A Paradoxical Evolutionary Mechanism in Stochastically Switching Environments

Organisms with environmental sensors that guide survival are considered more likely to be favored by natural selection if they possess more accurate sensors. In this paper, we develop a theoretical model which shows that under certain conditions of environmental stochasticity, selection actually favors sensors of lower accuracy. An analogy between this counter-intuitive phenomenon and the well-known Parrondo’s paradox is suggested.

The results in this paper are corroborated by both numerical simulations and analytical derivations. Detailed analytical derivations are given below for completeness.

Proof A
Let X n be the random variable that represents the population vector after n generations. X n can be expressed as where each S i is an independent random variable representing the switching matrix for generation i: We can express X n recursively in terms of X n−1 as Since X n−1 and S n are independent random variables, it follows that: where S is the expected switching matrix, as defined in Equation 20. We proceed by induction on n and Equation 21 follows (restated here):

Proof B
For the following analysis, we define for convenience the overall growth matrix for a single generation, T = MSG, as well as the complementary probabilities q = 1 − p and t i = 1 − s i . T can be expressed in terms of A and B from Equation 6 and G 1 and G 2 from Equation 15: Proof.
From this, we can derive Equation 29, restated here: where K can be expressed as Proof.
When mutation is absent, we have G 1 = E 1 I L and G 2 = E 2 I L . Given this, we can derive Equations 30 and 31: Proof. Since A, B, G 1 and G 2 are all diagonal matrices when mutation is absent, so are C, D, and K = C + D. Recall that the ith diagonal entries of A and B respectively are s i and t i = 1 − s i . Computing the ith diagonal entry of K, κ i = C ii + D ii , we have . When mutation is present and the number of sensor levels L = 3, K can be expressed as where we define the overall growth and mutation rates and where α * i are β * i are defined using the averages and the standard deviations of the mutation rates Proof. We define the matrices G and G σ as: We can thus write G 1 as G + G σ and G 2 as G − G σ . It follows from K = C + D that

Proof C
Substituting p = 0.5 into the general expression for K gives us Equation 33. As shown earlier, when p = 0.5, the sub-populations y 1 and y 3 grow at equal rates. We can now also see why y 1 grows more slowly than y 3 if p < 0.5 and why y 1 grows more quickly than y 3 if p > 0.5, even if y 1 = y 3 initially. Expanding the matrix notation in Equation 29 using K from Equation S6 gives us the following recurrence relations: From Equations S7 to S9, it is clear that when p < 0.5, κ i , α * i and β * i all increase with increasing s i , whereas when p > 0.5, κ i , α * i and β * i decrease with increasing s i . Since s 3 > s 1 , it follows from the recurrence relations that even if y 3 and y 1 start off being equal, y 3 will be greater than y 1 in the next generation if p < 0.5, whereas the opposite will occur if p > 0.5.