How can I replicate this plot using these differential equations and parameters?

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How can I replicate the plot above given these parameters?
Sam Chak
Sam Chak on 9 May 2022
Aha.. I see. You are looking for an example that has the same number of ODEs and lots of rational terms.
The second example "Solve Non-stiff ODEs" is relevant. Since you want 5, this example has 5.

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Accepted Answer

William Rose
William Rose on 9 May 2022
@Torsten is exactly correct, as usual. But since you are still stuck, here's some additional help.
Define a state vector x with 5 elements. The five elements correspond to the 5 variables:
, , , , ,
Then write the five differential equations in terms of the elements of the state vector. For example, the first diff. eq, for dm/dt, becomes
and the last diff.eq., for pN(t), becomes
and you can work out the three in between. Use the differential equations you have written to create a function that returns a column vector of length 5, called dxdt. You must define the various constants inside the function, or make them global. (I recommend defining them inside the function, because global variables can cause slow execution.)
function dxdt = circOsc(t,x)
vs=0.76; vm=0.65; KI=1; Km1=0.5;
k1=1.9; k2=1.3; n=4;
%Add definitions of additional constants which will be used below.
ks=0; V1=0; V2=0; %fix these and add the others
dxdt = zeros(5,1); %allocate column vector dxdt
dxdt(1) = vs/(1+(x(5)/KI)^n)-vm*x(1)/(Km1+x(1));
dxdt(2) = 0; %fill this in with the right equation
dxdt(3) = 0; %fill this in with the right equation
dxdt(4) = 0; %fill this in with the right equation
dxdt(5) = k1*x(4)-k2*x(5);
You can save this function as a file, circOsc.m, or you can include circOsc() as code after the body of your main script, if you prefer to have everything in a single file. The main script calls ode45(). You must pass to ode45() the initial condition (vector x0), the time span (0 to 70 hours, to reproduce Fig.7.19A), and the name of the function that computes the derivative of the state vector (circOsc, in the example above). All of this is illustrated clearly in the many examples associated with the ode45() help. The van der Pol oscillator example in the ode45 help is probably the most similar to your model.
%Simulate the Goldbeter circadian oscillator
x0=[2.5; 0.5; 0.5; 0.5; 0.7]; %estimated from Fig. 7.19A, assuming p0=p1=p2 at t=0, which may not be true
%% Display results graphically
legend('m','p_0','p_1','p_2','p_N'); grid on
xlabel('Time (h)'); ylabel('Concentration (\muM)');
Try it. Check my equations and the constants, since I might have made mistakes. Check everything and add the missing bits.

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