Wald’s martingale and the conditional distributions of absorption time in the Moran process

Abstract

Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth–death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth–death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.

Fixation probabilities in network structured meta-populations

“…(iii) For very high migration probability we calculate the fixation probability analytically using the Martingales introduced in 44,48 .…”

Martingales and the characteristic functions of absorption time on bipartite graphs

“…The fixation probability is then PrðS T ¼ aÞ ; a and the extinction probability is PrðS T ¼ bÞ ¼ 1 À a. We also want to find the conditional distributions PrðT ¼ tjS T ¼ aÞ 1 t¼0 and PrðT ¼ tjS T ¼ bÞ 1 t¼0 of T. It is very difficult to calculate PrðT ¼ tjS T ¼ aÞ 1 t¼0 and PrðT ¼ tjS T ¼ bÞ 1 t¼0 , even for simpler birthdeath processes like the fully connected, one-dimensional Moran process [16][17][18]. Instead, we consider the number of times that the mutant population size has changed upon absorption C T .…”

“…We repeat this birth-death selection procedure until the entire population comprises either mutants or residents. Our goals are to find the 'fixation probability' of the initial mutant population, and the (conditional) distribution of the number of time steps required to do so [16][17][18].…”

“…Very few general analytical results exist for fixation times on evolutionary graphs [ 27 ], even for graphs as simple as the complete bipartite graph. In a previous paper, we showed that we can apply Wald’s martingale [ 28 ] to the original, fully connected Moran process if we eliminate time steps where the mutant population size does not change [ 16 ]. We can therefore obtain tractable expressions for the full conditional characteristic functions (CCFs) of the number of times that the mutant population size changes before going extinct or fixing, i.e.…”

“…We will establish conditions for that approximation to be particularly accurate. Our analysis demonstrates that martingales are a powerful tool to solve fundamental problems in evolutionary graph theory, often within a few lines of mathematics [ 16 , 21 , 30 – 32 ].…”