Team:XJTU-China/Model

Modelling

Math Modelling

•  Protein Modelling
•  Summary
•  Model Establishment
•  Result and Conclusion
  • · The Population Density of E. coli
  • · The Effect of Toggle Switch
  • · The Product of Genetic Circuits
  • · The Output of Tryptophan
  • · The Best Production Strategy
•  Reference
•  Code

Mathematical Modelling

Protein Modelling

To predict the experimental result of our project, we constructed a set of mathematical models in five different aspects. Here is the showcase of our math. Additionally, to investigate the mechanism of allosteric inhibition on AroG by Phe and the alleviation in S211F mutant, it is proposed to be quantified and visualized using PyMOL, Gaussian16.0W, GaussView6.0, Swiss, AutoDockTools software. To see the result of our protein model, plese check our wiki of protein modelling.

View:  Protein Modelling 

Summary

Our modeling includes five steps:

• Establish the model of population dynamics, which displays the population change of E. coli;
• Establish the model of toggle switch, where the production of red fluorescent protein (RFP) and green fluorescent protein (GFP) shows the effect of toggle switch;
• Establish the model of genetic circuits based on the model of toggle switch;
• Establish the model of synthesis of tryptophan based on Michaelis-Menten equation;
• Finally, integrate the above models to establish the model of production.

Establishment of model

Due to the size limit of iGEM wiki, please click the following buttons to view our models establishment!

The Model of Population Dynamics The Model of Toggle Switch The Model of Genetic Circuits The Model of Synthesis of Tryptophan The Model of Production

Result and conclusion

The population density of E. coli

Here, $K=6.08\times {10}^9\mathrm{CFU}/\mathrm{ml}$ $K=6.08×10^9\mathrm{CFU/ml}$. The figure below shows the population density of E. coli.

• When $N<\frac{K}{2}$ $N<\dfrac{K}{2}$, the population density grows exponentially. And when $N>\frac{K}{2}$ $N>\dfrac{K}{2}$, the environmental resources have a restrictive effect on E. coli. Finally the population density approaches $K$ $K$;
• The population density reach balance at about $33\mathrm{h}$ $33\mathrm{h}$.

The effect of toggle switch

When $t=1000\min$ $t=1000\mathrm{min}$, add IPTG. When $t=2000\min$ $t=2000\mathrm{min}$, remove IPTG and raise temperature. The results are shown in the figure below.

• The change rates of $[rlacI]$ $[rlacI]$ and $[rGFP]$ $[rGFP]$ decrease as $[pcI857]$ $[pcI857]$ increases. The change rates of $[rcI857]$ $[rcI857]$ and $[rRFP]$ $[rRFP]$ decrease as $[placI]$ $[placI]$ increases. $[IPTG]$ $[IPTG]$ restrains the production of RFP when its concentration is low, and promotes the production when its concentration is high;
• At first, the concentration of GFP is more than the concentration of RFP, and green fluorescence appears. After adding IPTG, the concentration of RFP outnumbered GFP, and red fluorescence appears. After removing IPTG and raising temperature, the rank of RFP and GFP exchanged again, and green fluorescence appears.

The product of genetic circuits

Add IPTG at the beginning, and when $t=1170\min$ $t=1170\mathrm{min}$, remove IPTG and raise temperature. The figure below shows the concentration change of gene product.

• There are two stable states during the period of time;
• After raising temperature at $1170\min$ $1170\mathrm{min}$, the concentrations of cl857 and pykA go down, while the concentrations of lacI, aroG, trpA and trpB go up.

The output of tryptophan

Add IPTG at the beginning, and when $t=1170\min$ $t=1170\mathrm{min}$, remove IPTG and raise temperature. The figure below shows the concentration change during the synthesis of tryptophan.

• When reaction starts, Glc begin to convert to PEP, and PEP immediately turns into Pyr and DAHP;
• The concentration of DAHP reaches maximum at about $1900\min$ $1900\mathrm{min}$, and after that it goes down;
• The product of DAHP is 3IGP, and 3IGP immediately converts to Trp;
• The final products of reactions are Pyr and Trp, whose concentrations are stable at $3000\min$ $3000\mathrm{min}$.

The best production strategy

Let the initial value of $[Glc]$ $[Glc]$ be $20$ $20$. Add IPTG at the beginning, and change the time of raising the temperature. Let $t_2$ $t_2$ be the time of raising temperature. The final outputs of $[Pyr]$ $[Pyr]$ and $[Trp]$ $[Trp]$ are shown in the figure below.

• When $t_2<1170\min$ $t_2<1170\mathrm{min}$, the output of tryptophan increases slowly as $t_2$ $t_2$ increases;
• When $t_2=1170\min$ $t_2=1170\mathrm{min}$, the maximun output of tryptophan is $15.43$ $15.43$ ( $77.15\mathrm{\%}$ $77.15\%$ of $20$ $20$);
• When $1170\min <t_2<2000\min$ $1170\mathrm{min}<t_2<2000\mathrm{min}$, the output of tryptophan drops sharply as $t_2$ $t_2$ increases;
• Finally, when $t_2>2000\min$ $t_2>2000\mathrm{min}$, the output of tryptophan is around $8$ $8$ ( $40\mathrm{\%}$ $40\%$ of $20$ $20$).

Reference

[1] Verhulst, P.-F. "Recherches mathématiques sur la loi d'accroissement de la population." Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles 18, 1-41, 1845.

[2] Verhulst, P.-F. "Deuxième mémoire sur la loi d'accroissement de la population." Mém. de l'Academie Royale des Sci., des Lettres et des Beaux-Arts de Belgique 20, 1-32, 1847.

[3] XIAN YIN, HYUN-DONG SHIN, et al. 2017. P gas, a Low-pH-Induced Promoter, as a Tool for Dynamic Control of Gene Expression for Metabolic Engineering of Aspergillus niger. Appl Environ Microbiol. [J/OL], 2;83(6):e03222-16.

[4] https://www.vedantu.com/chemistry/michaelis-menten-kinetics

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Code

Code

Using MATLAB R2019b.

The model of population dynamics

main.m

% initialization
                                    
                                    
clc;
                                    
close all;
                                    
clear all;
                                    

            
                                    
% numerical solution
                                    
[t, N] = ode45(@(t, N) odefcn(t, N)', [0 3000], 6.08*10^9*0.0001);
                                    

            
                                    
% draw figure
                                    
figure();
                                    
plot(t, N, 'LineWidth', 2);
                                    

                                        
xlabel('Time/min');
                                    
                                    
ylabel('Population Density')
                                    
grid on;
                                
                            
                    
                
            
            

            
            

        
        
        

odefun.m

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function dN = odefcn(t, N)
            
            
    r = 0.0073;
            
    K = 6.08*10^9;
            
    dN = r*N*(1-N/K);
        
        
        
        
        
        

        
        

        
        
        

The model of toggle switch

main.m

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% initialization
            
            
clc;
            
close all;
            
clear all;
            

            
            
% initial value
            
tspan = [0 3000];
            
c0 = [0.1 0.1 0.1 0.1 0.01 0.01 0.01 0.01];
            

            
            
% add IPTG
            
t1 = 1000;
            
iptg1 = 10;
            

            
            
% raise temperature
            
t2 = 2000;
            
alpha = 1.2;
            

            
            
% numerical solution
            
[t, c] = ode45(@(t, c) odefcn(t, c, t1, iptg1, t2, alpha)', tspan, c0);
            

            
            
% draw figure
            
figure();
            
plot(t, c(:, 6), 'g', t, c(:, 8), 'r', 'LineWidth', 2);
            
legend('[pGFP]', '[pRFP]');
            
xlabel('Time/s');
            
ylabel('Concentration');
            
grid on;
        
        
        
        
        
        

        
        

        
        
        

odefun.m

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function dc = odefcn(t, c, t1, iptg1, t2, alpha)
            
            
    % lambdaPR: lacI, GFP
            
    lambdaPR = 0.5;
            
    alphal0 = 0.5;
            
    alphal1 = 1;
            
    
            
    % Plac: cl857, RFP
            
    Plac = 1.3;
            
    alphap0 = 0.5;
            
    alphap1 = 1;
            
    
            
    % RFP
            
    ksynr = 0.019;
            
    kder = 0.0013*10;
            
    kpsynr = 0.47;
            
    kpder = 0.136*10;
            

            
            
    % GFP
            
    kg = 0.877;
            
    ksyng = kg*ksynr;
            
    kdeg = kg*kder;
            
    kpsyng = kg*kpsynr;
            
    kpdeg = kg*kpder;
            
    
            
    % cI857
            
    kc = 0.95;
            
    ksync = kc*ksynr;
            
    kdec = kc*kder;
            
    kpsync = kc*kpsynr;
            
    kpdec = kc*kpder;
            
    
            
    % lacI
            
    kl = 0.626;
            
    ksynl = kl*ksynr;
            
    kdel = kl*kder;
            
    kpsynl = kl*kpsynr;
            
    kpdel = kl*kpder;
            
    
            
    % add IPTG
            
    iptg = 0;
            
    if (t>t1)
            
        iptg = iptg1;
            
    end
            
    
            
    % raise temperature
            
    if (t>t2)
            
        iptg = 0;
            
        alphal1 = alpha;
            
    end
            
    
            
    % differential equations
            
    dc(1) = ksynl*alphal0*lambdaPR + ksynl*alphal1*(1/(c(7)))*lambdaPR - kdel*c(1); % [rlacI]
            
    dc(2) = ksyng*alphal0*lambdaPR + ksyng*alphal1*(1/(c(7)))*lambdaPR - kdeg*c(2); % [rGFP]
            
    dc(3) = ksync*alphap0*Plac + ksync*alphap1*(iptg/(iptg+c(5)))*Plac - kdec*c(3); % [rcI857]
            
    dc(4) = ksynr*alphap0*Plac + ksynr*alphap1*(iptg/(iptg+c(5)))*Plac - kder*c(4); % [rRFP]
            
    dc(5) = kpsynl*c(1) - kpdel*c(5); % [placI]
            
    dc(6) = kpsyng*c(2) - kpdeg*c(6); % [pGFP]
            
    dc(7) = kpsync*c(3) - kpdec*c(7); % [pcI857]
            
    dc(8) = kpsynr*c(4) - kpder*c(8); % [pRFP]
        
        
        
        
        
        

        
        

        
        
        

The model of genetic circuits

main.m

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% initialization
            
            
clc;
            
close all;
            
clear all;
            

            
            
% initial value
            
tspan = [0 3000];
            
c0 = [0.1 0.1 0.1 0.1 0.1 0.1 0.01 0.01 0.01 0.01 0.01 0.01];
            

            
            
% add IPTG
            
t1 = 1000;
            
iptg1 = 10;
            

            
            
% raise temperature
            
t2 = 2000;
            
alpha = 1.2;
            

            
            
% numerical solution
            
[t, c] = ode45(@(t, c) odefcn(t, c, t1, iptg1, t2, alpha)', tspan, c0);
            

            
            
% draw figure
            
figure();
            
plot(t, c(:, 7), t, c(:, 8), t, c(:, 9), t, c(:, 10), t, c(:, 11), t, c(:, 12), 'LineWidth', 2)
            
legend('[placI]', '[paroG]', '[ptrpB]', '[ptrpA]', '[pcI857]', '[ppykA]');
            
grid on;
        
        
        
        
        
        

        
        

        
        
        

odefun.m

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function dc = odefcn(t, c, t1, iptg1, t2, alpha)
            
            
    % lambdaPR: lacI, aroG, trpB, trpA
            
    lambdaPR = 0.5;
            
    alphal0 = 0.5;
            
    alphal1 = 1;
            
    
            
    % Plac: cl857, pykA
            
    Plac = 1.3;
            
    alphap0 = 0.5;
            
    alphap1 = 1;
            
    
            
    % RFP
            
    ksynr = 0.019;
            
    kder = 0.0013*10;
            
    kpsynr = 0.47;
            
    kpder = 0.136*10;
            
    
            
    % lacI
            
    kl = 0.626;
            
    ksynl = kl*ksynr;
            
    kdel = kl*kder;
            
    kpsynl = kl*kpsynr;
            
    kpdel = kl*kpder;
            
  
            
    % aroG
            
    ka = 0.644;
            
    ksyna = ka*ksynr;
            
    kdea = ka*kder;
            
    kpsyna = ka*kpsynr;
            
    kpdea = ka*kpder;
            
    
            
    % trpB
            
    kB = 0.568;
            
    ksynB = kB*ksynr;
            
    kdeB = kB*kder;
            
    kpsynB = kB*kpsynr;
            
    kpdeB = kB*kpder;
            
    
            
    % trpA
            
    kA = 0.84;
            
    ksynA = kA*ksynr;
            
    kdeA = kA*kder;
            
    kpsynA = kA*kpsynr;
            
    kpdeA = kA*kpder;
            
    
            
    % cI857
            
    kc = 0.95;
            
    ksync = kc*ksynr;
            
    kdec = kc*kder;
            
    kpsync = kc*kpsynr;
            
    kpdec = kc*kpder;
            
    
            
    % pykA
            
    kp = 0.47;
            
    ksynp = kp*ksynr;
            
    kdep = kp*kder;
            
    kpsynp = kp*kpsynr;
            
    kpdep = kp*kpder;
            
    
            
    % add IPTG
            
    iptg = 0;
            
    if (t>t1)
            
        iptg = iptg1;
            
    end
            
    
            
    % raise temperature
            
    if (t>t2)
            
        iptg = 0;
            
        alphal1 = alpha;
            
    end
            
    
            
    % differential equations
            
    dc(1) = ksynl*alphal0*lambdaPR + ksynl*alphal1*(1/(c(11)))*lambdaPR - kdel*c(1); % [rlacI]
            
    dc(2) = ksyna*alphal0*lambdaPR + ksyna*alphal1*(1/(c(11)))*lambdaPR - kdea*c(2); % [raroG]
            
    dc(3) = ksynB*alphal0*lambdaPR + ksynB*alphal1*(1/(c(11)))*lambdaPR - kdeB*c(3); % [rtrpB]
            
    dc(4) = ksynA*alphal0*lambdaPR + ksynA*alphal1*(1/(c(11)))*lambdaPR - kdeB*c(4); % [rtrpA]
            
    dc(5) = ksync*alphap0*Plac + ksync*alphap1*(iptg/(iptg+c(7)))*Plac - kdec*c(5); % [rcI857]
            
    dc(6) = ksynp*alphap0*Plac + ksynp*alphap1*(iptg/(iptg+c(7)))*Plac - kder*c(6); % [rpykA]
            
    dc(7) = kpsynl*c(1) - kpdel*c(7); % [placI]
            
    dc(8) = kpsyna*c(2) - kpdea*c(8); % [paroG]
            
    dc(9) = kpsynB*c(3) - kpdeB*c(9); % [ptrpB]
            
    dc(10) = kpsynA*c(4) - kpdeA*c(10); % [ptrpA]
            
    dc(11) = kpsync*c(5) - kpdec*c(11); % [pcI857]
            
    dc(12) = kpsynp*c(6) - kpdep*c(12); % [ppykA]
        
        
        
        
        
        

        
        

        
        
        

The model of synthesis of tryptophan

main.m

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% initialization
            
            
clc;
            
close all;
            

            
            
% initial value
            
tspan = [0 3000];
            
c0 = [0 0 0 0 0 20];
            
aroG = 1;
            
pykA = 0.15;
            
trpAB = 2.5;
            

            
            
% numerical solution
            
[t, c] = ode45(@(t, c) odefcn(t, c, aroG, pykA, trpAB)', tspan, c0);
            

            
            
% draw figure
            
figure();
            
plot(t, c(:, 6), t, c(:, 1), t, c(:, 2), t, c(:, 3), t, c(:, 4), t, c(:, 5), 'LineWidth', 2)
            
legend('[Glc]',  '[PEP]', '[DAHP]', '[Pyr]', '[3IGP]', '[Trp]');
            
xlabel('Time/min');
            
ylabel('Concetration');
            
grid on;
            
axis([0 3000 0 20]);
        
        
        
        
        
        

        
        

        
        
        

odefun.m

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function dc = odefcn(t, c, aroG, pykA, trpAB)
            
            
    % reaction 1
            
    vmax1 = 0.1;
            
    km1 = 0.005;
            
    
            
    % reaction 2
            
    vmax2 = 0.03;
            
    km2 = 0.002;
            
    
            
    % reaction aroG
            
    kcataroG = 0.1;
            
    kmaroG = 0.009;
            
    
            
    % reaction pykA
            
    kcatpykA = 0.15;
            
    kmpykA = 0.05;
            
    
            
    % reaction trpAB
            
    kcattrpAB = 0.009;
            
    kmtrpAB = 0.1;
            
    
            
    % differential equations
            
    dc(1) = vmax1*c(6)/(c(6)+km1)-kcataroG*aroG*c(1)/(c(1)+kmaroG)-kcatpykA*pykA*c(1)/(c(1)+kmpykA); % [PEP]
            
    dc(2) = kcataroG*aroG*c(1)/(c(1)+kmaroG)-vmax2*c(2)/(c(2)+km2); % [DAHP]
            
    dc(3) = kcatpykA*pykA*c(1)/(c(1)+kmpykA); % [Pyr]
            
    dc(4) = vmax2*c(2)/(c(2)+km2)-kcattrpAB*trpAB*c(4)/(c(4)+kmtrpAB); % [3IGP]
            
    dc(5) = kcattrpAB*trpAB*c(4)/(c(4)+kmtrpAB); % [Trp]
            
    dc(6) = -vmax1*c(6)/(c(6)+km1); % [Glc]
        
        
        
        
        
        

        
        

        
        
        

The model of production

main.m

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% initialization
            
            
clc;
            
close all;
            
clear all;
            

            
            
% initial value
            
tspan = [0 3000];
            
c0 = [0.1 0.1 0.1 0.1 0.1 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0 0 0 0 0 20 6.08*10^9*0.001];
            

            
            
% adjust t2
            
n = 10;
            
maxt = 3000;
            
mint = 0;
            
for i = 1:n
            
    clc;
            
    fprintf('%d/%d\n',i,n);
            
    t2 = mint + (maxt-mint)*i/n;
            
    [t, c] = ode45(@(t, c) odefcn(t, c, 0, 10, t2, 1.2)', tspan, c0);
            
    result(i,1) = c(size(t,1),15);
            
    result(i,2) = c(size(t,1),17);
            
end
            

            
            
% figure 1
            
tspan2 = mint+(maxt-mint)/n:(maxt-mint)/n:maxt;
            
plot(tspan2, result(:,1), tspan2, result(:,2), 'LineWidth', 2);
            
xlabel('Time of raising temperature/min');
            
ylabel('Output')
            
legend('[Pyr]', '[Trp]');
            
grid on;
            

            
            
% find the maximun
            
clc;
            
[m, i] = max(result(:,2));
            
t2 = mint + (maxt-mint)*i/n;
            
fprintf('When raising temperature at %dmin, the maximun output of Trp is %f. \n', t2, m);
            
[t, c] = ode45(@(t, c) odefcn(t, c, 0, 10, t2, 1.2)', tspan, c0);
            

            
            
% figure 2
            
figure();
            
plot(t, c(:, 19), 'LineWidth', 2)
            
legend('N');
            
xlabel('Time/min');
            
ylabel('Population Density');
            
grid on;
            

            
            
% figure 3
            
figure();
            
plot(t, c(:, 7), t, c(:, 8), t, c(:, 9), t, c(:, 10), t, c(:, 11), t, c(:, 12), 'LineWidth', 2)
            
legend('[placI]', '[paroG]', '[ptrpB]', '[ptrpA]', '[pcI857]', '[ppykA]');
            
xlabel('Time/min');
            
ylabel('Concetration');
            
grid on;
            

            
            
% figure 4
            
figure();
            
plot(t, c(:, 18), t, c(:, 13), t, c(:, 14), t, c(:, 15), t, c(:, 16), t, c(:, 17), 'LineWidth', 2)
            
legend('[Glc]', '[PEP]', '[DAHP]', '[Pyr]', '[3IGP]', '[Trp]');
            
xlabel('Time/min');
            
ylabel('Concetration');
            
axis([0 3000 0 20]);
            
grid on;
        
        
        
        
        
        

        
        

        
        
        

odefun.m

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function dc = odefcn(t, c, t1, iptg1, t2, alpha)
            
            
    % lambdaPR: lacI, aroG, trpB, trpA
            
    lambdaPR = 0.5;
            
    alphal0 = 0.5;
            
    alphal1 = 1;
            
    
            
    % Plac: cl857, pykA
            
    Plac = 1.3;
            
    alphap0 = 0.5;
            
    alphap1 = 1;
            
    
            
    % RFP
            
    ksynr = 0.019;
            
    kder = 0.0013*10;
            
    kpsynr = 0.47;
            
    kpder = 0.136*10;
            
    
            
    % lacI
            
    kl = 0.626;
            
    ksynl = kl*ksynr;
            
    kdel = kl*kder;
            
    kpsynl = kl*kpsynr;
            
    kpdel = kl*kpder;
            
  
            
    % aroG
            
    ka = 0.644;
            
    ksyna = ka*ksynr;
            
    kdea = ka*kder;
            
    kpsyna = ka*kpsynr;
            
    kpdea = ka*kpder;
            
    
            
    % trpB
            
    kB = 0.568;
            
    ksynB = kB*ksynr;
            
    kdeB = kB*kder;
            
    kpsynB = kB*kpsynr;
            
    kpdeB = kB*kpder;
            
    
            
    % trpA
            
    kA = 0.84;
            
    ksynA = kA*ksynr;
            
    kdeA = kA*kder;
            
    kpsynA = kA*kpsynr;
            
    kpdeA = kA*kpder;
            
    
            
    % cI857
            
    kc = 0.95;
            
    ksync = kc*ksynr;
            
    kdec = kc*kder;
            
    kpsync = kc*kpsynr;
            
    kpdec = kc*kpder;
            
    
            
    % pykA
            
    kp = 0.47;
            
    ksynp = kp*ksynr;
            
    kdep = kp*kder;
            
    kpsynp = kp*kpsynr;
            
    kpdep = kp*kpder;
            

            
            
    % reaction 1
            
    vmax1 = 0.1;
            
    km1 = 0.005;
            
    
            
    % reaction 2
            
    vmax2 = 0.03;
            
    km2 = 0.002;
            
    
            
    % reaction aroG
            
    kcataroG = 0.1;
            
    kmaroG = 0.009;
            
    
            
    % reaction pykA
            
    kcatpykA = 0.15;
            
    kmpykA = 0.05;
            
    
            
    % reaction trpAB
            
    kcattrpAB = 0.009;
            
    kmtrpAB = 0.1;
            
    
            
    % population dynamics
            
    K = 6.08*10^9;
            
    r = (3.1364*10^(-04)*c(15)+0.0052);
            
    k = 5*10^(-11);
            
    
            
    % add IPTG
            
    iptg = 0;
            
    if (t>t1)
            
        iptg = iptg1;
            
    end
            
    
            
    % raise temperature
            
    if (t>t2)
            
        iptg = 0;
            
        alphal1 = alpha;
            
    end
            
    
            
    % differential equations
            
    dc(1) = (ksynl*alphal0*lambdaPR + ksynl*alphal1*(1/(c(11)))*lambdaPR - kdel*c(1)); % [rlacI]
            
    dc(2) = (ksyna*alphal0*lambdaPR + ksyna*alphal1*(1/(c(11)))*lambdaPR - kdea*c(2)); % [raroG]
            
    dc(3) = (ksynB*alphal0*lambdaPR + ksynB*alphal1*(1/(c(11)))*lambdaPR - kdeB*c(3)); % [rtrpB]
            
    dc(4) = (ksynA*alphal0*lambdaPR + ksynA*alphal1*(1/(c(11)))*lambdaPR - kdeB*c(4)); % [rtrpA]
            
    dc(5) = (ksync*alphap0*Plac + ksync*alphap1*(iptg/(iptg+c(7)))*Plac - kdec*c(5)); % [rcI857]
            
    dc(6) = (ksynp*alphap0*Plac + ksynp*alphap1*(iptg/(iptg+c(7)))*Plac - kder*c(6)); % [rpykA]
            
    dc(7) = (kpsynl*c(1) - kpdel*c(7)); % [placI]
            
    dc(8) = (kpsyna*c(2) - kpdea*c(8)); % [paroG]
            
    dc(9) = (kpsynB*c(3) - kpdeB*c(9)); % [ptrpB]
            
    dc(10) = (kpsynA*c(4) - kpdeA*c(10)); % [ptrpA]
            
    dc(11) = (kpsync*c(5) - kpdec*c(11)); % [pcI857]
            
    dc(12) = (kpsynp*c(6) - kpdep*c(12)); % [ppykA]
            
    dc(13) = k*c(19)*(vmax1*c(18)/(c(18)+km1)-kcataroG*c(8)*c(13)/(c(13)+kmaroG)-kcatpykA*c(12)*c(13)/(c(13)+kmpykA)); % [PEP]
            
    dc(14) = k*c(19)*(kcataroG*c(8)*c(13)/(c(13)+kmaroG)-vmax2*c(14)/(c(14)+km2)); % [DAHP]
            
    dc(15) = k*c(19)*(kcatpykA*c(12)*c(13)/(c(13)+kmpykA)); % [Pyr]
            
    dc(16) = k*c(19)*(vmax2*c(14)/(c(14)+km2)-kcattrpAB*(c(3)+c(4))*c(16)/(c(16)+kmtrpAB)); % [3IGP]
            
    dc(17) = k*c(19)*(kcattrpAB*(c(3)+c(4))*c(16)/(c(16)+kmtrpAB)); % [Trp]
            
    dc(18) = k*c(19)*(-vmax1*c(18)/(c(18)+km1)); % [Glc]
            
    dc(19) = r*c(19)*(1-c(19)/K); % N
        
        
        
        
        
        

        
        

        
        
        

 

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