Writing out a system of equations for an enzyme reaction

I have the given file, where the rate of reaction is evaluated and gives F as a function of the concentration of the product P. The file evaluates the function f (y, k) and computes the Jacobians ∇y f and ∇k f. y is the concentration of the enzyme while k is the reaction rate konstant.

We consider a system with the following reactions:

in which the constants μ and γ are unknown, and where ∅ → S denotes that substrates (S) are formed without using any products (P). A chemist has done a series of experiments with different initial concentrations (at time t0 = 0). At time T = 1, the chemist has measured the concentration of S and P again. The results of the measurements are summarized in Table 1

I have to write down the system of ordinary differential equations that describes system (2) (Notation: dydt = f (y, k)).

However, I am not sure, since the file uses dfdy and dfdk. What is the correct method to write up (2) in MATLAB code?

Thanks

[Sam Chak]

type enzymeRhs.m

function [f,dfdy,dfdk] = enzymeRhs(y,k)
%ENZYMERHS evaluates the right hand side of the enzyme ODE system.
%
%  [F,DFDY,DFDK] = ENZYMERHS(Y,K) evaluates the right hand side for the
%    enzyme system with rates K and current concentration Y and returns F,
%    the value of the right hand side as well as DFDY and DFDK, the
%    gradient of the right hand side function with respect to 
%    concentrations Y and rates K, respectively.
%
C = 0.1; %constant creation of substrate
f = zeros(2,1);
f(1) =  y(2) - k(1)*y(1)/(k(2)+y(1)) + C;
f(2) = -y(2) + k(1)*y(1)/(k(2)+y(1));
if nargout>1
    dfdy = [-k(1)/(k(2)+y(1))+k(1)*y(1)/(k(2)+y(1))^2  1;
             k(1)/(k(2)+y(1))-k(1)*y(1)/(k(2)+y(1))^2 -1];
    dfdk = [-y(1)/(k(2)+y(1))  k(1)*y(1)/(k(2)+y(1))^2;
             y(1)/(k(2)+y(1)) -k(1)*y(1)/(k(2)+y(1))^2];
end

Let y^T = ([S], [P]) and k^T = (μ, γ). Write down the system of ordinary differential equations that
describes system (2) (Notation: dydt = f (y, k)). The file enzymeRhs.m contains the Matlab function
enzymeRhs that evaluates the function f (y, k) and computes the Jacobians ∇y f and ∇k f. Verify the
correctness of your derivation by comparing with the system in enzymeRhs.m!