Why Do Phage Play Dice?

Abstract

Phage lambda is among the simplest organisms that make a developmental decision. An infected bacterium goes either into the lytic state, where the phage particles rapidly replicate and eventually lyse the cell, or into a lysogenic state, where the phage goes dormant and replicates along with the cell. Experimental observations by P. Kourilsky are consistent with a single phage infection deterministically choosing lysis and double infection resulting in a stochastic choice. We argue that the phage are playing a “game” of minimizing the chance of extinction and that the shift from determinism to stochasticity is due to a shift from a single-player to a multiplayer game. Crucial to the argument is the clonal identity of the phage. Organisms typically use information from the environment to suitably modify their behavior. Some of these can be considered “strategic” decisions for maximizing the chances of success of the population. For example, many organisms are able to adjust their reproductive strategies according to environmental conditions. A typical signal that often triggers changes in the reproductive strategy is population density (1, 6). Temperate bacteriophages are among the simplest organisms that are able to sense population density and choose their reproductive strategy accordingly. In general, temperate phage choose to stay dormant and replicate along with the host (lysogeny) rather than making many virions and killing the host (lysis) when larger numbers of phage attack a bacterial population (4, 7, 10). Phage lambda's choice between lysis and lysogeny has become a paradigm for developmental decisions (12). For this lysis-lysogeny decision, we use game theory to understand under what conditions different strategies might be optimal. In particular, we focus on the determinism versus the stochasticity of the strategy. We show that deterministic strategies are best when the phage has minimal information and must consider itself as the only “player” in the game. In contrast, having multiple identical players can make a stochastic strategy the best. We consider the phage to be playing a game whose purpose is to minimize the chance of extinction. For single-player games of this kind, where the player has several options but limited information, the optimal strategies are typically deterministic (13). And it is indeed the case that the lysis-lysogeny decision is often deterministic. This statement is somewhat at odds with the general perception that stochasticity plays an important role in the decision, a view initiated by reference 2, which invoked stochasticity to explain Kourilsky's measurements (7) of the frequency of lysogenization in lambda. However, our analysis of Kourilsky's data (7, 8) (Fig. 1; see also Materials and Methods) shows that (i) when a single phage infects a bacterium (i.e., when the multiplicity of infection [MOI] is 1), it invariably goes lytic and (ii) when the MOI is 2, the decision is stochastic, with a slight preference toward lysogeny (this preference increases as the MOI increases). The stochasticity in the strategy for an MOI of 2 refers to the fact that seemingly identical infection events lead to different developmental paths. This inhomogeneity in the decision could reflect either true randomness (for example, due to the stochasticity of individual molecular events) or inhomogeneity across different infected cells (for example, the cell size could affect the decision [14]). The determinism in the phage decision for an MOI of 1 fits the game-theoretic expectation, as we will show, but then, the conundrum is this: why is the decision stochastic for an MOI of 2? In this paper, we argue that the shift from determinism at an MOI of 1 to stochasticity at an MOI of 2 is analogous to the shift from deterministic to stochastic strategies in single-player versus multiplayer games.