The steady-state assumption in oscillating and growing systems

The steady-state assumption, which states that the production and consumption of metabolites inside the cell are balanced, is one of the key aspects that makes an efficient analysis of genome-scale metabolic networks possible. It can be motivated from two different perspectives. In the time-scales perspective, we use the fact that metabolism is much faster than other cellular processes such as gene expression. Hence, the steady-state assumption is derived as a quasi-steady-state approximation of the metabolism that adapts to the changing cellular conditions. In this article we focus on the second perspective, stating that on the long run no metabolite can accumulate or deplete. In contrast to the first perspective it is not immediately clear how this perspective can be captured mathematically and what assumptions are required to obtain the steady-state condition. By presenting a mathematical framework based on the second perspective we demonstrate that the assumption of steady-state also applies to oscillating and growing systems without requiring quasi-steady-state at any time point. However, we also show that the average concentrations may not be compatible with the average fluxes. In summary, we establish a mathematical foundation for the steady-state assumption for long time periods that justifies its successful use in many applications. Furthermore, this mathematical foundation also pinpoints unintuitive effects in the integration of metabolite concentrations using nonlinear constraints into steady-state models for long time periods

Mathematical modeling, simulation and analysis of metabolic oscillations in Bacillus subtilis biofilms

Metabolic oscillations in biofilms of Bacillus subtilis have been reported as periodic halting of growth in the expansion of the colony growing in a microfluidics chamber by Liu et al (2015). This thesis is aimed at understanding these oscillations through minimal dynamic model involving three ordinary differential equations (ODEs). The model is first applied in its basic form in order to describe the oscillations. Next, various modifications of the model are discussed in detail and the results of each modification are viewed in light of the underlying biology. The four modifications investigate the mechanism of oscillations with respect to spatial effects, reversible reactions and more robust reaction kinetics. Finally, we apply the minimal model in a broader perspective in order to understand population dynamics in a typical community of a social organism. We consider three interacting subpopulations of a species that have their own distinct phenotypes. None of the subpopulations have an absolute advantage over the other two. This gives rise to cyclic dynamics like the rock paper scissors game which is analysed using evolutionary game theory. We also present an asymmetrical two-player two- strategy game describing the same system, where the phenotype of each subpopulation is considered as a strategy. This investigation tests the ideal strategies for three different levels of antibiotic stress. We observe bet-hedging in the form of production of resistant cells which are a costly choice in the absence of the antibiotic stress. Although the population dynamics study is described with a broad range of applicability, we also discuss its applications in the B. subtilis biofilm