Moran Process Version of the Tug-of-War Model: Complex Behavior Revealed by Mathematical Analysis and Simulation Studies

In a series of publications McFarland and co-authors introduced the tug-of-warmodel of evolution of cancer cell populations. The model is explaining the joint effect ofrare advantageous and frequent slightly deleterious mutations, which may be identifiable withdriver and passenger mutations in cancer. In this paper, we put the Tug-of-War model inthe framework of a denumerable-type Moran process and use mathematics and simulationsto understand its behavior. The model is associated with a time-continuous Markov Chain(MC), with a generator that can be split into a sum of the drift and selection process partand of the mutation process part. Operator semigroup theory is then employed to prove thatthe MC does not explode, as well as to characterize a strong-drift limit version of the MCwhich displays instant fixation effect, which was an assumption in the original McFarlandsmodel. Mathematical results are fully confirmed by simulations of the complete and limitversions. They also visualize complex stochastic transients and genealogies of clones arising inthe model.

Martingales and the characteristic functions of absorption time on bipartite graphs

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald’s martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.

The Moran process on 2-chromatic graphs

Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where “properly colored” means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring.

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

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.