# Mathematical analysis, modelling and simulation of microbial population dynamics

Quedeville, Vincent. Mathematical analysis, modelling and simulation of microbial population dynamics. PhD, Dynamique des fluides, Institut National Polytechnique de Toulouse, 2020, 203 p.

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## Abstract

The physiology of unicellular organisms results from a central metabolism which input-output balance accounts for both the cells’ state and their culture medium’s abundance. When bacteria are cultivated in a locally fed fermenter and transported in a turbulent flow, they have to deal with concentration gradients throughout their trajectory in the reactor. Simulating this physics in a multiscale modelling approach requires taking into account not only the well-known laws of hydrodynamics, but also the cells’ biochemistry which is still ill-understood to date. Moreover, the prohibitive cost of the numerics forces to reduce the models to constrain the duration of the experiments to a few weeks. In this context, special consideration has been given to the biological phase. The bacteria population dynamics is given by an integro-differential transport-rupture equation in the space of the particles’ inner coordinates. Picking the most appropriate variables is of paramount importance to best report the time evolution of the cells’ state throughout their history in the fermenter, the latter being comparable to a markovian process. The microorganisms’ length testifies to their morphology and their progress in the cell cycle, whereas the uptake rate of the surrounding resources leads to an evaluation of the material transfer between the liquid and biotic phases. The result is the estimation of the source term in the organisms’ central metabolism which outputs are the apparent rate of anabolism and, if over-uptake, activation of peripheral reactions to combust the surplus in organic compounds. Beyond their own history, the individuals’ metabolic yields can be impacted by the substrate availability at their neighbourhood, which stems from the feeding and the level of mixing in the reactor. The state variables have a compact support, what raises the question of the mathematical problem’s wellposedness, similarly as solving a PDE over a bounded set is traditionally more difficult than over $\mathbb{R}^{n}$, $n \in \mathbb{N}$. It is shown that the Malthus eigenfunction associated with the transport-rupture equation is $\mathcal{C}^{1}$ as soon as fragmentation trumps cell growth near the right-hand edge of the size-distribution’s support. All in all, the solution is continuous at each time in the state space. These results allow the implementation of numerical codes to solve (in this work, by Monte-Carlo, Finite Volume, or Quadrature of MOMents methods) the well-posed problem, the algorithms being exploited to simulate five biochemical engineering experiments which conclusions are detailed in the literature.

Item Type: PhD Thesis Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE) Fox, Rodney O. and Morchain, Jérôme download 13 Oct 2020 10:34

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