427 research outputs found
Approximating Fixation Probabilities in the Generalized Moran Process
We consider the Moran process, as generalized by Lieberman, Hauert and Nowak
(Nature, 433:312--316, 2005). A population resides on the vertices of a finite,
connected, undirected graph and, at each time step, an individual is chosen at
random with probability proportional to its assigned 'fitness' value. It
reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly
at random, replacing the individual that was there. The initial population
consists of a single mutant of fitness placed uniformly at random, with
every other vertex occupied by an individual of fitness 1. The main quantities
of interest are the probabilities that the descendants of the initial mutant
come to occupy the whole graph (fixation) and that they die out (extinction);
almost surely, these are the only possibilities. In general, exact computation
of these quantities by standard Markov chain techniques requires solving a
system of linear equations of size exponential in the order of the graph so is
not feasible. We show that, with high probability, the number of steps needed
to reach fixation or extinction is bounded by a polynomial in the number of
vertices in the graph. This bound allows us to construct fully polynomial
randomized approximation schemes (FPRAS) for the probability of fixation (when
) and of extinction (for all ).Comment: updated to the final version, which appeared in Algorithmic
Approximating Fixation Probabilities in the Generalized Moran Process
We consider the Moran process, as generalized by Lieberman et al. (Nature 433:312â316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned âfitnessâ value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness r>0 placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when râ„1) and of extinction (for all r>0)
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
Universality of fixation probabilities in randomly structured populations
The stage of evolution is the population of reproducing individuals. The structure of the population is known to affect the dynamics and outcome of evolutionary processes, but analytical results for generic random structures have been lacking. The most general result so far, the isothermal theorem, assumes the propensity for change in each position is exactly the same, but realistic biological structures are always subject to variation and noise. We consider a finite population under constant selection whose structure is given by a variety of weighted, directed, random graphs; vertices represent individuals and edges interactions between individuals. By establishing a robustness result for the isothermal theorem and using large deviation estimates to understand the typical structure of random graphs, we prove that for a generalization of the ErdĆs-RĂ©nyi model, the fixation probability of an invading mutant is approximately the same as that of a mutant of equal fitness in a well-mixed population with high probability. Simulations of perturbed lattices, small-world networks, and scale-free networks behave similarly. We conjecture that the fixation probability in a well-mixed population, (1 â râ1)/(1 â rân), is universal: for many random graph models, the fixation probability approaches the above function uniformly as the graphs become large
Strong Amplifiers of Natural Selection: Proofs
We consider the modified Moran process on graphs to study the spread of
genetic and cultural mutations on structured populations. An initial mutant
arises either spontaneously (aka \emph{uniform initialization}), or during
reproduction (aka \emph{temperature initialization}) in a population of
individuals, and has a fixed fitness advantage over the residents of the
population. The fixation probability is the probability that the mutant takes
over the entire population. Graphs that ensure fixation probability of~1 in the
limit of infinite populations are called \emph{strong amplifiers}. Previously,
only a few examples of strong amplifiers were known for uniform initialization,
whereas no strong amplifiers were known for temperature initialization.
In this work, we study necessary and sufficient conditions for strong
amplification, and prove negative and positive results. We show that for
temperature initialization, graphs that are unweighted and/or self-loop-free
have fixation probability upper-bounded by , where is a
function linear in . Similarly, we show that for uniform initialization,
bounded-degree graphs that are unweighted and/or self-loop-free have fixation
probability upper-bounded by , where is the degree bound and
a function linear in . Our main positive result complements these
negative results, and is as follows: every family of undirected graphs with
(i)~self loops and (ii)~diameter bounded by , for some fixed
, can be assigned weights that makes it a strong amplifier, both
for uniform and temperature initialization
Fixation of a Deleterious Allele under Mutation Pressure and Finite Selection Intensity
The mean fixation time of a deleterious mutant allele is studied beyond the
diffusion approximation. As in Kimura's classical work [M. Kimura, Proc. Natl.
Acad. Sci. U.S.A. Vol.77, 522 (1980)], that was motivated by the problem of
fixation in the presence of amorphic or hypermorphic mutations, we consider a
diallelic model at a single locus comprising a wild-type A and a mutant allele
A' produced irreversibly from A at small uniform rate v. The relative fitnesses
of the mutant homozygotes A'A', mutant heterozygotes A'A and wild-type
homozygotes AA are 1-s, 1-h and 1, respectively, where it is assumed that v<<
s. Here, we adopt an approach based on the direct treatment of the underlying
Markov chain (birth-death process) obeyed by the allele frequency (whose
dynamics is prescribed by the Moran model), which allows to accurately account
for the effects of large fluctuations. After a general description of the
theory, we focus on the case of a deleterious mutant allele (i.e. s>0) and
discuss three situations: when the mutant is (i) completely dominant (s=h);
(ii) completely recessive (h=0), and (iii) semi-dominant (h=s/2). Our
theoretical predictions for the mean fixation time and the quasi-stationary
distribution of the mutant population in the coexistence state, are shown to be
in excellent agreement with numerical simulations. Furthermore, when s is
finite, we demonstrate that our results are superior to those of the diffusion
theory that is shown to be an accurate approximation only when N_e s^2 << 1,
where N_e is the effective population size.Comment: 26 pages, 5 figures. Accepted by the Journal of Theoretical Biolog
IST Austria Technical Report
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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