I am attempting to include the Michaelis-Menten equation in (continuous) variables $S$ and $v$
$$v=\frac{V_{\max} S}{K_m + S}$$
where $V_{\max}$ and $K_m$ are given constants, as a linear programming constraint.
Similar inquiries about linearizing non-linear expressions suggest that binary variables and variable splitting are potential strategies, but I cannot visualize how to apply these techniques (or if these are applicable) to the Michaelis-Menten equation without introducing a non-linear equality.
How can the equation be adapted into a linear system that is compatible with linear programming?
I appreciate your suggestions :)
Consider the function $$f(t) = \frac{at}{b+t}$$ Taking the reciprocal, you get $$\begin{align*} \frac{1}{f(t)} &= \frac{b+t}{at} \\ &= (\frac{b}{a})(\frac{1}{t}) + \frac{1}{a} \\ \end{align*}$$ Letting $\tilde{y}(t) = \frac{1}{f(t)}$, $\tilde{m} = \frac{b}{a}$, $\tilde{t} = \frac{1}{t}$, and $\tilde{b} = \frac{1}{a}$, you get $$\tilde{y}(t) = \tilde{m}\tilde{t} + \tilde{b}$$ You now have a linear equation where you can optimize for the parameters $\tilde{m}$ and $\tilde{b}$.