Optimal control of ribosome population for gene expression under periodic nutrient intake.

Translation of proteins is a fundamental part of gene expression that is mediated by ribosomes. As ribosomes significantly contribute to both cellular mass and energy consumption, achieving efficient management of the ribosome population is also crucial to metabolism and growth. Inspired by biological evidence for nutrient-dependent mechanisms that control both ribosome-active degradation and genesis, we introduce a dynamical model of protein production, that includes the dynamics of resources and control over the ribosome population. Under the hypothesis that active degradation and biogenesis are optimal for maximizing and maintaining protein production, we aim to qualitatively reproduce empirical observations of the ribosome population dynamics. Upon formulating the associated optimization problem, we first analytically study the stability and global behaviour of solutions under constant resource input, and characterize the extent of oscillations and convergence rate to a global equilibrium. We further use these results to simplify and solve the problem under a quasi-static approximation. Using biophysical parameter values, we find that optimal control solutions lead to both control mechanisms and the ribosome population switching between periods of feeding and fasting, suggesting that the intense regulation of ribosome population observed in experiments allows to maximize and maintain protein production. Finally, we find some range for the control values over which such a regime can be observed, depending on the intensity of fasting.

Differential equation model for central-place foragers with memory: implications for bumble bee crop pollination.

Bumble bees provide valuable pollination services to crops around the world. However, their populations are declining in intensively farmed landscapes. Understanding the dispersal behaviour of these bees is a key step in determining how agricultural landscapes can best be enhanced for bumble bee survival. Here we develop a partial integro-differential equation model to predict the spatial distribution of foraging bumble bees in dynamic heterogeneous landscapes. In our model, the foraging population is divided into two subpopulations, one engaged in an intensive search mode (modeled by diffusion) and the other engaged in an extensive search mode (modeled by advection). Our model considers the effects of resource-dependent switching rates between movement modes, resource depletion, central-place foraging behaviour, and memory. We use our model to investigate how crop pollination services are affected by wildflower enhancements. We find that planting wildflowers such that the crop is located in between the wildflowers and the nest site can benefit crop pollination in two different scenarios. If the bees do not have a strong preference for wildflowers, a small or low density wildflower patch is beneficial. If, on the other hand, the bees strongly prefer the wildflowers, then a large or high density wildflower patch is beneficial. The increase of the crop pollination services in the later scenario is of remarkable magnitude.

Dynamics of a general model of host-symbiont interaction.

We consider a model of host-symbiont interactions, in which symbionts can only live in association with their host and are transmitted both vertically from associated hosts to their offspring and horizontally from associated hosts to nearby unassociated hosts. The effect of the symbiont is modelled by a change in the birth rate of associated hosts. We analyze the two-dimensional dynamics in the resulting four-dimensional parameter space, and determine the qualitative behaviour for all parameter values. We find that for all but one choice of parameter values, solutions in the feasible region, apart from a 0- or 1-dimensional set of initial conditions, tend either to a unique equilibrium, or to one of two distinct equilibria. Moreover, the bistable case occurs only when the symbiont is a mutualist whose horizontal spread rate through the host population exceeds the positive change in the birth rate of associated hosts.

Tragedy of the commons in the chemostat.

We present a proof of principle for the phenomenon of the tragedy of the commons that is at the center of many theories on the evolution of cooperation. Whereas the tragedy is commonly set in a game theoretical context, and attributed to an underlying Prisoner's Dilemma, we take an alternative approach based on basic mechanistic principles of species growth that does not rely on the specification of payoffs which may be difficult to determine in practice. We establish the tragedy in the context of a general chemostat model with two species, the cooperator and the cheater. Both species have the same growth rate function and yield constant, but the cooperator allocates a portion of the nutrient uptake towards the production of a public good -the "Commons" in the Tragedy- which is needed to digest the externally supplied nutrient. The cheater on the other hand does not produce this enzyme, and allocates all nutrient uptake towards its own growth. We prove that when the cheater is present initially, both the cooperator and the cheater will eventually go extinct, hereby confirming the occurrence of the tragedy. We also show that without the cheater, the cooperator can survive indefinitely, provided that at least a low level of public good or processed nutrient is available initially. Our results provide a predictive framework for the analysis of cooperator-cheater dynamics in a powerful model system of experimental evolution.

Survival and extinction results for a patch model with sexual reproduction.

We consider a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents' patch or are sent to a nearby patch with some fixed probabilities. As the patch size tends to infinity, we identify a mean-field limit consisting of an infinite set of coupled differential equations. For the mean-field equations, we find explicit conditions for survival and extinction that we call expansion and retreat. Using duality techniques to compare the stochastic model to its mean-field limit, we find that expansion and retreat are also precisely the conditions needed to ensure survival and extinction of the stochastic model when the patch size is large. In addition, we study the dependence of survival on the dispersal range. We find that, with probability close to one and for a certain set of parameters, the metapopulation survives in the presence of nearest neighbor interactions while it dies out in the presence of long range interactions, suggesting that the best strategy for the population to spread in space is to use intermediate dispersal ranges.

A scaling law for random walks on networks.

The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.