The Dynamics of Microbial Growth on Mixtures of Substrates in Batch Reactors

A quantitative understanding of microbial growth on mixtures of substrates is of considerable biological and practical interest. Microbes growing in mixed-substrate environments display a rich spectrum of dynamics. The goal of this work is to show that it is possible to capture these dynamics by a judicious choice of only a few physiological variables. The central idea of the mathematical model is that the autocatalytic nature of enzyme induction is the basis of the dynamical behavior associated with mixed-substrate systems. We formulate a model taking due account of the cyclic structure of enzyme synthesis. By applying the quasisteadystate approximation to the model, we show how this cyclic structure makes enzyme synthesis autocatalytic. Simulations of the model show that it successfully captures the experimental data for both the diauxic and the simultaneous substrate utilization patterns. Further insight into these simulations is obtained by performing a complete analysis of the dynamics of the model. The model is first reduced by appealing to the theory of regular and singular perturbations. The reduced equations thus obtained are then analysed by the method of null clines.