Latest revision as of 22:24, 19 October 2016

Modeling - iGEM Peshawar 2016

Modeling and Simulation

Why model?

Before building a part or a device in lab, it is probably a good idea to simulate it in a simulation package — such as SimBiology, Cello, or GenoCAD — to check if the part or device one intends to make would work the way one wants it to. Although simulation can't solve all the problems that one might encounter during the actual physical design, it can certainly reduce them, and help in getting a better understanding of how a design behaves under different parameters and conditions.

Our project has two biological ystems: one for detection of corbon monoxide, and another for detection for oxides of nitrogen. Below we show you how we modelled and simulated both these systems.

Modeling of the CO-sensing system

In our CO-sensing system, the CO binds to CooA (an activator). It activates the activator protein, and so the activated activator protein (CooA)2CO binds to promoter and initiates transcription.

The reactions which take place can be given as:

$$\textrm{[CO]}+\textrm{[CooA]} \longleftrightarrow \textrm{[(CooA)2CO]} \label{R1} \tag{R1}$$

The equation above represents the first reaction that takes place in our system in which the CO present in the environment is taken up into the cell where it binds to the CooA protein. The CooA protein is constantly expressed in our system using a constitutive promoter. The coefficient 2 represents the number of CO which bind to CooA; this number was taken from the literature.

To develop a dissociation constant of this equation, we can write it as:

$$KI=\frac{\textrm{[CO].[CooA]}}{\textrm{[(CooA)2CO]}} \label{1} \tag{1}$$

Note that the concentrations of $\textrm{[CO]}$, $\textrm{[CooA]}$, and $\textrm{[(CooA)2CO]}$ are at equilibrium before the start of the reaction R1.

The second reaction that takes place in our system is the binding of the activated activator protein complex of $\textrm{[(CooA)2CO]}$ to the promoter Pr.

It can be described as:

$$\textrm{[(CooA)2CO]} \longleftrightarrow \textrm{[(CooA)2CO.Pr]} \label{R2} \tag{R2}$$

Here, $\textrm{[(CooA)2CO.Pr]}$ represents the activated protein $\textrm{[(CooA)2CO]}$ bound to the promoter.

To develop a dissociation constant of this equation, we can write as:

$$KP=\frac{\textrm{[(CooA)2CO]}}{\textrm{[(CooA)2CO.Pr]}} \label{2} \tag{2}$$

Now, in order to get an equation which describes the change in the concentration of mRNA with time we develop the following differential equation:

$$\frac{d(mRNA)}{dt}=k_{on1}.P_{exp}-\gamma_1.[mRNA] \label{3} \tag{3}$$

where, $k_{on1}$ is represents the formation rate constant of the mRNA, and $\gamma_1$ represents the degradation rate constant of mRNA.

In Eq. (\ref{3}) it can be seen that the change in the $\textrm{[mRNA]}$ with time is equal to the difference of the formation rate of mRNA and its degradation rate.

The initial conditions for the Eq. (\ref{3}) are assumed to be:

$$\textrm{[mRNA(t=0)] = 0 molar}$$

This means that at the time $t=0$ sec (initial time), there are no molecules of mRNA encoding for our desired chromoprotein in the cell.

In Eq. (\ref{3}), $P_{exp}$ represents the probability that the promoter will be activated by the activated activator protein complex $\textrm{[(CooA)2CO]}$. It can be calculated by the following equation:

$$P_{exp} = \frac{\textrm{number of situations in which the promoter will be activated}}{\textrm{total number of situations}} = \frac{w}{z} \label{4} \tag{4}$$

In our modeling of the CO-sensing system, we have assumed that at any moment only one molecule of the activated activator protein complex $\textrm{[(CooA)2CO]}$ can bind to the promoter, and that the \ref{R1} reaches equilibrium before the transcription of the system begins.

To develop our equations further, we say that the number of situations in which a promoter will be activated is equal to:

$$ w=\textrm{[(CooA)2CO.Pr]}=\frac{\textrm{[(CooA)2CO]}}{K_p} \label{5} \tag{5}$$

Eq. (\ref{5}) demonstrates the relation between the binding of the activated activator protein complex $\textrm{[(CooA)2CO]}$, showing that it is equal to the amount of $\textrm{[(CooA)2CO]}$ upon the dissociation constant of the binding of $\textrm{[(CooA)2CO]}$ to the promoter.

To simplify this equation, we assume that the number of situations in which the promoter will not be activated by $\textrm{[(CooA)2CO]}$ is equal to $1$.

Note that this assumption is made because the only situation in which transcription will not take place is when there are no molecules bound to the promoter.

Substituting the assumptions in Eq. (\ref{5}), we get:

$$ z=w+1=\frac{\textrm{[(CooA)2CO]}}{K_p}+1 \label{6} \tag{6}$$

where, $z$ represents the total number of situations, $w$ represents the number of situations in which the promoter will be activated, and $1$ represents the number of situations in which the promoter will not be activated.

Sustituting Eq. (\ref{5}) and (\ref{6}) into Eq. (\ref{4}), we get:

$$P_{exp} = \frac{w}{z} = \frac{\frac{\textrm{[(CooA)2CO]}}{K_p}}{\frac{\textrm{[(CooA)2CO]}}{K_p}+1} \label{7} \tag{7}$$

As we want our $P_{exp}$ to be expressed in the terms of $\textrm{[CO]}$ and $\textrm{[CooA]}$ separately instead of $\textrm{[(CooA)2CO]}$, we repeat these above equations by taking into consideration \ref{R1} and its dissociation constant $\textrm{KI}$.

By doing so, we get:

$$P_{exp} = \frac{\frac{\textrm{[CooA]}.\frac{\frac{\textrm{CO}}{K_I}}{1+\frac{\textrm{CO}}{K_I}}}{K_p}} {1+\frac{\textrm{[CooA]}.\frac{\frac{\textrm{CO}}{K_I}}{1+\frac{\textrm{CO}}{K_I}}}{K_p}} \label{8} \tag{8}$$

As the Eq. (\ref{8}) demonstrates, $P_{exp}$ depends only upon the concentration of $\textrm{[CO]}$ and $\textrm{[CooA]}$.

Recall Eq. (\ref{3})

$$\frac{d(\textrm{mRNA})}{dt}=k_{on1}.P_{exp}-\gamma_1.[\textrm{mRNA}]$$

The initial condition for this equation was $\textrm{[mRNA(t=0)] = 0 molar}$.

The analytical solution of this differential equation is equal to:

$$[mRNA]=\frac{k_{on1}.P_{exp}}{\gamma_1}(1-e^{-\gamma_1t})$$

Now that we have got the equation describing the change in concentration of mRNA, we can use the same method to derive an equation which describes the change in the concentration of the chromoprotein in time as following:

$$\frac{d\textrm{[CP]}}{dt} = k_{on2}.\textrm{[mRNA]}-\gamma_2\textrm{[CP]} \label{9} \tag{9}$$

Here, $k_{on2}$ is the formation rate constant of the chromoprotein and $\gamma_2$ is the degradation rate constant of the chromoprotein.

**Figure 1:** The central theme considered for mathematical modeling of CO system, where the chromoprotein DNA leads to the production of mRNA of chromoprotein and then subsequently of chromoprotein. $K_{on1}$ is the rate of transcription of chromoprotein DNA, while $K_{on2}$ is the translation rate of mRNA of chromoprotein. Also, $\gamma_1$ is the degradation rate of mRNA of cheomoprotein and $\gamma_2$ is the degradation rate of chromoprotein.

Again, the initial conditions for this equation are assumed to be:

$$\textrm{[CP(t=0)] = 0 molar}$$

which states that at the initial time, the amount of chromoprotein in the cell is equal to $0$.

Here on forth, we will repeat the steps from Eq. (\ref{4}) to Eq. (\ref{8}) and arrive to the analytical solution of the differential equation of Eq. (\ref{9}), which is:

$$\textrm{[CP]} = k_{on2}.\frac{k_{on1}.P_{exp}}{\gamma_1.\gamma_2}(1-e^{-\gamma_2t}) + k_{on2}.\frac{k_{on1}.P_{exp}}{\gamma_1(\gamma_1-\gamma_2)}(e^{-\gamma_2t}-e^{-\gamma_1t}) \label{10} \tag{10}$$

Eq. (\ref{10}) is the final equation which describes the amount of $\textrm{[CP]}$ along with the amount of $\textrm{[mRNA]}$.

Assumptions

1. Reactions \ref{R1} and \ref{R2} reach equilibrium before the transcription begins.
2. Only one molecule of the complex $\textrm{[(CooA)2CO]}$ can be bound to the promoter at a time.
3. $\textrm{[mRNA(t=0)] = 0 molar}$
4. $\textrm{[CO]}$ and $\textrm{[CooA]}$ concentrations inside the cell are constant.

Simulation of the CO-sensing system

Simulation parameters

**Table 1:** Parameters, their description, and their values used in MATLAB simulation of CO-sensing system.
Parameter	Description	Value
$\textrm{[CO]}$	Carbon monoxide concentration at equilibrium	$\textrm{10}$
$\textrm{[CooA]}$	CooA protein concentration at equilibrium	$11$
$\textrm{KI}$	Dissociation constant of the binding of CO to CooA	$\textrm{0.011 mM}$
$\textrm{KP}$	Dissociation Constant of the binding of $\textrm{(CO2CooA)}$ to the promoter	$\textrm{0.000009}$ $\textrm{min}^{-1}$
Kon1	Formation rate constant of mRNA	$\textrm{0.001}$ $\textrm{min}^{-1}$
Kon2	Formation rate constant of the chromoprotein	0.006 $\textrm{min}^{-1}$ / 0.0051 $\textrm{min}^{-1}$
Gama1 ($\gamma$)	Degradation rate constant of the mRNA	$\textrm{0.00039}$ $\textrm{min}^{-1}$
Gama2 ($\gamma$)	Degradation rate constant of the chromoprotein	$\textrm{0.001}$ $\textrm{min}^{-1}$

Simulation results

Figure 2: Activity of the different promoters after stimulation by carbon monoxide. The strong promoter pCooF, shown in blue, is yielding higher level of expression than the weak promoter pCooM, shown in violet. The difference between the Strong promoter (pCooF) and the weak promoter (pCooM) is meaningful, and both of these promoters will express enough chromoprotein to be visual. Hence, our goal is achieved.

Conclusion

It is clear from the results that our model will achieve the steady state and give us the amount of chromoprotein that can be visualized.

MATLAB Code


function dx = ODEfun_CooM(t,x)
    dx      = zeros(1,2);
    %x is a vector (x(1),x(2))
    %x(1) = [mRNA]
    %x(2) = [Chromoprotein]
    KI      = 0.011;    %Dissociation constant of the binding of CO to CooA
    KP      = 0.1;      %Dissociation constant of the binding of 
                        %(CO2CooA) to the promoter.
    Kon1    = 0.001;    %Formation rate constant of mRNA
    Kon2    = 0.0051;   %Formation rate constant of the CP
    gama1   = 0.00039;  %Degradation rate constant of mRNA
    gama2   = 0.002;    %Degradation rate constant of the CP
    co      = 10;       %Concentration of CO
    cooa    = 11;       %Concentration of CooA
    w       = (cooa*co/(KI*(1+co/KI)))/KP;
    z       = (cooa*co/(KI*(1+co/KI)))/KP+1;
    P_exp   = w/z;
    dx1     = (Kon1*P_exp)-(gama1*x(1));    %Rate equation of mRNA
    dx2     = (Kon2*x(1))-(gama2*x(2));     %Rate equation of CP
    dx      = [dx1;dx2];
end

%Save this code as ODEfun_CooM.m file


function dx = ODEfun_CooF(t,x)
    dx      = zeros(1,2);
    %x is a vector [x(1),x(2)]
    %x(1) = [mRNA]
    %x(2) = [chromoprotein]
    KI      = 0.011;    %Dissociation constant of the binding of CO to CooA
    KP      = 0.1;      %Dissociation constant of the binding of 
                        %(CO2CooA) to the promoter.
    Kon1    = 0.001;    %Formation rate constant of mRNA
    Kon2    = 0.006;    %Formation rate constant of the CP
    gama1   = 0.00039;  %Degradation rate constant of mRNA
    gama2   = 0.002;    %Degradation rate constant of the CP
    co      = 10;       %Concentration of CO
    cooa    = 11;       %Concentration of CooA
    w       = (cooa*co/(KI*(1+co/KI)))/KP;
    z       = (cooa*co/(KI*(1+co/KI)))/KP+1;
    P_exp   = w/z;
    dx1     = (Kon1*P_exp)-(gama1*x(1));    %Rate equation of mRNA
    dx2     = (Kon2*x(1))-(gama2*x(2));     %Rate equation of CP
    dx      = [dx1;dx2];
end

%Save this code as ODEfun_CooF.m file


% Clear worskpace
clc 
clear all
% Code for calling the previous function:
time_range      = 18000; 
[T1,X1]         = ode45(@ODEfun_CooF,[0 time_range],[0 0]); 
[T2,X2]         = ode45(@ODEfun_CooM,[0 time_range],[0 0]); 
figure 
plot(T1,X1(:,2),'r', T2,X2(:,2),'b', T1,X1(:,1),'g')
legend('Conc. of chromoprotein with pCooF promoter ',...
    'Conc. of chromoprotein with pCooM promoter',...
    'mRNA concentration') 
xlabel('Time [sec]'); 
ylabel('Concentration [molar]'); 
title('Chromoprotein concentration with different promoters (pCooF, pCooM)')

%Save this code as co_sensing_system_simulation.m file

Modeling of the NOx-sensing system

To gain an insight into the stability of NO-sensing circuit, we consider the cooperativity of NO binding with NsrR and the law of mass action. The goal of the modeling is to determine the stable expression of chromoprotein by considering a specified concentration of different parameters of the dynamic system.

Rate of change of chromoprotein expression

The following ordinary differential equations are used for determining the expression levels of chromoproteins (CP) and its mRNA (mCP):

$$\frac{d\textrm{[mCP]}}{dt} = k_{transcription}\textrm{[PyeaR]} - \gamma_m \textrm{[mCP]} \label{11} \tag{11}$$ $$\frac{d\textrm{[CP]}}{dt}=k_{translation}\textrm{[mCP]}-\gamma_p\textrm{[CP]} \label{12} \tag{12}$$

**Figure 3:** The core idea used in mathematical modeling of NOx system, where the activation of PyeaR promotor leads to the production of mRNA of chromoprotein and then subsequently of chromoprotein. $K_{transcription}$ is the transcription rate of PyeaR promotor, while $K_{translation}$ is the translation rate of mRNA of chromoprotein. Additionally, $\gamma_m$ is the degradation rate of mCP (mRNA of cheomoprotein) and $\gamma_p$ is the degradation rate of CP (chromoprotein).

Binding cooperativity betweeen NO AND NsrR

The cooperativity of $\textrm{NO}$ binding with NsrR can be found using the Hill Equation, where free $\textrm{NO}^*$ combines with the $\textrm{n}$ amount of NsrR, leading to the formation of $\textrm{n[NsrR][NO]}$ complex. The total amount of inducer $NO_T$ concentration can be described by the following equation:

$$\textrm{NO}_T = \textrm{n[NsrR][NO]} + [\textrm{NO}^*]$$

Additionally, $\textrm{NsrR}$ dimerization with respect to unbound $\textrm{NsrR}$ can be represented by the following equation by considering the degree of cooperativity of $\textrm{NO}$ binding:

$$\frac{[\textrm{NO}^*]}{[\textrm{NO}_T]} = \frac{1}{1+\left(\frac{NsrR}{K_{NO}} \right)^n} \label{13} \tag{13}$$

Equation for PyeaR activity

The correlation between PyeaR activity and rate of transcription of mCP (\beta) can represented as:

$$\textrm{[PyeaR]}activity = \frac{\beta}{1+ \left( \frac{\textrm{NO}^*}{K_{d(\textrm{NsrR})}} \right)} \label{14} \tag{14}$$

Solving equations for chromoprotein expressions

Combining the Eqs. (\ref{13}) and (\ref{14}) into equation (\ref{11}), yields:

$$\frac{d[\textrm{mCP}]}{dt} = \frac{k_{transcription}[\textrm{PyeaR}]}{1 + \left(\frac{[\textrm{NsrR}]}{\left(1+\left[\frac{[\textrm{NO}]}{K_{NO}}\right]^n \right)}\right) k_{d(\textrm{NsrR})} } - \gamma_m [\textrm{mCP}]$$

In order to further simplify the system, let

$$K=\frac{k_{transcription}[\textrm{PyeaR}]}{1 + \left(\frac{[\textrm{NsrR}]}{\left(1+\left[\frac{[\textrm{NO}]}{K_{NO}}\right]^n \right)}\right) k_{d(\textrm{NsrR})} }$$

Additionally, the value of $\textrm{K}$ is inversely proportional to the amount of $\textrm{NsrR}$ and directly proportional to the $\textrm{NO}$ concentration in the system. Substituting $\textrm{K}$ into the differential equation of $\textrm{mCP}$ generates the following equation:

$$\frac{d[\textrm{mCP}]}{dt} = K - \gamma_m [\textrm{mCP}]$$

After solving the equation through homogenous and particular solution, the following equation is obtained for steady—state concentration of mCP generated after PyeaR is triggered:

$$[\textrm{mCP}]=\frac{K}{\gamma_m} - \frac{K}{\gamma_m}e^{-\gamma_mt}$$

The steady—state concentration of mCP is then substituted into Eq. (\ref{12}) and is also solved through homogenous and particular solution. The resulting equation gives the steady—state concentration of CP in the desired system according to the specified formation and degradation rates of CP in response to promotor stimulation:

$$[\textrm{CP}]=\frac{K k_{translation}}{\gamma_p \gamma_m} + \frac{K k_{translation}}{{\gamma_m}^2-\gamma_p \gamma_m} e^{-\gamma_mt} - \left(\frac{K k_{translation}}{\gamma_p \gamma_m} + \frac{K k_{translation}}{{\gamma_m}^2-\gamma_p \gamma_m} \right)e^{-\gamma_pt} $$

Simulation of the NOx-sensing system

Simulation parameters

**Table 2:** Parameters, their description, and their values used in MATLAB simulation of the NOx-sensing system.
Parameter	Description	Value
$[\textrm{PyeaR}]$	Concentration of PyeaR	1 nM
$k_{transcription}$	Rate of chromoprotein mRNA synthesis	0.167$\textrm{min}^{-1}$
$k_{translation}$	Rate of chromoprotein synthesis	$0.0011 \textrm{min}^{-1}$
$\gamma_m$	mRNA degradation rate	$0.19 \textrm{min}^{-1}$
$\gamma_p$	Chromoprotein degradation rate	$0.18 \textrm{min}^{-1}$
$K_{\textrm{NO}}$	Dissociation constant of NO	$0.12 \textrm{min}^{-1}$
$K_{\textrm{d(NsrR)}}$	Dissociation constant of NsrR	$\textrm{35}$ nM
n	Cooperativity of NO binding to NsrR	2

Simulation results

Figure 4. The graph shows the steady–state concentration/expression of mCP over time with respect to the mRNA degradation rates of chromoprotein. It is clear from the graph that presence of 0.87 nM of mCP is important in the cell in order to achieve the steady–state concentration of mCP.

Figure 5. The graph shows the relationship between the steady–state concentration over time of CP with respect to its degradation rate. It can be seen from the graph that 0.005 nM of CP is required in the cell in order to acquire the steady–state concentration of CP.

Matlab Code

    
%Defining the parameters
ktr     = 0.167;     %mRNA synthesis rate of 
                     %Chromoprotein per minute (rate of transcription)
ktrans  = 0.0011;    %Chromoprotein synthesis rate 
                     %per minute (rate of translation)
ycp     = 0.18;      %Chromoproteins degradation rate per minute
ym      = 0.19;      %mRNA degradation rate of chromoproteins per minute 
Kno     = 0.12;      %Dissociation constant for inducer (NO)
Kdn     = 35;        %Dissociation constant for repressor (NsrR)
pyear   = 1;         %Number of plasmids per cell
n       = 2;         %Cooperativity of inducer (NO)

%Assume that the repressor is removed 
NsrR    = 0;
NO      = 2;
t       = 0:0.1:240; 

%Calculate mRNA and Chromoprotein values
K       = ktr*pyear/(1+(NsrR/((1+(NO/Kno)^n)*Kdn)));
mCP     = K/ym-K/ym*exp(-ym*t);
CP      = K*ktrans/(ycp*ym)+K*ktrans/(ym^2-ycp*ym)*exp(-ym*t)...
    -(K*ktrans/(ycp*ym)+K*ktrans/(ym^2-ycp*ym))*exp(-ycp*t);

disp('Steady State mCP Concentration:')
disp(mCP(end)) %Steady state mRNA value
disp('Steady State CP Concentration:')
disp(CP(end)) %Steady state chromoprotein value
 
%Plotting mRNA concentration over time
figure(1)
plot(t, mCP,'r','LineWidth',2)
hold on
grid on
xlabel('Time (minutes)')
ylabel('mCP per cell (nM)')
title('Expression of mRNA of chromoprotein (mCP)')

%Plotting the chromoprotein concentrations over time
figure(2)
plot(t,CP,'r', 'LineWidth',2)
grid on
hold on
xlabel('Time (minutes)')
ylabel('CP per cell (nM)')
title('Expression of chromoprotein (CP)')

%Save this as nox_sensing_system_simulation.m file

Conclusion

The system is considered as stable, because when NsrR detaches itself from pYeaR, the promoter then initiate the production of mCP and CP, until a steady–state concentration is reached. This steady–state concentration is the maximum production rate of mCP and CP with respect to their degradation rates. However, it is clear from the graphs that the degradation rates are only related to the maximum production rate/ number of molecules in the cell; it is not responsible for altering the time span required to reach the steady–state concentration. Furthermore, this simplified solution for NOx–sensing system demonstrates the migration of system towards steady–state under assumed parameters and prescribed conditions. Therefore, this mathematical model can be considered as an appropriate model for determining the behavior of NOx–sensing system.

References

Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F.L. and Scire, J.J. (2007) 'A comprehensive kinetic mechanism for CO, CH2O, and CH3OH combustion', International Journal of Chemical Kinetics, 39(3), pp. 109–136. doi: 10.1002/kin.20218.
Chen, H., Shiroguchi, K., Ge, H. and Xie, X.S. (2015) 'Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli', Molecular Systems Biology, 11(1), pp. 781–781. doi: 10.15252/msb.20145794.
Leduc, J., Thorsteinsson, M.V., Gaal, T. and Roberts, G.P. (2001) 'Mapping CooA.RNA polymerase interactions: Identification of activating regions 2 and 3 in CooA, the co–sensing transcriptional activator', Journal of Biological Chemistry, 276(43), pp. 39968–39973. doi: 10.1074/jbc.m105758200.
Benabbas, A., Karunakaran, V., Youn, H., Poulos, T.L. and Champion, P.M. (2012) 'Effect of DNA binding on Geminate CO recombination Kinetics in CO-sensing transcription factor CooA', Journal of Biological Chemistry, 287(26), pp. 21729–21740. doi: 10.1074/jbc.m112.345090.
Kuchinskas, M., Li, H., Conrad, M., Roberts, G. and Poulos, T.L. (2006) 'The role of the DNA-Binding domains in CooA activation', Biochemistry, 45(23), pp. 7148–27153. doi: 10.1021/bi052609o.
Bergmann, John E, and Harvey F Lodish. 'A kinetic model of protein synthesis. Application to hemoglobin synthesis and translational control.' The Journal of biological chemistry 254.23 (1979) : 11927–37.
Nick Csicsery, and Ricky O'Laughlin. 'A Mathematical Model of a Synthetically Constructed Genetic Toggle Switch' (2015)

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Difference between revisions of "Team:Peshawar/Model"

Latest revision as of 22:24, 19 October 2016

Modeling and Simulation

Why model?

Modeling of the CO-sensing system

Assumptions

Simulation of the CO-sensing system

Simulation parameters

Simulation results

Conclusion

MATLAB Code

Modeling of the NOx-sensing system

Rate of change of chromoprotein expression

Binding cooperativity betweeen NO AND NsrR

Equation for PyeaR activity

Solving equations for chromoprotein expressions

Simulation of the NOx-sensing system

Simulation parameters

Simulation results

Matlab Code

Conclusion

References

Reach Us

Institute of Integrative Biosciences,
CECOS University of IT & Emerging Sciences,
Phase 6, F-5, Hayatabad, Peshawar, Pakistan

iib@cecos.edu.pk

+92 91 5860291

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+            <li><a href="https://2016.igem.org/Team:Peshawar/Wiki">How to design an iGEM Wiki</a></li>
-<p> Delete this box in order to be evaluated for this medal. See more information at <a href="https://2016.igem.org/Judging/Pages_for_Awards/Instructions"> Instructions for Pages for awards</a>.</p>
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+                        <div id="js-parallax-window" class="parallax-window">
+                          <div class="parallax-static-content">
+                            <h1>Modeling and Simulation</h1>
+                            <div class="video-overlay" style="background-image: url(&quot;https://d3k5xyayaartr5.cloudfront.net/_assets/pattern-overlays/patterns/black-medium-checks.png&quot;);"></div>
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+                </div>
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+                <!--End of full-width-content-wrapper-container-->
+                <div class="centered-content-wrapper-container model">
+                    <div class="centered-content-wrapper">
+                        <h2>Why model?</h2>
+                        <p>Before building a part or a device in lab, it is probably a good idea to simulate it in a simulation package &mdash; such as SimBiology, Cello, or GenoCAD &mdash; to check if the part or device one intends to make would work the way one wants it to. Although simulation can't solve all the problems that one might encounter during the actual physical design, it can certainly reduce them, and help in getting a better understanding of how a design behaves under different parameters and conditions.</p>
+                        <p>Our project has two biological ystems: one for detection of corbon monoxide, and another for detection for oxides of nitrogen. Below we show you how we modelled and simulated both these systems.</p>
+                        <h2>Modeling of the CO-sensing system</h2>
+                        <p>In our CO-sensing system, the CO binds to CooA (an activator). It activates the activator protein, and so the activated activator protein (CooA)2CO binds to promoter and initiates transcription.</p>
+                        <p>The reactions which take place can be given as:</p>
+                        $$\textrm{[CO]}+\textrm{[CooA]} \longleftrightarrow \textrm{[(CooA)2CO]} \label{R1} \tag{R1}$$
+                        <p>The equation above represents the first reaction that takes place in our system in which the CO present in the environment is taken up into the cell where it binds to the CooA protein. The CooA protein is constantly expressed in our system using a constitutive promoter. The coefficient 2 represents the number of CO which bind to CooA; this number was taken from the literature.</p>
+                        <p>To develop a dissociation constant of this equation, we can write it as:</p>
+                        $$KI=\frac{\textrm{[CO].[CooA]}}{\textrm{[(CooA)2CO]}} \label{1} \tag{1}$$
+                        <p>Note that the concentrations of $\textrm{[CO]}$, $\textrm{[CooA]}$, and $\textrm{[(CooA)2CO]}$ are at equilibrium before the start of the reaction R1.</p>
+                        <p>The second reaction that takes place in our system is the binding of the activated activator protein complex of $\textrm{[(CooA)2CO]}$ to the promoter Pr.</p>
+                        <p>It can be described as:</p>
+                        $$\textrm{[(CooA)2CO]} \longleftrightarrow \textrm{[(CooA)2CO.Pr]} \label{R2} \tag{R2}$$
+                        <p>Here, $\textrm{[(CooA)2CO.Pr]}$ represents the activated protein $\textrm{[(CooA)2CO]}$ bound to the promoter.</p>
+                        <p>To develop a dissociation constant of this equation, we can write as:</p>
+                        $$KP=\frac{\textrm{[(CooA)2CO]}}{\textrm{[(CooA)2CO.Pr]}} \label{2} \tag{2}$$
+                        <p>Now, in order to get an equation which describes the change in the concentration of mRNA with time we develop the following differential equation:</p>
+                        $$\frac{d(mRNA)}{dt}=k_{on1}.P_{exp}-\gamma_1.[mRNA] \label{3} \tag{3}$$
+                        <p>where, $k_{on1}$ is represents the formation rate constant of the mRNA, and $\gamma_1$ represents the degradation rate constant of mRNA.</p>
+                        <p>In Eq. (\ref{3}) it can be seen that the change in the $\textrm{[mRNA]}$ with time is equal to the difference of the formation rate of mRNA and  its degradation rate.</p>
+                        <p>The initial conditions for the Eq. (\ref{3}) are assumed to be:</p>
+                        $$\textrm{[mRNA(t=0)] = 0 molar}$$
+                        <p>This means that at the time $t=0$ sec (initial time), there are no molecules of mRNA encoding for our desired chromoprotein in the cell.</p>
+                        <p>In Eq. (\ref{3}), $P_{exp}$ represents the probability that the promoter will be activated by the activated activator protein complex $\textrm{[(CooA)2CO]}$. It can be calculated by the following equation:</p>
+                        $$P_{exp} = \frac{\textrm{number of situations in which the promoter will be activated}}{\textrm{total number of situations}} = \frac{w}{z} \label{4} \tag{4}$$
+                        <p>In our modeling of the CO-sensing system, we have assumed that at any moment only one molecule of the activated activator protein complex $\textrm{[(CooA)2CO]}$ can bind to the promoter, and that the \ref{R1} reaches equilibrium before the transcription of the system begins.</p>
+                        <p>To develop our equations further, we say that the number of situations in which a promoter will be activated is equal to:</p>
+                        $$ w=\textrm{[(CooA)2CO.Pr]}=\frac{\textrm{[(CooA)2CO]}}{K_p} \label{5} \tag{5}$$
+                        <p>Eq. (\ref{5}) demonstrates the relation between the binding of the activated activator protein complex $\textrm{[(CooA)2CO]}$, showing that it is equal to the amount of $\textrm{[(CooA)2CO]}$ upon the dissociation constant of the binding of $\textrm{[(CooA)2CO]}$ to the promoter.</p>
+                        <p>To simplify this equation, we assume that the number of situations in which the promoter will not be activated by $\textrm{[(CooA)2CO]}$ is equal to $1$.</p>
+                        <p>Note that this assumption is made because the only situation in which transcription will not take place is when there are no molecules bound to the promoter.</p>
+                        <p>Substituting the assumptions in Eq. (\ref{5}), we get:</p>
+                        $$ z=w+1=\frac{\textrm{[(CooA)2CO]}}{K_p}+1  \label{6} \tag{6}$$
+                        <p>where, $z$ represents the total number of situations, $w$ represents the number of situations in which the promoter will be activated, and $1$ represents the number of situations in which the promoter will not be activated.</p>
+                        <p>Sustituting Eq. (\ref{5}) and (\ref{6}) into Eq. (\ref{4}), we get:</p>
+                        $$P_{exp} = \frac{w}{z} = \frac{\frac{\textrm{[(CooA)2CO]}}{K_p}}{\frac{\textrm{[(CooA)2CO]}}{K_p}+1} \label{7} \tag{7}$$
+                        <p>As we want our $P_{exp}$ to be expressed in the terms of $\textrm{[CO]}$ and $\textrm{[CooA]}$ separately instead of $\textrm{[(CooA)2CO]}$, we repeat these above equations by taking into consideration \ref{R1} and its dissociation constant $\textrm{KI}$.</p>
+                        <p>By doing so, we get:</p>
+                        $$P_{exp} = \frac{\frac{\textrm{[CooA]}.\frac{\frac{\textrm{CO}}{K_I}}{1+\frac{\textrm{CO}}{K_I}}}{K_p}} {1+\frac{\textrm{[CooA]}.\frac{\frac{\textrm{CO}}{K_I}}{1+\frac{\textrm{CO}}{K_I}}}{K_p}} \label{8} \tag{8}$$
+                        <p>As the Eq. (\ref{8}) demonstrates, $P_{exp}$ depends only upon the concentration of $\textrm{[CO]}$ and $\textrm{[CooA]}$.</p>
+                        <p>Recall Eq. (\ref{3})</p>
+                        $$\frac{d(\textrm{mRNA})}{dt}=k_{on1}.P_{exp}-\gamma_1.[\textrm{mRNA}]$$
+                        <p>The initial condition for this equation was $\textrm{[mRNA(t=0)] = 0 molar}$.</p>
+                        <p>The analytical solution of this differential equation is equal to:</p>
+                        $$[mRNA]=\frac{k_{on1}.P_{exp}}{\gamma_1}(1-e^{-\gamma_1t})$$
+                        <p>Now that we have got the equation describing the change in concentration of mRNA, we can use the same method to derive an equation which describes the change in the concentration of the chromoprotein in time as following:</p>
+                        $$\frac{d\textrm{[CP]}}{dt} = k_{on2}.\textrm{[mRNA]}-\gamma_2\textrm{[CP]} \label{9} \tag{9}$$
+                        <p>Here, $k_{on2}$  is the formation rate constant of the chromoprotein and $\gamma_2$  is the degradation rate constant of the chromoprotein.</p>
+                        <br>
+                        <figure id="co-block">
+                            <img src="https://static.igem.org/mediawiki/2016/f/f2/T--Peshawar--co-block-diagram.png">
+                            <figcaption>
+                            <b>Figure 1: </b> The central theme considered for mathematical modeling of CO system, where the chromoprotein DNA leads to the production of mRNA of chromoprotein and then subsequently of chromoprotein. $K_{on1}$ is the rate of transcription of chromoprotein DNA, while $K_{on2}$ is the translation rate of mRNA of chromoprotein. Also, $\gamma_1$ is the degradation rate of mRNA of cheomoprotein and $\gamma_2$ is the degradation rate of chromoprotein.
+                            </figcaption>
+                        </figure>
+                        <br>
+                        <p>Again, the initial conditions for this equation are assumed to be:</p>
+                        $$\textrm{[CP(t=0)] = 0 molar}$$
+                        <p>which states that at the initial time, the amount of chromoprotein in the cell is equal to $0$.</p>
+                        <p>Here on forth, we will repeat the steps from Eq. (\ref{4}) to Eq. (\ref{8}) and arrive to the analytical solution of the differential equation of Eq. (\ref{9}), which is:</p>
+                        $$\textrm{[CP]} = k_{on2}.\frac{k_{on1}.P_{exp}}{\gamma_1.\gamma_2}(1-e^{-\gamma_2t}) + k_{on2}.\frac{k_{on1}.P_{exp}}{\gamma_1(\gamma_1-\gamma_2)}(e^{-\gamma_2t}-e^{-\gamma_1t}) \label{10} \tag{10}$$
+                        <p>Eq. (\ref{10}) is the final equation which describes the amount of $\textrm{[CP]}$ along with the amount of $\textrm{[mRNA]}$.</p>
+                        <h3>Assumptions</h3>
+                        <ol>
+                            <li>1. Reactions \ref{R1} and \ref{R2} reach equilibrium before the transcription begins.</li>
+                            <li>2. Only one molecule of the complex $\textrm{[(CooA)2CO]}$ can be bound to the promoter at a time.</li>
+                            <li>3. $\textrm{[mRNA(t=0)] = 0 molar}$</li>
+                            <li>4. $\textrm{[CO]}$ and $\textrm{[CooA]}$ concentrations inside the cell are constant.</li>
+                        </ol>
+                        <h2>Simulation of the CO-sensing system</h2>
+                        <h3>Simulation parameters</h3>
+                        <br>
+                        <table class="table-minimal">
+                            <caption><b>Table 1:</b> Parameters, their description, and their values used in MATLAB simulation of CO-sensing system.</caption>
+                            <thead>
+                                <tr>
+                                    <th>Parameter</th>
+                                    <th>Description</th>
+                                    <th>Value</th>
+                                </tr>
+                            </thead>
+                            <tbody>
+                                <tr>
+                                    <td>$\textrm{[CO]}$</td>
+                                    <td>Carbon monoxide concentration at equilibrium</td>
+                                    <td>$\textrm{10}$</td>
+                                </tr>
+                                <tr>
+                                    <td>$\textrm{[CooA]}$</td>
+                                    <td>CooA protein concentration at equilibrium</td>
+                                    <td>$11$</td>
+                                </tr>
+                                <tr>
+                                    <td>$\textrm{KI}$</td>
+                                    <td>Dissociation constant of the binding of CO to CooA</td>
+                                    <td>$\textrm{0.011 mM}$</td>
+                                </tr>
+                                <tr>
+                                    <td>$\textrm{KP}$</td>
+                                    <td>Dissociation Constant of the binding of $\textrm{(CO2CooA)}$ to the promoter</td>
+                                    <td>$\textrm{0.000009}$ $\textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>Kon1</td>
+                                    <td>Formation rate constant of mRNA</td>
+                                    <td>$\textrm{0.001}$ $\textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>Kon2</td>
+                                    <td>Formation rate constant of the chromoprotein</td>
+                                    <td>0.006 $\textrm{min}^{-1}$ / 0.0051 $\textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>Gama1 ($\gamma$)</td>
+                                    <td>Degradation rate constant of the mRNA</td>
+                                    <td>$\textrm{0.00039}$ $\textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>Gama2 ($\gamma$)</td>
+                                    <td>Degradation rate constant of the chromoprotein</td>
+                                    <td>$\textrm{0.001}$ $\textrm{min}^{-1}$</td>
+                                </tr>
+                            </tbody>
+                        </table>
+                        <br>
+                        <h3>Simulation results</h3>
+                        <br><br>
+                        <figure>
+                         <!--Demogrpahics-->
+<script type="text/javascript">
+google.charts.load('current', {
+    packages: ['corechart', 'line']
+});
+google.charts.setOnLoadCallback(drawCurveTypes);
+function drawCurveTypes() {
+    var data = new google.visualization.DataTable();
+    data.addColumn('number', 'Time');
+    data.addColumn('number', "Conc. of chromoprotein with pCooF promoter");
+    data.addColumn('number', "Conc. of chromoprotein with pCooM promoter");
+    data.addColumn('number', "Conc. of mRNA");
+    data.addRows([
+        [0, 0, 0, 0],
+        [0.00084492, 7.64009e-09, 6.49407e-09, 5.02372e-05],
+        [0.0016898, 3.05591e-08, 2.59752e-08, 0.000100473],
+        [0.0025347, 6.87552e-08, 5.84419e-08, 0.000150709],
+        [0.0033797, 1.22227e-07, 1.03893e-07, 0.000200943],
+        [0.0076042, 6.18647e-07, 5.2585e-07, 0.000452099],
+        [0.011829, 1.49667e-06, 1.27217e-06, 0.000703231],
+        [0.016053, 2.75607e-06, 2.34266e-06, 0.000954338],
+        [0.020278, 4.3966e-06, 3.73711e-06, 0.00120542],
+        [0.041401, 1.83083e-05, 1.55621e-05, 0.00246046],
+        [0.062524, 4.1714e-05, 3.54569e-05, 0.00371487],
+        [0.083647, 7.45849e-05, 6.33972e-05, 0.00496867],
+        [0.10477, 0.000116892, 9.93584e-05, 0.00622185],
+        [0.21038, 0.000468979, 0.000398632, 0.0124785],
+        [0.316, 0.00105272, 0.000894814, 0.0187196],
+        [0.42161, 0.00186462, 0.00158493, 0.0249454],
+        [0.52723, 0.00290122, 0.00246604, 0.0311558],
+        [1.0048, 0.0103016, 0.00875638, 0.0590466],
+        [1.4823, 0.0219241, 0.0186355, 0.0866276],
+        [1.9599, 0.0374825, 0.0318601, 0.113902],
+        [2.4375, 0.0567059, 0.0482, 0.140873],
+        [3.2995, 0.0998687, 0.0848884, 0.188803],
+        [4.1616, 0.15278, 0.129863, 0.235775],
+        [5.0236, 0.21421, 0.182079, 0.281809],
+        [5.8856, 0.283037, 0.240581, 0.326924],
+        [7.1129, 0.391805, 0.333035, 0.389605],
+        [8.3402, 0.511068, 0.434408, 0.450511],
+        [9.5675, 0.638707, 0.542901, 0.509693],
+        [10.7948, 0.772827, 0.656903, 0.5672],
+        [12.4036, 0.95574, 0.812379, 0.640122],
+        [14.0124, 1.14424, 0.972602, 0.71035],
+        [15.6212, 1.33605, 1.13564, 0.777984],
+        [17.2299, 1.52913, 1.29976, 0.843118],
+        [19.2775, 1.77411, 1.50799, 0.922551],
+        [21.3251, 2.0164, 1.71394, 0.998267],
+        [23.3727, 2.2544, 1.91624, 1.07044],
+        [25.4203, 2.48659, 2.1136, 1.13924],
+        [28.0161, 2.77081, 2.35519, 1.22185],
+        [30.6119, 3.04327, 2.58678, 1.29959],
+        [33.2078, 3.30361, 2.80807, 1.37275],
+        [35.8036, 3.55121, 3.01853, 1.4416],
+        [39.1395, 3.85025, 3.27271, 1.52415],
+        [42.4754, 4.12908, 3.50972, 1.60051],
+        [45.8113, 4.38884, 3.73051, 1.67113],
+        [49.1472, 4.63002, 3.93552, 1.73645],
+        [53.5666, 4.92212, 4.1838, 1.81549],
+        [57.986, 5.18634, 4.40839, 1.88677],
+        [62.4054, 5.42549, 4.61167, 1.95104],
+        [66.8248, 5.64133, 4.79513, 2.009],
+        [72.9821, 5.90656, 5.02057, 2.0804],
+        [79.1394, 6.13655, 5.21607, 2.14221],
+        [85.2967, 6.33635, 5.3859, 2.1957],
+        [91.4539, 6.50928, 5.53289, 2.24202],
+        [98.9539, 6.68778, 5.68461, 2.29017],
+        [106.4539, 6.83798, 5.81228, 2.33056],
+        [113.9539, 6.96511, 5.92034, 2.36441],
+        [121.4539, 7.07154, 6.01081, 2.3928],
+        [128.9539, 7.15888, 6.08504, 2.41667],
+        [136.4539, 7.23294, 6.148, 2.43669],
+        [143.9539, 7.29714, 6.20257, 2.45346],
+        [151.4539, 7.35063, 6.24803, 2.46753],
+        [158.9539, 7.39199, 6.28319, 2.47936],
+        [166.4539, 7.42804, 6.31384, 2.48929],
+        [173.9539, 7.46188, 6.3426, 2.4976],
+        [181.4539, 7.48963, 6.36619, 2.50458],
+        [188.9539, 7.50688, 6.38085, 2.51044],
+        [196.4539, 7.52364, 6.3951, 2.51536],
+        [203.9539, 7.54382, 6.41225, 2.51948],
+        [211.4539, 7.55968, 6.42573, 2.52294],
+        [218.4274, 7.5641, 6.42948, 2.52565],
+        [225.4009, 7.5706, 6.43501, 2.52796],
+        [232.3744, 7.58281, 6.44539, 2.52992],
+        [239.3479, 7.59222, 6.45339, 2.53158],
+        [245.7054, 7.59267, 6.45377, 2.53288],
+        [252.0629, 7.59504, 6.45578, 2.534],
+        [258.4203, 7.60131, 6.46111, 2.53497],
+        [264.7778, 7.6063, 6.46535, 2.5358],
+        [271.1393, 7.60623, 6.46529, 2.53651],
+        [277.5008, 7.60741, 6.4663, 2.53713],
+        [283.8624, 7.61107, 6.46941, 2.53766],
+        [290.2239, 7.61396, 6.47186, 2.53812],
+        [292.6679, 7.61412, 6.472, 2.53828],
+        [295.1119, 7.61436, 6.47221, 2.53843],
+        [297.556, 7.61466, 6.47246, 2.53857],
+        [300, 7.61498, 6.47273, 2.5387]
+    ]);
+    var options = {
+        title: 'Chromoprotein concentration with different promoters (pCooF, pCooM)',
+        colors: ['#185FB1', '#702EA6', '#ED2B72'],
+        lineWidth: 5,
+        hAxis: {
+            gridlines: {color: '#F1F1F1'},
+            title: 'Time (minutes)',
+            textStyle: {
+                fontSize: 16,
+                //fontName: 'Arial',
+                bold: false,
+                italic: false
+            },
+            titleTextStyle: {
+                fontSize: 17,
+                bold: false,
+                italic: false
+            }
+        },
+        vAxis: {
+            gridlines: {color: '#F1F1F1'},
+            title: 'Concentration [nM]',
+            textStyle: {
+                fontSize: 16,
+                //fontName: 'Arial',
+                bold: false,
+                italic: false
+            },
+            titleTextStyle: {
+                fontSize: 17,
+                bold: false,
+                italic: false
+            }
+        },
+        titleTextStyle: {
+            fontSize: 19
+        },
+    };
+    var chart = new google.visualization.LineChart(document.getElementById('chart16_div'));
+    chart.draw(data, options);
+}
+</script>
+<div id="chart16_div" style="width: 900px; height: 600px;"></div>
+                            <figcaption>
+                                <b>Figure 2:</b> Activity of the different promoters after stimulation by carbon monoxide. The strong promoter pCooF, shown in blue, is yielding higher level of expression than the weak promoter pCooM, shown in violet. The difference between the Strong promoter (pCooF) and the weak promoter (pCooM) is meaningful, and both of these promoters will express enough chromoprotein to be visual. Hence, our goal is achieved.
+                            </figcaption>
+                        </figure>
+                        <h3>Conclusion</h3>
+                        <p>It is clear from the results that our model will achieve the steady state and give us the amount of chromoprotein that can be visualized.</p>
+                        <h3>MATLAB Code</h3>
+<pre>
+<code class="language-matlab">
+function dx = ODEfun_CooM(t,x)
+    dx      = zeros(1,2);
+    %x is a vector (x(1),x(2))
+    %x(1) = [mRNA]
+    %x(2) = [Chromoprotein]
+    KI      = 0.011;    %Dissociation constant of the binding of CO to CooA
+    KP      = 0.1;      %Dissociation constant of the binding of
+                        %(CO2CooA) to the promoter.
+    Kon1    = 0.001;    %Formation rate constant of mRNA
+    Kon2    = 0.0051;   %Formation rate constant of the CP
+    gama1   = 0.00039;  %Degradation rate constant of mRNA
+    gama2   = 0.002;    %Degradation rate constant of the CP
+    co      = 10;       %Concentration of CO
+    cooa    = 11;       %Concentration of CooA
+    w       = (cooa*co/(KI*(1+co/KI)))/KP;
+    z       = (cooa*co/(KI*(1+co/KI)))/KP+1;
+    P_exp   = w/z;
+    dx1     = (Kon1*P_exp)-(gama1*x(1));    %Rate equation of mRNA
+    dx2     = (Kon2*x(1))-(gama2*x(2));     %Rate equation of CP
+    dx      = [dx1;dx2];
+end
+%Save this code as ODEfun_CooM.m file
+</code>
+</pre>
+<pre>
+<code class="language-matlab">
+function dx = ODEfun_CooF(t,x)
+    dx      = zeros(1,2);
+    %x is a vector [x(1),x(2)]
+    %x(1) = [mRNA]
+    %x(2) = [chromoprotein]
+    KI      = 0.011;    %Dissociation constant of the binding of CO to CooA
+    KP      = 0.1;      %Dissociation constant of the binding of
+                        %(CO2CooA) to the promoter.
+    Kon1    = 0.001;    %Formation rate constant of mRNA
+    Kon2    = 0.006;    %Formation rate constant of the CP
+    gama1   = 0.00039;  %Degradation rate constant of mRNA
+    gama2   = 0.002;    %Degradation rate constant of the CP
+    co      = 10;       %Concentration of CO
+    cooa    = 11;       %Concentration of CooA
+    w       = (cooa*co/(KI*(1+co/KI)))/KP;
+    z       = (cooa*co/(KI*(1+co/KI)))/KP+1;
+    P_exp   = w/z;
+    dx1     = (Kon1*P_exp)-(gama1*x(1));    %Rate equation of mRNA
+    dx2     = (Kon2*x(1))-(gama2*x(2));     %Rate equation of CP
+    dx      = [dx1;dx2];
+end
+%Save this code as ODEfun_CooF.m file
+</code>
+</pre>
+<pre>
+<code class="language-matlab">
+% Clear worskpace
+clc
+clear all
+% Code for calling the previous function:
+time_range      = 18000;
+[T1,X1]         = ode45(@ODEfun_CooF,[0 time_range],[0 0]);
+[T2,X2]         = ode45(@ODEfun_CooM,[0 time_range],[0 0]);
+figure
+plot(T1,X1(:,2),'r', T2,X2(:,2),'b', T1,X1(:,1),'g')
+legend('Conc. of chromoprotein with pCooF promoter ',...
+    'Conc. of chromoprotein with pCooM promoter',...
+    'mRNA concentration')
+xlabel('Time [sec]');
+ylabel('Concentration [molar]');
+title('Chromoprotein concentration with different promoters (pCooF, pCooM)')
+%Save this code as co_sensing_system_simulation.m file
+</code>
+</pre>
+                        <!--NOx circuit-->
+                        <h2>Modeling of the NOx-sensing system</h2>
+                        <p>To gain an insight into the stability of NO-sensing circuit, we consider the cooperativity of NO binding with NsrR and the law of mass action. The goal of the modeling is to determine the stable expression of chromoprotein by considering a specified concentration of different parameters of the dynamic system.</p>
+                        <h3>Rate of change of chromoprotein expression</h3>
+                        <p>The following ordinary differential equations are used for determining the expression levels of chromoproteins (CP) and its mRNA (mCP):</p>
+                        $$\frac{d\textrm{[mCP]}}{dt} = k_{transcription}\textrm{[PyeaR]} - \gamma_m \textrm{[mCP]} \label{11} \tag{11}$$
+                        $$\frac{d\textrm{[CP]}}{dt}=k_{translation}\textrm{[mCP]}-\gamma_p\textrm{[CP]} \label{12} \tag{12}$$
+                        <br>
+                        <figure id="nox-block">
+                            <img src="https://static.igem.org/mediawiki/2016/f/f7/T--Peshawar--nox-block-diagram.png">
+                            <figcaption>
+                            <b>Figure 3: </b> The core idea used in mathematical modeling of NOx system, where the activation of PyeaR promotor leads to the production of mRNA of chromoprotein and then subsequently of chromoprotein. $K_{transcription}$ is the transcription rate of PyeaR promotor, while $K_{translation}$ is the translation rate of mRNA of chromoprotein. Additionally, $\gamma_m$ is the degradation rate of mCP (mRNA of cheomoprotein) and $\gamma_p$ is the degradation rate of CP (chromoprotein).
+                            </figcaption>
+                        </figure>
+                        <br>
+                        <h3>Binding cooperativity betweeen NO AND NsrR</h3>
+                        <p>The cooperativity of $\textrm{NO}$ binding with NsrR can be found using the Hill Equation, where free $\textrm{NO}^*$ combines with the $\textrm{n}$ amount of NsrR, leading to the formation of $\textrm{n[NsrR][NO]}$ complex. The total amount of inducer $NO_T$ concentration can be described by the following equation:</p>
+                        $$\textrm{NO}_T = \textrm{n[NsrR][NO]} + [\textrm{NO}^*]$$
+                        <p>Additionally, $\textrm{NsrR}$ dimerization with respect to unbound $\textrm{NsrR}$ can be represented by the following equation by considering the degree of cooperativity of $\textrm{NO}$ binding:</p>
+                        $$\frac{[\textrm{NO}^*]}{[\textrm{NO}_T]} = \frac{1}{1+\left(\frac{NsrR}{K_{NO}} \right)^n} \label{13} \tag{13}$$
+                        <h3>Equation for PyeaR activity</h3>
+                        <p>The correlation between PyeaR activity and rate of transcription of mCP (\beta) can represented as: </p>
+                        $$\textrm{[PyeaR]}activity = \frac{\beta}{1+ \left( \frac{\textrm{NO}^*}{K_{d(\textrm{NsrR})}} \right)} \label{14} \tag{14}$$
+                        <h3>Solving equations for chromoprotein expressions</h3>
+                        <p>Combining the Eqs. (\ref{13}) and (\ref{14}) into equation (\ref{11}), yields:</p>
+                        $$\frac{d[\textrm{mCP}]}{dt} = \frac{k_{transcription}[\textrm{PyeaR}]}{1 + \left(\frac{[\textrm{NsrR}]}{\left(1+\left[\frac{[\textrm{NO}]}{K_{NO}}\right]^n \right)}\right) k_{d(\textrm{NsrR})} } - \gamma_m [\textrm{mCP}]$$
+                        <p>In order to further simplify the system, let</p>
+                        $$K=\frac{k_{transcription}[\textrm{PyeaR}]}{1 + \left(\frac{[\textrm{NsrR}]}{\left(1+\left[\frac{[\textrm{NO}]}{K_{NO}}\right]^n \right)}\right) k_{d(\textrm{NsrR})} }$$
+                        <p>Additionally, the value of $\textrm{K}$ is inversely proportional to the amount of $\textrm{NsrR}$ and directly proportional to the $\textrm{NO}$ concentration in the system. Substituting $\textrm{K}$ into the differential equation of $\textrm{mCP}$ generates the following equation:</p>
+                        $$\frac{d[\textrm{mCP}]}{dt} = K - \gamma_m [\textrm{mCP}]$$
+                        <p>After solving the equation through homogenous and particular solution, the following equation is obtained for steady&mdash;state concentration of mCP generated after PyeaR is triggered:</p>
+                        $$[\textrm{mCP}]=\frac{K}{\gamma_m} - \frac{K}{\gamma_m}e^{-\gamma_mt}$$
+                        <p>The steady&mdash;state concentration of mCP is then substituted into Eq. (\ref{12}) and is also solved through homogenous and particular solution. The resulting equation gives the steady&mdash;state concentration of CP in the desired system according to the specified formation and degradation rates of CP in response to promotor stimulation:</p>
+                        $$[\textrm{CP}]=\frac{K k_{translation}}{\gamma_p \gamma_m} + \frac{K k_{translation}}{{\gamma_m}^2-\gamma_p \gamma_m} e^{-\gamma_mt} - \left(\frac{K k_{translation}}{\gamma_p \gamma_m} + \frac{K k_{translation}}{{\gamma_m}^2-\gamma_p \gamma_m} \right)e^{-\gamma_pt} $$
+                        <br><br>
+                        <h2>Simulation of the NOx-sensing system</h2>
+                        <h3>Simulation parameters</h3>
+                        <table class="table-minimal">
+                            <caption><b>Table 2:</b> Parameters, their description, and their values used in MATLAB simulation of the NOx-sensing system.</caption>
+                            <thead>
+                                <tr>
+                                    <th>Parameter </th>
+                                    <th>Description</th>
+                                    <th>Value</th>
+                                </tr>
+                            </thead>
+                            <tbody>
+                                <tr>
+                                    <td>$[\textrm{PyeaR}]$</td>
+                                    <td>Concentration of PyeaR</td>
+                                    <td>1 nM</td>
+                                </tr>
+                                <tr>
+                                    <td>$k_{transcription}$</td>
+                                    <td>Rate of chromoprotein mRNA synthesis</td>
+                                    <td>0.167$\textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>$k_{translation}$</td>
+                                    <td>Rate of chromoprotein synthesis</td>
+                                    <td>$0.0011 \textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>$\gamma_m$</td>
+                                    <td>mRNA degradation rate</td>
+                                    <td>$0.19 \textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>$\gamma_p$</td>
+                                    <td>Chromoprotein degradation rate</td>
+                                    <td>$0.18 \textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>$K_{\textrm{NO}}$</td>
+                                    <td>Dissociation constant of NO</td>
+                                    <td>$0.12 \textrm{min}^{-1}$</td>
+                                </tr>
+                                <tr>
+                                    <td>$K_{\textrm{d(NsrR)}}$</td>
+                                    <td>Dissociation constant of NsrR</td>
+                                    <td>$\textrm{35}$ nM</td>
+                                </tr>
+                                <tr>
+                                    <td>n</td>
+                                    <td>Cooperativity of NO binding to NsrR</td>
+                                    <td>2</td>
+                                </tr>
+                            </tbody>
+                        </table>
+                        <h3>Simulation results</h3>
+                        <br><br>
+                        <figure>
+                            <!--Demogrpahics-->
+<script type="text/javascript">
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+    packages: ['corechart', 'line']
+});
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+function drawBasic() {
+    var data = new google.visualization.DataTable();
+    data.addColumn('number', 'Time (minutes)');
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+    data.addRows([
+        [0, 0],
+        [1, 0.15209],
+        [2, 0.27787],
+        [3, 0.38188],
+        [4, 0.46789],
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+        [6, 0.59784],
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+        [20, 0.85928],
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+        [22, 0.8655],
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+        [238, 0.87895],
+        [239, 0.87895],
+        [240, 0.87895]
+    ]);
+    var options = {
+        title: 'Expression of mRNA of chromoprotein (mCP)',
+        colors: ['#833789'],
+        lineWidth: 5,
+        hAxis: {
+            gridlines: {color: '#F1F1F1'},
+            title: 'Time (minutes)',
+            textStyle: {
+                fontSize: 16,
+                //fontName: 'Arial',
+                bold: false,
+                italic: false
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+            gridlines: {color: '#F1F1F1'},
+            title: 'mCP per cell (nM)',
+            textStyle: {
+                fontSize: 16,
+                //fontName: 'Arial',
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+                fontSize: 17,
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+            position: "none"
+        },
+        titleTextStyle: {
+            fontSize: 19
+        },
+    };
+    var chart = new google.visualization.LineChart(document.getElementById('chart_div'));
+    chart.draw(data, options);
+}
+</script>
+<div id="chart_div" style="width: 900px; height: 400px;"></div>
+                            <figcaption>
+                                <b>Figure 4.</b> The graph shows the steady&ndash;state concentration/expression of mCP over time with respect to the mRNA degradation rates of chromoprotein. It is clear from the graph that presence of 0.87 nM of mCP is important in the cell in order to achieve the steady&ndash;state concentration of mCP.
+                            </figcaption>
+                        </figure>
+                        <br><br>
+                        <figure>
+                        <!--Demogrpahics-->
+<script type="text/javascript">
+google.charts.load('current', {
+    packages: ['corechart', 'line']
+});
+google.charts.setOnLoadCallback(drawBasic);
+function drawBasic() {
+    var data = new google.visualization.DataTable();
+    data.addColumn('number', 'Time (minutes)');
+    data.addColumn('number', 'CP per cell (nM)');
+    data.addRows([
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+        [1, 8.1271e-05],
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+        [5, 0.0012704],
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+    var options = {
+        title: 'Expression of chromoprotein (CP)',
+        colors: ['#248689'],
+        lineWidth: 5,
+        hAxis: {
+            gridlines: {color: '#F1F1F1'},
+            title: 'Time (minutes)',
+            textStyle: {
+                fontSize: 16,
+                //fontName: 'Arial',
+                bold: false,
+                italic: false
+            },
+            titleTextStyle: {
+                fontSize: 17,
+                bold: false,
+                italic: false
+            }
+        },
+        vAxis: {
+            gridlines: {color: '#F1F1F1'},
+            title: 'CP per cell (nM)',
+            textStyle: {
+                fontSize: 16,
+                //fontName: 'Arial',
+                bold: false,
+                italic: false
+            },
+            titleTextStyle: {
+                fontSize: 17,
+                bold: false,
+                italic: false
+            }
+        },
+        legend: {
+            position: "none"
+        },
+        titleTextStyle: {
+            fontSize: 19
+        },
+    };
+    var chart = new google.visualization.LineChart(document.getElementById('chart2_div'));
+    chart.draw(data, options);
+}
+</script>
+<div id="chart2_div" style="width: 900px; height: 400px;"></div>
+                            <figcaption>
+                                <b>Figure 5.</b> The graph shows the relationship between the steady&ndash;state concentration over time of CP with respect to its degradation rate. It can be seen from the graph that 0.005 nM of CP is required in the cell in order to acquire the steady&ndash;state concentration of CP.
+                            </figcaption>
+                        </figure>
+                        <h3>Matlab Code</h3>
+<pre>
+    <code class="language-matlab">
+%Defining the parameters
+ktr     = 0.167;     %mRNA synthesis rate of
+                     %Chromoprotein per minute (rate of transcription)
+ktrans  = 0.0011;    %Chromoprotein synthesis rate
+                     %per minute (rate of translation)
+ycp     = 0.18;      %Chromoproteins degradation rate per minute
+ym      = 0.19;      %mRNA degradation rate of chromoproteins per minute
+Kno     = 0.12;      %Dissociation constant for inducer (NO)
+Kdn     = 35;        %Dissociation constant for repressor (NsrR)
+pyear   = 1;         %Number of plasmids per cell
+n       = 2;         %Cooperativity of inducer (NO)
+%Assume that the repressor is removed
+NsrR    = 0;
+NO      = 2;
+t       = 0:0.1:240;
+%Calculate mRNA and Chromoprotein values
+K       = ktr*pyear/(1+(NsrR/((1+(NO/Kno)^n)*Kdn)));
+mCP     = K/ym-K/ym*exp(-ym*t);
+CP      = K*ktrans/(ycp*ym)+K*ktrans/(ym^2-ycp*ym)*exp(-ym*t)...
+    -(K*ktrans/(ycp*ym)+K*ktrans/(ym^2-ycp*ym))*exp(-ycp*t);
+disp('Steady State mCP Concentration:')
+disp(mCP(end)) %Steady state mRNA value
+disp('Steady State CP Concentration:')
+disp(CP(end)) %Steady state chromoprotein value
+%Plotting mRNA concentration over time
+figure(1)
+plot(t, mCP,'r','LineWidth',2)
+hold on
+grid on
+xlabel('Time (minutes)')
+ylabel('mCP per cell (nM)')
+title('Expression of mRNA of chromoprotein (mCP)')
-<div class="column full_size">
+%Plotting the chromoprotein concentrations over time
-<h2> Modeling</h2>
+figure(2)
-<p>Mathematical models and computer simulations provide a great way to describe the function and operation of BioBrick Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior in conjunction with experiments in the wetlab.</p>
+plot(t,CP,'r', 'LineWidth',2)
+grid on
+hold on
+xlabel('Time (minutes)')
+ylabel('CP per cell (nM)')
+title('Expression of chromoprotein (CP)')
+%Save this as nox_sensing_system_simulation.m file
+    </code>
+</pre>
+                        <h3>Conclusion</h3>
+                        <p>The system is considered as stable, because when NsrR detaches itself from pYeaR, the promoter then initiate the production of mCP and CP, until a steady&ndash;state concentration is reached. This steady&ndash;state concentration is the maximum production rate of mCP and CP with respect to their degradation rates. However, it is clear from the graphs that the degradation rates are only related to the maximum production rate/ number of molecules in the cell; it is not responsible for altering the time span required to reach the steady&ndash;state concentration. Furthermore, this simplified solution for NOx&ndash;sensing system demonstrates the migration of system towards steady&ndash;state under assumed parameters and prescribed conditions. Therefore, this mathematical model can be considered as an appropriate model for determining the behavior of NOx&ndash;sensing system. </p>
+                        <h2>References</h2>
+                        <ol>
+                        <li><p>Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F.L. and Scire, J.J. (2007) 'A comprehensive kinetic mechanism for CO, CH2O, and CH3OH combustion', International Journal of Chemical Kinetics, 39(3), pp. 109&ndash;136. doi: 10.1002/kin.20218.</p></li>
+                        <li><p>Chen, H., Shiroguchi, K., Ge, H. and Xie, X.S. (2015) 'Genome-wide study of mRNA degradation and transcript elongation in Escherichia coli', Molecular Systems Biology, 11(1), pp. 781&ndash;781. doi: 10.15252/msb.20145794.</p></li>
+                        <li><p>Leduc, J., Thorsteinsson, M.V., Gaal, T. and Roberts, G.P. (2001) 'Mapping CooA.RNA polymerase interactions: Identification of activating regions 2 and 3 in CooA, the co&ndash;sensing transcriptional activator', Journal of Biological Chemistry, 276(43), pp. 39968&ndash;39973. doi: 10.1074/jbc.m105758200.</p></li>
+                        <li><p>Benabbas, A., Karunakaran, V., Youn, H., Poulos, T.L. and Champion, P.M. (2012) 'Effect of DNA binding on Geminate CO recombination Kinetics in CO-sensing transcription factor CooA', Journal of Biological Chemistry, 287(26), pp. 21729&ndash;21740. doi: 10.1074/jbc.m112.345090.</p></li>
+                        <li><p>Kuchinskas, M., Li, H., Conrad, M., Roberts, G. and Poulos, T.L. (2006) 'The role of the DNA-Binding domains in CooA activation', Biochemistry, 45(23), pp. 7148&ndash;27153. doi: 10.1021/bi052609o.</p></li>
+                        <li><p>Bergmann, John E, and Harvey F Lodish. 'A kinetic model of protein synthesis. Application to hemoglobin synthesis and translational control.' The Journal of biological chemistry 254.23 (1979) : 11927&ndash;37.</p></li>
+                        <li><p>Nick Csicsery, and Ricky O'Laughlin. 'A Mathematical Model of a Synthetically Constructed Genetic Toggle Switch' (2015) </p></li>
+                        </ol>
-<h5> Inspiration </h5>
-<p>
-Here are a few examples from previous teams:
-</p>
-<ul>
-<li><a href="https://2014.igem.org/Team:ETH_Zurich/modeling/overview">ETH Zurich 2014</a></li>
-<li><a href="https://2014.igem.org/Team:Waterloo/Math_Book">Waterloo 2014</a></li>
-</ul>
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Difference between revisions of "Team:Peshawar/Model"

Latest revision as of 22:24, 19 October 2016

Modeling and Simulation

Why model?

Modeling of the CO-sensing system

Assumptions

Simulation of the CO-sensing system

Simulation parameters

Simulation results

Conclusion

MATLAB Code

Modeling of the NOx-sensing system

Rate of change of chromoprotein expression

Binding cooperativity betweeen NO AND NsrR

Equation for PyeaR activity

Solving equations for chromoprotein expressions

Simulation of the NOx-sensing system

Simulation parameters

Simulation results

Matlab Code

Conclusion

References

Reach Us

Institute of Integrative Biosciences, CECOS University of IT & Emerging Sciences, Phase 6, F-5, Hayatabad, Peshawar, Pakistan

iib@cecos.edu.pk

+92 91 5860291

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Connect with us

Institute of Integrative Biosciences,
CECOS University of IT & Emerging Sciences,
Phase 6, F-5, Hayatabad, Peshawar, Pakistan