Mathematical Models of Quorum Sensing
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Mathematical models of biological phenomena are constructed in order to further the understanding of the known and unknown interactions that result in the behaviors of dynamical systems. We present mathematical models dealing with quorum sensing, which is the biological process of communication in bacteria. The density-dependent means of communication are mediated by molecules called autoinducers that are both synthesized and collected by quorum sensing bacteria.
We employ differential equations to investigate and understand the dynamics of underlying signaling processes. The first two models of this study were constructed with the idea of relating flocking movements observed in birds to gene expression in quorum sensing. To this end, modified Cucker-Smale flocking equations which do not require detailed knowledge of signal transductions mechanisms or regulatory proteins are used to represent quorum sensing and chemotaxis. The dynamical behaviors of these models are analyzed and approximated using asymptotic analysis and simulations. The coupling of quorum sensing and chemotaxis systems results in the formation of two groups of cells during the migration towards the attractant, which is similar to behavior observed in experiments of chemotaxing E. coli. This consequence of density influencing the velocity of bacteria suggests the possibility that density (or a density-dependent system such as quorum sensing) affects the chemotaxis system. We also show an application of this coupled model that produces qualitatively similar results with experimental data.
To further analyze collective behavior emerging from the interactions of quorum sensing and chemotaxis, this study uses statistical physics to derive a partial differential equation that tracks the time evolution in phase space of the distribution of these cells. Lastly, this study combines theory and experimental data to present a compartmental model that predicts p-aminophenol (PAP) response to various autoinducer concentrations in quorum sensing cells. The use of compartments allows for the model to be customized for constructs that do not use autoinducer-mediated production of the beta-galactosidase enzyme.