Geometric analysis of a model for cross‐feeding in the chemostat

Abstract

A model for competition between bacterial strains in a chemostat is studied using a model reduction argument and phase portrait analysis for the specific case of trophic chain. The first strain feeds on glucose but secretes a metabolic intermediate (acetate), which the second strain consumes. However, the metabolic intermediate imposes a cost to growth to both strains. The geometric and asymptotic analysis of the reduced model allows a rigourous treatment of the role of this assumption in the emergence of coexistence (cross-feeding) equilibria. Conditions for non-existence of cross-feeding equilibria are given for certain parameter ranges. In the case of trophic chains, existence of cross-feeding equilibria does not rely only on the presence of the acetate specialist scavenging for the intermediate metabolite inside the resource-limited chemostat environment. The glucose specialist must, in addition, possess a certain degree of tolerance to the metabolic intermediate. Relaxing the assumption of the cost of growth helps establish a cross-feeding equilibrium.KEYWORDS bifurcation analysis, chemostat, cross-feeding INTRODUCTIONMetabolic interactions affect the function and stability of microbial communities and obtaining understanding on them is crucial in many biological applications. 1 When multiple microbial strains share a single limiting resource in a well-stirred chemostat, the outcome is competitive exclusion where at most one strain could be stably sustained over time. [2][3][4][5][6] However, when one strain consumes metabolites produced by the other (via cross-feeding), stable coexistence of both strains may be possible. 1,7,8 The study 9 shows that locally asymptotically stable coexistence between 2 strains of Escherichia coli occurs because of cross-feeding for different ranges of the glucose input and appropriate initial population densities.The mathematical model proposed by Gudelj et al 9 builds upon 10 and describes the dynamics of competition between the strains, which metabolise glucose in a 2-step process: glycolysis and tricarboxylic acid (TCA) cycle. In the first step, the bacterial cell partially converts the glucose to a metabolic intermediate (acetate), and in the second step it, either fully consumes the intermediate or releases it in the environment. The excreted metabolic intermediate creates additional resource niches, where other strains can grow, and the model 9 bypasses the assumptions of the competitive exclusion principle. The models 9,10 assume both strains feed on either resource but differ in the respective uptake kinetics as observed by Rosenzweig et al. 8 Thus, the strain with higher affinity for glucose shall be called glucose specialist, and the second strain with higher affinity for acetate acetate specialist.As a possible biological mechanism behind the observed outcome, 8 mentions changes in glucose uptake kinetics by the glucose specialist resulting from scavenging for the limiting resource. However, the upregulated glucose fermenta-Math Meth Appl Sci. 2018;41:8765-8783.wi...