How to use Matlab for non linear least squares Michaelis–Menten parameters estimation

Question

I have a set of measurements and I started making a linear approximation (as in this plot). A linear least squares estimation of the parameters V_{max} and K_{m} from this code in Matlab:

data=[2.0000 0.0615
2.0000 0.0527
0.6670 0.0334
0.6670 0.0334
0.4000 0.0138
0.4000 0.0258
0.2860 0.0129
0.2860 0.0183
0.2220 0.0083
0.2200 0.0169
0.2000 0.0129
0.2000 0.0087 ];
x = 1./data(:,1);
y = 1./data(:,2);
J = [x,ones(length(x),1)];
k = J\y;
vmax = 1/k(2);
km = k(1)*vmax;
lse = (vmax.*data(:,1))./(km+data(:,1));
plot(data(:,1),data(:,2),'o','color','red','linewidth',1)
line(data(:,1),lse,'linewidth',2)

This yields a fit that looks alright. Next, I wanted to do the same thing but with non-linear least squares. However, the fit always looks wrong, here is the code for that attempt:

options =  optimset('MaxIter',10000,'MaxFunEvals',50000,'FunValCheck',...
    'on','Algorithm',{'levenberg-marquardt',.00001});
p=lsqnonlin(@myfun,[0.1424,2.5444]);
lse = (p(1).*data(:,1))./(p(2)+data(:,1));
plot(data(:,1),data(:,2),'o','color','red','linewidth',1)
line(data(:,1),lse,'linewidth',2)

which requires this function in an M-File:

function F = myfun(x)
    F = data(:,2)-(x(1).*data(:,1))./x(2)+data(:,1);

If you run the code you will see my problem. But hopefully, unlike me, you see what I'm doing wrong.

Answer 1

I think that you forgot some parentheses (some others are superfluous) in your nonlinear function. Using an anonymous function:

myfun = @(x)data(:,2)-x(1).*data(:,1)./(x(2)+data(:,1)); % Parentheses were missing
options = optimset('MaxIter',10000,'MaxFunEvals',50000,'FunValCheck','on',...
                   'Algorithm',{'levenberg-marquardt',.00001});
p = lsqnonlin(myfun,[0.1424,2.5444],[],[],options);
lse = p(1).*data(:,1)./(p(2)+data(:,1));
plot(data(:,1),data(:,2),'o','color','red','linewidth',1)
line(data(:,1),lse,'linewidth',2)

You also weren't actually applying any of your options.

You might look into using lsqcurvefit instead as it was designed for data fitting problems:

myfun = @(x,dat)x(1).*dat./(x(2)+dat);
options = optimset('MaxIter',10000,'MaxFunEvals',50000,'FunValCheck','on',...
                   'Algorithm',{'levenberg-marquardt',.00001});
p = lsqcurvefit(myfun,[0.1424,2.5444],data(:,1),data(:,2),[],[],options);
lse = myfun(p,data(:,1));
plot(data(:,1),data(:,2),'o','color','red','linewidth',1)
line(data(:,1),lse,'linewidth',2)