Team:XJTU-CHINA/Model
Modeling
Multiple Biological Reactions Related to GPP Synthesis
Introduction
The mevalonate pathway for GPP (isopentenyl diphosphate) biosynthesis was first identified in eukaryotic cells. In this pathway, three acetyl coenzyme A (acetyl-CoA) units are joined successively to form 3-hydroxy-3-methylglutaryl coenzyme A (HMG-CoA). HMG-CoA is then reduced to mevalonate, which is subsequently phosphorylated, decarboxylated, and dehydrated to form IPP (Fig. 1).
Then, Isopentenyl diphosphate isomerase (IPP isomerase) is responsible for isomerization of the carbon-carbon double bond of IPP to create the potent electrophile DMAPP(dimethylallyl disphosphate)(Fig. 2). Finally, IPP and DMAPP would be catalysed into geranyl pyrophosphate (GPP) by a geranyl pyrophosphate synthase (GPPS). Hence, to get more reliable data of the concentration of GPP, we construct a chemical kinetic model based on biological processes mentioned before.
Model Organisation
We have built up a regulatory model for single cells which mainly contains ordinary differential equations (ODE), ignoring the communication and quorum-sensing in colonies. We assummed that those genes in the same operon which encodes different key syntases(called protein in equations shown below) share the same expressing rate, so that all derivatives of those key enzymes can be described in the same way. \[\begin{align*} acetyl-CoA & \stackrel{atoB}{\longrightarrow} Aceto-acetyl-CoA \\& \stackrel{HMGS}{\longrightarrow}HMG-CoA \\& \stackrel{HMGR}{\longrightarrow}Mevalonate \\& \stackrel{MK}{\longrightarrow}Mevalonate-P \\& \stackrel{PMK}{\longrightarrow}Mevalonate-PP \\& \stackrel{PMD}{\longrightarrow}IPP \\& \stackrel{idi}{\longrightarrow}DMAPP \end{align*}\] \[\begin{equation*} \left\{ \begin{aligned} \frac{d[mRNA_{PlacUV5}]}{dt}& =copynumber \cdot k_{transcription}-[mRNA_{PlacUV5}]\cdot k_{dil-mRNA}\\ \frac{d[enzyme]}{dt}&=k_{transcription}\cdot[mRNA_{PlacUV5}]-[enzyme]\cdot k_{dil-enzyme} \\&=\frac{d[atoB]}{dt}=\frac{d[HMGS]}{dt}=\frac{d[HMGR]}{dt}=\frac{d[MK]}{dt} \\&=\frac{d[PMK]}{dt}= \frac{d[PMD]}{dt}= \frac{d[idi]}{dt} \end{aligned} \right. \end{equation*}\]
ODE system which indicates the synthesis of those key enzymes, which catalyse multiple biological reactions.
kdil-mRNA and kdil-enzyme refer to dilution rate constant of mRNA and synthases respectively.
The \(k_{transcription}\) refers to transcription rate constant of DNA,
while the \(k_{translation}\) refers to translation rate constant of mRNA.
The term copynumber refers to the statistical figure of plasmids pSB1K3 carrying this operator in one single bacteria cell.
The catalysing processes determine the produce of GPP.
In Figure 4, the ODEs are organised in a similar way as chemical kinetic equations to describe those key catalyzing processes.
It is worth mentioning that we have get significant parametres' values from published documents.(Tab. 1)
| Parameters | Figure | Unit |
|---|---|---|
| kcat-atoB | 308050 | 1/(minute*umol) |
| Km-atoB | 188150 | 1 |
| kcat-HMGR | 0.32*60 | 1/(minute*umol) |
| Km-HMGR | 178590 | 1 |
| kcat-PMD | 4.9*60 | 1/(minute*umol) |
| Km-PMD | 2268900 | 1 |
| kcat-idi | 1389.5 | 1/(minute*umol) |
| Km-idi | 138950 | 1 |
\[\begin{equation*} \left\{ \begin{aligned} \frac{d[Aceto-acetyl-CoA]}{dt}&=\frac{k_{cat-atoB}\cdot [enzeyme]\cdot[acetyl-CoA]}{[acetyl-CoA]+k_{m-atoB}} \\&-2\cdot \frac{k_{cat-HMGS}\cdot[enzyme]\cdot[Aceto-acetyle-CoA]}{[Aceto-acetyl-CoA]+k_{m-HMGS}} \\\frac{d[HMG-CoA]}{dt}& =\frac{k_{cat-HMGS}\cdot[enzyme]\cdot[Aceto-acetyl-CoA]}{[Aceto-acetyl-CoA]+k_{m-HMGS}\cdot[enzyme]} \\&-\frac{k_{cat-HMGR}\cdot[enzyme]\cdot[HMG-CoA]}{[HMG-CoA]+k_{m-HMGR}\cdot[enzyme]} \\\frac{d[Mevalonate]}{dt}&=\frac{k_{cat-HMGR}\cdot[enzyme]\cdot[Mevalonate-P]}{[Mevalonate-P]+k_{m-PMK}\cdot[enzyme]} \\&-\frac{k_{cat-PMK}\cdot[enzyme]\cdot[MEvalonate-PP]}{[Mevalonate-PP]+k_{m-PMD}\cdot[enzyme]} \\\frac{Mevalonate-P}{dt}&=\frac{k_{cat-MK}\cdot[enzyme]\cdot[Mevalonate]}{[Mevalonate]+k_{m-MK}\cdot[enzyme]} \\&-\frac{k_{cat-PMK}\cdot[enzyme]\cdot[Mevalonate-P]}{[Mevalonate-P]+k_{m-PMK}\cdot[enzyme]} \\\frac{d[Mevalonta-PP]}{dt}&=\frac{k_{cat-PMK}\cdot[enzyme]\cdot[MEvalonate-P]}{[MEvalonta-P]+k_{m-PMK}\cdot[enzyme]} \\-&\frac{k_{cat-PMD}\cdot[enzyme]\cdot[Mevalonate-PP]}{[Mevalonate-PP]+k_{m-PMD}\cdot[enzyme]} \\\frac{d[IPP]}{dt}&=\frac{[k_{cat-PMD}\cdot[enzyme]\cdot[Mevalonate-PP]]}{[Mevalonate-PP]+k_{m-PMD}\cdot[enzyme]} \\-&\frac{k_{cat-idi}\cdot[enzyme]\cdot[IPP]}{[IPP]+k_{m-idi}\cdot[enzyme]} \\\frac{d[DMAPP]}{dt}&=\frac{k_{cat-idi}\cdot[enzyme]\cdot[IPP]}{[IPP]+k_{m-idi}\cdot[enzyme]} \\&-\frac{k_{cat-idi}\cdot[enzyme]\cdot[IPP]}{[IPP]+k_{m-idi}\cdot[enzyme]} \\\frac{d[GPP]}{dt}&=k_{GPP}\cdot[DMAPP] \end{aligned} \right. \end{equation*} \] A set of model assumptions are compulsory conditions.
- Proteins coded by the same mRNA have same producing rate;
- Build a predictive model for single cells;
- 60 per cents of glucose is tranferred to acetyl-CoA which maintains a steady concentration;
- Some quantative parameter values help to build up the model. (Tab. 2)
| Parameters | Figure | Unit |
|---|---|---|
| kcat-atoB | 308050 | 1/(minute*umol) |
| copynumber | 30 | 1 |
| Acetyl-CoA | 4.87e-9 | umol |
| k-transcription-lac | 1.0108e-19 | umol/(minute*umol) |
| kdil-mRNA | 0.02372 | 1/(minute*cell) |
| kdil-protein | 0.347 | 1/(minute*cell) |
| k-translation | 10.02 | 1/(minute*cell) |
| kcat-HMGS | 400,000 | 1/(minute*umol) |
| Km-atoB | 180,000 | 1 |
| kcat-MK | 40,000 | 1/(minute*umol) |
| Km-MK | 4.9*50,000 | 1 |
| kcat-PMK | 400,000 | 1/(minute*umol) |
| Km-PMK | 1.1*100,000 | 1 |
Realising the Model
The MATLAB programme is set with initial value in zero (all variables are assumed as 0 at 0 min because the tenuous expression amount), and performance period from 0 min to 300 min. The expression of the mRNA and the synthases (protein) is shown in Figure 5 and Figure 6 respectively. IPP and DMAPP synthesis are simulated as Figure 5 and Figure 6 indicate.
In short, according to the predictive modelling results, we are able to achieve a deeper comprehension towards these biological reactions related to GPP synthesis. Meanwhile, our modelling results can help to construct following models which invovle specific GPP quantities.
Light-Control Switch Regulates the Expression of Linalool and Limonene
Introduction
The Cph8-Omp optogenetics system existing in our E.coli DH5 α strain can function as a `Switch' sensitive to light at the frequency range around 650nm and 705nm, which is shown in Figure 7.
It halts expression of promoter Omp(pomp) under 650nm light, and starts it again when wave length is switched to 705nm. To achieve our goals that produce of linalool and limonene is able to be controlled separately under various conditions, we constructed a linalool-and-limonene-controlling-circuit in the vector pSB1K3. It is clear from Figure 2 that in 650nm light the impressed pomp reduces CI and limonene synthase (short for CS in following paragraphs)'s expression. The function of decreased CI as inhibition towards \(rm P\lambda\) is damaged, so that expression of linalool synthase (short for LS in following text) declines after a period when light-control switch is activated. Though theoretically the circuit regulates genes carried on different operators obviously, the sensitivity, accuracy and the response time of it remains unclear. Hence, in our modelling, we try to investigate features related of this circuit by the mathematical methodology.
We have built up a regulatory model for single cells which mainly contains ordinary differential equations (ODE), ignoring the communication and quorum-sensing in colonies. Firstly, we established one ODE system to elucidate the biochemical reactions through light signal to product secretion, as shown below. According to Evan J Olson's light-sensing model, we assume the conversion between cph8 and cph8k are reactions with unchanged rate constants (\(k_{positive}\) and \(k_{reverse}\) for cph8 to cph8k and cph8k back to cph8 respectively), so as the dilution reactions, producing reactions and dark reversing reactions. The quantitative change of cph8 and cph8k is described by the use of master equations. Hill function can illustrate the impressing effects associated with pomp by replacing the Michaelis-Menten constant with the excited ratio of cph8 (Y in following text, which equals the proportion of cph8k in total cph8 and cph8k). In this way, we can calculate the expressing rate constant for pomp (\(k_{pomp}\)) in a numerical method. \[\begin{equation} \left\{ \begin{aligned} \frac{d[cph8]}{dt} =& k_{positive}+(k_{positive}+k_{dark-reverse})\cdot [cph8k] \\&-(k_{positive}+k_{dilution})\cdot[cph8k] \\ \frac{d[cph8]}{dt} =& k_{reverse}\cdot[cph8]-(k_{reverse}+k_{dilution}+k_{dark-reverse})\cdot[cph8k]\\ \end{aligned} \right. \end{equation}\]
Let\[Y=\frac{[cph8k]}{[cph8]+[cph8k]}\] Then\[\frac{dY}{dt}=k_{produce}-(k_{produce}+k_{reverse}+k_{dilution}+k_{dark-reverse)}\cdot Y\] \[\frac{[cph8k]}{[cph8]}=\frac{Y}{1-Y}\] \[k_{pomp}=(b_{pomp}+(a_{pomp}-b_{pomp}))\]
Let\[Y=\frac{[cph8k]}{[cph8]+[cph8k]}\] Then\[\frac{dY}{dt}=k_{produce}-(k_{produce}+k_{reverse}+k_{dilution}+k_{dark-reverse)}\cdot Y\] \[\frac{[cph8k]}{[cph8]}=\frac{Y}{1-Y}\] $$k_{pomp}=(b_{pomp}+(a_{pomp}-b_{pomp}))\cdot (\frac{Km_{pomp}^{n_{pomp}}{Y^{n_{pomp}}+Km_{pomp}^{n_{pomp}}}-k_diu\cdot mRNA_{pomp})\cdot copynumber$$
The expressing system determine the produce of synthases of our target product, limonene synthase (CS) and linalool synthase (LS). CI works as a repressor for promoter \(\rm \lambda\) and affects whether LS will be synthesised as a root factor. In the following equations, the ODEs are organised in a similar way as chemical kinetic equations, where all mRNA translation and dilution of mRNA as well as protein are considered first-order reactions. The Hill function in the third line indicates the concentration change of \(mRNA_{plambda}\) in a similar way as above. A set of model assumptions are compulsory conditions.
- Proteins coded by the same mRNA have same producing rate;
- Light-control system cannot be perturbed by other genes, proteins or compounds in cell;
- All mRNAs share a same dilution rate constant.
Realising the Model
The Matlab programme is set with initial value in zero
(all variables are assumed as 0 at 0 min because the tenuous expression amount),
and performance period from 0 min to 600 min
(at 0 min the light-control switch system exposed in dark is applied for 650nm light).
Initial frequency of the entire light-control switch system is 705nm at 0 min.
During running period the frequency is switched from 650nm to 705nm at 200 min and 400 min,
and from 705nm to 650nm at 100 min and 300 min.
The transcription from occupied promoter with 0.0005/second,
transcription from unoccupied promoter with 0.5/second,
translation with 0.167/(mRNA*second), protein degradation with a half-life of 10 min,
and mRNA degradation with a half-life of 2 min.
The phosphorylation ratio is shown in Figure 9.
The system response to frequency change sensitively and rapidly,
from the highest point at 0.611 to the lowest point at 0.0237.
CS, CI and LS expressions are simulated as Figure 10 and Figure 11 indicate.
The highest concentration of CS and CS reaches 5.5e-15 umol in one single cell while the lowest is 0.6e-15,
while LS concentration is constantly one order magnitude lower than CS.
CS and CI response to frequency-change slowlier and with a delay which does not exist while Y responsing.
LS expression even shows a heavier delay than CS and CI as about 60 minutes when 650nm light is applied,
and about 40 minutes when 705nm light is applied.
Long time delay illustrates the relatively low sensitivity towards light-control.
As CS, CI and LS's dilution rates affect the concentration considerably,
especially the dilution rate of CI.
So we have taken the protein fast degradation labels into consideration,
which decrease the half-time of protein from 100 min to 10 min.
The results related are shown in Figure 12.
By contrast, when fast degradation labels are applied, the protein expression shows more rapid response. As Figure 12 shows, when light frequency is switched between 650nm and 705nm, LS concentration soars to the peak or plummets to the bottom for about 50 minutes then levels off until the next turn of light frequency changing, though the delay (less than 10 minutes) cannot be eliminated totally. CS and CI concentration keeps a similar trend, while the concentration increase is not as rapid as LS. It takes about 100 minutes for CS and CI to reach the maximum, twofold longer than LS does. Interestingly, the CS and CI concentration is persistently one order of magnitude lower compared to Figure 11, while LS concentration is sixfold higher compared with Figure 12. Hence, it is likely for us to change the expression of LS and CS according to practical requirements.
In summary, in common conditions (100 min half-life for proteins), the light-control system works though the response of downstream promoters are insensitive to some extent, while when fast degradation labels are applied (10 min half-life for proteins) the increament in sensitivity and shorter delay can be observed.
Stability Analysis
In the processes of bio-dynamics, part of the parameters are difficult to determine, which leads to the significance of stability of solutions as the parametres change. Meanwhile, when other parametres remain constant, the changing values of one certain parametre lead to various results in terms of functions of the entire system. Inspired by the basis of Liapunov's stability criteria, we define the stability with the parameters as follows, in order to check whether the models own a strong robustness.
Definition: Stability Suppose that \[F=F(\alpha_{1},\alpha_{2},\cdots,\alpha_{n},x)\] is a real function of variable \(x\) with parameters \(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\). For each parameter \(\alpha_{i}\), \(F\) is defined to be stable for \(\alpha_{i}\) if and only if for \[\forall \epsilon >0, \exists \delta>0, \lim_{x \to +\infty}\left|F(\alpha_{1},\cdots,\alpha_{i},\alpha_{n},x)-F(\alpha_{1},\cdots,\alpha_{i_{0}},\alpha_{n},x)\right|,\] for \(\left|\alpha_{i}-\alpha_{i_{0}}\right|<\delta\).
Stability analysis has been utilised to demonstrate how LS concentration changes with a changing \(\rm Km-P\lambda\) (mentioned above as the Michaelis-Menten constant for promoter lambda. The \(\rm Km-P\lambda\) measured by former researchers may fluctuate in a board range because of different measuring methods and various experimental conditions. Fixing other parameters, let parameter \(\rm Km-P\lambda\) change from 1e-14 to 2e-14 with step of 1e-15 (in the unit of micromole per micromole of enzyme). The graphs of the solution to LS expression changes as the following Figure 13 and Figure 14 show.
Furthermore, we calculate the date of LS expression at the time point 100s, 200s and 300s when \(\rm Km-P\lambda\) is changing and the results are listed in table 1.
LS expression at time 100s,200s and 300s under different value of \(\rm Km-P\lambda\).
LS expression at time 100s,200s and 300s under different value of \(\rm Km-P\lambda\).
| \(\rm Km-P\lambda/10^-{14}\) | t=100s | t=200s | t=300s | |||
|---|---|---|---|---|---|---|
| \(y_{i}\times10^{-15}\) | \(y_{i}-y_{i-1}\times10^{-15}\) | \(y_{i}\times10^{-15}\) | \(y_{i}-y_{i-1}\times10^{-15}\) | \(y_{i}\times10^{-15}\) | \(y_{i}-y_{i-1}\times10^{-15}\) | |
| 1.0 | 0.6185 | 5.6980 | 0.6439 | |||
| 1.1 | 0.7242 | 0.1057 | 5.7911 | 0.0931 | 0.7578 | 0.1139 |
| 1.2 | 0.8502 | 0.1261 | 5.8576 | 0.0665 | 0.8938 | 0.1360 |
| 1.3 | 0.9986 | 0.1484 | 5.9061 | 0.0485 | 1.0538 | 0.1600 |
| 1.4 | 1.1707 | 0.1721 | 5.9422 | 0.0361 | 1.2392 | 0.1854 |
| 1.5 | 1.3680 | 0.1973 | 5.9697 | 0.0275 | 1.4517 | 0.2125 |
| 1.6 | 1.5919 | 0.2237 | 5.9908 | 0.0211 | 1.6927 | 0.2410 |
| 1.7 | 1.8536 | 0.2617 | 6.0074 | 0.0166 | 1.9634 | 0.2707 |
| 1.8 | 2.1241 | 0.2705 | 6.0206 | 0.0132 | 2.2650 | 0.3016 |
| 1.9 | 2.4345 | 0.3104 | 6.0312 | 0.0106 | 2.5983 | 0.3333 |
| 2.0 | 2.7755 | 0.3410 | 6.0398 | 0.0086 | 2.9643 | 0.3660 |
| Maximum | 0.3410 | 0.0931 | 0.3660 | |||
From the table above, we obtain that when the step of \(\rm Km-P\lambda\) reaches \(1\times 1\times 10^{-15}\) (unit), the changing of the value of LS expression is no more than \(0.3660\times 10^{-15}\) (unit), which can also be observed visually. And that exactly fits the definition of stability. Furthermore, the change of LS expression is even less when it responses to the light-control switch, suggesting that we may come up with the opinion that light-control strengthens the stability.
When fast degradation labels cannot be applied, the modelling part of LS expression has a weak robustness. According to Figure 13, the concentration varies dramatically from 1.3e-14 micromole per cell to lower than 0.5e-14 as \(\rm Km-P\lambda\) changes in the range of 1e-14 micromole to 2e-14 micromole. In comparison, if the degradation labels are linked to the CS, CI and LS while expressing, the concentration of LS expressed, which varies in the range of about 5.6e-14 micromole per cell to 6 micromole, shows a much stronger stability, as shown in Figure 14.
Finally, we have testified the function of fast degradation labels by processing stability analysis procedures towards half-life of both proteins(CS, LS) as following figures indicate. It can be clear from the figures that we can operate our light-switch system in the wide range of response sensitivity of LS and CS, as well as varying LS and CS expression levels, to adapt our hardware to different working environments and satisfy the users’ desire that control the expressing time and expression levels of both LS and CS.
Summary
Generally, according to the modelling results and stability analysis, there come several practical instructions for experiments in laboratories. Firstly, higher dilution rate of CS,CI and LS contributes substantially towards the sensitivity of the whole linalool-and-limonene-controlling-circuit, with more rapid CS and LS produce and shorter delay. Furthermore, higher robustness of LS concentration changing can be observed in stability analysis for $\rm Km-P\lambda$ while fast degradation labels are applied. Intrigued, the maximum concentration of CS, CI and LS changes dramatically in different dilution rates. Hence, we can optimise our initiative designs and the functions of the entire system by applying functional fast degradation labels with various dilution rates.
Cell Growth and Product Inhibition
Introduction
The model based on experimental data provides an approach to ascertain unknown parametres in certain scales and distributions corresponding to cell growth and product inhibition mechanism according to Zeng and Deckwer's modelling work. Based on the theory of statistics and functional analysis, the cell concentration function's interval is determined by obtaining the expectation of the parameter through the parametres' distribution. After this, the norm of function interval and parameter interval is defined. The ratio of norm change is taken as parameter affect index.
Cell Growth and Product Inhibition Mechanism
Cell Growth Expressing
By assuming the total substrate uptake follows Michaelis-Menten kinetics and the substrate is used for non-growth associated metabolic maintenance during stationary phase, the substrate uptake rate is expressed as: \[\begin{equation} frac{dS}{dt}=K_{max S/X}\cdot\frac{S}{K_{X-S}+S}\cdot E=\frac{dE}{dt}\cdot Y_{X/S}\cdot E+m\cdot E \end{equation}\] In equation (1), \(E\) refers to cell concentration, as well as \(S\) is substrate concentration. \(y\) is the yield coefficient of the biomass to the substrate consumed. \(K_{max}\) is the maximum substrate uptake rate by cells. \(K_{X-S}\) means the substrate affinity. \(m\) is cell maintenance constant. In normal bacterial living conditions, \(S\) is much greater than \(E\) and equation (1) can be reformed into equation (2) as follows. \[\begin{equation} \frac{dE}{dt_{max}}=\frac{1}{Y_{X/S}(k_{max S/X}-m)} \end{equation}\] Furthermore, the rate of cell growth can be calculated into equation (3). \[\begin{equation} \frac{dE}{dt}=(\frac{dE}{dt_{max}+\frac{m}{Y_{X/S}}})\cdot\frac{S}{K_{X-S}-\frac{m}{Y_{X/S}}} \end{equation}\]
Product Inhibition
While inhibitory effects of products are accounted, equation (3) can be rewritten into equation(4). \[\frac{d[E]}{dt}=\left\{\left[\left(\frac{d[E]}{dt}\right)_{max}+\frac{m_{E-S}}{Y_{E/S}}\right]\cdot \frac{[S]}{K_{E-S}+[S]}-\frac{m_{E-S}}{Y_{E/S}}\right\}\cdot I_{C-E}\cdot I_{L-E}\] \[\begin{equation} \\I_{C-E}= \left\{ \begin{aligned} 1-(\frac{[C]}{I^*_{C-E}})^{n_{C}},while [C]\leq I^*_{C-E} \\0,while [C]\geq I^*_{C-E} \end{aligned} \right. \end{equation}\] \[\begin{equation*} \\I_{L-E}= \left\{ \begin{aligned} 1-(\frac{[L]}{I^*_{L-E}})^{n_{L}},while [L]\leq I^*_{L-E} \\0,while [L]\geq I^*_{L-E} \end{aligned} \right. \end{equation*}\] Where \(I_{C-E}\) and \(I_{L-E}\) refer to inhibitory terms by limonene and linalool, in which inhibitory terms with '*' is product concentration thresholds. When product concentration reaches \(I\) with '*', cell growth will be halted.
The Parametres' Distributions Obtaining Process
The variables within equations above (including \(E\), \(S\), \(C\), \(L\)) can be measured during experiments. Functions related to \(E\), \(S\), \(C\) and \(L\) with \(t\) as the independent variable can be transformed into polynomial forms by curve fitting. After this, by sampling 300 sets of data of \(E\), \(S\), \(C\) and \(L\) with a equal time interval, we built 300 systems of nonlinear equations wherein parametres(\(m_{E-S}\), \(K_{E-S}\) and \(Y_{E-S}\)) are unknown to be solved. Based on solutions it can be feasible to obtain distributions of each parametre as Figure 17, Figure 18 and Figure 19 mentioned below.
Measurements of Parametres' Effect
During this part, we are aiming to compare the effect of three paremetres
(\(m_{E-S}\), \(K_{E-S}\) and \(Y_{E-S}\)) on the results \(E(t)\).
First and foremost, \(\Theta\) is the parametre space,
\(\forall \theta \in \Theta\), the norm \(\Vert \theta\Vert\) is defined the same as Euclidean Norm.
Then we define the norm of the function space \(F\) as follows,
\[\forall f(x;\theta_1,\cdots,\theta_{n}) \in F,\quad \Vert f\Vert =\max \limits_{x\in V}|f(x)|\]
where \(V\) is the value range of \(x\).
When \(\theta\) changes, we can use the ratio of change of \(f(x;\theta_1,\cdots,\theta_{n})\)
to the change of parametres in the sense of norm defined above to measure its effect on the results.
In the situation of our model, based on the density function of parametres
(\(m_{E-S}\), \(K_{E-S}\) and \(Y_{E-S}\)),
expected values are chosen to be the midpoint of value range.
Since \(E(t)\) cannot be displayed by elementary functions,
we fit \(I_{C}(t),I_{L}(t)\) and \(S_{t}\) with polynomial
based on Least Squares and the result perform well in simplifying the function without much errors.
Obtaining the function \(E(t)\), we change the parametres (\(m_{E-S}\), \(K_{E-S}\) and \(Y_{E-S}\))
one by one with others fixed, and the change of \(E(t)\) is displayed as follows.
According to the graph, we claim that \(Y_{E-S}\) can affect the results far more in a visual way; then we can also prove it with more data, the ratio of the change of results to the parametres' disturbance in the sense of norm define above.
Effect on results of \(m_{E-S}\), \(K_{E-S}\) and \(Y_{E-S}\)
| \(m_{E-S}\) | \(K_{E-S}\) | \(Y_{E-S}\) | ||||||
|---|---|---|---|---|---|---|---|---|
| \(\Vert\Delta\theta\Vert\) | \(\Vert\Delta f\Vert\) | Ratio | \(\Vert\Delta\theta\Vert\) | \(\Vert\Delta f\Vert\) | Ratio | \(\Vert\Delta\theta\Vert\) | \(\Vert\Delta f\Vert\) | Ratio |
| 1.0 | 0.2080 | 0.2080 | 0.2 | 2.4169 | 12.0854 | 10 | 0.0271 | 0.0027 |
| 2.0 | 1.0995 | 0.5497 | 0.5 | 4.9419 | 9.8838 | 30 | 0.0914 | 0.0030 |
| 3.0 | 0.7709 | 0.2570 | 1.0 | 7.6528 | 7.6528 | 50 | 0.1885 | 0.0038 |
| 4.0 | 1.0644 | 0.2661 | 1.5 | 9.3816 | 6.2544 | 80 | 0.3277 | 0.0041 |
According to the quantitative analysis, \(Y_{E_S}\) affects cell growth most dramatically, two magnitude orders larger than m which affects one order larger than \(K_{E_S}\). \(Y_{E_S}\) has a biological meaning as substrate uptake rate for bacterial appreciation, which can be theoretical foundation for the significant affect by varying \(Y_{E_S}\) towards cell growth, while m means substrate uptake rate to maintain non-growth related metabolism. Hence stability of aroma molecules production can be improved by constrain the air transmission from the medium stationary phase to early decline phase when \(Y_{E_S}\) is considerably lower than exponential phase. Methods to change \(m_{E-S}\) can be implemented by genetic editing or genetic engineering, which is hard to realised and achieve quantitative measurement. For \(K_{E_S}\), the affinity coefficient for specific substrate (carbon sources for anaerobic microorganism), shows cells' preference to various substrate, affects most slightly between the three parametres. Hence, the substitution of carbon sources (even nitrogen sources) cannot influence bacterial behaviour associated with cell growth to a large extent. The feasibility of changing the constitution and ratio of different non-growth related ingredients for specific improvement has been guaranteed. Meanwhile, this method can also be applied to analyze other parameters, such as $I^*$, $n$ and $\mu$ in this particular model.
Hardware
Introduction
This sector is built to evaluate functionality of our hardware for aroma molecule production in indoor environments. It is worth mentioning that we have applied some key physics and chemistry laws to our modelling with ODE, such as Lambert Bill's law ,and Henry's law. The results will show how much products transmitted into air.
Model Organisation
Here are some compulsory assumptions we have made:
- The fugacity coefficient constant is nearly 1 under atmospheric pressure to apply some key laws;
- Concentration of bacteria in the medium is the same to guarantee culture medium even;
- Two products follow simple diffusion with a diffusion ratio of 1:100 (1 for intracellular concentration while 100 for extracellular);
- Two products follow Henry's law of ideal dilute solution with coefficient of 10 to describe the two-phase balance;
- Culture medium's optical properties are similar to pure water's to apply Lambert Bill's law.
Since light absorption by cultural medium, we have considered different optical intensity in different culture medium depth with Lambert Bill's law by calculus methodology. \[\begin{equation} lg\frac{I_0}{I}=A=\epsilon*b*c \end{equation}\] $I$ in equation (1) refers to various optical intensity related to medium depth b in (1). $c$ in (1) refers to different product or substance concentration in culture medium. Then (1) can be rewritten into (2) as following. \[\begin{equation} I_n=\frac{I_0}{e^{b*(\epsilon_sC_s+\epsilon_cC_c+\epsilon_lC_l+\epsilon_EC_E+Const)}} \end{equation}\] The products which can be utilised by users remains in gas phase, hence Herry's Law can illustrates the product accumulation with rough results as the form in (3) as follows. \[\begin{equation} \left\{ \begin{aligned} P_L&=\frac{K_L*C}{C^\theta} P_C=\frac{K_C*L}{C^\theta} \end{aligned} \right. \end{equation}\] $C$ and $L$ in (3) refer to concentration of limonene and linalool in liquid phase respectively. $P_L$ and $P_C$ refer to equilibrium partial pressure of limonene and linalool in gas phase.
Model Realisation
According to experimental data related to cell growth during the first 12 hours as following(table 5), the cell growth equation has been obtained by fitting methods.
| 0h | 1h | 2h | 4h | 5h | 6h | 7h | 8h | 9h | 10h | 11h | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| OD600 | 0.00425 | 0.006125 | 0.010625 | 0.015375 | 0.036375 | 0.092 | 0.36025 | 0.4105 | 0.767 | 0.8965 | 0.986 |
| bacteria population | 141525 | 46019039 | 79828945 | 115517180 | 273296742 | 691224750 | 2706670828 | 3084214781 | 5762710688 | 6735684656 | 7408126125 |
The Matlab programme is set with initial value in zero (all variables are assumed as 0 at 0 h because the tenuous expression amount), and performance period from 0 h to 6 h (at 0 h the light-control switch system exposed in dark is applied for 650nm light). Initial frequency of the entire light-control switch system is 705nm at 0 h. During running period the frequency is switched from 705nm to 650nm at 3 h. With all processes mentioned above, we are able to provide the results equilibrium partial pressure of limonene and linalool in gas phase.
As Figure shows, linalool's production is later than limonene's production, which symbols the regulatory functions of light-control switch designed before. However, limonene's pressure is still increasing steadily after 3 h.Based on modelling phenomena, fast degradation labels and other potential methods' application can ensure higher sensitivity by improving our experimental conditions and procedures.
Advanced Light-control switch
Though the cph8-cph8k conversion illustrated above shows high sensitivity towards light induce, the phosphorylation processes through OmpR to OmpR-Pi’ dimer ,which is catalysed by cph8k as kinase, have been ignored. The previous work about versatile regulation of multisite protein phosphorylation (Salazar, C.,& Hofer, W., 2006) provides an approach to simulate light-control switch in a accurate way based on phosphorylation mechanism. Here, we build up a advanced light-control switch model through phosphorylating and dephosphorating reactions to help develop accuracy.
Model Elaboration
To simplify the modelling problems, several assumptions are set as compulsory conditions. Based on these assumptions reactions associated with light-control switch can be explained with substrates binding and dissociating with enzymes, coupled with phosphorylation and dephosphorylation.
Model assumptions
- Total amount of cph8 and cph8k remains constant;
- OmpR-Pi’s amount is constant as an intermediate;
- Furthermore all biochemical reactions are held in one single cell.
Mathematical expressions
The phosphorylation-process-simulated model is mainly consisted of ODEs showing below. P represents concentration of dephosphorylatase in this case. L0 and Q0 are the binding and dissociating equilibrium constants between OmpR and phosphoratase, OmpR and dephosphorytase respectively in terms of reversible binding between enzymes and substrates. Wherein kinetic parametres related to cph8 are the same as mentioned above,.as well as Hill constant, maximum expression, leakage expression and copy number for promoter pomp. L1 and Q1 refer to similar biological meaning as L0 and Q0. a and b are rate constant in processes of phosphorylating and depohsphorylating.
Model Running
The running results are shown in Figure 22 in biological conditions with one single cell. The values of parametres related are close to the real intracellular conditions. As concentration of dimer and cph8 increases, the expression of mRNA_pomp ascends first and then is hardly expressed at the point around 4min, which is earlier than the previous light-control switch model. In this way, higher sensitivity and accuracy is achieved.
