UC Berkeley

An interesting way to study biological systems is from the perspective of control theory. Most of these systems can only operate within a narrow range of conditions, and external disturbances typically conspire to push them out of their normal operating domains. This has necessitated the development of mechanisms for physiological and environmental sensing to ensure that vital internal variables are regulated. Without these feedback control systems, life itself would be impossible due to daily and seasonal variations in environmental conditions. Survival therefore depends largely on the effectiveness of these control systems, and it is commonly assumed that natural selection has led to the evolution of optimal control strategies. This framework of evolutionary optimality is used in this dissertation to examine sporulation, a bacterial survival strategy employed by Bacillus subtilis.

Sporulation is the process of forming a morphologically-distinct, dormant structure that is able to withstand severe environmental insults. This dormant cell structure, called an endospore (or spore), is typically formed in response to nutrient limitation or other environmental signals associated with low food supply. A signal transduction pathway integrates intra- and extra-cellular signals to arrive at a decision to sporulate, effectively acting as a control mechanism that allows survival in the face of environmental nutrient limitation and other serious disturbances. In the framework of evolutionary optimality, this control strategy is hypothesized to be optimal in the sense of maximizing a fitness reward function for the B. subtilis colony.

In this dissertation, sporulation is analyzed from this perspective to uncover some interesting characteristics of the decision policy. It is shown that sporulation provides a higher fitness than a simpler bacterial survival strategy with many of the same benefits (called dormancy in this study). The particular environment in which this is verified features a catastrophic event several generations into the future, after which only the survival structures remain. This result agrees with the morphological differences between a spore and a simple dormant cell. Moreover, the optimal sporulation policy qualitatively agrees with experimental data, while the optimal dormancy strategy is significantly different from observed behavior in literature studies. These results offer a candidate evolutionary justification for the existence of sporulation.

Fitness-maximizing sporulation policies are then studied in the contexts of simple population-level models with food-per-growing-cell rate dependencies. Since the results for a single population model are not consistent with expected results, a second model is postulated based on resource competition between two bacterial populations. The competitive exclusion principle is shown to hold for the proposed model structure, which provides an extension of previous resource competition results due to the sporulation model's "storage-like" spore states. For specialized cases of this competing populations model (birth and death rates equal for both populations), an approximation of the steady state solution is derived and input-output stability is analytically proved using perturbation methods. Both of these are notable due to the model's non-unique steady state solutions and the nonlinearity of the proposed model. Though the steady state approximation imposes nutrient influx changes to be small, it is intended to model adiabatic system trajectories for slowly-changing nutrient conditions. In response to these changes, a game theoretic analysis yields two policies that cannot be invaded by rare mutants: 100% steady state sporulation efficiency if nutrient influx decreases on average, and 0% steady state sporulation efficiency if nutrient influx increases on average. Evolutionary dynamics are introduced to model changes between these two optimal policies, and the nutrient influx is assumed to randomly switch between positive and negative values based on a two-state Markov chain. A non-equilibrium policy is derived for these modeling and environmental conditions, which is much closer to experimental data than the optimal policies. The "choice" of the non-equilibrium policy over the optimal policies is then examined in a prospect theory framework, and it is shown that the preference of the non-optimal policy could be explained with appropriate shifts of reference: when nutrient influx increases on average, the bacterium expects the worst (pessimistic), and when nutrient influx decreases on average, the bacterium expects the best (optimistic). While the non-equilibrium policy appears to fall within the class of bet-hedging (risk-spreading) strategies, this is not the case as the probability of population extinction is not decreased.

The work presented in this dissertation yields some interesting results on B. subtilis behavior, some of which may generalize to other organisms that are capable of entering an inactive state.