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Regulatory
dCas9 & Growth Rate
Plasmid Copy Number
Noise Control
Productivity
Quorum Sensing

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 bacterium. 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 applied a model to explain how our bacteria can improve the productivity of some specific bioproducts. Fifth, 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 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{cases} \displaystyle \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 \\ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}c = \beta_c m - \gamma_c c + k_-b - k_+cr\\ \displaystyle \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\\ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}b = -k_- b + k_+ c r - \gamma_b b \end{cases}

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

parameter	description	value
$n_m$	Hill's coefficient of pBAD promoter^[1]	1.427
$K_m$	Half-activate Arabinose concentration of pBAD promoter^[1]	0.5736%
$\alpha_m$	Maximum dCas9 mRNA expression rate^[2]	0.0011 nM·min^-1
$n_r$	Hill's coefficient of pTac promoter^[1]	3.859
$K_r$	Half-activate Arabinose concentration of pTac promoter^[1]	0.05137
$\alpha_r$	Maximum sgRNA expression rate^[2]	0.0011 nM·min^-1
$\beta_c$	dCas9 protein synthesis rate from dCas9 mRNA^[2]	0.0057 protein·transcript^-1·min^-1
$\gamma_c$	dCas9 degradation rate^[2]	5.6408×10^-4min^-1
$k_+$	Rate of dimerization of dCas9 and sgRNA^[2]	1.0

By munerically solving the ODE system under different arabinose and IPTG concentration, we can show that the expression output follows the AND gate logic: the concentration of dCas9-sgRNA is only considerable when the concentration of both IPTG and Arabinose are high enough.

Steady State of the Bi-regulation model

To study the steady state of the system more carefully, 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{cases} \displaystyle \frac{[\mathrm{Ara}]^{n_m}\alpha_m}{[\mathrm{Ara}]^{n_m}+K_m^{n_m}} - \gamma_m m = 0 \\ \displaystyle \beta_c m - \gamma_c c + k_-b - k_+cr = 0\\ \displaystyle \frac{[\mathrm{IPTG}]^{n_r}\alpha_r}{[\mathrm{IPTG}]^{n_r}+K_r^{n_r}} - \gamma_r r + k_- b - k_+ c r = 0\\ \displaystyle -k_- b + k_+ c r - \gamma_b b = 0 \end{cases}

$b$ ).

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

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

$\gamma_m \rightarrow +\infty$ $(\gamma_c \gamma_r k_- + \gamma_c \gamma_b \gamma_r + k_+ \gamma_b \alpha_r)/(k_+ \gamma_b^2)$ $\gamma_m$ 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{cases} \displaystyle A = \frac{k_+ \gamma_b^2}{\gamma_c\gamma_r}b^2 \\ \displaystyle 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}} \\ \displaystyle 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{cases}

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.

Here, the AND gate logic picture is more clear. We get a dCas9-sgRNA concentration over 10^-3nM only when the concentrations of both IPTG and sgRNA are high enough. It can also be seen that the Arabinose regulation has a wider dynamic range. This guided us to use more Arabinose regulation in our actual experiments.

Conclusion

This model theoretically explains the mechanisms of our system's regulation parts. Regulation parts in our system, and also in all synthetic systems, are essential both for experiments and for practical usage. By modelling the regulation, we gained an insight of the ways to improve the performance of the "switch" in our system.

References and comments

Fitted from our CRISPRi experiment data
http://2014.igem.org/Team:Waterloo/Math_Book/CRISPRi
Bakshi S, Siryaporn A, Goulian M, Weisshaar JC. Superresolution imaging of ribosomes and RNA polymerase in live Escherichia coli cells. Mol Microbiol. 2012 Jul85(1): 21-38. doi: 10.1111/j.1365-2958.2012.08081.x. p.22 right column top paragraph

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

Introduction

In this section we show that our system hacks the bacteria's replication rate. 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.

The bacteria's replication is initiated by an accummulation of DnaA on the genome's OriC. To simplify the cases but still remaining the essential mechanics, we regard the dCas9's replication hacking is because of competing binding to OriC with DnaA. Once OriC is occupied by dCas9, it cannot bind DnaA-ATP, and hence cannot replicate.

List of hypotheses

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

Stochastic model of dCas9 binding and replication initiation

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.

Conclusion

This model provides the core mechanism of our system - that the dCas9 hacks the replication by binding to OriC, keeping the genome from replication initiation.

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^[1]. After a cell's division, the plasmids are equivalently distributed to both of the children 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 (pic plasmid_dilution.png). 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. This corresponds to "dilution" on a macro scale.

plasmid_transition $k_N$ $k_C$ , the frequency that a plasmid and dCas9 decouple. shows all the probable transition from one state to another.

plasmid_transition

Another factor taken into account is 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 of plasmid replication 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 representing the maximal replication 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})

$T, 2T, 3T \cdots$ . The parameters are given as follows

parameter	description	value
$T$	time of cell cycle^[2]	100 min
$k_N$	dCas9 unbinding frequency to the plasmids	0.067 min^-1
$k_C$	dCas9 binding frequency to the plasmids	0min ~ 0.05 min^-1
$\kappa$	the plasmid replication rate	0.02857 min^-1
$n_\max$	maximal cell copy number in the c^[3]	20

$k_C$ $k_C$ $k_C$ 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)

$k_C=0$ $k_C>0$ $k_C$ , dcas9 can clear plasmids earlier.

plasmid_clearance_generation.png $k_C$ , the number of generations spent to clear plasmids in cell lines will decrease.

It should be noted that in our system, the elimination of the plasmid by one cell line does not mean that the plasmid is completely eliminated in the whole bacterial culture system. Because we did not introduce the degradation and loss of plasmids into our model, the only factor to reduce the concentration of plasmids was dilution due to division, so the number of plasmids in the whole bacterial culture system has been increasing. But its increasing speed is not as fast as that of bacteria. The plasmids are always diluted until the concentration is very close to 0, so we can think that the plasmids are eliminated. The significance of our model is that when the average algebraic value of a cell line to clear plasmids is very small, it can be predicted that plasmids will be diluted to a very thin concentration in a few algebras. In general, under the condition of limited nutrition, cells will inevitably have a certain mortality rate, and plasmids will also have a certain probability of loss. Taking these factors into account, the removal effect will be significant.

References

ZHAO Yue-e, ZHU Shun-ya, MA Yong- ping. Control Mechanism of Bacterial Plasmid Replication. LETTERS IN BIOTECHNOLOGY Vol.18 No.3 May, 2007
Michelsen O, Teixeira de Mattos MJ, Jensen PR, Hansen FG. Precise determinations of C and D periods by flow cytometry in Escherichia coli K-12 and B/r. Microbiology. 2003 Apr149(Pt 4):1001-10.
https://www.qiagen.com/cn/service-and-support/learning-hub/technologies-and-research-topics/plasmid-resource-center/growth-of-bacterial-cultures/#tab2

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"^[1]. 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 exceed ten. In these cases, genes near the OriC may express more than genes far from OriC. Ting Lu^[2] deduced from Helmstetter-Copper^[3] 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.

Gene copy number fluctuation and its control

$T$ $T$ $T$ 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.

$k$ $x$ becomes smaller. To further demonstrate this relation, we numerically calculate the analytical results and visualize them. The parameters' values are given as follows:

parameter	description	value
$T$	Bacteria replication time^[4]	41 min
$k$	binding frequency of dCas9 to the genome	0~0.04 min^-1

$T$ ^[2] $k$ $k$ noise_errplot.png $k$ $k$ noise_XK.png $k$ .

In this section, we explore the potential of our system as a bacterial noise control component. The change of gene copy number is an important part of expression noise. By controlling the replication of bacterial genome, our system can reduce the fluctuation of gene copy number to control the noise of bacterial transcription, so as to increase the stability of various synthetic biological gene circuits.

References

Elowitz, Michael B., et al. Science 297.5584 (2002): 1183-1186.
Chen Liao, Andrew E. Blanchard and Ting Lu. An integrative circuit–host modelling framework for predicting synthetic gene network behaviours. Nature Microbiology volume 2, pages1658–1666 (2017)
Cooper S, Helmstetter CE. Chromosome replication and the division cycle of Escherichia coli B/r. J Mol Biol. 1968 Feb 14;31(3):519-40.
Helmstetter CE. DNA synthesis during the division cycle of rapidly growing Escherichia coli B/r. J Mol Biol. 1968 Feb 14 31(3):507-18. P.518 top paragraph
Blattner FR1, Plunkett G 3rd, Bloch CA, Perna NT, Burland V, Riley M, Collado-Vides J, Glasner JD, Rode CK, Mayhew GF, Gregor J, Davis NW, Kirkpatrick HA, Goeden MA, Rose DJ, Mau B, Shao Y. The complete genome sequence of Escherichia coli K-12. Science. 1997 Sep 5;277(5331):1453-62.
Méchali M. Eukaryotic DNA replication origins: many choices for appropriate answers. Nat Rev Mol Cell Biol. 2010 Oct11(10):728-38. doi: 10.1038/nrm2976. p.728 left column 2nd paragraph

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 counterintuitive to think 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 biological production 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^[1] 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^[2] 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 [1]. 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{cases} \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}n = -\frac{k_n n}{n+K_n}m\\ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}m = \frac{k_m n}{n+K_n}m\alpha\\ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}p = \frac{k_p n}{n+K_n}m(1-\alpha)\\ \end{cases}

$n$ $m$ $p$ $k_n$ $k_m$ $k_p$ $K_n$ $\alpha$ 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{cases} \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}n = -\frac{k_n n}{n+K_n}m\\ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}m = \frac{n\lambda}{n+K_n}m\\ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}p = \frac{k_p n}{n+K_n}\left(1-\frac{\lambda}{k_m}\right)m\\ \end{cases}

the equation explicitly includes the growth rate. The parameter values are given as follows

parameter	description	value
$k_m$	the theoretical maximal growth rate of E. coli^[1]	2.85 h^-1
$k_n$	the maximal nutrient uptake rate of E. coli ^[3]	5.70 h^-1
$k_p$	the maximal producting rate of the bio-product	0.1425 h^-1
$K_n$	the nutrient concentration corresponding to half-maximal growth rate	4.0 kg/ml
$\lambda$	the growth rate	$k_m$

The ODE is difficult to solve but we can analyze the steady states of 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 entire 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 accordance with the empirical formula provided in [1].

In this section, we discussed the possibility of our system being used to increase production of biological products. Our experiments also confirmed that in our system, the yield of GFP and indigo would increase with the increase of the concentration of inducer. It is consistent with the prediction of our model. In general, we can control the growth rate of bacteria by reducing nutrition, and our system has successfully controlled the growth rate of bacteria under the condition of sufficient nutrition. At this time, the energy of bacteria will be allocated to more other metabolic activities, specifically, those metabolic activities that are not so closely related to their core life activities. This is the core mechanism that our model will explain.

References

Matthew Scott, Carl W. Gunderson, Eduard M. Mateescu, Zhongge Zhang, Terence Hwa. Interdependence of Cell Growth and Gene Expression: Origins and Consequences. Science 330, 1099 (2010)
Chen Liao, Andrew E. Blanchard and Ting Lu. An integrative circuit–host modelling framework for predicting synthetic gene network behaviours. Nature Microbiology volume 2, pages1658–1666 (2017)
Smil, 1998, Energies: An illustrated guide to the biosphere and civilization, MIT Press

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 "recipient". The recipient 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}}

$x, y$ $\sigma$ $C$ $D$ denotes the diffusion constant. The randomly distributed recipients 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.

Reference

Lupp C1, Ruby EG. Vibrio fischeri uses two quorum-sensing systems for the regulation of early and late colonization factors. J Bacteriol. 2005 Jun;187(11):3620-9.

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-<div id="leftm" style="position: fixed; top: 150px; left:150px;">
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+<div id="leftm" style="position: fixed; top: 120px; left:100px;">
 					<ul class="menuleft" >
-						<li class="menunew"><a href="#Ragulatory">Ragulatory</a></li>
+						<li class="menunew"><a href="#Regulatory">Regulatory</a></li>
 						<li class="menunew"><a href="#dCas">dCas9 & Growth Rate</a></li>
 						<li class="menunew"><a href="#Plasmid">Plasmid Copy Number</a></li>
-                                                <li class="menunew"><a href="#Collection">Parts Collection</a></li>
                                                  <li class="menunew"><a href="#Noise">Noise Control</a></li>
+                                                <li class="menunew"><a href="#Productivity">Productivity</a></li>
                                                  <li class="menunew"><a href="#Quorum Sensing">Quorum Sensing</a></li>
 					</ul>
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-<body class='typora-export' >
 <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 bacterium. 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 applied a model to explain how our bacteria can improve the productivity of some specific bioproducts. Fifth, 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">
@@ Line 49: / Line 50: @@
 \displaystyle 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}} \\
 \displaystyle 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{cases}</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.</span></p><p><img src='https://2019.igem.org/wiki/images/4/46/T--Peking--bireg_plot.png' alt='' referrerPolicy='no-referrer' /></p><p><span>Here, the AND gate logic picture is more clear. We get a dCas9-sgRNA concentration over 10</span><sup><span>-3</span></sup><span>nM only when the concentrations of both IPTG and sgRNA are high enough. It can also be seen that the Arabinose regulation has a wider dynamic range. This guided us to use more Arabinose regulation in our actual experiments.</span></p><h3><a name="conclusion" class="md-header-anchor"></a><span>Conclusion</span></h3><p><span>This model theoretically explains the mechanisms of our system&#39;s regulation parts. Regulation parts in our system, and also in all synthetic systems, are essential both for experiments and for practical usage. By modelling the regulation, we gained an insight of the ways to improve the performance of the &quot;switch&quot; in our system.</span></p><h3><a name="references-and-comments" class="md-header-anchor"></a><span>References and comments</span></h3><ol start='' ><li><span>Fitted from our CRISPRi experiment data</span></li><li><a href='http://2014.igem.org/Team:Waterloo/Math_Book/CRISPRi' target='_blank' class='url'>http://2014.igem.org/Team:Waterloo/Math_Book/CRISPRi</a></li><li><span>Bakshi S, Siryaporn A, Goulian M, Weisshaar JC. Superresolution imaging of ribosomes and RNA polymerase in live Escherichia coli cells. Mol Microbiol. 2012 Jul85(1): 21-38. doi: 10.1111/j.1365-2958.2012.08081.x. p.22 right column top paragraph</span></li></ol><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><h3><a name="introduction" class="md-header-anchor"></a><span>Introduction</span></h3><p><span>In this section we show that our system hacks the bacteria&#39;s replication rate. 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.</span></p><p><span>The bacteria&#39;s replication is initiated by an accummulation of DnaA on the genome&#39;s OriC. To simplify the cases but still remaining the essential mechanics, we regard the dCas9&#39;s replication hacking is because of competing binding to OriC with DnaA. Once OriC is occupied by dCas9, it cannot bind DnaA-ATP, and hence cannot replicate.</span></p><p><img src='https://2019.igem.org/wiki/images/6/60/T--Peking--mechanism.png' alt='' referrerPolicy='no-referrer' /></p><h3><a name="list-of-hypotheses" class="md-header-anchor"></a><span>List of hypotheses</span></h3><ul><li><span>The bingding process of both 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><h3><a name="stochastic-model-of-dcas9-binding-and-replication-initiation" class="md-header-anchor"></a><span>Stochastic model of dCas9 binding and replication initiation</span></h3><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="E58-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="E58-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="#E58-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E58-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="E61-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="E61-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="#E61-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-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="E83-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="#E83-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-n109" cid="n109" mdtype="math_block">
+\end{cases}</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.</span></p><p><img src='https://2019.igem.org/wiki/images/4/46/T--Peking--bireg_plot.png' alt='' referrerPolicy='no-referrer' /></p><p><span>Here, the AND gate logic picture is more clear. We get a dCas9-sgRNA concentration over 10</span><sup><span>-3</span></sup><span>nM only when the concentrations of both IPTG and sgRNA are high enough. It can also be seen that the Arabinose regulation has a wider dynamic range. This guided us to use more Arabinose regulation in our actual experiments.</span></p><h3><a name="conclusion" class="md-header-anchor"></a><span>Conclusion</span></h3><p><span>This model theoretically explains the mechanisms of our system&#39;s regulation parts. Regulation parts in our system, and also in all synthetic systems, are essential both for experiments and for practical usage. By modelling the regulation, we gained an insight of the ways to improve the performance of the &quot;switch&quot; in our system.</span></p><h3><a name="references-and-comments" class="md-header-anchor"></a><span>References and comments</span></h3><ol start='' ><li><span>Fitted from our CRISPRi experiment data</span></li><li><a href='http://2014.igem.org/Team:Waterloo/Math_Book/CRISPRi' target='_blank' class='url'>http://2014.igem.org/Team:Waterloo/Math_Book/CRISPRi</a></li><li><span>Bakshi S, Siryaporn A, Goulian M, Weisshaar JC. Superresolution imaging of ribosomes and RNA polymerase in live Escherichia coli cells. Mol Microbiol. 2012 Jul85(1): 21-38. doi: 10.1111/j.1365-2958.2012.08081.x. p.22 right column top paragraph</span></li></ol>
+<div class="clear extra_space" id="dCas"></div>
+<div class="clear extra_space"></div>
+<div class="clear extra_space"></div>
+<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><h3><a name="introduction" class="md-header-anchor"></a><span>Introduction</span></h3><p><span>In this section we show that our system hacks the bacteria&#39;s replication rate. 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.</span></p><p><span>The bacteria&#39;s replication is initiated by an accummulation of DnaA on the genome&#39;s OriC. To simplify the cases but still remaining the essential mechanics, we regard the dCas9&#39;s replication hacking is because of competing binding to OriC with DnaA. Once OriC is occupied by dCas9, it cannot bind DnaA-ATP, and hence cannot replicate.</span></p><p><img src='https://2019.igem.org/wiki/images/6/60/T--Peking--mechanism.png' alt='' referrerPolicy='no-referrer' /></p><h3><a name="list-of-hypotheses" class="md-header-anchor"></a><span>List of hypotheses</span></h3><ul><li><span>The bingding process of both 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><h3><a name="stochastic-model-of-dcas9-binding-and-replication-initiation" class="md-header-anchor"></a><span>Stochastic model of dCas9 binding and replication initiation</span></h3><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="E58-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="E58-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="#E58-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E58-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="E61-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="E61-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="#E61-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-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="E83-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="#E83-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-n109" cid="n109" mdtype="math_block">
 		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="93.152ex" height="9.114ex" viewBox="0 -2213.4 40106.9 3924.2" role="img" focusable="false" style="vertical-align: -3.974ex; max-width: 100%;"><defs><path stroke-width="0" id="E9-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="E9-MJMAIN-38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z"></path><path stroke-width="0" id="E9-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="E9-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><path stroke-width="0" id="E9-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="E9-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E9-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="E9-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="E9-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><path stroke-width="0" id="E9-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="E9-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="E9-MJSZ4-239B" d="M837 1154Q843 1148 843 1145Q843 1141 818 1106T753 1002T667 841T574 604T494 299Q417 -84 417 -609Q417 -641 416 -647T411 -654Q409 -655 366 -655Q299 -655 297 -654Q292 -652 292 -643T291 -583Q293 -400 304 -242T347 110T432 470T574 813T785 1136Q787 1139 790 1142T794 1147T796 1150T799 1152T802 1153T807 1154T813 1154H819H837Z"></path><path stroke-width="0" id="E9-MJSZ4-239D" d="M843 -635Q843 -638 837 -644H820Q801 -644 800 -643Q792 -635 785 -626Q684 -503 605 -363T473 -75T385 216T330 518T302 809T291 1093Q291 1144 291 1153T296 1164Q298 1165 366 1165Q409 1165 411 1164Q415 1163 416 1157T417 1119Q417 529 517 109T833 -617Q843 -631 843 -635Z"></path><path stroke-width="0" id="E9-MJSZ4-239C" d="M413 -9Q412 -9 407 -9T388 -10T354 -10Q300 -10 297 -9Q294 -8 293 -5Q291 5 291 127V300Q291 602 292 605L296 609Q298 610 366 610Q382 610 392 610T407 610T412 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-1 0 0)"><g transform="translate(38828,0)"><g id="mjx-eqn-8"><use xlink:href="#E9-MJMAIN-28"></use><use xlink:href="#E9-MJMAIN-38" x="389" y="0"></use><use xlink:href="#E9-MJMAIN-29" x="889" y="0"></use></g></g><g transform="translate(14152,0)"><g transform="translate(-15,0)"><use xlink:href="#E9-MJMAINB-41" x="0" y="0"></use><use xlink:href="#E9-MJMAIN-3D" x="1146" y="0"></use><g transform="translate(2202,0)"><g transform="translate(0,2150)"><use xlink:href="#E9-MJSZ4-239B" x="0" y="-1155"></use><g transform="translate(0,-2036.022544283414) scale(1,0.4524959742351047)"><use xlink:href="#E9-MJSZ4-239C"></use></g><use xlink:href="#E9-MJSZ4-239D" x="0" y="-3156"></use></g><g transform="translate(1042,0)"><g transform="translate(-15,0)"><g transform="translate(0,1350)"><use xlink:href="#E9-MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#E9-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E9-MJMAIN-31" x="736" y="-213"></use></g><use xlink:href="#E9-MJMAIN-2212" x="1974" y="0"></use><use xlink:href="#E9-MJMATHI-3BA" x="2974" y="0"></use></g><g transform="translate(1288,-50)"><use xlink:href="#E9-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E9-MJMAIN-32" x="736" y="-213"></use></g><use xlink:href="#E9-MJMAIN-30" x="1525" y="-1450"></use></g><g transform="translate(4536,0)"><g transform="translate(389,1350)"><use xlink:href="#E9-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E9-MJMAIN-31" x="736" y="-213"></use></g><g transform="translate(0,-50)"><use xlink:href="#E9-MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#E9-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E9-MJMAIN-32" x="736" y="-213"></use></g></g><use xlink:href="#E9-MJMAIN-30" x="626" y="-1450"></use></g><g transform="translate(7288,0)"><use xlink:href="#E9-MJMATHI-3BA" x="0" y="1350"></use><use xlink:href="#E9-MJMAIN-30" x="38" y="-50"></use><use xlink:href="#E9-MJMAIN-30" x="38" y="-1450"></use></g></g><g transform="translate(9073,2150)"><use xlink:href="#E9-MJSZ4-239E" x="0" y="-1154"></use><g transform="translate(0,-2036.0048309178746) scale(1,0.45410628019323673)"><use xlink:href="#E9-MJSZ4-239F"></use></g><use xlink:href="#E9-MJSZ4-23A0" x="0" y="-3156"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-8">{\bf A} = \begin{pmatrix}
@@ Line 67: / Line 73: @@
 \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-n128" cid="n128" mdtype="math_block">
-		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="93.152ex" height="5.963ex" viewBox="0 -1509.8 40106.9 2567.2" role="img" focusable="false" style="vertical-align: -2.456ex; max-width: 100%;"><defs><path stroke-width="0" id="E14-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="E14-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 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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 3L600 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666ZM137 341Q137 131 136 89T130 37Q129 36 129 35H235Q233 41 231 48L226 61V623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM557 603V648H504Q504 646 515 639Q527 634 542 619L557 603ZM420 317V397L406 383Q394 370 380 363L366 355Q373 350 382 346Q400 333 409 328L420 317ZM582 61L586 88Q585 88 582 83Q557 61 526 46L511 37L542 35H577Q577 36 578 39T580 49T582 61Z"></path><path stroke-width="0" id="E14-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 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xlink:href="#E14-MJMATHI-63" x="974" y="0"></use><use xlink:href="#E14-MJMAIN-2B" x="1629" y="0"></use><g transform="translate(2629,0)"><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 transform="translate(1086,-686)"><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><use xlink:href="#E14-MJMATHI-3BA" x="974" y="0"></use></g></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-13">{\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><h3><a name="conclusion" class="md-header-anchor"></a><span>Conclusion</span></h3><p><span>This model provides the core mechanism of our system - that the dCas9 hacks the replication by binding to OriC, keeping the genome from replication initiation. </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</span><sup><span>[1]</span></sup><span>. After a cell&#39;s division, the plasmids are equivalently distributed to both of the children cells. Here we discuss a cell line&#39;s behavior over time.</span></p><p><img src='https://2019.igem.org/wiki/images/c/cb/T--Peking--plasmid_cell_line.png' alt='plasmid_cell_line' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/2/28/T--Peking--plasmid_dilution.png' alt='plasmid_dilution.png' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/7/78/T--Peking--plasmid_process.png' alt='the simulation process' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/8/82/T--Peking--plasmid_clearance.png' alt='' 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 (pic </span><code>plasmid_dilution.png</code><span>). 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. This corresponds to &quot;dilution&quot; on a macro scale. </span></p><p><span>Besides, the plasmids may also replicate itself and bind or unbind to dCas9, These process, together with dilution, make up all the plasmids&#39; behaviors we are interested in (pic </span><code>plasmid_transition</code><span> ). When bound to dCas9, the plasmid cannot replicate. 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xlink:href="#E14-MJMATHI-63" x="974" y="0"></use><use xlink:href="#E14-MJMAIN-2B" x="1629" y="0"></use><g transform="translate(2629,0)"><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 transform="translate(1086,-686)"><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><use xlink:href="#E14-MJMATHI-3BA" x="974" y="0"></use></g></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-13">{\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><h3><a name="conclusion" class="md-header-anchor"></a><span>Conclusion</span></h3><p><span>This model provides the core mechanism of our system - that the dCas9 hacks the replication by binding to OriC, keeping the genome from replication initiation. </span></p>
+<div class="clear extra_space" id="Plasmid"></div>
+<div class="clear extra_space"></div>
+<div class="clear extra_space"></div>
+<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</span><sup><span>[1]</span></sup><span>. After a cell&#39;s division, the plasmids are equivalently distributed to both of the children cells. Here we discuss a cell line&#39;s behavior over time.</span></p><p><img src='https://2019.igem.org/wiki/images/c/cb/T--Peking--plasmid_cell_line.png' alt='plasmid_cell_line' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/2/28/T--Peking--plasmid_dilution.png' alt='plasmid_dilution.png' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/7/78/T--Peking--plasmid_process.png' alt='the simulation process' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/8/82/T--Peking--plasmid_clearance.png' alt='' 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 (pic </span><code>plasmid_dilution.png</code><span>). 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. This corresponds to &quot;dilution&quot; on a macro scale. </span></p><p><span>Besides, the plasmids may also replicate itself and bind or unbind to dCas9, These process, together with dilution, make up all the plasmids&#39; behaviors we are interested in (pic </span><code>plasmid_transition</code><span> ). 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="E81-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="E81-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="#E81-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-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. shows all the probable transition from one state to another.</span></p><p><img src='https://2019.igem.org/wiki/images/3/36/T--Peking--plasmid_transition.png' alt='plasmid_transition' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/6/67/T--Peking--modified_gillespie.png' alt='' referrerPolicy='no-referrer' /></p><p><span>Another factor taken into account is  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 of plasmid replication is</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n144" cid="n144" mdtype="math_block">
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transform="translate(7308,0)"><use xlink:href="#E15-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMATHI-4E" x="848" y="-213"></use></g><use xlink:href="#E15-MJMAIN-22C5" x="8859" y="0"></use><g transform="translate(9359,0)"><use xlink:href="#E15-MJSZ3-28"></use><use xlink:href="#E15-MJMAIN-31" x="736" y="0"></use><use xlink:href="#E15-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="#E15-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMATHI-4E" x="848" y="-213"></use><use xlink:href="#E15-MJMAIN-2B" x="1550" y="0"></use><g transform="translate(2550,0)"><use xlink:href="#E15-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMATHI-43" x="848" y="-218"></use></g></g><g transform="translate(945,-686)"><use xlink:href="#E15-MJMATHI-6E" x="0" y="0"></use><g transform="translate(600,-150)"><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-6D"></use><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-61" x="833" y="0"></use><use transform="scale(0.707)" xlink:href="#E15-MJMAIN-78" x="1333" y="0"></use></g></g></g></g><use xlink:href="#E15-MJSZ3-29" x="6606" y="-1"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-14">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="E72-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 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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="#E72-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E72-MJMATHI-4E" x="848" y="-213"></use></g></svg></span><script type="math/tex">n_N</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="2.874ex" height="1.76ex" viewBox="0 -504.6 1237.4 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E73-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="E73-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="#E73-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E73-MJMATHI-43" x="848" y="-218"></use></g></svg></span><script type="math/tex">n_C</script><span> are the number of plasmids unbound or bound to 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="4.682ex" height="1.76ex" viewBox="0 -504.6 2015.9 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E84-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 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402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E84-MJMAIN-61" d="M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z"></path><path stroke-width="0" id="E84-MJMAIN-78" d="M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E84-MJMATHI-6E" x="0" y="0"></use><g transform="translate(600,-150)"><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-6D"></use><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-61" x="833" y="0"></use><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-78" x="1333" y="0"></use></g></g></svg></span><script type="math/tex">n_\max</script><span> is the maximum number of the plasmid allowed in a single 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="1.338ex" 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="E83-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="#E83-MJMATHI-3BA" x="0" y="0"></use></g></svg></span><script type="math/tex">\kappa</script><span> is a constant representing the maximal replication rate.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n146" cid="n146" mdtype="math_block">
@@ Line 78: / Line 89: @@
 \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-n150" cid="n150" mdtype="math_block">
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550T191 615Q191 616 190 616Q188 616 179 611T157 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 693Q295 686 288 635T278 575Q278 572 287 582Q336 635 402 669T540 704Q603 704 633 673T664 599Q664 559 638 523T580 462Q553 440 504 413L491 407L504 402Q566 381 596 338T627 244Q627 172 575 110T444 13T284 -22Q208 -22 158 28Q144 42 146 50Q150 67 178 85T230 103Q236 103 246 95T267 75T302 56T357 47Q436 47 486 93Q526 136 526 198V210Q526 228 518 249T491 292T436 330T350 345Q335 345 321 344T304 342Z"></path><path stroke-width="0" id="E18-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="E18-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 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transform="translate(5320,0)"><g transform="translate(286,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E18-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E18-MJMAIN-32" x="60" y="-686"></use></g></g><use xlink:href="#E18-MJMAIN-29" x="6347" y="0"></use><use xlink:href="#E18-MJMAIN-2C" x="6736" y="0"></use><g transform="translate(7930,0)"><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><use xlink:href="#E18-MJMAIN-223C" x="9445" y="0"></use><use xlink:href="#E18-MJCAL-42" x="10501" y="0"></use><use xlink:href="#E18-MJMAIN-28" x="11165" y="0"></use><g transform="translate(11554,0)"><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></g><use xlink:href="#E18-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="#E18-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E18-MJMAIN-32" x="60" y="-686"></use></g></g><use xlink:href="#E18-MJMAIN-29" x="14096" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-17">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 (similar to traditional Gillespie algorithm except that the dilution happens exactly 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="12.403ex" height="2.344ex" viewBox="0 -755.9 5340 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" 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y="0"></use></g></svg></span><script type="math/tex">T, 2T, 3T \cdots</script><span>. The parameters are given as follows</span></p><figure><table><thead><tr><th><span>parameter</span></th><th><span>description</span></th><th><span>value</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="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="E135-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="#E135-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script></td><td><span>time of cell cycle</span><sup><span>[2]</span></sup></td><td><span>100 min</span></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="2.901ex" 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="E81-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="E81-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="#E81-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-MJMATHI-4E" x="736" y="-213"></use></g></svg></span><script type="math/tex">k_N</script></td><td><span>dCas9 unbinding frequency to the plasmids</span></td><td><span>0.067 min</span><sup><span>-1</span></sup></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="2.69ex" 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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script></td><td><span>dCas9 binding frequency to the plasmids</span></td><td><span>0min ~ 0.05 min</span><sup><span>-1</span></sup></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="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="E83-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="#E83-MJMATHI-3BA" x="0" y="0"></use></g></svg></span><script type="math/tex">\kappa</script></td><td><span>the plasmid replication rate</span></td><td><span>0.02857 min</span><sup><span>-1</span></sup></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="4.682ex" height="1.76ex" viewBox="0 -504.6 2015.9 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E84-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="E84-MJMAIN-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E84-MJMAIN-61" d="M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z"></path><path stroke-width="0" id="E84-MJMAIN-78" d="M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E84-MJMATHI-6E" x="0" y="0"></use><g transform="translate(600,-150)"><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-6D"></use><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-61" x="833" y="0"></use><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-78" x="1333" y="0"></use></g></g></svg></span><script type="math/tex">n_\max</script></td><td><span>maximal cell copy number in the c</span><sup><span>[3]</span></sup></td><td><span>20</span></td></tr></tbody></table></figure><p><span>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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-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://2019.igem.org/wiki/images/9/97/T--Peking--plasmid_time_series.png' alt='image-20191008012315005' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/2/2b/T--Peking--plasmid_clearance_generation.png' alt='image-20191008012315005' referrerPolicy='no-referrer' /></p><p><span>It can be seen from Figure 1 that when dcas9 is not expressed (</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.949ex" height="2.344ex" viewBox="0 -755.9 2992 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E88-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="E88-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="E88-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="E88-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="#E88-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E88-MJMATHI-43" x="736" y="-218"></use><use xlink:href="#E88-MJMAIN-3D" x="1436" y="0"></use><use xlink:href="#E88-MJMAIN-30" x="2491" y="0"></use></g></svg></span><script type="math/tex">k_C=0</script><span>), the number of plasmid copies remains around 20. However, after the introduction 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="6.949ex" height="2.344ex" viewBox="0 -755.9 2992 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E89-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="E89-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="E89-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E89-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="#E89-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E89-MJMATHI-43" x="736" y="-218"></use><use xlink:href="#E89-MJMAIN-3E" x="1436" y="0"></use><use xlink:href="#E89-MJMAIN-30" x="2491" y="0"></use></g></svg></span><script type="math/tex">k_C>0</script><span>), the number of plasmid copies begins to decrease and the proportion of plasmid combined with dcas9 increases. With the increase 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="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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span>, dcas9 can clear plasmids earlier.</span></p><p><span>To more accurately describe this process, we have simulated many times to get a figure </span><code>plasmid_clearance_generation.png</code><span>. It can be seen that with the increase 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="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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span>, the number of generations spent to clear plasmids in cell lines will decrease.</span></p><p><span>It should be noted that in our system, the elimination of the plasmid by one cell line does not mean that the plasmid is completely eliminated in the whole bacterial culture system. Because we did not introduce the degradation and loss of plasmids into our model, the only factor to reduce the concentration of plasmids was dilution due to division, so the number of plasmids in the whole bacterial culture system has been increasing. But its increasing speed is not as fast as that of bacteria. The plasmids are always diluted until the concentration is very close to 0, so we can think that the plasmids are eliminated. The significance of our model is that when the average algebraic value of a cell line to clear plasmids is very small, it can be predicted that plasmids will be diluted to a very thin concentration in a few algebras. In general, under the condition of limited nutrition, cells will inevitably have a certain mortality rate, and plasmids will also have a certain probability of loss. Taking these factors into account, the removal effect will be significant.</span></p><h3><a name="references" class="md-header-anchor"></a><span>References</span></h3><ol start='' ><li><span>ZHAO Yue-e, ZHU Shun-ya, MA Yong- ping. Control Mechanism of Bacterial Plasmid Replication. LETTERS IN BIOTECHNOLOGY Vol.18 No.3 May, 2007</span></li><li><span>Michelsen O, Teixeira de Mattos MJ, Jensen PR, Hansen FG. Precise determinations of C and D periods by flow cytometry in Escherichia coli K-12 and B/r. Microbiology. 2003 Apr149(Pt 4):1001-10.</span></li><li><a href='https://www.qiagen.com/cn/service-and-support/learning-hub/technologies-and-research-topics/plasmid-resource-center/growth-of-bacterial-cultures/#tab2' target='_blank' class='url'>https://www.qiagen.com/cn/service-and-support/learning-hub/technologies-and-research-topics/plasmid-resource-center/growth-of-bacterial-cultures/#tab2</a></li></ol><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;</span><sup><span>[1]</span></sup><span>. 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 exceed ten. In these cases, genes near the OriC may express more than genes far from OriC. Ting Lu</span><sup><span>[2]</span></sup><span> deduced from Helmstetter-Copper</span><sup><span>[3]</span></sup><span> 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-n194" cid="n194" mdtype="math_block">
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transform="translate(13070,0)"><g transform="translate(286,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E18-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E18-MJMAIN-32" x="60" y="-686"></use></g></g><use xlink:href="#E18-MJMAIN-29" x="14096" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-17">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 (similar to traditional Gillespie algorithm except that the dilution happens exactly 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="12.403ex" height="2.344ex" viewBox="0 -755.9 5340 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" 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y="0"></use></g></svg></span><script type="math/tex">T, 2T, 3T \cdots</script><span>. The parameters are given as follows</span></p><figure><table><thead><tr><th><span>parameter</span></th><th><span>description</span></th><th><span>value</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="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="E135-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="#E135-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script></td><td><span>time of cell cycle</span><sup><span>[2]</span></sup></td><td><span>100 min</span></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="2.69ex" 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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script></td><td><span>dCas9 binding frequency to the plasmids</span></td><td><span>0min ~ 0.05 min</span><sup><span>-1</span></sup></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="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="E83-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="#E83-MJMATHI-3BA" x="0" y="0"></use></g></svg></span><script type="math/tex">\kappa</script></td><td><span>the plasmid replication rate</span></td><td><span>0.02857 min</span><sup><span>-1</span></sup></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="4.682ex" height="1.76ex" viewBox="0 -504.6 2015.9 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E84-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="E84-MJMAIN-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E84-MJMAIN-61" d="M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z"></path><path stroke-width="0" id="E84-MJMAIN-78" d="M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E84-MJMATHI-6E" x="0" y="0"></use><g transform="translate(600,-150)"><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-6D"></use><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-61" x="833" y="0"></use><use transform="scale(0.707)" xlink:href="#E84-MJMAIN-78" x="1333" y="0"></use></g></g></svg></span><script type="math/tex">n_\max</script></td><td><span>maximal cell copy number in the c</span><sup><span>[3]</span></sup></td><td><span>20</span></td></tr></tbody></table></figure><p><span>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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-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://2019.igem.org/wiki/images/9/97/T--Peking--plasmid_time_series.png' alt='image-20191008012315005' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/2/2b/T--Peking--plasmid_clearance_generation.png' alt='image-20191008012315005' referrerPolicy='no-referrer' /></p><p><span>It can be seen from Figure 1 that when dcas9 is not expressed (</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.949ex" height="2.344ex" viewBox="0 -755.9 2992 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E88-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="E88-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="E88-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="E88-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="#E88-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E88-MJMATHI-43" x="736" y="-218"></use><use xlink:href="#E88-MJMAIN-3D" x="1436" y="0"></use><use xlink:href="#E88-MJMAIN-30" x="2491" y="0"></use></g></svg></span><script type="math/tex">k_C=0</script><span>), the number of plasmid copies remains around 20. However, after the introduction 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="6.949ex" height="2.344ex" viewBox="0 -755.9 2992 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E89-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="E89-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="E89-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E89-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="#E89-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E89-MJMATHI-43" x="736" y="-218"></use><use xlink:href="#E89-MJMAIN-3E" x="1436" y="0"></use><use xlink:href="#E89-MJMAIN-30" x="2491" y="0"></use></g></svg></span><script type="math/tex">k_C>0</script><span>), the number of plasmid copies begins to decrease and the proportion of plasmid combined with dcas9 increases. With the increase 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="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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span>, dcas9 can clear plasmids earlier.</span></p><p><span>To more accurately describe this process, we have simulated many times to get a figure </span><code>plasmid_clearance_generation.png</code><span>. It can be seen that with the increase 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="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="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-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="#E91-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E91-MJMATHI-43" x="736" y="-218"></use></g></svg></span><script type="math/tex">k_C</script><span>, the number of generations spent to clear plasmids in cell lines will decrease.</span></p><p><span>It should be noted that in our system, the elimination of the plasmid by one cell line does not mean that the plasmid is completely eliminated in the whole bacterial culture system. Because we did not introduce the degradation and loss of plasmids into our model, the only factor to reduce the concentration of plasmids was dilution due to division, so the number of plasmids in the whole bacterial culture system has been increasing. But its increasing speed is not as fast as that of bacteria. The plasmids are always diluted until the concentration is very close to 0, so we can think that the plasmids are eliminated. The significance of our model is that when the average algebraic value of a cell line to clear plasmids is very small, it can be predicted that plasmids will be diluted to a very thin concentration in a few algebras. In general, under the condition of limited nutrition, cells will inevitably have a certain mortality rate, and plasmids will also have a certain probability of loss. Taking these factors into account, the removal effect will be significant.</span></p><h3><a name="references" class="md-header-anchor"></a><span>References</span></h3><ol start='' ><li><span>ZHAO Yue-e, ZHU Shun-ya, MA Yong- ping. Control Mechanism of Bacterial Plasmid Replication. LETTERS IN BIOTECHNOLOGY Vol.18 No.3 May, 2007</span></li><li><span>Michelsen O, Teixeira de Mattos MJ, Jensen PR, Hansen FG. Precise determinations of C and D periods by flow cytometry in Escherichia coli K-12 and B/r. Microbiology. 2003 Apr149(Pt 4):1001-10.</span></li><li><a href='https://www.qiagen.com/cn/service-and-support/learning-hub/technologies-and-research-topics/plasmid-resource-center/growth-of-bacterial-cultures/#tab2' target='_blank' class='url'>https://www.qiagen.com/cn/service-and-support/learning-hub/technologies-and-research-topics/plasmid-resource-center/growth-of-bacterial-cultures/#tab2</a></li></ol>
+<div class="clear extra_space" id="Noise"></div>
+<div class="clear extra_space"></div>
+<div class="clear extra_space"></div>
+<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;</span><sup><span>[1]</span></sup><span>. 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 exceed ten. In these cases, genes near the OriC may express more than genes far from OriC. Ting Lu</span><sup><span>[2]</span></sup><span> deduced from Helmstetter-Copper</span><sup><span>[3]</span></sup><span> 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-n194" cid="n194" mdtype="math_block">
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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><path stroke-width="0" id="E19-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(38328,0)"><g id="mjx-eqn-17" transform="translate(0,-7)"><use xlink:href="#E19-MJMAIN-28"></use><use xlink:href="#E19-MJMAIN-31" x="389" y="0"></use><use xlink:href="#E19-MJMAIN-37" x="889" y="0"></use><use xlink:href="#E19-MJMAIN-29" x="1389" y="0"></use></g></g><g transform="translate(12576,0)"><g transform="translate(-15,0)"><g transform="translate(0,-7)"><use xlink:href="#E19-MJMATHI-67" x="0" y="0"></use><use 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transform="translate(1138,0)"><use xlink:href="#E19-MJMAIN-5B"></use><g transform="translate(278,0)"><use xlink:href="#E19-MJMAIN-28"></use><use xlink:href="#E19-MJMAIN-31" x="389" y="0"></use><use xlink:href="#E19-MJMAIN-2212" x="1111" y="0"></use><g transform="translate(2111,0)"><use xlink:href="#E19-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E19-MJMATHI-71" x="808" y="-213"></use></g><use xlink:href="#E19-MJMAIN-29" x="3108" y="0"></use></g><use xlink:href="#E19-MJMATHI-43" x="3942" y="0"></use><use xlink:href="#E19-MJMAIN-2B" x="4924" y="0"></use><use xlink:href="#E19-MJMATHI-44" x="5924" y="0"></use><use xlink:href="#E19-MJMAIN-5D" x="6752" y="0"></use></g><use xlink:href="#E19-MJMAIN-29" x="8169" y="0"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-18">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="E92-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 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="0" id="E92-MJMATHI-71" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E92-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E92-MJMATHI-71" x="674" y="-213"></use></g></svg></span><script type="math/tex">g_q</script><span> is the gene&#39;s 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="E162-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="#E162-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="E94-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="E94-MJMATHI-71" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E94-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E94-MJMATHI-71" x="808" y="-213"></use></g></svg></span><script type="math/tex">x_q</script><span> is the gene&#39;s relative location to OriC, and ohter parameters are constants. 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="E140-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="#E140-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="E135-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="#E135-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="E97-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="E97-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="#E97-MJMATHI-6B" x="0" y="0"></use><use xlink:href="#E97-MJMATHI-54" x="521" y="0"></use></g></svg></span><script type="math/tex">kT</script><span>.</span></p><p><img src='https://2019.igem.org/wiki/images/3/33/T--Peking--replication_fork_formation.gif' alt='' referrerPolicy='no-referrer' /></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="E140-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="#E140-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://2019.igem.org/wiki/images/3/3f/T--Peking--noise_ctrl_mech.png' 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="E140-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="#E140-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="E135-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="#E135-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="E140-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="#E140-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="gene-copy-number-fluctuation-and-its-control" class="md-header-anchor"></a><span>Gene copy number fluctuation and its control</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="E135-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="#E135-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="E135-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="#E135-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="E135-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="#E135-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="E140-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="#E140-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="E149-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="#E149-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="E107-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="E107-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="E107-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="E107-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="E107-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="E107-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="#E107-MJMATHI-74" x="0" y="0"></use><use xlink:href="#E107-MJMAIN-28" x="611" y="0"></use><use xlink:href="#E107-MJMAIN-30" x="1000" y="0"></use><use xlink:href="#E107-MJMAIN-2264" x="1777" y="0"></use><use xlink:href="#E107-MJMATHI-74" x="2833" y="0"></use><use xlink:href="#E107-MJMAIN-2264" x="3472" y="0"></use><use xlink:href="#E107-MJMATHI-54" x="4528" y="0"></use><use xlink:href="#E107-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-n213" cid="n213" mdtype="math_block">
@@ Line 90: / Line 106: @@
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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="E126-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="E126-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="E126-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="E126-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="0" id="E126-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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E126-MJAMS-56" x="0" y="0"></use><use xlink:href="#E126-MJMAIN-61" x="722" y="0"></use><use xlink:href="#E126-MJMAIN-72" x="1222" y="0"></use><use xlink:href="#E126-MJMAIN-5B" x="1614" y="0"></use><use xlink:href="#E126-MJMATHI-4E" x="1892" y="0"></use><use xlink:href="#E126-MJMAIN-28" x="2780" y="0"></use><use xlink:href="#E126-MJMAIN-30" x="3169" y="0"></use><use xlink:href="#E126-MJMAIN-29" x="3669" y="0"></use><use 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0)"><use xlink:href="#E127-MJAMS-45" x="0" y="0"></use><use xlink:href="#E127-MJMAIN-5B" x="667" y="0"></use><use xlink:href="#E127-MJMATHI-4E" x="945" y="0"></use><use xlink:href="#E127-MJMAIN-28" x="1833" y="0"></use><use xlink:href="#E127-MJMAIN-30" x="2222" y="0"></use><use xlink:href="#E127-MJMAIN-29" x="2722" y="0"></use><use xlink:href="#E127-MJMAIN-5D" x="3111" y="0"></use><use xlink:href="#E127-MJMAIN-3D" x="3666" y="0"></use><use xlink:href="#E127-MJMAIN-31" x="4722" y="0"></use></g></svg></span><script type="math/tex">\mathbb{E}[N(0)] = 1</script><span> corresponds to the model hypothesis that the copy number of Ter is invariantly 1.</span></p><p><span>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:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n225" cid="n225" mdtype="math_block">
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To further demonstrate this relation, we numerically calculate the analytical results and visualize them. The parameters&#39; values are given as follows:</span></p><figure><table><thead><tr><th><span>parameter</span></th><th><span>description</span></th><th><span>value</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="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="E135-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="#E135-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script></td><td><span>Bacteria replication time</span><sup><span>[4]</span></sup></td><td><span>41 min</span></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="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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script></td><td><span>binding frequency of dCas9 to the genome</span></td><td><span>0~0.04 min</span><sup><span>-1</span></sup></td></tr></tbody></table></figure><p><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="E135-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="#E135-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> is relatively stable</span><sup><span>[2]</span></sup><span>. However, </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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>, as stated before, can be hacked by our system. Therefore we are extremely interested in figuring out how the noise changes with </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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>. Fig</span><code>noise_errplot.png</code><span> shows resulting noise amplitude at 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="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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span> value. It turns out that the noise amplitude is significantly greater 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="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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span> is bigger. Fig </span><code>noise_XK.png</code><span> shows the copy number noise of gene at the middle of the entire genome varying with </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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>.</span></p><p><img src='https://2019.igem.org/wiki/images/7/79/T--Peking--noise_errplot.png' alt='' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/f/f4/T--Peking--noise_XK.png' alt='' referrerPolicy='no-referrer' /></p><p><span>In this section, we explore the potential of our system as a bacterial noise control component. The change of gene copy number is an important part of expression noise. By controlling the replication of bacterial genome, our system can reduce the fluctuation of gene copy number to control the noise of bacterial transcription, so as to increase the stability of various synthetic biological gene circuits.</span></p><h3><a name="references" class="md-header-anchor"></a><span>References</span></h3><ol start='' ><li><span>Elowitz, Michael B., et al. Science 297.5584 (2002): 1183-1186.</span></li><li><span>Chen Liao, Andrew E. Blanchard and Ting Lu. An integrative circuit–host modelling framework for predicting synthetic gene network behaviours. Nature Microbiology volume 2, pages1658–1666 (2017)</span></li><li><span>Cooper S, Helmstetter CE. Chromosome replication and the division cycle of Escherichia coli B/r. J Mol Biol. 1968 Feb 14;31(3):519-40.</span></li><li><span>Helmstetter CE. DNA synthesis during the division cycle of rapidly growing Escherichia coli B/r. J Mol Biol. 1968 Feb 14 31(3):507-18. P.518 top paragraph</span></li><li><span>Blattner FR1, Plunkett G 3rd, Bloch CA, Perna NT, Burland V, Riley M, Collado-Vides J, Glasner JD, Rode CK, Mayhew GF, Gregor J, Davis NW, Kirkpatrick HA, Goeden MA, Rose DJ, Mau B, Shao Y. The complete genome sequence of Escherichia coli K-12. Science. 1997 Sep 5;277(5331):1453-62.</span></li><li><span>Méchali M. Eukaryotic DNA replication origins: many choices for appropriate answers. Nat Rev Mol Cell Biol. 2010 Oct11(10):728-38. doi: 10.1038/nrm2976. p.728 left column 2nd paragraph</span></li></ol><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 counterintuitive to think 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 biological production 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><sup><span>[1]</span></sup><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-n264" cid="n264" mdtype="math_block">
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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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E132-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span>, indicating that the copy number of genes near OriC in sequence tend to vary more than that of genes far from OriC in sequence. </span></p><p><span>It is obvious that the CV value decreases when 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="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="E140-MJMATHI-6B" d="M121 647Q121 657 125 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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="E132-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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E132-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span> becomes smaller. To further demonstrate this relation, we numerically calculate the analytical results and visualize them. The parameters&#39; values are given as follows:</span></p><figure><table><thead><tr><th><span>parameter</span></th><th><span>description</span></th><th><span>value</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="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="E135-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="#E135-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script></td><td><span>Bacteria replication time</span><sup><span>[4]</span></sup></td><td><span>41 min</span></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="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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script></td><td><span>binding frequency of dCas9 to the genome</span></td><td><span>0~0.04 min</span><sup><span>-1</span></sup></td></tr></tbody></table></figure><p><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="E135-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="#E135-MJMATHI-54" x="0" y="0"></use></g></svg></span><script type="math/tex">T</script><span> is relatively stable</span><sup><span>[2]</span></sup><span>. However, </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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>, as stated before, can be hacked by our system. Therefore we are extremely interested in figuring out how the noise changes with </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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>. Fig</span><code>noise_errplot.png</code><span> shows resulting noise amplitude at 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="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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span> value. It turns out that the noise amplitude is significantly greater 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="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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span> is bigger. Fig </span><code>noise_XK.png</code><span> shows the copy number noise of gene at the middle of the entire genome varying with </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="E140-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="#E140-MJMATHI-6B" x="0" y="0"></use></g></svg></span><script type="math/tex">k</script><span>.</span></p><p><img src='https://2019.igem.org/wiki/images/7/79/T--Peking--noise_errplot.png' alt='' referrerPolicy='no-referrer' /></p><p><img src='https://2019.igem.org/wiki/images/f/f4/T--Peking--noise_XK.png' alt='' referrerPolicy='no-referrer' /></p><p><span>In this section, we explore the potential of our system as a bacterial noise control component. The change of gene copy number is an important part of expression noise. By controlling the replication of bacterial genome, our system can reduce the fluctuation of gene copy number to control the noise of bacterial transcription, so as to increase the stability of various synthetic biological gene circuits.</span></p><h3><a name="references" class="md-header-anchor"></a><span>References</span></h3><ol start='' ><li><span>Elowitz, Michael B., et al. Science 297.5584 (2002): 1183-1186.</span></li><li><span>Chen Liao, Andrew E. Blanchard and Ting Lu. An integrative circuit–host modelling framework for predicting synthetic gene network behaviours. Nature Microbiology volume 2, pages1658–1666 (2017)</span></li><li><span>Cooper S, Helmstetter CE. Chromosome replication and the division cycle of Escherichia coli B/r. J Mol Biol. 1968 Feb 14;31(3):519-40.</span></li><li><span>Helmstetter CE. DNA synthesis during the division cycle of rapidly growing Escherichia coli B/r. J Mol Biol. 1968 Feb 14 31(3):507-18. P.518 top paragraph</span></li><li><span>Blattner FR1, Plunkett G 3rd, Bloch CA, Perna NT, Burland V, Riley M, Collado-Vides J, Glasner JD, Rode CK, Mayhew GF, Gregor J, Davis NW, Kirkpatrick HA, Goeden MA, Rose DJ, Mau B, Shao Y. The complete genome sequence of Escherichia coli K-12. Science. 1997 Sep 5;277(5331):1453-62.</span></li><li><span>Méchali M. Eukaryotic DNA replication origins: many choices for appropriate answers. Nat Rev Mol Cell Biol. 2010 Oct11(10):728-38. doi: 10.1038/nrm2976. p.728 left column 2nd paragraph</span></li></ol>
+<div class="clear extra_space" id="Productivity"></div>
+<div class="clear extra_space"></div>
+<div class="clear extra_space"></div>
+<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 counterintuitive to think 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 biological production 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><sup><span>[1]</span></sup><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-n264" cid="n264" mdtype="math_block">
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xlink:href="#E25-MJMATHI-55" x="842" y="-213"></use></g><use xlink:href="#E25-MJMAIN-29" x="2210" y="0"></use><use xlink:href="#E25-MJMAIN-3D" x="2877" y="0"></use><use xlink:href="#E25-MJMATHI-3BB" x="3932" y="0"></use><use xlink:href="#E25-MJMAIN-28" x="4515" y="0"></use><g transform="translate(4904,0)"><use xlink:href="#E25-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E25-MJMATHI-55" x="842" y="-213"></use></g><use xlink:href="#E25-MJMAIN-3D" x="6421" y="0"></use><use xlink:href="#E25-MJMAIN-30" x="7476" y="0"></use><use xlink:href="#E25-MJMAIN-29" x="7976" y="0"></use><g transform="translate(8532,0)"><use xlink:href="#E25-MJSZ3-28"></use><use xlink:href="#E25-MJMAIN-31" x="736" y="0"></use><use xlink:href="#E25-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="#E25-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E25-MJMATHI-55" x="842" y="-213"></use></g><g transform="translate(178,-686)"><use xlink:href="#E25-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E25-MJMATHI-63" x="842" y="-213"></use></g></g></g><use xlink:href="#E25-MJSZ3-29" x="4056" y="-1"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-24">\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="E143-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="E143-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="#E143-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E143-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="E162-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="#E162-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="E143-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="E143-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="#E143-MJMATHI-3D5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E143-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><sup><span>[2]</span></sup><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 [1]. 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://2019.igem.org/wiki/images/b/be/T--Peking--prodictivity_mech.png' 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="E156-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="#E156-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="E145-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="E145-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E145-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="#E145-MJMAIN-31" x="0" y="0"></use><use xlink:href="#E145-MJMAIN-2212" x="722" y="0"></use><use xlink:href="#E145-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="E146-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="E146-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="E146-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="#E146-MJMATHI-3B1" x="0" y="0"></use><use xlink:href="#E146-MJMAIN-3D" x="917" y="0"></use><use xlink:href="#E146-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="E147-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="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-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="#E147-MJMATHI-3B1" x="0" y="0"></use><use xlink:href="#E147-MJMAIN-3D" x="917" y="0"></use><use xlink:href="#E147-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-n269" cid="n269" mdtype="math_block">
@@ Line 123: / Line 144: @@
 \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>It&#39;s worth noting  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="E170-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="E170-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="#E170-MJMATHI-70" x="0" y="0"></use><g transform="translate(503,362)"><use transform="scale(0.707)" xlink:href="#E170-MJMAIN-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E170-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 accordance with the empirical formula provided in [1].</span></p><p><span>In this section, we discussed the possibility of our system being used to increase production of biological products. Our experiments also confirmed that in our system, the yield of GFP and indigo would increase with the increase of the concentration of inducer. It is consistent with the prediction of our model. In general, we can control the growth rate of bacteria by reducing nutrition, and our system has successfully controlled the growth rate of bacteria under the condition of sufficient nutrition. At this time, the energy of bacteria will be allocated to more other metabolic activities, specifically, those metabolic activities that are not so closely related to their core life activities. This is the core mechanism that our model will explain.</span></p><h3><a name="references" class="md-header-anchor"></a><span>References</span></h3><ol start='' ><li><span>Matthew Scott, Carl W. Gunderson, Eduard M. Mateescu, Zhongge Zhang, Terence Hwa. Interdependence of Cell Growth and Gene Expression: Origins and Consequences. Science 330, 1099 (2010)</span></li><li><span>Chen Liao, Andrew E. Blanchard and Ting Lu. An integrative circuit–host modelling framework for predicting synthetic gene network behaviours. Nature Microbiology volume 2, pages1658–1666 (2017)</span></li><li><span>Smil, 1998, Energies: An illustrated guide to the biosphere and civilization, MIT Press</span></li></ol><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;recipient&quot;. The recipient 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 recipients are 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="E172-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="E172-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="E172-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="E172-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="#E172-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="389" y="0"></use><use xlink:href="#E172-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="1333" y="0"></use><use xlink:href="#E172-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="E172-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="E172-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="E172-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="E172-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="#E172-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="389" y="0"></use><use xlink:href="#E172-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="1333" y="0"></use><use xlink:href="#E172-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-n326" cid="n326" mdtype="math_block">
+\end{cases}</script></div></div><p><span>It&#39;s worth noting  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="E170-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="E170-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="#E170-MJMATHI-70" x="0" y="0"></use><g transform="translate(503,362)"><use transform="scale(0.707)" xlink:href="#E170-MJMAIN-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E170-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 accordance with the empirical formula provided in [1].</span></p><p><span>In this section, we discussed the possibility of our system being used to increase production of biological products. Our experiments also confirmed that in our system, the yield of GFP and indigo would increase with the increase of the concentration of inducer. It is consistent with the prediction of our model. In general, we can control the growth rate of bacteria by reducing nutrition, and our system has successfully controlled the growth rate of bacteria under the condition of sufficient nutrition. At this time, the energy of bacteria will be allocated to more other metabolic activities, specifically, those metabolic activities that are not so closely related to their core life activities. This is the core mechanism that our model will explain.</span></p><h3><a name="references" class="md-header-anchor"></a><span>References</span></h3><ol start='' ><li><span>Matthew Scott, Carl W. Gunderson, Eduard M. Mateescu, Zhongge Zhang, Terence Hwa. Interdependence of Cell Growth and Gene Expression: Origins and Consequences. Science 330, 1099 (2010)</span></li><li><span>Chen Liao, Andrew E. Blanchard and Ting Lu. An integrative circuit–host modelling framework for predicting synthetic gene network behaviours. Nature Microbiology volume 2, pages1658–1666 (2017)</span></li><li><span>Smil, 1998, Energies: An illustrated guide to the biosphere and civilization, MIT Press</span></li></ol>
+<div class="clear extra_space" id="Quorum Sensing"></div>
+<div class="clear extra_space"></div>
+<div class="clear extra_space"></div>
+<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;recipient&quot;. The recipient 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 recipients are 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="E172-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="E172-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="E172-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="E172-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="#E172-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="389" y="0"></use><use xlink:href="#E172-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="1333" y="0"></use><use xlink:href="#E172-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="E172-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="E172-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="E172-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="E172-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="#E172-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="389" y="0"></use><use xlink:href="#E172-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E172-MJMAIN-30" x="1333" y="0"></use><use xlink:href="#E172-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-n326" cid="n326" mdtype="math_block">
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The randomly distributed recipients 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://2019.igem.org/wiki/images/7/79/T--Peking--im.png' alt='' referrerPolicy='no-referrer' /></p><h3><a name="reference" class="md-header-anchor"></a><span>Reference</span></h3><ol start='' ><li><span>Lupp C1, Ruby EG. Vibrio fischeri uses two quorum-sensing systems for the regulation of early and late colonization factors. J Bacteriol. 2005 Jun;187(11):3620-9.</span></li></ol><p>&nbsp;</p></div>

Difference between revisions of "Team:Peking/Model"

Revision as of 14:11, 19 October 2019

Modeling

regulatory part

Introduction

Mono-regulation

Bi-regulation

Steady State of the Bi-regulation model

Conclusion

References and comments

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

Introduction

List of hypotheses

Stochastic model of dCas9 binding and replication initiation

Conclusion

Plasmid Copy number Hacking

Introduction

References

Gene Expression Noise Control

Hypothesis

Gene copy number fluctuation and its control

References

Productivity

References

Quorum Sensing

A Simulation of Donor-Receptor Experiment

Reference