Fixed points and limit cycles in the population dynamics of lysogenic viruses and their hosts

Starting with stochastic rate equations for the fundamental interactions between microbes and their viruses, we derive a mean field theory for the population dynamics of microbe-virus systems, including the effects of lysogeny. In the absence of lysogeny, our model is a generalization of that proposed phenomenologically by Weitz and Dushoff. In the presence of lysogeny, we analyze the possible states of the system, identifying a novel limit cycle, which we interpret physically. To test the robustness of our mean field calculations to demographic fluctuations, we have compared our results with stochastic simulations using the Gillespie algorithm. Finally, we estimate the range of parameters that delineate the various steady states of our model.Comment: 20 pages, 16 figures, 4 table

Giant number fluctuations in microbial ecologies

Statistical fluctuations in population sizes of microbes may be quite large depending on the nature of their underlying stochastic dynamics. For example, the variance of the population size of a microbe undergoing a pure birth process with unlimited resources is proportional to the square of its mean. We refer to such large fluctuations, with the variance growing as square of the mean, as Giant Number Fluctuations (GNF). Luria and Delbruck showed that spontaneous mutation processes in microbial populations exhibit GNF. We explore whether GNF can arise in other microbial ecologies. We study certain simple ecological models evolving via stochastic processes: (i) bi-directional mutation, (ii) lysis-lysogeny of bacteria by bacteriophage, and (iii) horizontal gene transfer (HGT). For the case of bi-directional mutation process, we show analytically exactly that the GNF relationship holds at large times. For the ecological model of bacteria undergoing lysis or lysogeny under viral infection, we show that if the viral population can be experimentally manipulated to stay quasi-stationary, the process of lysogeny maps essentially to one-way mutation process and hence the GNF property of the lysogens follows. Finally, we show that even the process of HGT may map to the mutation process at large times, and thereby exhibits GNF.Comment: 18 pages, 5 figure

Epigenetics as a first exit problem

We develop a framework to discuss stability of epigenetic states as first exit problems in dynamical systems with noise. We consider in particular the stability of the lysogenic state of the lambda prophage, which is known to exhibit exceptionally large stability. The formalism defines a quantative measure of robustness of inherited states. In contrast to Kramers' well-known problem of escape from a potential well, the stability of inherited states in our formulation is not a numerically trivial problem. The most likely exit path does not go along a steepest decent of a potential -- there is no potential. Instead, such a path can be described as a zero-energy trajectory between two equilibria in an auxiliary classical mechanical system. Finding it is similar to e.g. computing heteroclinic orbits in celestial mechanics. The overall lesson of this study is that an examination of equilibria and their bifurcations with changing parameter values allow us to quantify both the stability and the robustness of particular states of a genetic control system.Comment: 6 pages, 3 figures, in REVTe

<i>In silico</i> evolution of lysis-lysogeny strategies reproduces observed lysogeny propensities in temperate bacteriophages

Bacteriophages are the most abundant organisms on the planet and both lytic and temperate phages play key roles as shapers of ecosystems and drivers of bacterial evolution. Temperate phages can choose between (i) lysis: exploiting their bacterial hosts by producing multiple phage particles and releasing them by lysing the host cell, and (ii) lysogeny: establishing a potentially mutually beneficial relationship with the host by integrating their chromosome into the host cell's genome. Temperate phages exhibit lysogeny propensities in the curiously narrow range of 5–15%. For some temperate phages, the propensity is further regulated by the multiplicity of infection, such that single infections go predominantly lytic while multiple infections go predominantly lysogenic. We ask whether these observations can be explained by selection pressures in environments where multiple phage variants compete for the same host. Our models of pairwise competition, between phage variants that differ only in their propensity to lysogenize, predict the optimal lysogeny propensity to fall within the experimentally observed range. This prediction is robust to large variation in parameters such as the phage infection rate, burst size, decision rate, as well as bacterial growth rate, and initial phage to bacteria ratio. When we compete phage variants whose lysogeny strategies are allowed to depend upon multiplicity of infection, we find that the optimal strategy is one which switches from full lysis for single infections to full lysogeny for multiple infections. Previous attempts to explain lysogeny propensity have argued for bet-hedging that optimizes the response to fluctuating environmental conditions. Our results suggest that there is an additional selection pressure for lysogeny propensity within phage populations infecting a bacterial host, independent of environmental conditions

Well-temperate phage: optimal bet-hedging against local environmental collapses

Upon infection of their bacterial hosts temperate phages must chose between lysogenic and lytic developmental strategies. Here we apply the game-theoretic bet-hedging strategy introduced by Kelly to derive the optimal lysogenic fraction of the total population of phages as a function of frequency and intensity of environmental downturns affecting the lytic subpopulation. "Well-temperate" phage from our title is characterized by the best long-term population growth rate. We show that it is realized when the lysogenization frequency is approximately equal to the probability of lytic population collapse. We further predict the existence of sharp boundaries in system's environmental, ecological, and biophysical parameters separating the regions where this temperate strategy is optimal from those dominated by purely virulent or} dormant (purely lysogenic) strategies. We show that the virulent strategy works best for phages with large diversity of hosts, and access to multiple independent environments reachable by diffusion. Conversely, progressively more temperate or even dormant strategies are favored in the environments, that are subject to frequent and severe temporal downturns.Comment: 26 pages, 3 figure

An Abstraction Theory for Qualitative Models of Biological Systems

Multi-valued network models are an important qualitative modelling approach used widely by the biological community. In this paper we consider developing an abstraction theory for multi-valued network models that allows the state space of a model to be reduced while preserving key properties of the model. This is important as it aids the analysis and comparison of multi-valued networks and in particular, helps address the well-known problem of state space explosion associated with such analysis. We also consider developing techniques for efficiently identifying abstractions and so provide a basis for the automation of this task. We illustrate the theory and techniques developed by investigating the identification of abstractions for two published MVN models of the lysis-lysogeny switch in the bacteriophage lambda.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005

Calculating Biological Behaviors of Epigenetic States in Phage lambda Life Cycle

Gene regulatory network of lambda phage is one the best studied model systems in molecular biology. More 50 years of experimental study has provided a tremendous amount of data at all levels: physics, chemistry, DNA, protein, and function. However, its stability and robustness for both wild type and mutants has been a notorious theoretical/mathematical problem. In this paper we report our successful calculation on the properties of this gene regulatory network. We believe it is of its first kind. Our success is of course built upon numerous previous theoretical attempts, but following 3 features make our modeling uniqu: 1) A new modeling method particular suitable for stability and robustness study; 2) Paying a close attention to the well-known difference of in vivo and in vitro; 3) Allowing more important role for noise and stochastic effect to play. The last two points have been discussed by two of us (Ao and Yin, cond-mat/0307747), which we believe would be enough to make some of previous theoretical attempts successful, too. We hope the present work would stimulate a further interest in the emerging field of gene regulatory network.Comment: 16 pages, 3 figures, 1 tabl

Transcriptional delay stabilizes bistable gene networks

Transcriptional delay can significantly impact the dynamics of gene networks. Here we examine how such delay affects bistable systems. We investigate several stochastic models of bistable gene networks and find that increasing delay dramatically increases the mean residence times near stable states. To explain this, we introduce a non-Markovian, analytically tractable reduced model. The model shows that stabilization is the consequence of an increased number of failed transitions between stable states. Each of the bistable systems that we simulate behaves in this manner