Finding min and max of ODEs

The task was to model an enzymatic reaction which I did with the written code. The obtained plot and the equations are in the attached picture. I need to find the the time when the enzyme concentration is minimal and enzyme-substrate concentration maximal. I know that the concentration change in the minimum/maximum should be zero ( d[E]/dt=d[C]/dt=0 ) but I don't know how I would write the code to find the time.

function dydt = reaction(~, cse)
% Defining reaction parameters:
k1 = 6;
k2 = 1;
kminus1 = 2;
% Defining reactions
reaction=zeros(4,1);
reaction(1) = k1*cse(2)*cse(3)-kminus1*cse(1)-k2*cse(1);
reaction(2) = -k1*cse(2)*cse(3)+kminus1*cse(1);
reaction(3) = -k1*cse(2)*cse(3)+kminus1*cse(1)+k2*cse(1);
reaction(4) = k2*cse(1);
dydt=[reaction(1); reaction(2); reaction(3); reaction(4)];

end
_____
% defining the time range:
T=12;
trange=[0 T];
% defining initial values (concentrations) for c,s,e i p:
cs0 = [0; 1; 1; 0];
  % [t,x] = ode45(@reaction, trange, initial values)
[t,unknown]=ode45(@reaction,trange,cs0);
c = unknown(:, 1);
s = unknown(:, 2);
e = unknown(:, 3);
p = unknown(:, 4);

  % plot drawing
plot(t,c,'m',t,s,'c',t,e,'r',t,p,'k')
ylim([0 1.1])
xlabel('time, t')
ylabel('concentration')
legend({' enzyme-substrate complex concentration',' substrate concentration',' enzyme concentration ',' product concentration'}, "Location","best")

Answer 1

[Ameer Hamza]

See this example

% defining the time range:
T=12;
trange=[0 T];
% defining initial values (concentrations) for c,s,e i p:
cs0 = [0; 1; 1; 0];
  % [t,x] = ode45(@reaction, trange, initial values)
[t,unknown]=ode45(@reaction,trange,cs0);
c = unknown(:, 1);
s = unknown(:, 2);
e = unknown(:, 3);
p = unknown(:, 4);

[c_max, idx_c_max] = max(c);
t_c_max = t(idx_c_max);

[e_min, idx_e_min] = min(e);
t_e_min = t(idx_e_min);

% plot drawing
hold on
plot(t,c,'m',t,s,'c',t,e,'r',t,p,'k')
plot(t_c_max, c_max, 'm+', t_e_min, e_min, 'r+');
ylim([0 1.1])
xlabel('time, t')
ylabel('concentration')
legend({' enzyme-substrate complex concentration',' substrate concentration',' enzyme concentration ',' product concentration'}, "Location","best")

function dydt = reaction(~, cse)
% Defining reaction parameters:
k1 = 6;
k2 = 1;
kminus1 = 2;
% Defining reactions
reaction=zeros(4,1);
reaction(1) = k1*cse(2)*cse(3)-kminus1*cse(1)-k2*cse(1);
reaction(2) = -k1*cse(2)*cse(3)+kminus1*cse(1);
reaction(3) = -k1*cse(2)*cse(3)+kminus1*cse(1)+k2*cse(1);
reaction(4) = k2*cse(1);
dydt=[reaction(1); reaction(2); reaction(3); reaction(4)];

end

[Sara]

Thank you!! :)