scholarly journals Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

2018 ◽  
Author(s):  
David Hathcock ◽  
Steven H. Strogatz
Keyword(s):  
Complete Graph ◽  
Time Distribution ◽  
Evolutionary Dynamics ◽  
Structured Populations ◽  
Relative Fitness ◽  
Death Process ◽  
Fixation Time ◽  
Large Network ◽  
Moran Process ◽  
Fitness Dependence

Evolutionary graph theory models the effects of natural selection and random drift on structured populations of mutant and non-mutant individuals. Recent studies have shown that fixation times, which determine the rate of evolution, often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, each of which admits an exact solution in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a weighted convolution of two Gumbel distributions, with a weight depending on the relative fitness. When fitness is neutral, however, the Moran process has a highly skewed fixation-time distribution on both the complete graph and the ring. In this sense, the case of neutral fitness is singular. Even on these simple network structures, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Applications of our methods to a multi-fitness Moran model, times to partial fixation, and evolution on random networks are discussed.

scholarly journals Counterintuitive properties of the fixation time in network-structured populations

2014 ◽  
Vol 11 (99) ◽  
pp. 20140606 ◽  
Author(s):  
Laura Hindersin ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Transition Probabilities ◽  
Structured Populations ◽  
Death Process ◽  
Fixation Time ◽  
Fixation Probability ◽  
Mutant Type ◽  
Starting Conditions ◽  
Fixation Probabilities ◽  
Regular Networks

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth–death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.

scholarly journals Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

2019 ◽  
Vol 100 (1) ◽  
Author(s):  
David Hathcock ◽  
Steven H. Strogatz
Keyword(s):  
Time Distribution ◽  
Evolutionary Dynamics ◽  
Fixation Time ◽  
Fitness Dependence

scholarly journals On the fixation probability of superstars

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130193 ◽  
Author(s):  
Josep Díaz ◽  
Leslie Ann Goldberg ◽  
George B. Mertzios ◽  
David Richerby ◽  
Maria Serna ◽  
...  
Keyword(s):  
Computer Algebra ◽  
Evolutionary Dynamics ◽  
Process Models ◽  
Relative Fitness ◽  
Genetic Mutations ◽  
Fixation Probability ◽  
Mutant Population ◽  
Moran Process ◽  
Limiting Behaviour ◽  
Symbolic Matrix

The Moran process models the spread of genetic mutations through populations. A mutant with relative fitness r is introduced and the system evolves, either reaching fixation (an all-mutant population) or extinction (no mutants). In a widely cited paper, Lieberman et al. (2005 Evolutionary dynamics on graphs. Nature 433 , 312–316) generalize the model to populations on the vertices of graphs. They describe a class of graphs (‘superstars’), with a parameter k and state that the fixation probability tends to 1− r − k as the graphs get larger: we show that this is untrue as stated. Specifically, for k =5, we show that the fixation probability (in the limit, as graphs get larger) cannot exceed 1−1/ j ( r ), where j ( r )= Θ ( r 4 ), contrary to the claimed result. Our proof is fully rigorous, though we use a computer algebra package to invert a 31×31 symbolic matrix. We do believe the qualitative claim of Lieberman et al. —that superstar fixation probability tends to 1 as k increases—and that it can probably be proved similarly to their sketch. We were able to run larger simulations than the ones they presented. Simulations on graphs of around 40 000 vertices do not support their claim but these graphs might be too small to exhibit the limiting behaviour.

scholarly journals Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150334 ◽  
Author(s):  
Karan Pattni ◽  
Mark Broom ◽  
Jan Rychtář ◽  
Lara J. Silvers
Keyword(s):  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Process ◽  
Structured Populations ◽  
Fixation Probability ◽  
Finite Populations ◽  
Moran Model ◽  
Moran Process ◽  
Evolutionary Graph Theory ◽  
Associated Matrix

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

scholarly journals Changes in the distribution of fitness effects and adaptive mutational spectra following a single first step towards adaptation

2020 ◽  
Author(s):  
Dimitra Aggeli ◽  
Yuping Li ◽  
Gavin Sherlock
Keyword(s):  
Evolutionary Dynamics ◽  
Common Ancestor ◽  
Single Point ◽  
Relative Fitness ◽  
Single Point Mutation ◽  
Historical Contingency ◽  
Mutational Spectra ◽  
Fitness Effects ◽  
Mutational Step ◽  
Diminishing Returns

AbstractThe fitness effects of random mutations are contingent upon the genetic and environmental contexts in which they occur, and this contributes to the unpredictability of evolutionary outcomes at the molecular level. Despite this unpredictability, the rate of adaptation in homogeneous environments tends to decrease over evolutionary time, due to diminishing returns epistasis, causing relative fitness gains to be predictable over the long term. Here, we studied the extent of diminishing returns epistasis and the changes in the adaptive mutational spectra after yeast populations have already taken their first adaptive mutational step. We used three distinct adaptive clones that arose under identical conditions from a common ancestor, from which they diverge by a single point mutation, to found populations that we further evolved. We followed the evolutionary dynamics of these populations by lineage tracking and determined adaptive outcomes using fitness assays and whole genome sequencing. We found compelling evidence for diminishing returns: fitness gains during the 2nd step of adaptation are smaller than those of the 1st step, due to a compressed distribution of fitness effects in the 2nd step. We also found strong evidence for historical contingency at the genic level: the beneficial mutational spectra of the 2nd-step adapted genotypes differ with respect to their ancestor and to each other, despite the fact that the three founders’ 1st-step mutations provided their fitness gains due to similar phenotypic improvements. While some targets of selection in the second step are shared with those seen in the common ancestor, other targets appear to be contingent on the specific first step mutation, with more phenotypically similar founding clones having more similar adaptive mutational spectra. Finally, we found that disruptive mutations, such as nonsense and frameshift, were much more common in the first step of adaptation, contributing an additional way that both diminishing returns and historical contingency are evident during 2nd step adaptation.

scholarly journals Cucumber mosaic virus satellite RNAs that induce similar symptoms in melon plants show large differences in fitness

2011 ◽  
Vol 92 (8) ◽  
pp. 1930-1938 ◽  
Author(s):  
Mónica Betancourt ◽  
Aurora Fraile ◽  
Fernando García-Arenal
Keyword(s):  
Cucumber Mosaic Virus ◽  
Mosaic Virus ◽  
Evolutionary Dynamics ◽  
Plant Biomass ◽  
Aphid Transmission ◽  
Helper Virus ◽  
Relative Fitness ◽  
Single Infection ◽  
Trade Offs ◽  
Satellite Rnas

Two groups of Cucumber mosaic virus (CMV) satellite RNAs (satRNAs), necrogenic and non-necrogenic, can be differentiated according to the symptoms they cause in tomato plants, a host in which they also differ in fitness. In most other CMV hosts these CMV-satRNA cause similar symptoms. Here, we analyse whether they differ in traits determining their relative fitness in melon plants, in which the two groups of CMV-satRNAs cause similar symptoms. For this, ten necrogenic and ten non-necrogenic field satRNA genotypes were assayed with Fny-CMV as a helper virus. Neither type of CMV-satRNA modified Fny-CMV symptoms, and both types increased Fny-CMV virulence similarly, as measured by decreases in plant biomass and lifespan. Necrogenic and non-necrogenic satRNAs differed in their ability to multiply in melon tissues; necrogenic satRNAs accumulated to higher levels both in single infection and in competition with non-necrogenic satRNAs. Indeed, multiplication of some non-necrogenic satRNAs was undetectable. Transmission between hosts by aphids was less efficient for necrogenic satRNAs as a consequence of a more severe reduction of CMV accumulation in leaves. The effect of CMV accumulation on aphid transmission was not compensated for by differences in satRNA encapsidation efficiency or transmissibility to CMV progeny. Thus, necrogenic and non-necrogenic satRNAs differ in their relative fitness in melon, and trade-offs are apparent between the within-host and between-host components of satRNA fitness. Hence, CMV-satRNAs could have different evolutionary dynamics in CMV host-plant species in which they do not differ in pathogenicity.

scholarly journals Calculating Evolutionary Dynamics in Structured Populations

2009 ◽  
Vol 5 (12) ◽  
pp. e1000615 ◽  
Author(s):  
Charles G. Nathanson ◽  
Corina E. Tarnita ◽  
Martin A. Nowak
Keyword(s):  
Evolutionary Dynamics ◽  
Structured Populations

scholarly journals The effect of population structure on the rate of evolution

2013 ◽  
Vol 280 (1762) ◽  
pp. 20130211 ◽  
Author(s):  
Marcus Frean ◽  
Paul B. Rainey ◽  
Arne Traulsen
Keyword(s):  
Population Structure ◽  
Evolutionary Process ◽  
Ecological Factors ◽  
Structured Populations ◽  
Fixation Time ◽  
Fixation Probability ◽  
Star Graphs ◽  
Rate Of Evolution ◽  
Population Structures ◽  
Evolution Proceeds

Ecological factors exert a range of effects on the dynamics of the evolutionary process. A particularly marked effect comes from population structure, which can affect the probability that new mutations reach fixation. Our interest is in population structures, such as those depicted by ‘star graphs’, that amplify the effects of selection by further increasing the fixation probability of advantageous mutants and decreasing the fixation probability of disadvantageous mutants. The fact that star graphs increase the fixation probability of beneficial mutations has lead to the conclusion that evolution proceeds more rapidly in star-structured populations, compared with mixed (unstructured) populations. Here, we show that the effects of population structure on the rate of evolution are more complex and subtle than previously recognized and draw attention to the importance of fixation time. By comparing population structures that amplify selection with other population structures, both analytically and numerically, we show that evolution can slow down substantially even in populations where selection is amplified.

scholarly journals Fixation time in evolutionary graphs: a mean field approach

2018 ◽  
Author(s):  
Mahdi Hajihashemi ◽  
Keivan Aghababaei Samani
Keyword(s):  
Analytical Method ◽  
Transition Matrix ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Fixation Time ◽  
Field Approach ◽  
Moran Process ◽  
Structured Population ◽  
Population Structures

A novel analytical method is proposed for calculation of average fixation and extinction times of mutants in a general structured population of two types of species. The method is based on Markov chains and uses a mean field approximation to calculate the corresponding transition matrix. Analytical results are compared with the results of simulation of the Moran process on a number of population structures.

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