The Monod growth function is used extensively in models from the life sciences. It was proposed by Jacques Monod to describe the rate of microbial growth as a function of nutrient density. Some of Monod’s original data are plotted below, in this case for the bacteria M. tuberculosis when grown at different glucose concentrations. The animation allows you to explore the Monod function for different values of the constants S and c. Convince yourself that the constant S can can be interpreted as the maximum possible growth rate predicted by the function and c is the glucose concentration at which M. tuberculosis is predicted to be growing at 50% of its maximum possible rate.
The Monod function has the same mathematical form as another important function in the life sciences called the Michaelis-Menten function. The Michaelis-Menten function was developed to model the rate at which an enzymatic reaction generates a product, as a function of the substrate concentration. This function is straightforward to derive from first principles and doing so helps to reveal why functions like the Monod function are reasonable descriptions of biochemical processes.
Consider an enzymatic reaction in which a substrate N and an enzyme E bind to form a complex C. The complex C can then either undergo a reaction, generating a new product molecule, or it can instead dissociate back into the original substrate and enzyme components. The chemical equation for the binding and dissociation processes is
Let’s use [N], [E], and [C] to denote the concentrations of each component of the reaction. We will use R to denote the rate at which the complex C undergoes the reaction to generate the new product molecule, and we assume that this is proportional to the concentration of C; that is, R=a [C] where a is a positive constant. Therefore, if we can find a formula that describes [C] as a function of [N] then this will also tell us how the rate of generation of the product depends on the substrate concentration [N]. We now seek such a formula.
Suppose that the rate of binding of N and E is proportional to their concentrations; that is, the rate of binding is b [N] [E] where b is a positive constant. Likewise, suppose the rate of dissociation of C back into the components N and E (instead of proceeding through the reaction) is proportional to the concentration of C. In other words, the rate of dissociation is d [C] where d is a positive constant. If the binding and dissociation rates are very fast relative to the rate at which C undergoes the reaction, then the concentration of C at any time will be determined primarily by the balance between binding and dissociation. In other words, the rate of binding will equal the rate of dissociation, giving the balance equation:
Now enzymatic reactions neither create nor destroy the enzyme, and therefore the total concentration of the enzyme T must remain constant at T=[E]+[C] (recall that the enzyme is present in both free and bound form). Solving this equation for [E] gives [E]=T-[C], and substituting this result into the above balance equation then gives
We can now solve this equation for [C] to obtain
or
where k is the ratio of the dissociation and binding constants, k=d/b. This function makes intuitive sense from a biochemical standpoint; the maximum possible concentration of the complex is T, which is the total concentration of the enzyme, and this maximum concentration is reached asymptotically as the substrate concentration [N] increases.
Putting the above result together with the assumption that the rate of generation of the product is R=a [C] we obtain the Michaelis-Menten function
You can see that the Michaelis-Menten function has the same mathematical form as the Monod function.
References
Monod, J. 1949. The growth of bacterial cultures. Annual Review of Microbiology 3:371-394