Coevolutionary arms races between bacteria and bacteriophage

Abstract

We propose a computational and theoretical framework for analyzing rapid coevolutionary dynamics of bacteriophage and bacteria in their ecological context. Bacteriophage enter host cells via membrane-bound surface receptors often responsible for nutrient uptake. As such, a selective pressure will exist for the bacteria to modify its receptor configuration and, in turn, for the phage to modify its tail fiber. A mathematical model of these trait adaptations is developed by using the framework of adaptive dynamics. Host strains differ in their efficiency of resource uptake and resistance to phage, whereas phage strains differ in their host preference for adsorption. We solve the evolutionary ecology model and find the conditions for coevolutionary branching and relevant dimensionless parameters leading to distinct quasispecies. We confirm these calculations using stochastic Monte Carlo simulations of populations evolving in a chemostat with fixed washout rate and inflow resource density. We find that multiple quasispecies of bacteria and phage can coexist in a homogeneous medium with a single resource. When diversification occurs, quasispecies of phage adsorb effectively to only a limited subset of the total number of quasispecies of bacteria, i.e., functional differences between quasispecies arise endogenously within the evolutionary ecology framework. Finally, we discuss means to relate predictions of this model to experimental studies in the chemostat, using the model organisms Escherichia coli and the virulent strain of lambda phage.

Fig. 1.

A schematic of the linkages between the ecological model of population dynamics, the trait model describing resource uptake and adsorption of phage, and the theoretical and numerical approaches for analyzing the evolutionary ecology. Note that adaptive dynamics is the limit of the evolutionary ecology in the small-mutation limit, results from which are combined with biological parameters to guide stochastic simulations of coevolutionary dynamics of bacteriophage and bacteria in the chemostat.

Fig. 2.

An evolutionary-rate controlled switch between cyclical trait changes and convergence to the fixed point (0, 0) in a dimensionless x - y trait-space domain (40). The coupled ordinary differential equations of Eqs. 4 and 5 were numerically integrated with ξ_n/ξ_v = 4, ϕ₀V_c/ω = 0.25, along with k_n/k_v = 6.25 (dashed) and k_n/k_v = 4.56 (solid).

Fig. 3.

Results from stochastic simulations of an evolutionary chemostat model with parameters as described in Table 1. In this case, the ratio of the stable uptake range of hosts to the host-range of phage, ξ_n/ξ_v, varies from 2 to 3.6. The dimensionless ratios are ϕ₀V_c/ω = 0.25 and k_n/k_v = 0.056. The y axis depicts the steady-state trait values for bacteria (circles) and phage (diamonds). The depicted strains are those with at least 1% of the total bacteria or phage population, respectively. A succession of bifurcations leading to multistrain coexistence is shown. The strains group naturally into distinct clusters of quasispecies.

Fig. 4.

Results from stochastic simulations of an evolutionary chemostat model with parameters as described in Fig. 3 using ξ_n/ξ_v = 3.6. Relative population density of bacteria N (circles) and phage V (diamonds) are plotted as a function of their trait values, x and y respectively. There are three quasispecies of bacteria and three of phage, despite the presence of 27 bacteria strains and 185 phage strains.