A free boundary problem for a couple trace-metals precipitation-complexation process in granular biofilms

Zirhumanana Balike

University of Naples Federico II

"A free boundary problem for a couple trace-metals precipitation-complexation process in granular biofilms"

Biofilms are colonies of microorganisms embedded in a matrix of extracellular polymeric substances (EPS). They play major roles in many fields such as biotechnology and health, cite{flemming2016biofilms}. Mathematical modelling is an essential tool in understanding biofilms and their interactions with the media in which they evolve and in particular with inorganic materials because it reduces experimental testing and scale up,cite{delavar2022advanced}. In this work, we present a mathematical model that describes the growth of a granular biofilm and accounts for the two major interactions between trace metals and biofilms, namely precipitation and complexation. Indeed, experimental results show that in most cases precipitation is not an isolated phenomenon; and complexation is the other major process occurring simultaneously with it. To our knowledge, our model is the first to consider simultaneously these two phenomena inside a granule. More precisely, the general formulation of the model following the mass conservation includes: begin{itemize} item A system of first order quasi-linear hyperbolic equations that describes the growth of the biofilm. In this system, we have $n$ equations for the growth of biomasses within the biofilm, $m$ equations for the accumulation of precipitates throughout the life of the biofilm, and one equation that takes into account the evolution of porosity over time and space. The source terms of the porosity and precipitation equations are formulated so that the space occupied by the precipitates and porosity remains constant over time. item A system of diffusion-reaction equations of soluble components in the biofilm. The first system comprises $p$ equations for biomass nutrients, $q$ equations for cations in the biofilm, $r$ equations for anions which combine with cations to form precipitates, and $k$ equations for complexes. item A nonlinear ordinary differential equation which is the free boundary of the problem and takes into account the temporal evolution of the biofilm thickness. end{itemize} The entire model is therefore a free boundary problem that can be adapted to any type of biofilms (including all the evolution phases of the biofilm), ligands, and trace-metals. An existence and uniqueness theorem was proved and numerical application of the model is proposed. begin{thebibliography}{} bibitem{delavar2022advanced} Delavar, Mojtaba Aghajani, and Junye Wang. Advanced Methods and Mathematical Modeling of Biofilms: Applications in Health Care, Medicine, Food, Aquaculture, Environment, and Industry. Academic Press, 2022. bibitem{flemming2016biofilms} Flemming, Hans-Curt, et al. 'Biofilms: an emergent form of bacterial life.' Nature Reviews Microbiology 14.9 (2016): 563-575. end{thebibliography}
Additional authors: Maria Rosaria Mattei, University of Naples Federico II; Vincenzo Luongo, Unversity of Naples Federico II; Véronique Deluchat Antony, University of Limoges

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