Inhibited enzyme reactions often follow what are known as Michaelis--Menten kinetics, in which a reaction rate v follows a law of the form
where x is the concentration of a substrate. The real values V and 
are parameters that are free to fit to data. For this example we cook up some artificial data with 
and 
.
are parameters that are free to fit to data. For this example we cook up some artificial data with and .m = 25;
x = linspace(0.05,6,m)';
y = 2*x./(0.5+x); % exactly on the curve
y = y + 0.15*cos(2*exp(x/16).*x); % noise added
plot(x,y,'.')
title('Data') % ignore this line
xlabel('x'), ylabel('v') % ignore this line
The idea is to pretend that we know nothing of the origins of this data and use nonlinear least squares on the misfit to find the parameters in the theoretical model function 
. Note in the Jacobian that the derivatives are not with respect to x, but with respect to the two parameters, which are contained in the vector c. The function coding the equations and Jacobian is found at the end of this file.
. Note in the Jacobian that the derivatives are not with respect to x, but with respect to the two parameters, which are contained in the vector c. The function coding the equations and Jacobian is found at the end of this file. c1 = [1; 0.75];
c = newtonsys(@misfit,c1);
V = c(1,end), Km = c(2,end) % final values
model = @(x) V*x./(Km+x);
The final values are close to the noise-free values of , 
that we used to generate the data. We can calculate the amount of misfit at the end, although it's not completely clear what a "good" value would be. Graphically, the model looks reasonable.
, that we used to generate the data. We can calculate the amount of misfit at the end, although it's not completely clear what a "good" value would be. Graphically, the model looks reasonable.final_misfit_norm = norm(model(x)-y)
hold on
fplot( model, [0 6] )
title('Michaelis-Menten fitting') % ignore this line
xlabel('concentration'), ylabel('reaction rate') % ignore this line
For this model, we also have the option of linearizing the fit process. Rewrite the model as 
for the new parameters 
and 
. This corresponds to the misfit function whose entries are
for the new parameters and . This corresponds to the misfit function whose entries are
for 
. Although the misfit is nonlinear in x and y, it's linear in the unknown parameters α and β, and so can be posed and solved as a linear least-squares problem.
. Although the misfit is nonlinear in x and y, it's linear in the unknown parameters α and β, and so can be posed and solved as a linear least-squares problem.A = [ x.^(-1), x.^0 ]; u = 1./y;
z = A\u;
alpha = z(1); beta = z(2);
The two fits are different, because they do not optimize the same quantities.
linmodel = @(x) 1 ./ (beta + alpha./x);
final_misfit_linearized = norm(linmodel(x)-y)
fplot(linmodel, [0 6] )
function [f,J] = misfit(c)
V = c(1); Km = c(2);
f = V*x./(Km+x) - y;
J(:,1) = x./(Km+x); % d/d(V)
J(:,2) = -V*x./(Km+x).^2; % d/d(Km)
end