Overview
In order to further understand the role of temperature-controlling device below and try to find a more suitable temperature-controlling device for our project, we established the Temperature-Controlled Genetic Expression Model.
Figure 1: The Genetic Pathways
We described our device with equilibrium conditions for reversible reactions, chemical rate equation, and additionally, a function with several meaningful parameters for the \(\mathrm{TcI_2}\) inactivation. At last, we obtained the model to show the intensity of gene expression at different temperature. $$ P'(T)=\frac{K_d}{K_d+(1-\alpha_d(T))\cdot\frac{1}{2}(c-\frac{\sqrt{{K_\mathrm{dim}}^2(T)+8K_\mathrm{dim}(T)c}-K_\mathrm{dim}(T)}{4})} $$ What's more, after analysis and optimization, we finally illustrated technical feasibility and a promising prospect of our genetic device.
Process of Building Temperature-Controlled Genetic Expression Model
1 Introduction
The plasmid we constructed is temperature-controlled.
We established this model to further understand the role of temperature control device and try to find more suitable device for our project.
The genetic pathways are shown below:
Transcription and translation of \(\mathrm{TcI}\) (produce \(\mathrm{TcI}\) monomer protein); degradation of \(\mathrm{TcI}\)monomer protein;
\(\mathrm{TcI}\) monomers form dimers (reversible process) \(\mathrm{TcI\ +TcI\rightleftharpoons TcI_2}\);
As temperature rises up, \(\mathrm{TcI_2}\) is gradually inactivated and loses its inhibitory effect on transcription \(\mathrm{TcI_2 \rightarrow\varnothing}\);
\(\mathrm{TcI_2}\) binding to promoter inhibits transcription (reversible process) \(\mathrm{TcI_2+DNA\rightleftharpoons DNA\cdot TcI_2}\);
Promoter initiates transcription of the target gene, which in turn translates the target products \(\mathrm{DNA\xrightarrow {Transcription,Translation} Pro}\)
2 Model Assumptions
All reversible reactions are in equilibrium;
We were going to introduce temperature variable into our model. To simplify our model, we didn't use ode but equilibrium conditions for reversible reactions, in which the chemical equilibrium constant is relative with temperature.
Compared with the process of dimerization and depolymerization of TcI, the background expression and degradation of the monomer were much slower and could be neglected;
Furthermore, the total amount of \(\mathrm{TcI}\) can be viewed as constant (including TcI, free \(\mathrm{TcI_2}\) and \(\mathrm{TcI_2}\) binding to promoter).
The number of target genes in the strain was sufficiently large, but far less than \(\mathrm{TcI_2}\).
The influence of temperature on all transcription and translation processes could be neglected;
Transcription and translation of target gene was described by chemical rate equation, and chemical rate constant would be eliminated by normalization.
3 Temperature-Controlled Genetic Expression Model
3.1 The relationship between product expression and temperature
The most important part of this model is calculating the concentration of active \(\mathrm{TcI_2}\), so that we could calculate how much DNA is still active.
Firstly, we obtained the total concentration of \(\mathrm{TcI_2}\) through the process of dimerization and depolymerization of \(\mathrm{TcI}\).
1)\(\mathrm{TcI\ +TcI\rightleftharpoons TcI_2}\):
Let the concentration of \(\mathrm{TcI}\) monomer and dimer be \(\mathrm{X}\) and \(\mathrm{Y}\), respectively;
Assume the molar mass of \(\mathrm{TcI}\) is \(\mathrm{M}\), then that of \(\mathrm{TcI_2}\) is \(\mathrm{2M}\);
\(\mathrm{V}\) is the total reaction volume.
Conservation of materials ( \(\mathrm{TcI_2}\) binding with promoter was neglected by assumption 3) :
$$ (MX+2MY)V=m $$
$$ \hphantom{\qquad(3)}\Rightarrow X+2Y=\frac m{MV}:=c \qquad(1) $$
- \(\mathrm{c}\) is equivalent to the concentration of \(\mathrm{TcI}\) protein when it exists entirely in monomer form.
Equilibrium condition:
$$ \hphantom{\qquad(3)}\frac{X^2}Y=K_\mathrm{dim}(T)=e^{\Delta G_\mathrm{dim}/RT}\qquad(2) $$
\(\mathrm{K_\mathrm{dim}}\) is the equilibrium constant of dimer depolymerization;
\(\mathrm{\Delta G_\mathrm{dim}}\) is the change in gibbs free energy for dimer formation;
\(\mathrm{R=8.314\mathrm{\ J\cdot mol^{-1}\cdot K^{-1}}}\) is gas constant;
\(\mathrm{T}\) is thermodynamic temperature of system.
2)\(\mathrm{TcI_2 \rightarrow\varnothing}\):
Assume the ratio of inactivated \(\mathrm{TcI_2}\) is \(\alpha_d(T)\). (The exact expression of \(\alpha_d(T)\) will be discuss in a later part)
So the concentration of active \(\mathrm{TcI_2}\) is \(\mathrm{(1-\alpha_d)\cdot Y}\)
3)\(\mathrm{TcI_2+DNA\leftrightarrows DNA\cdot TcI_2}\):
Let the total concentration of DNA be \(\mathrm{D_t}\) (including inactive DNA and active DNA) and concentration of active DNA be \(\mathrm{D}\)
\(\mathrm{K_d}\) is the equilibrium constant of depolymerization of \(\mathrm{DNA\cdot TcI2}\) compound
We had equilibrium condition:
$$ \mathrm{\frac{D\cdot(1-\alpha_d)\cdot Y}{D_t-D}=K_d} $$
We could calculate \(\mathrm{D}\) from the equation above:
$$ D(T)=\frac{K_dD_t}{K_d+(1-\alpha_d(T))\cdot\ Y(T)}\qquad(4) $$
4)\(\mathrm{DNA\xrightarrow {Transcription,Translation} Pro}\):
There is the chemical rate equation for this process: $$ \hphantom{\qquad(5)}P(T)=k\cdot D(T)\cdot t\qquad(5) $$
\(\mathrm{k}\) is the rale constants of target gene expression;
\(\mathrm{t}\) is reaction time;
\(\mathrm{P}\) is the quantity of target protein expressed within reaction time.
5)The Final Model
With equation (3~5), we could get following function describing the amount of products varying with temperature:
$$ P(T)=\frac{K_d}{K_d+(1-\alpha_d(T))\cdot\frac{1}{2}(c-\frac{\sqrt{{K_\mathrm{dim}}^2(T)+8K_\mathrm{dim}(T)c}-K_\mathrm{dim}(T)}{4})}\cdot kD_tt $$
Within this equation:
\(\mathrm{K_\mathrm{dim}(T)=e^{\Delta D_\mathrm{dim}/RT}}\).
\(\mathrm{\alpha_d(T)}\) is the ratio of inactivated \(\mathrm{TcI_2}\)
In order to normalize P and make it easy to visualize, make \(\mathrm{(P_{\mathrm{max}}=k\cdot\ D_t\cdot\ t}\), equivalent to the product gene expression in the absence of repressor
$$ P(T)=P/P_\mathrm{max}=\frac{K_d}{K_d+(1-\alpha_d(T))\cdot\frac{1}{2}(c-\frac{\sqrt{{K_\mathrm{dim}}^2(T)+8K_\mathrm{dim}(T)c}-K_\mathrm{dim}(T)}{4})}\qquad(6) $$
Variable \(\mathrm{P'}\) represents the intensity of expression at different temperature.
2 Ratio of inactivated \(\mathrm{TcI_2}\) \(\alpha_d(T)\)
Here, we discussed the form of \(\alpha_d(T)\).
Without the data of inactivation of \(\mathrm{TcI_2}\), we can't fit \(\alpha_d(T)\) directly. But we find that of protein \(\mathrm{Cro}\), which is similar to \(\mathrm{TcI}\)[notion] ~ They both can form dimer, and dimer can binding to promoter and will be inactivated at high temperature. Fit \(\alpha_d=\frac{1}{1+e^{aT+b}}\) into data of \(\mathrm{Cro}\), we get \(\mathrm{a=-0.1893}\), \(\mathrm{b=7.848}\). \(\mathrm{R^2=0.9943}\), which means satisfactory fitting effect.
The fitted curve is shown below.
Figure 2: fitted curve
We assumed that the relationship between the mole fraction of inactivated \(\mathrm{TcI}\) dimer and temperature satisfies the same form, \(\alpha_d=\frac{1}{1+e^{A(T_c-T)}}\), in which \(\mathrm{T_c}\) represents the temperature when half of dimer is inactivated, and \(\mathrm{A}\) is related to the length of temperature interval of inactivation (for example, when 1%<\(\alpha_d\)<99%, \(\mathrm{T\in(T_c-\frac{\ln99}A,T_c+\frac{\ln99}A)}\).
According to the property of temperature control device we use, we could assume:
$$ \mathrm{T_c=38^\circ C , }\ \mathrm{A=0.8}\ $$
So, we give the function and curve of \(\mathrm{TcI}\) below:
$$ \alpha_d=\frac{1}{1+e^{0.8(38-T)}}\qquad (7) $$
Figure 3: inactivation curve for Cro2 and TcI2
Application and Optimization of Model
1 Result
With equation (6) and (7), we can obtain the following figure:
Figure 4: P'-Temperature
Analysis:
Quantity of product arrives 88.3% of maximum at \(\mathrm{45^\circ C}\).
It's worth noting that it arrives half of maximum at \(\mathrm{42.5^\circ C}\), the temperature where \(\alpha_d\) arrives a higher value than 90% and there are almost no active \(\mathrm{TcI_2}\).
This phenomenon occurs because promoter and \(\mathrm{TcI_2}\) have a large binding constant.
2 Robustness Analysis
Figure 5: Robustness Analysis of Kd
Figure 6: Robustness Analysis of c
Ordinates \(\frac{\partial[P]/[P]}{\partial x/x}\ (x=K_d\mathrm{\ or\ }c)\) represent the influence of small change of variate \(\mathrm{x}\) to amount of product \(\mathrm{P}\). The closer they are to zero, the smaller the influence is.
According to graphs, the influence is small enough, and will get even weaker as temperature rises up.
Thus, we can know that even though those two parameters won't be exact, the result is still reliable.
3 Optimization of our device
To get a better clinical effect , we tried to find a more satisfactory repressor \(\mathrm{TcI}\) by optimizing parameters of repressor in this part.
We had these requirements:
- Gene expression is almost nonexistent at \(\mathrm{37^\circ C}\);
- High expression at \(\mathrm{45^\circ C}\);
- More gradual change of \(\mathrm{P'}\) with \(\mathrm{T}\) for user can adjust the heating temperature in a suitable interval.
We used the function below to describe how quickly \(\mathrm{P'}\) changes with \(\mathrm{T}\) for \(\mathrm{P'}\in(80\)%\(,95\)%\()\) (We assumed this interval is acceptable for treatment):
$$ L(A,T_c)=P^{-1}_{A,\ T_c}(0.95)-P^{-1}_{A,\ T_c}(0.80) $$
\(A\) and \(T_c\) are parameters in \(\mathrm{\alpha_d(T)}\)
\(\mathrm{P^{-1}}\) is the inverse of \(\mathrm{P'}\).
\(\mathrm{L}\) is the interval length of temperature when \(\mathrm{P'}\) is between 80% and 95%
Then we had an optimization problem:
Figure 7: Optimization Problem
After calculation, we got graph below:
Figure 8: Optimization Result
The region between the two black curves is the feasible region. We found an optimal solution \({L=1.74}\) at \({(A,T_c)=(0.88,38.42)}\). This means when the treatment temperature varies from \(\mathrm{43.26^\circ C}\) to \(\mathrm{45^\circ C}\), we can have perfect effect.
All in all, after carefully researching and optimizing, we achieved our ultimate goal.
Conclusion
Congratulations!
The Temperature-Controlled Genetic Expression Model we established exceeded expectations and achieved our goals:
Show the entire temperature-controlled gene expression pathway more clearly
It obviously reflects the temperature control effect of \(\mathrm{TcI_2}\)
Prove the robustness of the model under high temperature conditions, thereby confirming the reliability of the results
Prove the feasibility of achieving the purpose of our device through mutation
References
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