Team:NWU-China/Model

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iGEM Mathematical Model

* All the following parameters are from the literature, please see the parameter table at the bottom of the page for details.

    Our model is to find the quantitative relationship between the measured amino acid concentration and the expression level of the fluorescent protein in the final steady state. It is hoped that the feasibility and detection sensitivity of our experiment are proved to some extent and the data trend is predicted for the experimental group.
   The whole system will go through a transient process and finally reach a stable state. When the steady state, the data is relatively stable, and the data can be considered as a fixed value at this time. The data of the transient process is unstable and has no practical research value, so there is no need for analysis. The purpose of our model study is: the quantitative relationship between amino acids and fluorescent proteins in the system at steady state.
    We assume that the bacterial liquid is relatively uniform during steady state, and the density of the bacteria combined with the internal characteristics of the individual bacteria can reflect the characteristics of the whole bacterial liquid.
    For the bacterial population density, we looked for literature to obtain the growth curve of E. coli under the M9 medium (as shown below). We can find the microbial density at steady state is 3.67*10^7 /mL, which is a constant from the graph.

For individual bacteria, we need to perform further modeling analysis, so we focus on a single strain at steady state. Within the bacteria, gene expression will also reach a steady state over time, and we need to model the data characteristics at this time. Because the components inside and outside the cell are very complex and cannot be quantitatively studied, and the relevant data indicate that the amino acid concentrations inside and outside the cell are not much different, we assume that the concentration of amino acids inside and outside the cell is equal.
According to the order in which the various biochemical reactions in the model occur, we divide the model into three parts:

Model one: TyrR constant expression：

According to the experimental principle , we need to model and predict the concentration of TyrR in bacteria. Therefore, the biochemical reactions involved in model one include the transcriptional translation of TyrR gene itself and the degradation of mRNA and protein:

$\begin{align} & Tyr{{R}_{DNA}}\xrightarrow{{{v}_{1}}}mRN{{A}_{TyrR}} \notag \\ & mRN{{A}_{TyrR}}\xrightarrow{{{v}_{2}}}Tyr{{R}_{\Pr }} \notag \\ & mRNA\xrightarrow{{{D}_{1}}}\phi \notag \\ & Protein\xrightarrow{{{D}_{2}}}\phi \notag \\ \notag \end{align}$

The TyrR protein is a transcription factor, which is composed by 513 amino acids. Providing sufficient IPTG in the growth environment of the bacteria, under such conditions, its induced expression mechanism can be ignored, and it is considered to be frequently expressed. Combined with the data we reviewed, we can get the following equations to represent the process of constant expression. The rate of generation of TyrR in a single bacterium is the difference between the rate of transcriptional translation and the rate of mRNA and protein itself.

$\begin{align} & \frac{d{{n}_{mRNA}}}{dt}=\frac{\alpha {{v}_{1}}}{l}-{{D}_{1}}{{n}_{mRNA}} \notag \\ & \frac{d{{n}_{\Pr }}}{dt}={{v}_{2}}{{n}_{mRNA}}-{{D}_{2}}{{n}_{\Pr }} \notag \\ & {{n}_{mRNA}}{{|}_{t=0}}=42 \notag \\ & {{n}_{\Pr }}{{|}_{t=0}}=500 \notag \\ \notag \end{align}$

    The protein is an intracellular protein that is not secreted extracellularly, assuming no TyrR protein in the medium.
    The change in cell volume is ignored, and the effect of dilution rate on degradation is not considered.
    Initial value conditions: At zero time, the expression level of the foreign gene is zero, the number of mRNA expressed in the background is 42 molecules, and the protein expressed in the background is 500 molecules.
    Boundary conditions: We do not set the boundary condition first, and set the time as long as possible to reach a steady state. Substituting each of the above parameters into our established TyrR-induced expression model, and the functional equation of the expression of TyrR (time) n_pras a function of time (seconds) t is solved by Matlab:

${{n}_{pr}}=\frac{\left( 3102360000*exp\left( -\left( 4*t \right)/25 \right) \right)}{31637471\text{ }-\text{ }\left( 1783533624500*exp\left( -t/3000 \right) \right)/31637471\text{ }+\text{ }3750000000/66049}$

Figure 1-1 shows the change in the expression level of TyrR over time (seconds): As can be seen from the figure, the amount of TyrR is stable in about four hours, and gene expression enters a steady state. We speculate that the bacteria accumulate nutrients during this time to prepare for reproduction. The separation of prokaryotic cells such as Escherichia coli is characterized by a substantially constant concentration of cells before and after division, so the amount of TyrR in the figure can be used as a reference amount for TyrR of a single cell at steady state. Combined with our experimental results, it is found that there is indeed a four-hour silent period during the actual measurement process. During this period, the optical density of the bacterial liquid hardly changes, which is consistent with the previous inference. The magnitude of the molecules of the protein is also symbolic, indicating that our model makes sense. The magnitude of the molecules of the protein is also symbolic, indicating that our model makes sense.
After a simple calculation, when the internal gene expression of a single bacteria reaches a steady state, the concentration of TyrR is $0.16\times {{10}^{-3}}umol/l$

Model two: TyrR polymerization simulation:

According to the experimental principle , we need to model the results of TyrR and amino acid calculations, and the concentration of each amino acid after TyrR polymerization is expressed by the concentration of the amino acid to be tested. Often expressed TyrR molecules have a total of four destinations: binding to phenylalanine to form a living dimer, binding to tyrosine to form a hexamer, self-binding to form an inactive dimer, and unchanged monomer.

$\begin{align} & 2TyrR\overset{{{k}_{1}}}{\leftrightarrows}[T-T] \notag \\ & [T-T]+2Phe\overset{{{k}_{2}}}{\leftrightarrows}{{[2T]}_{ac}} \notag \\ & 3[T-T]+6Tyr\overset{{{k}_{3}}}{\leftrightarrows}{{[6T]}_{re}} \notag \\ \notag \end{align}$

The three formulas briefly show the general chemical reaction equation. The purpose of our test is to set the concentration of tyrosine and phenylalanine to x and y. By combining the chemical equilibrium constants of these three reactions, four TyrR molecules can be expressed in terms of amino acid concentration.

$\left\{ \begin{matrix} {{k}_{1}}=\frac{{{C}_{2}}}{C_{1}^{2}} \\ {{k}_{2}}=\frac{{{C}_{3}}}{{{x}^{2}}{{C}_{2}}} \\ {{k}_{3}}=\frac{{{C}_{4}}}{{{y}^{6}}C_{2}^{3}} \\ {{C}_{1}}=\frac{{{n}_{\Pr }}}{{{N}_{A}}{{V}_{0}}}-2{{C}_{2}}-2{{C}_{3}}-6{{C}_{4}} \\ \end{matrix} \right.$

Manually simplify the equations and use constants to express the concentration of the four molecules as follows:

$\left\{ \begin{matrix} 6{{y}^{6}}k_{1}^{3}{{k}_{3}}C_{1}^{6}+2{{k}_{1}}\text{(1}+{{x}^{2}}{{k}_{2}})C_{1}^{2}+{{C}_{1}}=0.16 \\ {{C}_{2}}={{k}_{1}}C_{1}^{2} \\ {{C}_{3}}={{x}^{2}}{{k}_{1}}{{k}_{2}}C_{1}^{2} \\ {{C}_{4}}={{y}^{6}}k_{1}^{3}{{k}_{3}}C_{1}^{6} \\ \end{matrix} \right.$

The solved expression of C₁ can get the expression of the component. According to the mathematical theorem, there is no general solution for the equation of more than five times. Our equation is not solved for the sixth-order equation, we need to use some special methods to get its analytical solution.
The numerical solution of the equation can be obtained by Matlab calculation, and then a suitable method can be used to obtain a relational expression as follows. And we substitute the result C₁ into the third model.

${{C}_{1}}=\frac{917.1102}{8196.825+0.8247{{e}^{4.0237y}}+8826.6522x+x{{e}^{4.0237y}}}$

Almost all solutions of this formula fall on the sixth-order equation, so this formula can be approximated as an analytical solution of the sixth-order equation and substituted into the third model.

Model three: fluorescent protein expression

According to the experimental principle , I finally need to simulate the expression process of fluorescent protein after TyrR effect, combined with the first two models, construct the mathematical relationship between the amino acid to be tested and the concentration of fluorescent protein, and finally make a visual image. The biochemical reactions included in the expression of fluorescent proteins are:

$\begin{align} & GF{{P}_{DNA}}\xrightarrow{{{v}_{3}}}mRN{{A}_{GFP}} \notag \\ & mRN{{A}_{GFP}}\xrightarrow{{{v}_{2}}}GF{{P}_{\Pr }} \notag \\ & RF{{P}_{DNA}}\xrightarrow{{{v}_{3}}}mRN{{A}_{RFP}} \notag \\ & mRN{{A}_{RFP}}\xrightarrow{{{v}_{2}}}RF{{P}_{\Pr }} \notag \\ & mRNA\xrightarrow{{{D}_{1}}}\phi \notag \\ & Protein\xrightarrow{{{D}_{2}}}\phi \notag \\ \notag \end{align}$

Our fluorescent proteins have two systems of green fluorescent protein and red fluorescent protein, and their expression does not affect each other, and the cells are two independent systems. Assuming uniform flow within the cell, all molecules and promoters have the same probability of encounter, but only two forms of TyrR molecules can bind to the promoter for activation or inhibition, and one is a TyrR dimer that binds to phenylalanine. PtyrP can be activated to control GFP green fluorescent protein expression, and the other is TyrR hexamer that binds to tyrosine to inhibit ParoF activity and control red fluorescent protein expression. In the presence of TyrR, the strength of the original promoter is inhibited to varying degrees. The ratio of the TyrR dimer hexamer to all components of TyrR was used as the probability of binding to the promoter, and the average promoter strength was calculated. The ratio of the TyrR dimer hexamer to all components of TyrR was used as the probability of binding to the promoter, and the average promoter strength was calculated.
Green fluorescent protein expression:

$\left\{ \begin{matrix} {{v}_{3}}={{v}_{1}}{{\beta }_{3}}\frac{{{C}_{3}}}{{{C}_{1}}+{{C}_{2}}+{{C}_{3}}+{{C}_{4}}}+(1-\frac{{{C}_{3}}}{{{C}_{1}}+{{C}_{2}}+{{C}_{3}}+{{C}_{4}}}){{\beta }_{1}}{{v}_{1}} \\ {{C}_{2}}={{k}_{1}}C_{1}^{2} \\ {{C}_{3}}={{x}^{2}}{{k}_{1}}{{k}_{2}}C_{1}^{2} \\ {{C}_{4}}={{y}^{6}}{{k}_{3}}k_{1}^{3}C_{1}^{6} \\ \end{matrix} \right.$

$\left\{ \begin{matrix} {{v}_{3}}=(1-\frac{{{x}^{2}}{{k}_{1}}{{k}_{2}}C_{1}^{2}{{\beta }_{3}}}{{{C}_{1}}+{{k}_{1}}C_{1}^{2}+{{x}^{2}}{{k}_{1}}{{k}_{2}}C_{1}^{2}+{{y}^{6}}{{k}_{3}}k_{1}^{3}C_{1}^{6}})({{\beta }_{3}}{{v}_{1}}-{{\beta }_{1}}{{v}_{1}})+{{\beta }_{1}}{{v}_{1}} \\ \frac{d{{n}_{mRNA}}}{dt}=\frac{\alpha {{v}_{3}}}{{{l}_{3}}}-{{D}_{1}}{{n}_{mRNA}} \\ \frac{d{{n}_{\Pr }}}{dt}={{v}_{2}}{{n}_{mRNA}}-{{D}_{2}}{{n}_{\Pr }} \\ \end{matrix} \right.$

Red fluorescent protein expression:

$\left\{ \begin{matrix} {{v}_{4}}=(1-\frac{{{y}^{6}}{{k}_{3}}k_{1}^{3}C_{1}^{6}}{{{C}_{1}}+{{k}_{1}}C_{1}^{2}+{{x}^{2}}{{k}_{1}}{{k}_{2}}C_{1}^{2}+{{y}^{6}}{{k}_{3}}k_{1}^{3}C_{1}^{6}})({{\beta }_{4}}{{v}_{1}}-{{\beta }_{2}}{{v}_{1}})+{{\beta }_{2}}{{v}_{1}} \\ \frac{d{{n}_{mRNA}}}{dt}=\frac{\alpha {{v}_{3}}}{{{l}_{3}}}-{{D}_{1}}{{n}_{mRNA}} \\ \frac{d{{n}_{\Pr }}}{dt}={{v}_{2}}{{n}_{mRNA}}-{{D}_{2}}{{n}_{\Pr }} \\ \end{matrix} \right.$

The population density at steady state under the M9 medium condition is:

$\rho =3.67\times {{10}^{7}}/mL$

The total bacterial solution has a fluorescent protein concentration of:

$C=\frac{{{n}_{pr}}}{{{N}_{A}}}\rho$

The transcription rate was adjusted according to the action of TyrR, and the normal expression model of model one was modeled to adjust the parameters. Combined with the growth state of the flora. Drawing with Matlab:

$\text{c(GFP)=3}\text{.535}\times \text{1}{{\text{0}}^{\text{-15}}}\cdot (\frac{2.363\times {{10}^{6}}{{x}^{2}}{{\left( \sqrt{21.55{{x}^{2}}+4.879}-1 \right)}^{2}}}{{{\left( 5.556{{x}^{2}}+1 \right)}^{2}}(\frac{0.02062{{\left( \sqrt{21.55{{x}^{2}}+4.879}-1 \right)}^{2}}}{{{\left( 5.556{{x}^{2}}+1 \right)}^{2}}}+\frac{0.1146{{x}^{2}}{{\left( \sqrt{21.55{{x}^{2}}+4.879}-1 \right)}^{2}}}{{{\left( 5.556{{x}^{2}}+1 \right)}^{2}}}+\frac{33{{\left( \sqrt{21.55{{x}^{2}}+4.879}-1 \right)}^{2}}}{2222{{x}^{2}}+400}}+2.812\times {{10}^{6}})\cdot ({{e}^{-0.16t}}-480{{e}^{-0.0003333t}}+479)$

$c(RFP)=-2.427\times {{10}^{-18}}(\frac{3.625\times {{10}^{16}}{{y}^{6}}}{\frac{9.15\times {{10}^{10}}}{{{(0.8247{{e}^{4.024y}}+8197)}^{2}}}+\frac{2.265\times {{10}^{18}}}{5.668\times {{10}^{10}}{{e}^{4.024y}}+5.633\times {{10}^{14}}}+4.167\times {{10}^{7}}{{y}^{6}}}-9\times {{10}^{8}})({{e}^{-0.16t}}-480{{e}^{-0.0003333t}}+479)$

The concentration of the fluorescent protein is the z-axis, the time is the x-axis, and the concentration of the amino acid to be tested is the y-axis. In order to facilitate observation, we cut out the time-saturated two-dimensional map from the three-dimensional map, that is, the relationship between the concentration of the amino acid to be measured and the concentration of the fluorescent protein at steady state.

$c(GFP,t=15000)=\frac{3.975\times {{10}^{6}}{{x}^{2}}{{(\sqrt{21.55{{x}^{2}}+4.879}-1)}^{2}}}{(5.556{{x}^{2}}+1)(\frac{0.02062{{\left( \sqrt{21.55{{x}^{2}}+4.879}-1 \right)}^{2}}}{{{\left( 5.556{{x}^{2}}+1 \right)}^{2}}}+\frac{0.1146{{x}^{2}}{{\left( \sqrt{21.55{{x}^{2}}+4.879}-1 \right)}^{2}}}{{{\left( 5.556{{x}^{2}}+1 \right)}^{2}}}+\frac{33{{\left( \sqrt{21.55{{x}^{2}}+4.879}-1 \right)}^{2}}}{2222{{x}^{2}}+400}}+4.731\times {{10}^{-6}}$

$c(RFP,t=15000)=1.039\times {{10}^{-6}}-\frac{41.86{{y}^{6}}}{\frac{9.15\times {{10}^{10}}}{{{(0.8247{{e}^{4.024y}}+8197)}^{2}}}+\frac{2.265\times {{10}^{18}}}{5.668\times {{10}^{10}}{{e}^{4.024y}}+5.633\times {{10}^{14}}}+4.167\times {{10}^{7}}{{y}^{6}}}$

    The relationship between green fluorescent protein and phenylalanine is very linear. We do semi-quantitative semi-deterministic sensors, so we can treat them as a linear relationship. The slope of the curve is obvious and the slope is large, indicating that the model has good sensitivity and proves the feasibility of our project.
    The concentration of red fluorescent protein is not very obvious under low tyrosine conditions, and a strong inhibition is produced after a certain amount, and finally completely inhibited. There are obvious slope changes in the graph, and the model has better sensitivity in this interval, which proves the feasibility of our project.
    Our model can well prove the feasibility of our experiment that adding a small amount of amino acid can cause changes in the expression of fluorescent protein. According to the prediction of the model, we can provide good guidance and prediction for the later experimental design, and save time for pre-experiment.
    Combined with our later experimental results, we found that the curves of the experiment and the model are basically consistent, indicating that the fit of our model and experiment is very high, and the range of the experiment without measurement can be predicted by the model.

Fig. comparison of experimental results with models

Parameter	Description	Value	Unite	Reference
$\alpha$	Plasmid copy number	100	copies
${v}_{1}$	Average transcription rate	600	Bp/s	U. Alon.,2007
$r$	Negative feedback regulator
${l}_{1}$	Length of TyrR gene	1542	bp	Pittard, J.2005
${l}_{2}$	Length of green fluorescent gene	720	bp	Tsien, R. Y. (1998).
${l}_{3}$	Length of red fluorescent gene	687	bp	Rostagni, G. (1995).
${D}_{1}$	Degration of mRNA	0.16	Molecules/s	U. Alon.,2007
${v}_{2}$	Average translation rate	120	Molecules/s	U. Alon.,2007
${D}_{2}$	Degration rate of TyrR protein	1/3000	Molecules/s	U. Alon.,2007
$K_{1}^{-1}$	Chemical Equilibrium Constants of Reaction 1	330	μmol	Wilson et al., 1995
$K_{2}^{-1}$	Chemical Equilibrium Constants of Reaction 2	180	μmol	Wilson et al., 1995
$K_{3}^{-1}$	Chemical Equilibrium Constants of Reaction 3	24	μmol	Wilson et al., 1995
${v}_{0}$	Volume of Escherichia coli	0.6~0.7	μL	U. Alon.,2007
${v}_{3}$	Affected transcription rate	caculated	bp/s
${\beta}_{1}$	PrimerEffect of TyrRdimers on Transcription of Promoters	0.3		J Pittard.,1996
${\beta}_{2}$	PrimerEffect of TyrRhexamer on Transcription of Promoters	0.06		J Pittard.1996
${\beta}_{3}$	Effect of TyrR active dimers on Transcription of Promoters	2.5		J Pittard.1996
${\beta}_{4}$	Effect of TyrR active hexamer on Transcription of Promoters	0.002		J Pittard.1996

Table. parameter comparison