3,175 research outputs found
The statistical mechanics of a polygenic characterunder stabilizing selection, mutation and drift
By exploiting an analogy between population genetics and statistical
mechanics, we study the evolution of a polygenic trait under stabilizing
selection, mutation, and genetic drift. This requires us to track only four
macroscopic variables, instead of the distribution of all the allele
frequencies that influence the trait. These macroscopic variables are the
expectations of: the trait mean and its square, the genetic variance, and of a
measure of heterozygosity, and are derived from a generating function that is
in turn derived by maximizing an entropy measure. These four macroscopics are
enough to accurately describe the dynamics of the trait mean and of its genetic
variance (and in principle of any other quantity). Unlike previous approaches
that were based on an infinite series of moments or cumulants, which had to be
truncated arbitrarily, our calculations provide a well-defined approximation
procedure. We apply the framework to abrupt and gradual changes in the optimum,
as well as to changes in the strength of stabilizing selection. Our
approximations are surprisingly accurate, even for systems with as few as 5
loci. We find that when the effects of drift are included, the expected genetic
variance is hardly altered by directional selection, even though it fluctuates
in any particular instance. We also find hysteresis, showing that even after
averaging over the microscopic variables, the macroscopic trajectories retain a
memory of the underlying genetic states.Comment: 35 pages, 8 figure
Behavior of different numerical schemes for population genetic drift problems
In this paper, we focus on numerical methods for the genetic drift problems,
which is governed by a degenerated convection-dominated parabolic equation. Due
to the degeneration and convection, Dirac singularities will always be
developed at boundary points as time evolves. In order to find a \emph{complete
solution} which should keep the conservation of total probability and
expectation, three different schemes based on finite volume methods are used to
solve the equation numerically: one is a upwind scheme, the other two are
different central schemes. We observed that all the methods are stable and can
keep the total probability, but have totally different long-time behaviors
concerning with the conservation of expectation. We prove that any extra
infinitesimal diffusion leads to a same artificial steady state. So upwind
scheme does not work due to its intrinsic numerical viscosity. We find one of
the central schemes introduces a numerical viscosity term too, which is beyond
the common understanding in the convection-diffusion community. Careful
analysis is presented to prove that the other central scheme does work. Our
study shows that the numerical methods should be carefully chosen and any
method with intrinsic numerical viscosity must be avoided.Comment: 17 pages, 8 figure
Numerical Complete Solution for Random Genetic Drift by Energetic Variational Approach
In this paper, we focus on numerical solutions for random genetic drift
problem, which is governed by a degenerated convection-dominated parabolic
equation. Due to the fixation phenomenon of genes, Dirac delta singularities
will develop at boundary points as time evolves. Based on an energetic
variational approach (EnVarA), a balance between the maximal dissipation
principle (MDP) and least action principle (LAP), we obtain the trajectory
equation. In turn, a numerical scheme is proposed using a convex splitting
technique, with the unique solvability (on a convex set) and the energy decay
property (in time) justified at a theoretical level. Numerical examples are
presented for cases of pure drift and drift with semi-selection. The remarkable
advantage of this method is its ability to catch the Dirac delta singularity
close to machine precision over any equidistant grid.Comment: 22 pages, 11 figures, 2 table
Evolution in range expansions with competition at rough boundaries.
When a biological population expands into new territory, genetic drift develops an enormous influence on evolution at the propagating front. In such range expansion processes, fluctuations in allele frequencies occur through stochastic spatial wandering of both genetic lineages and the boundaries between genetically segregated sectors. Laboratory experiments on microbial range expansions have shown that this stochastic wandering, transverse to the front, is superdiffusive due to the front's growing roughness, implying much faster loss of genetic diversity than predicted by simple flat front diffusive models. We study the evolutionary consequences of this superdiffusive wandering using two complementary numerical models of range expansions: the stepping stone model, and a new interpretation of the model of directed paths in random media, in the context of a roughening population front. Through these approaches we compute statistics for the times since common ancestry for pairs of individuals with a given spatial separation at the front, and we explore how environmental heterogeneities can locally suppress these superdiffusive fluctuations
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