Supplementary information for"The fixation probability of rare mutators in finite asexual populations"

In this technical section, we construct the Kolmogorov equations which determine the dynamics of the probability distribution function P (x, t). In order to do this, we first calculate the transition probabilities between the various states x ∈ {0, Let T ↑ (x) denote the probability that the system makes a transition from the state with a fraction x of mutators to the state with a fraction x + 1 N of mutators. This may occur in one of the following two ways: 1. A mutator is selected for birth, a wild-type is selected for death, and no mutation occurs. 2. A mutator is selected for birth, a wild-type is selected for death, a beneficial mutation occurs, and this mutation is part of the fraction 1 − s that is destined for loss by random drift. Computing these probabilities in the order listed, we arrive at the following expression for T ↑ (x) T ↑ (x) r = x(1 − x)(1 − µ +) + x(1 − x)µ + α e (1 − s) = x(1 − x) [1 − µ + (1 − α e (1 − s))] (1) The factor of r on the LHS is just the birth probability per time-step which, according to A1-A3 is common to all members of the population and will soon be scaled out. In a similar way we calculate T ↓ (x), the probability that the system makes a transition from the state with a fraction x mutators to the state with a fraction x − 1 N mutators. In fact, we may simply interchange x ↔ 1 − x and µ + ↔ µ − in Eq.1 which results in T ↓ (x) r = x(1 − x) [1 − µ − (1 − α e (1 − s))] (2) Within the framework of A1-A3, the population may also make large, non-local transitions to the " absorbing " x = 0 and x = 1 states if the mutator or wild-type strains produce an advantageous mutant which is marked for fixation. This