# Stochastic slowdown

### Jul;82(1 Pt 1):011925, 2010

Consider the following simple setup. There are only two states A and B. The whole population (of size $N$) currently resides in state B. An individual can move from state B to state A by acquiring a mutation but not vice versa. For the dynamics we make use of the Moran process (Moran, 1962). Each time step of the Moran process consists of a birth even and a death event. For the whole population to move from state B to state A means that all individuals currently in state B have to acquire the mutation. The mutation rate is given by $\mu$. We consider the limit of small mutation rates such that a mutant reaches its fate (either fixation or extinction) before a new one arises. There are no back mutations from A to B. Thus in our variant of the Moran process in each time step, one of the following three things can happen, The Moran process. To describe the dynamics of a population of finite size we resort to stochastic processes. The Moran process is a birth-death process. That is, in time each step a birth event and a death event occur. The population size is maintained constant by a random death. If the population consists of say $i$ A individuals then in one time step the number of A can increase by one (with probability $T_i^+$) or decrease by one (with probability $T_i^-$) or stay the same (with probability $1- T_i^+ - T_i^-$).
• Number of individuals in state $\mathbf{A}$ increases by 1. The number of individuals in state $A$ increases by $1$ with probability,

The increase can happen in two ways. The first term gives the probability when an $A$ state individual is chosen for reproduction and a $B$ for death. The second term gives the probability when a $B$ is chosen for reproduction, but mutates to $A$ and again a $B$ is chosen for death.

• Number of individuals in state $\mathbf{A}$ decreases by 1. The number in state $A$ decreases only if a $B$ is chosen for reproduction and it does not mutate and an $A$ is chosen for death. This event happens with probability, $\begin{equation} T_i^- = \frac{N-i}{N} (1-\mu) \frac{i}{N}. \end{equation}$
• No change in either state. This happens with probability $1-T_i^+ - T_i^-$.

Hence in each reproductive step there is a bias towards producing an individual with a mutation (see Fig. 2.5). Intuitively we expect that the average conditional time required for the fixation of a single $A$ individual in a population of $N-1$ $B$ individuals should be smaller than in a balanced process in which there is no bias. However, we observe that for a small bias the average conditional fixation time is larger than that of a balanced process (without bias). Bias towards A. Throughout the process there is a frequency dependent bias towards moving to an all A state. It diminishes in strength as the system gets closer to an all A state. Yet the time time required to get to the final state is greater than without such a bias.

To go to the heart of this counterintuitive observation we must dissect out the quantity of interest, the conditional fixation time. Conditional fixation time means given that the mutant does fix, what is the time required for the population to reach a state where all individuals are mutants. For the Moran process we can exactly calculate the conditional fixation time from a formula which is well known for such a birth-death process (Moran, 1962; Goel and Richter-Dyn, 1974; Ewens, 1979; Landauer and Büttiker, 1987; Antal and Scheuring, 2006; Traulsen and Hauert, 2009) (see Fig. 2.6). In the following publication we observe what happens to this quantity of interest as we introduce a small bias to the system. Even a simpler process than directed mutations can exhibit such counter-intuitive behaviour. All it requires is a slight asymmetry in the transition probabilities ($T_i^+$ and $T_i^-$). Effect of increasing bias on components of conditional fixation time $\mathbf{\tau_1^{N}}$. The expression gives us the exact conditional fixation time $\tau_1^{N}$ for the Moran process beginning with a single mutant. If we introduce a frequency dependent bias such that we have $T_i^+ > T_i^-$, then we see that the ratio of transition probabilities and the inverse of $T_l^+$ decrease. On the contrary, the fixation probability, $\phi_l^N$, increases. The effect of this tug of war is an increase in the conditional fixation time for a small bias.

P. A. P. Moran. The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford, 1962.

N. S. Goel and N. Richter-Dyn. Stochastic Models in Biology. Academic Press, New York, 1974.

W. J. Ewens. Mathematical Population Genetics. Springer, Berlin, 1979.

R. Landauer and M. Büttiker. Diffusive traversal time: Effective area in magnetically induced interference. Phys. Rev. B, 36:6255–6260, 1987.

T. Antal and I. Scheuring. Fixation of strategies for an evolutionary game in finite populations. Bull. Math. Biol., 68:1923–1944, 2006.

A. Traulsen and C. Hauert. Stochastic evolutionary game dynamics. In H.-G. Schuster, editor, Reviews of nonlinear dynamics and complexity, pages 25–61. Wiley-VCH, 2009.