766 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
Emergence of clones in sexual populations
In sexual population, recombination reshuffles genetic variation and produces
novel combinations of existing alleles, while selection amplifies the fittest
genotypes in the population. If recombination is more rapid than selection,
populations consist of a diverse mixture of many genotypes, as is observed in
many populations. In the opposite regime, which is realized for example in the
facultatively sexual populations that outcross in only a fraction of
reproductive cycles, selection can amplify individual genotypes into large
clones. Such clones emerge when the fitness advantage of some of the genotypes
is large enough that they grow to a significant fraction of the population
despite being broken down by recombination. The occurrence of this "clonal
condensation" depends, in addition to the outcrossing rate, on the heritability
of fitness. Clonal condensation leads to a strong genetic heterogeneity of the
population which is not adequately described by traditional population genetics
measures, such as Linkage Disequilibrium. Here we point out the similarity
between clonal condensation and the freezing transition in the Random Energy
Model of spin glasses. Guided by this analogy we explicitly calculate the
probability, Y, that two individuals are genetically identical as a function of
the key parameters of the model. While Y is the analog of the spin-glass order
parameter, it is also closely related to rate of coalescence in population
genetics: Two individuals that are part of the same clone have a recent common
ancestor.Comment: revised versio
Asymptotic power law of moments in a random multiplicative process with weak additive noise
It is well known that a random multiplicative process with weak additive
noise generates a power-law probability distribution. It has recently been
recognized that this process exhibits another type of power law: the moment of
the stochastic variable scales as a function of the additive noise strength. We
clarify the mechanism for this power-law behavior of moments by treating a
simple Langevin-type model both approximately and exactly, and argue this
mechanism is universal. We also discuss the relevance of our findings to noisy
on-off intermittency and to singular spatio-temporal chaos recently observed in
systems of non-locally coupled elements.Comment: 11 pages, 9 figures, submitted to Phys. Rev.
A simple mathematical model of gradual Darwinian evolution: Emergence of a Gaussian trait distribution in adaptation along a fitness gradient
We consider a simple mathematical model of gradual Darwinian evolution in
continuous time and continuous trait space, due to intraspecific competition
for common resource in an asexually reproducing population in constant
environment, while far from evolutionary stable equilibrium. The model admits
exact analytical solution. In particular, Gaussian distribution of the trait
emerges from generic initial conditions.Comment: 21 pages, 2 figures, as accepted to J Math Biol 2013/03/1
The genetics of mate preferences in hybrids between two young and sympatric Lake Victoria cichlid species
The genetic architecture of mate preferences is likely to affect significant evolutionary processes, including speciation and hybridization. Here, we investigate laboratory hybrids between a pair of sympatric Lake Victoria cichlid fish species that appear to have recently evolved from a hybrid population between similar predecessor species. The species demonstrate strong assortative mating in the laboratory, associated with divergent male breeding coloration (red dorsum versus blue). We show in a common garden experiment, using DNA-based paternity testing, that the strong female mate preferences among males of the two species are fully recovered in a large fraction of their F2 hybrid generation. Individual hybrid females often demonstrated consistent preferences in multiple mate choice trials (more than or equal to five) across a year or more. This result suggests that female mate preference is influenced by relatively few major genes or genomic regions. These preferences were not changed by experience of a successful spawning event with a male of the non-preferred species in a no-choice single-male trial. We found no evidence for imprinting in the F2 hybrids, although the F1 hybrid females may have been imprinted on their mothers. We discuss this nearly Mendelian inheritance of consistent innate mate preferences in the context of speciation theory
Monte carlo simulations of parapatric speciation
Parapatric speciation is studied using an individual--based model with sexual
reproduction. We combine the theory of mutation accumulation for biological
ageing with an environmental selection pressure that varies according to the
individuals geographical positions and phenotypic traits. Fluctuations and
genetic diversity of large populations are crucial ingredients to model the
features of evolutionary branching and are intrinsic properties of the model.
Its implementation on a spatial lattice gives interesting insights into the
population dynamics of speciation on a geographical landscape and the
disruptive selection that leads to the divergence of phenotypes. Our results
suggest that assortative mating is not an obligatory ingredient to obtain
speciation in large populations at low gene flow.Comment: submitted to Phys.Rev.
How Gaussian competition leads to lumpy or uniform species distributions
A central model in theoretical ecology considers the competition of a range
of species for a broad spectrum of resources. Recent studies have shown that
essentially two different outcomes are possible. Either the species surviving
competition are more or less uniformly distributed over the resource spectrum,
or their distribution is 'lumped' (or 'clumped'), consisting of clusters of
species with similar resource use that are separated by gaps in resource space.
Which of these outcomes will occur crucially depends on the competition kernel,
which reflects the shape of the resource utilization pattern of the competing
species. Most models considered in the literature assume a Gaussian competition
kernel. This is unfortunate, since predictions based on such a Gaussian
assumption are not robust. In fact, Gaussian kernels are a border case
scenario, and slight deviations from this function can lead to either uniform
or lumped species distributions. Here we illustrate the non-robustness of the
Gaussian assumption by simulating different implementations of the standard
competition model with constant carrying capacity. In this scenario, lumped
species distributions can come about by secondary ecological or evolutionary
mechanisms or by details of the numerical implementation of the model. We
analyze the origin of this sensitivity and discuss it in the context of recent
applications of the model.Comment: 11 pages, 3 figures, revised versio
Using evolutionary demography to link life history theory, quantitative genetics and population ecology
1. There is a growing number of empirical reports of environmental change simultaneously influencing population dynamics, life history and quantitative characters. We do not have a well-developed understanding of links between the dynamics of these quantities
A stochastic differential equation analysis of cerebrospinal fluid dynamics
<p>Abstract</p> <p>Background</p> <p>Clinical measurements of intracranial pressure (ICP) over time show fluctuations around the deterministic time path predicted by a classic mathematical model in hydrocephalus research. Thus an important issue in mathematical research on hydrocephalus remains unaddressed--modeling the effect of noise on CSF dynamics. Our objective is to mathematically model the noise in the data.</p> <p>Methods</p> <p>The classic model relating the temporal evolution of ICP in pressure-volume studies to infusions is a nonlinear differential equation based on natural physical analogies between CSF dynamics and an electrical circuit. Brownian motion was incorporated into the differential equation describing CSF dynamics to obtain a nonlinear stochastic differential equation (SDE) that accommodates the fluctuations in ICP.</p> <p>Results</p> <p>The SDE is explicitly solved and the dynamic probabilities of exceeding critical levels of ICP under different clinical conditions are computed. A key finding is that the probabilities display strong threshold effects with respect to noise. Above the noise threshold, the probabilities are significantly influenced by the resistance to CSF outflow and the intensity of the noise.</p> <p>Conclusions</p> <p>Fluctuations in the CSF formation rate increase fluctuations in the ICP and they should be minimized to lower the patient's risk. The nonlinear SDE provides a scientific methodology for dynamic risk management of patients. The dynamic output of the SDE matches the noisy ICP data generated by the actual intracranial dynamics of patients better than the classic model used in prior research.</p
- …