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

The system was modelled using the following differential equations:

Substrate-limited specific cell growth (Monod equation): $$r_g = x_v \mu_{max} \frac{S}{K_s + S}$$
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, with cell type-specific parameter values; the equations shown 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.