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.

FIG. 1.

Red circles show the fraction of bacteria that entered lysogeny as a function of the API (overall phage/bacterium ratio) in Kourilsky's experiments (7, 8, 9). Solid lines are theoretical estimates using different functions for Q(m), which is the probability of going lytic as a function of the MOI (m) (see Materials and Methods). As shown by the red curve, the best fits ± estimated 95% intervals (Table 1) for Q(m) are 0.004 ± 0.001, 0.70 ± 0.04, and 0.99 ± 0.08 for m values of 1, 2, and 3, respectively. Even a small amount of stochasticity in the decision for an MOI of 1 is inconsistent with the data: for the orange curve the Q(m) values are 0.05, 0.70, and 0.99 for m values of 1, 2, and 3, respectively.

FIG. 2.

Phage game payoff matrix. p₁ is the probability that all free phage die, and p₂ is the probability that each lysogenized bacterium and all its descendants die before the phage escapes. q is the probability that the infected bacteria go lytic. “Cost” represents the probability of phage extinction in a given situation and, for simplicity, is set to either 1 or 0.

FIG. 3.

(A) Umbrella game for one person, with 60% chance of rain and 40% chance of sunshine. The best strategy is to always carry an umbrella; all stochastic strategies result in more mistakes. (B) Umbrella game for two people with identical strategies. As there is a collective reward for carrying an umbrella, if the chance of rain is significant, the ideal situation is that in which one person brings an umbrella, leaving the other person free from carrying one. This is possible only with the use of a stochastic strategy, because both individuals are using the same strategy. If the chance of rain is below a certain threshold (p < 1/3), however, then the deterministic strategy of never carrying an umbrella is best.

FIG. 4.

(A) Optimal strategies for the phage game for one player. The q_min strategy, which minimizes the chance of phage extinction, is deterministic: always go lytic (q_min = 1) when p₁ is <p₂, and always go lysogenic (q_min = 0) when p₁ is >p₂. All stochastic strategies result in a greater chance of extinction. (B) Optimal strategy for the phage game with n identical players. Blue, n = 2; green, n = 4; yellow, n = 10; red, n = 100. As there is a collective reward for going lytic, if the chance of free phage dying is low, the ideal situation is that in which one infected bacterium is lysed, leaving the rest to go lysogenic. This is possible only with the use of a stochastic strategy, because each player is using the same strategy. If the danger of going lytic is above a certain threshold (p₁ > p₂), however, then the best strategy is to deterministically go lysogenic.