frac{n^{\mathrm{ss}}}{k_n} + \frac{m^{\mathrm{ss}}}{\lambda} = \frac{n_0}{k_n} + \frac{m_0}{\lambda} \\
+\displaystyle \frac{n^{\mathrm{ss}}}{k_n} + \frac{k_m p^{\mathrm{ss}}}{k_p(k_m-\lambda)} = \frac{n_0}{k_n} + \frac{k_m p_0}{k_p(k_m-\lambda)}
+\end{cases}</script></div></div><p><span>moreover, at the final stage, the nutrients are exhausted so </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.179ex" height="1.994ex" viewBox="0 -755.9 3090.8 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E147-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E147-MJMAIN-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path stroke-width="0" id="E147-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E147-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E147-MJMATHI-6E" x="0" y="0"></use><g transform="translate(600,362)"><use transform="scale(0.707)" xlink:href="#E147-MJMAIN-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E147-MJMAIN-73" x="394" y="0"></use></g><use xlink:href="#E147-MJMAIN-3D" x="1534" y="0"></use><use xlink:href="#E147-MJMAIN-30" x="2590" y="0"></use></g></svg></span><script type="math/tex">n^{\mathrm{ss}}=0</script><span>. Then we can solve</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n222" cid="n222" mdtype="math_block">
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y="220"></rect><g transform="translate(60,752)"><use xlink:href="#E34-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E34-MJMAIN-30" x="848" y="-213"></use><g transform="translate(1053,0)"><use xlink:href="#E34-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E34-MJMATHI-70" x="736" y="-213"></use></g></g><g transform="translate(552,-686)"><use xlink:href="#E34-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E34-MJMATHI-6E" x="736" y="-213"></use></g></g></g><use xlink:href="#E34-MJMAIN-28" x="7062" y="0"></use><use xlink:href="#E34-MJMAIN-31" x="7451" y="0"></use><use xlink:href="#E34-MJMAIN-2212" x="8174" y="0"></use><g transform="translate(8952,0)"><g transform="translate(342,0)"><rect stroke="none" width="1361" height="60" x="0" y="220"></rect><use xlink:href="#E34-MJMATHI-3BB" x="389" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E34-MJMATHI-6B" x="0" y="0"></use><use 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+\displaystyle m^{\mathrm{ss}} = m_0 + \frac{\lambda n_0}{k_n} \\
+\displaystyle p^{\mathrm{ss}} = p_0 + \frac{n_0k_p}{k_n}(1 - \frac{\lambda}{k_m})
+\end{cases}</script></div></div><p><span>noticing that </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.785ex" height="2.344ex" viewBox="-39 -755.9 1199.2 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex; margin-left: -0.091ex;"><defs><path stroke-width="0" id="E148-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="0" id="E148-MJMAIN-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E148-MJMATHI-70" x="0" y="0"></use><g transform="translate(503,362)"><use transform="scale(0.707)" xlink:href="#E148-MJMAIN-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E148-MJMAIN-73" x="394" y="0"></use></g></g></svg></span><script type="math/tex">p^{\mathrm{ss}}</script><span> is negatively and linearly related to the growth rate. This result is in consistent to the empirical equation provided by Terence Hwa.</span></p><h2><a name="quorum-sensing" class="md-header-anchor"></a><span>Quorum Sensing</span></h2><p><span>In this section we model to illustrate how our system works coupling with a quorum sensing system. We want out system to realize the population&#39;s auto-regulation - the cells stop growing fast when they sense a lot of other cells crowding around it. We realize this by applying LuxI-LuxR system, in which a kind of small molecule call AHL. This system was firstly discovered in </span><em><span>V. fischeri</span></em><span> as a means of intercellular communication. The cells produce exceeding amount of AHL which can move either into or out of the cells. When the population is large or dense in a region, the local AHL concentration increases and the cells sense this high concentration and respond to this by up- or down-regulating some genes&#39; expression. In our system, the sensing of AHL results in an increment of dCas9, and consequently a down regulation in growth or replication rate (see </span><a href='http://www.baidu.com'><span>design</span></a><span> for the design and experiments)</span></p><h3><a name="a-simulation-of-donor-receptor-experiment" class="md-header-anchor"></a><span>A Simulation of Donor-Receptor Experiment</span></h3><p><span>Our first series of experiments involve the testing of our fore-mentioned logic. We separately introduced the AHL producing parts and the AHL sensing parts to two strains of </span><em><span>E. coli</span></em><span>. The former is called &quot;donor&quot; and the latter is called &quot;receptor&quot;. The receptor cells are evenly coated onto the solid medium and the donor cells were dropped at the center of the medium. It is expected that the donors AHL, the AHL difusses around and inhibits the growth of the receptors around it (see </span><a href='http://www.baidu.com'><span>design</span></a><span>)</span></p><p><span>Here we use a simple diffusion model and visualized simulation to describe this process. We suppose that the solid media is a 2D plane, the receptor is uniformly distributed on the plane, and the donor is dense at the center </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.162ex" height="2.577ex" viewBox="0 -806.1 2222.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E150-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E150-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E150-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E150-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E150-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E150-MJMAIN-30" x="389" y="0"></use><use xlink:href="#E150-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E150-MJMAIN-30" x="1333" y="0"></use><use xlink:href="#E150-MJMAIN-29" x="1833" y="0"></use></g></svg></span><script type="math/tex">(0, 0)</script><span> of the media (subjected to a normal distribution), thus the AHL production is subjected to a normal distribution centered at </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.162ex" height="2.577ex" viewBox="0 -806.1 2222.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E150-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E150-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E150-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E150-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E150-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E150-MJMAIN-30" x="389" y="0"></use><use xlink:href="#E150-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E150-MJMAIN-30" x="1333" y="0"></use><use xlink:href="#E150-MJMAIN-29" x="1833" y="0"></use></g></svg></span><script type="math/tex">(0, 0)</script><span>. A partial differential equation (PDE) can be derived from these hypothethes:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n229" cid="n229" mdtype="math_block">
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xlink:href="#E35-MJMAIN-2B" x="1172" y="0"></use><g transform="translate(975,0)"><use transform="scale(0.5)" xlink:href="#E35-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E35-MJMAIN-32" x="499" y="362"></use></g></g><g transform="translate(529,-455)"><use transform="scale(0.5)" xlink:href="#E35-MJMATHI-3C3" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E35-MJMAIN-32" x="572" y="288"></use></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-34">\frac{\partial u}{\partial t} = D(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})u + C\mathrm{e}^{-\frac{x^2 + y^2}{\sigma^2}}</script></div></div><p><span>where </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.516ex" height="1.877ex" viewBox="0 -504.6 1513.7 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path 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id="E151-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E151-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E151-MJMAIN-2C" x="572" y="0"></use><use xlink:href="#E151-MJMATHI-79" x="1016" y="0"></use></g></svg></span><script type="math/tex">x, y</script><span> are coordinates, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E152-MJMATHI-3C3" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E152-MJMATHI-3C3" x="0" y="0"></use></g></svg></span><script type="math/tex">\sigma</script><span> represents the radius of the donor colony, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.765ex" height="2.11ex" viewBox="0 -806.1 760 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E153-MJMATHI-43" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E153-MJMATHI-43" x="0" y="0"></use></g></svg></span><script type="math/tex">C</script><span>  and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.923ex" height="1.877ex" viewBox="0 -755.9 828 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E154-MJMATHI-44" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E154-MJMATHI-44" x="0" y="0"></use></g></svg></span><script type="math/tex">D</script><span> [=2.0μM/(L·cm</span><sup><span>2</span></sup><span>)] denotes the diffusion constant. The randomly distributed cells stop growing when the local concentration reach 10</span><sup><span>-6</span></sup><span>μM/L.</span></p><p><span>Here we numerically solve the equation with finite difference method. We draw an animation visualize this dynamic process. Furthermore, we randomly place 400 receptors on the plane to show their colony formation. It can be seen that the receptors distant from the center grow into colonies, while most of the receptors nearby the center grow into relatively small colonies or cannot grow into colonies. </span></p><p><img src='https://tva1.sinaimg.cn/large/006y8mN6ly1g7xvuha4zbg30c409khdv.gif' alt='' referrerPolicy='no-referrer' /></p><p>&nbsp;</p></div>
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Difference between revisions of "Team:Peking/Model"

Revision as of 12:45, 19 October 2019

Modeling

regulatory part

Introduction

Mono-regulation

Bi-regulation

The relationship between the expression of dCas9 and the cell's growth rate

Plasmid Copy number Hacking

Introduction

Gene Expression Noise Control

Hypothesis

Analysis of gene copy number and its fluctuation

Analysis of dCas9 binding process

Productivity

Quorum Sensing

A Simulation of Donor-Receptor Experiment