Martingales and fixation probabilities of evolutionary graphs

Monk, Travis; Green, Peter; Paulin, Michael G.

doi:10.1098/rspa.2013.0730

Cited by 50 publications

(99 citation statements)

References 22 publications

(33 reference statements)

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“…3a, we consider Star graphs S N with N = 10, 20, …, 500. For the unweighted Star, there is an exact formula for fixation probability under both uniform and temperature initialization 32 . The values for weighted Star were computed by numerically solving large systems of linear equations.…”

Section: Resultsmentioning

confidence: 99%

Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory

et al. 2018

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Because of the intrinsic randomness of the evolutionary process, a mutant with a fitness advantage has some chance to be selected but no certainty. Any experiment that searches for advantageous mutants will lose many of them due to random drift. It is therefore of great interest to find population structures that improve the odds of advantageous mutants. Such structures are called amplifiers of natural selection: they increase the probability that advantageous mutants are selected. Arbitrarily strong amplifiers guarantee the selection of advantageous mutants, even for very small fitness advantage. Despite intensive research over the past decade, arbitrarily strong amplifiers have remained rare. Here we show how to construct a large variety of them. Our amplifiers are so simple that they could be useful in biotechnology, when optimizing biological molecules, or as a diagnostic tool, when searching for faster dividing cells or viruses. They could also occur in natural population structures.

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Section: Resultsmentioning

confidence: 99%

Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory

et al. 2018

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“…More precise formulas have been derived in various works 15, 30, 38 . As N → ∞, the fixation probability on the Star becomes ρ ( r , S ∞ ) = 1 − r −2 .…”

Section: Introductionmentioning

confidence: 99%

Amplification on Undirected Population Structures: Comets Beat Stars

Pavlogiannis

Tkadlec

Chatterjee

et al. 2017

Sci Rep

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The fixation probability is the probability that a new mutant introduced in a homogeneous population eventually takes over the entire population. The fixation probability is a fundamental quantity of natural selection, and known to depend on the population structure. Amplifiers of natural selection are population structures which increase the fixation probability of advantageous mutants, as compared to the baseline case of well-mixed populations. In this work we focus on symmetric population structures represented as undirected graphs. In the regime of undirected graphs, the strongest amplifier known has been the Star graph, and the existence of undirected graphs with stronger amplification properties has remained open for over a decade. In this work we present the Comet and Comet-swarm families of undirected graphs. We show that for a range of fitness values of the mutants, the Comet and Comet-swarm graphs have fixation probability strictly larger than the fixation probability of the Star graph, for fixed population size and at the limit of large populations, respectively.

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“…We can often 24 calculate absorption (i.e. fixation) probabilities [8,13] and/or times [17,18] from it. 25 Abraham Wald identified a powerful martingale for stochastic processes whose steps 26 are independent and identically distributed [17,18].…”

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confidence: 99%

“…Our CFs for the 35 number of mutant population size changes are novel, clean, and exact results. More 36 generally, Wald's methodology demonstrates an elegant approach to investigate classic 37 problems of evolutionary models [8,13,22]. 38 Results 39 Fig.…”

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Wald’s martingale and the Moran process

Monk

Schaik

2020

Preprint

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Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moran birth-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. The parameter dependence of the characteristic functions is explicit, so it is easy to explore their properties in parameter space. We also use them 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 we can quickly and tractably obtain absorption probabilities and times of evolutionary stochastic processes.February 18, 2020 1/21The Moran process is a probabilistic birth-death model of evolution. A mutant is introduced to an indigenous population, and we randomly choose organisms to live or die on subsequent time steps. Our goals are to calculate the probabilities that the mutant eventually dominates the population or goes extinct, and the distribution of time it requires to do so. The conditional distributions of time are difficult to obtain for the Moran process, so we consider a slightly different but related problem. We instead calculate the conditional distributions of the number of times that the mutant population size changes before it dominates the population or goes extinct. We use a martingale identified by Abraham Wald to obtain elegant and exact expressions for those distributions. We then use them to approximate conditional time distributions, and we show when that approximation is accurate. Our analysis outlines the basic concepts martingales and demonstrates why they are a formidable tool for studying probabilistic evolutionary models such as the Moran process. Introduction 1 The Moran birth-death process is a classic model of evolution [1, 2]. It considers a 2 population of two species that we call mutants and indigenous. On sequential time 3 steps, one organism is chosen to reproduce from either species. The newborn is the 4 same species as its parent, and it replaces another organism in the population uniformly 5 at random. The total population size then remains constant, but the numbers of mutant 6 and indigenous organisms in it are stochastic in time. The mutants will drive the 7 indigenous organisms to extinctio...

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Martingales and fixation probabilities of evolutionary graphs

Cited by 50 publications

References 22 publications

Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory

Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory

Amplification on Undirected Population Structures: Comets Beat Stars

Wald’s martingale and the Moran process

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