First-order unstable bioreactor model: from microscopic (statistical) to macroscopic (differential equation)

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 19:55

*Enlace a Spanish version

Summary:

This video describes the modelling of a bioreactor as an unstable first-order system, such that the model predicts exponential growth of the microorganism population, based on the conclusions of a ‘microscopic’ statistical model on the ‘probability of a cell division in a small time interval’. The result is a ‘macroscopic’ deterministic ODE model of type $\frac{dx}{dt}=0.4x$, whose solution is $x(t)=e^{0.4t}x(0)$. The underlying statistics has a close relationship with ‘exponential decay’ models in physics of radioactivity, as outlined in the video.

If, in addition to the ‘natural’ growth by cell division, there is an inflow of biomass $u$ (or extraction if the sign of $u$ were negative), then the model will be $\frac{dx}{dt}=0.4x+u$. This model is still open-loop unstable, and to maintain a constant population of microorganisms, it must be ‘stabilized’ by closed-loop control (not the objective of this video, which only deals with modelling).

It should be noted that this model is very approximate: population growth consumes nutrients, secondary metabolites appear that are sometimes toxic, there are mutations, etc. so it is only a first approximation to the considerations in this type of systems and also serves as motivation for the analysis of unstable systems in industrial control, given that the usual systems of electrical circuits, fluids, electromechanical, etc. in initial courses in automatic control are usually stable.

For example, as an aside, exponential growth with a growth rate that is itself random gives rise to Black-Scholes-Merton models in economics, under some particular assumptions.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.