The model consists of two organisms in co-culture, each dependant on one another for an essential substrate, ie without one the other would be unable to survive. The model predicts the concentration of the 2 organisms as well as the acetate which is the main product of the system.

The system was modelled using the following differential equations:

Substrate limited specific cell growth (Monad equation) $$r_g = \mu_{max} \frac{Sx_v}{K_s + S}$$
Cell death. First order with respect to viable cell concentration $$r_d = k_d x_v$$
Product formation (Luedeking–Piret equation) Product produced in cell growth and in cell maintenance $$r_p = \alpha r_g +\beta x_v $$ $$x_v = \int r_x dt \qquad , \text{ where } r_x = r_g - r_v$$
The substrate uptake: $$r_s = r_{s_x} + r_{s_p} + r_{s_m}$$ that is, the sum of rate of substrate utilisation in cell sythensis, maintenance and product formation

A MATLAB script was produced to solve the system of equations.

Note: each equation occurs twice, once for each cell and these are just the generic forms. Side reactions in the system were not considered at this stage.


Reference: McNeil, B. and Harvey, L. (2008). Practical fermentation technology. Chichester, England: Wiley.