Firstly, to design submodel 1, an ODE model was written and simulated for the growth of the bacterial volume depending on a constant growth rate. Here, it was assumed that the concentration of the nutrient does not influence bacterial growth.

Next, the ODE model was modified to take into account the effect the concentration of nutrient inside the bacteria has on the growth rate. This was achieved by using the Hill Function. The Hill function models the cooperativity between a ligand and a macromolecule. So in this case, it links the concentration of nutrients and growth together.

In the final part of step 1, it was assumed that the internal concentration of nutrients (i.e. the nutrient concentration inside the bacteria) varies with time in a way illustrated by a piecewise function. This resulted in a new ODE model.

Step 2

To incorporate the consumption of nutrients into the overall model, two more phenomena were embodied into the second submodel to help create an even more realistic model; the increase in bacterial volume, which consumes nutrients and energy, and the fact that there is only a finite amount of nutrients as the culture medium has a finite volume. These two phenomena were modelled.

Assumptions: The external concentration of nutrients remains constant.

The model (blue line), experimental data (red line)

The maths behind the model

Step 3

The modelling of growth would be continued, given a longer time period, by considering the evolution of the internal concentration of nutrients; the diffusion of nutrients through the membrane of the bacteria. This would further refine the model. Two things would be considered in terms of modelling: the geometric model for B-Subtilis and the diffusion model.
Several assumptions would be made. Firstly, the nutrients are not consumed by the metabolism of the bacteria. Secondly, the bacteria in a colony share the same shape. Therefore, their surface and volume are linked by a relation of the kind S = a V^{(2/3)}

Step 4

Furthermore, a more complex model can be built. To do this, the way that the replication machinery switches on can be modelled. This model is linked to the internal concentration of nutrients and the growth rate.