Context for the question

I am a mathematics researcher working on the mathematics of imaging and parameter estimation (inverse problems). I found an interesting diffusion equation that models the growth of biofilms. A question I am interested in: Given some measurements of the biofilm, and assuming the mathematical model is accurate, can we deduce the coefficients in the model? This, however, is not the question I want to ask here.

Please pardon nonstandard use and misuse of biological terminology and gaps in knowledge; this is by no means my field. I have never done biological experiments in a laboratory, for example, or studied any university-level biology.

The model

Let us assume a simple scenario where we have a biofilm and some substrate it consumes to grow. We consider a surface where the model should tell how fast and to where the biofilm grows. The parameters in the model are:

  • $d_1$ substrate diffusion coefficient;
  • $d_2$ biomass diffusion coefficient;
  • $K_1$ maximum specific consumption rate;
  • $K_2$ biomass decay rate;
  • $K_3$ maximum specific growth rate;
  • $K_4$ Monod half saturation constant (relative to initial data $S_0$);
  • $a$, $b$ biomass spreading parameters.

The quantities that are solved from the equations are the following, both of them scaled so that $0 \le M, S \le 1$. Both of them are functions of space $x$ and time $t$, which I omit from the notation.

  • $S$ is the concentration of the substrate that the biofilm feeds on
  • $M$ is the biomass density

The governing equations are the following, with some boundary conditions I will not specify here:

$$ \begin{cases} \partial_t S = d_1 \Delta_x S - K_1 \frac{SM}{K_4+S} \\ \partial_t M = d_2 \nabla_x \cdot \left( \frac{M^b}{(1-M)^a} \nabla_x M \right) - K_2 M + K_3 \frac{SM}{K_4+S} \end{cases} $$

The questions

Can both $M$ and $S$ be measured with fair accuracy in laboratory conditions? How is it done and how reliable is it? Can the experiments continue afterwards, or does the measurement significantly alter the experimental set-up in some way (by for example destroying the biofilm or mixing the entire thing etc.)?

Of the parameters in the model, which are well-known?

I suppose the answer to last question depends on whether the biofilm is of known or unknown nature, and in the case of an unknown substance, only $d_1$ might be known. What about a known kind of biofilm?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.