Model

​ To illustrate the change taken by the decompose of signal molecule, we can see following simulation curves:

​

​ X-axis refers to time. We find the efficiency will not be disturbed greatly at initial time, and will have a rapid decrease when the concentration equals to the half of origin. This property shows that we should control the reaction time otherwise the production will decay without production with the time going by. So the main purpose of this model is to predict when we dilute the input signal solution to obtain the maximum of protein to convert out signal.

​ We use matlab to obtain a rough curve of protein expression. X-axis refers to time.

​ This is a important result because it indicates that the production will not always increase with the time going. Actually, there exists a so-called "best time" to process next step in our system. For example, this peak can determine when we dilute input signal to get output signal as much as possible.

​ Red stars refers to "best time" according to different input concentration from upstream block.

*matlab code:

 
n = [];
fn = [];
for i=1:T/dt
n = [n i];
t = exp(-a*i*dt);
sum=0;
for j=0:i
sum = sum + (Vm-(j*dt)^n)*exp(a*dt*j)*dt/(k^n+(Vm-(dt*j)^n));
end
y = t*sum+\phi* (Vm - dt*i)^(n-1)/(k^n + (Vm - dt*i)^n)^2;
fn = [fn y];
end
plot(n,fn);
max(fn);

​ This matlab code shows how we draw the curves and how to find maximum.

Model of parameter fitting and simulation

Hill equation

​ To get the parameter of Hill equation through our data, we tranfer Hill equation to following form: