Decomposition kinetics as an optimal control problem

Microbial explicit models are constructed by linking decomposition (the process of organic matter break-down) and substrate uptake to microbial growth, respiration, and mortality. Therefore, the specific choice of the decomposition and/or uptake kinetics affects how in the model microbes grow and die, with consequences for carbon stabilization. There are well-established theories for extracellular enzymatic reactions and for substrate transport and uptake by cells, which allow deriving formulas for the decomposition and uptake kinetics, respectively. These laws account for microbial growth (e.g., in the Monod equation), but implicitly assume that microbial traits encoded in model parameters are static. Yet, microbes adapt to the environmental conditions they experience, resulting in temporally dynamic traits at both population and community levels. Adaptation is a result of natural selection for the fittest organisms. Therefore, we can describe adapted microbes by assuming they maximize their growth for given environmental conditions (e.g., limiting the amount of available resources) and given metabolic tradeoffs (e.g., decreasing efficiency of substrate to biomass conversion at high growth rates). In this contribution, we translate this assumption into a formulation of decomposition as an optimal control problem, where the objective is the maximization of cumulative growth, the constraint is imposed via a substrate mass balance, and the control parameter is the realized substrate uptake rate, assumed to be the outcome of optimally adapted production of extracellular enzymes and cellular uptake capacity. This optimal control problem is solved analytically for a simple case study (one substrate, homogeneous microbial community), leading to optimal decomposition kinetics that scale with the square root of substrate carbon content (different from Monod or Michaelis-Menten equations) and with a strong effect of maintenance respiration. If maintenance respiration is high, the kinetics flattens, and the optimal decomposition rate remains larger than zero even as the substrate is depleted. This means that the optimal decomposition rate approaches zero-order kinetics and exhibits increasingly high values as maintenance costs are increased. Interestingly, a tradeoff emerges between the rate of substrate consumption at the beginning of decomposition and microbial carbon use efficiency (ratio of growth over uptake). At high resource availability, efficient but slow-growing microbes are selected, whereas at low resource availability inefficient but fast-growing microbes are favored because they can more effectively compete for the limited resources. These results suggest that optimization methods offer an alternative way to define decomposition kinetics laws that account for microbial adaptation.