# Determine max of 2 parameter functions

I have 12 experimental tables with OD(substrate concentration, time), where OD is optical density and one table with $\mu$ (substrate), where $\mu$ is growth speed. OD and $\mu$ are functions. The experiment is "growth of proteins on a methanol substrate".

With passage of time, the concentration of substrate and oxygen is decreasing (substrate to zero, and $\ce{O2}$ to 10% from 100% and the process is terminated).

With substrate decrease the number of particles in bioreactor is growthing. I must determine the parameters $(s, \ce{O2})$ when the growth speed is $\rm max$ to $\mathrm{max }- x\%$. $x$ may be $10\%-15\%$ of $\rm max$.

I need a model to determine the functions $\mu(s)$, $\mu(\ce{O2})$ and $\mu(s, \ce{O2})$ or the max value of miu(s,o2), where $\mu$ = growth speed, $s$ = substrate concentration and $\ce{O2}$ = oxygen concentration.

I must simulate the function $\mu(s,\ce{O2})$, to observe the maximization surface(sized by $s$ and $\ce{O2}$) through variation of parameters $s$ and $\ce{O2}$. The surface formed by $s$ and $\ce{O2}$ when $\mu$ values are between $\mathrm{max}(\mu) - 10\%$ and $\mathrm{max}(\mu)$.

The volume of bioreactor is $15\:\mathrm{L}$ and the density inside is $\approx 1$.

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Have you tried methods to fit your data into an equation (linear/exponential/logistic etc.) - there are few applications which can be useful - Maxima - en.wikipedia.org/wiki/Maxima_%28software%29 , R - en.wikipedia.org/wiki/R_%28programming_language%29 or even simple solver of Excel or LibreOffice can do it's job –  java_xof Jan 15 '13 at 20:56
Have you looked into Michaelis Menten kinetics? With this you should be able to come up with some differential equations for your growth rates –  Michiel Apr 23 '13 at 20:25