Values of Miu_max and Ks from parameter estimation?

Background

I have this simple biomass growth model: $$\mu = \mu_{max}\cdot \left(\frac{S}{K_S+S}\right) \cdot \left(\frac{1}{1+S/K_{iS}}\right) \cdot \left(\frac{K_{iP}}{K_{iP}+P}\right) \\ \frac{dX}{dt} = \mu \cdot X \\ \frac{dS}{dt} = -1 \cdot ( \frac{\mu}{Y_{XS}} \cdot X + mS \cdot X) \\ \frac{dP}{dt} = a \cdot \frac{dX}{dt} + b \cdot X$$

Problem

I can fit the model on experimental data (see the Figure), but the values of $\mu_{max}$ or $K_S$ are often quite unrealistic. In this plot the $\mu_{max}$ is 3.9 and $K_S$ is 10. Question

Since $\mu_{max}$ and $K_S$ are bounded in Monod-based $\mu$ expressions, is it possible to get real values of both of these particular parameters through parameter estimation techniques? Or one of these parameters should be obtained by other methods?

• If you are getting unrealistic values in your model fitting then something is missing or wrong in your model or assumptions. That's the beauty of mathematical models, when they fail, they tell you something is missing :) Plus I don't know why do you think those are unrealistic ? – Dexter Dec 7 '15 at 14:48
• Can you give some more details on how you fit the model? A nls fit tends to create strong correlation between the fit parameters. A fit using heuristic rules of thumb can be sensitive to a few points. – Hans Dec 7 '15 at 19:35
• @Dexter : the biology professor I'm working for tells that $\mu_{max}$ can be no higher than 0.7 or 0.8. According to that, Ks=10 might be ok, but $\mu_{max}$ so high can never be ok. – MartinsM Dec 8 '15 at 9:18
• @Hans : I use summed RMSE from multiple experiments with different initial substrate concentration. The values are estimated for 8 parameters (miu_max, Yxs, mS, Ks, Kis, Kip, a, b), so they fit fairly well with all experiments. The parameter estimation is done by a global optimization algorithm Differential Evolution from SciPy. I have also set bounds for the parameter values according to the literature. The problem is only with values of $\mu_{max}$ and Ks. When I increase boundaries for them, they both tend to increase until one of them has reached the defined ceiling. – MartinsM Dec 8 '15 at 9:33
• @MartinsM My experience fitting time series tends to yield highly correlated parameters. You might look at the joint distribution of the $K_S$ and $\mu_{max}$ fits (that is a plot of the objective function for z, and the two parameters for x and y). It is possible they fit for a realistic pair, but the best fit is when both are way off. – Hans Dec 8 '15 at 14:16