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A model is presented in which alleles at a number of loci combine to influence the value of a quantitative trait that is subject to stabilizing selection. Mutations can occur to alleles at the loci under consideration. Some of these mutations will tend to increase the value of the trait, while others will tend to decrease it. In contrast to most previous models, we allow the mean effect of mutations to be nonzero. This means that, on average, mutations can have a bias, such that they tend to either increase or decrease the value of the trait. We find, unsurprisingly, that biased mutation moves the equilibrium mean value of the quantitative trait in the direction of the bias. What is more surprising is the behavior of the deviation of the equilibrium mean value of the trait from its optimal value. This has a nonmonotonic dependence on the degree of bias, so that increasing the degree of bias can actually bring the mean phenotype closer to the optimal phenotype. Furthermore, there is a definite maximum to the extent to which biased mutation can cause a difference between the mean phenotype and the optimum. For plausible parameter values, this maximum-possible difference is small. Typically, quantitative-genetics models assume an unconstrained model of mutation, where the expected difference in effect between a parental allele and a mutant allele is independent of the current state of the parental allele. Our results show that models of this sort can easily lead to biologically implausible consequences when mutations are biased. In particular, unconstrained mutation typically leads to a continual increase or decrease in the mean allelic effects at all trait-controlling loci. Thus at each of these loci, the mean allelic effect eventually becomes extreme. This suggests that some of the models of mutation most commonly used in quantitative genetics should be modified so as to introduce genetic constraints.

It has been observed repeatedly that the distribution of new mutations of a quantitative trait has a kurtosis (a statistical measure of the distribution's shape) that is systematically larger than that of a normal distribution. Here we suggest that rather than being a property of individual loci that control the trait, the enhanced kurtosis is highly likely to be an emergent property that arises directly from the loci being mutationally nonequivalent. We present a method of incorporating nonequivalent loci into quantitative genetic modeling and give an approximate relation between the kurtosis of the mutant distribution and the degree of mutational nonequivalence of loci. We go on to ask whether incorporating the experimentally observed kurtosis through nonequivalent loci, rather than at locus level, affects any biologically important conclusions of quantitative genetic modeling. Concentrating on the maintenance of quantitative genetic variation by mutation-selection balance, we conclude that typically nonequivalent loci yield a genetic variance that is of order 10% smaller than that obtained from the previous approaches. For large populations, when the kurtosis is large, the genetic variance may be

We use entropy to characterize intrinsic ageing properties of the human brain. Analysis of fMRI data from a large dataset of individuals, using resting state BOLD signals, demonstrated that a functional entropy associated with brain activity increases with age. During an average lifespan, the entropy, which was calculated from a population of individuals, increased by approximately 0.1 bits, due to correlations in BOLD activity becoming more widely distributed. We attribute this to the number of excitatory neurons and the excitatory conductance decreasing with age. Incorporating these properties into a computational model leads to quantitatively similar results to the fMRI data. Our dataset involved males and females and we found significant differences between them. The entropy of males at birth was lower than that of females. However, the entropies of the two sexes increase at different rates, and intersect at approximately 50 years; after this age, males have a larger entropy. [ABSTRACT FROM AUTHOR]

A fundamental result of population genetics states that a new mutation, at an unlinked neutral locus in a randomly mating diploid population, has a mean time of fixation of ~4/Ve generations, where Ne is the effective population size. This result is based on an assumption of fixed population size, which does not universally hold in natural populations. Here, we analyze such neutral fixations in populations of changing size within the framework of the diffusion approximation. General expressions are derived for the mean and variance of the fixation time in changing populations. Some explicit results are given for two cases: (i) the effective population size undergoes a sudden change, representing a sudden population expansion or a sudden bottleneck; (ii) the effective population changes linearly for a limited period of time and then remains constant. Additionally, a lower bound for the mean time of fixation is obtained for an effective population size that increases with time, and this is applied to exponentially growing populations. The results obtained in this work show, among other things, that for populations that increase in size, the mean time of fixation can be enhanced, sometimes substantially so, over 4Ne,o generations, where Ne,o is the effective population size at the time the mutation arises. Such an enhancement is associated with (i) an increased probability of neutral polymorphism in a population and (ii) an enhanced persistence of high-frequency neutral variation, which is the variation most likely to be observed. [ABSTRACT FROM AUTHOR]

The fixation probability is determined when population size and selection change over time and differs from Kimura's result, with long-term implications for a population. It is found that changes in population size are not equivalent to the corresponding changes in selection and can result in less drift than anticipated. [ABSTRACT FROM AUTHOR]

Abstract: A result is derived, in the form of a sum, for the time-dependent probability of fixation of an unlinked neutral locus. The result captures many of the key features of the probability of fixation in a highly compact form. For ‘small’ times () a single term of the sum accurately determines the time-dependent probability of fixation. This is in contrast to the well-known result of Kimura, which requires the contribution of many terms in a different sum, for ‘small’ times. Going beyond small times, an approximation is derived for the time-dependent probability of fixation which applies for all times when the initial relative allele frequency is small. [Copyright &y& Elsevier]

Abstract: We investigate the detailed connection between the Wright–Fisher model of random genetic drift and the diffusion approximation, under the assumption that selection and drift are weak and so cause small changes over a single generation. A representation of the mathematics underlying the Wright–Fisher model is introduced which allows the connection to be made with the corresponding mathematics underlying the diffusion approximation. Two ‘hybrid’ models are also introduced which lie ‘between’ the Wright–Fisher model and the diffusion approximation. In model 1 the relative allele frequency takes discrete values while time is continuous; in model 2 time is discrete and relative allele frequency is continuous. While both hybrid models appear to have a similar status and the same level of plausibility, the different nature of time and frequency in the two models leads to significant mathematical differences. Model 2 is mathematically inconsistent and has to be ruled out as being meaningful. Model 1 is used to clarify the content of Kimura''s solution of the diffusion equation, which is shown to have the natural interpretation as describing only those populations where alleles are segregating. By contrast the Wright–Fisher model and the solution of the diffusion equation of McKane and Waxman cover populations of all categories, namely populations where alleles segregate, are lost, or fix. [Copyright &y& Elsevier]

Abstract: This work presents a new approach to Muller''s ratchet, where Haigh''s model is approximately mapped into a simpler model that describes the behaviour of a population after a click of the ratchet, i.e., after loss of what was the fittest class. This new model predicts the distribution of times to the next click of the ratchet and is equivalent to a Wright–Fisher model for a population of haploid asexual individuals with one locus and two alleles. Within this model, the fittest members of a population correspond to carriers of one allele, while all other individuals have suboptimal fitness and are represented as carriers of the other allele. In this way, all suboptimal fitness individuals are amalgamated into a single “mutant” class. The approach presented here has some limitations and the potential for improvement. However, it does lead to results for the rate of the ratchet that, over a wide range of parameters, are accurate within one order of magnitude of simulation results. This contrasts with existing approaches, which are designed for only one or other of the two different parameter regimes known for the ratchet and are more accurate only in the parameter regime they were designed for. Numerical results are presented for the mean time between clicks of the ratchet for (i) the Wright–Fisher model, (ii) a diffusion approximation of this model and (iii) individually based simulations of a full model. The diffusion approximation is validated over a wide range of parameters by its close agreement with the Wright–Fisher model. The present work predicts that: (a) the time between clicks of the ratchet is insensitive to the value of the selection coefficient when the genomic mutation rate is large compared with the selection coefficient against a deleterious mutation, (b) the time interval between clicks of the ratchet has, approximately, an exponential distribution (or its discrete analogue). It is thus possible to determine the variance in times between clicks, given the expected time between clicks. Evidence for both (a) and (b) is seen in simulations. [Copyright &y& Elsevier]

Abstract: In this paper we investigate some of the mathematical properties of meiotic recombination. Working within the framework of a genetic model with n loci, where alleles are possible at each locus, we find that the proportion of all possible diploid parental genotypes that can produce a particular haploid gamete is . We show that this proportion connects recombination with a fractal geometry of dimension . The fractal dimension of a geometric object manifests itself when it is measured at increasingly smaller length scales. Decreasing the length scale of a geometric object is found to be directly analogous, in a genetics problem, to specifying a multilocus haplotype at a larger number of loci, and it is here that the fractal dimension reveals itself. [Copyright &y& Elsevier]

The safety of the blood supply depends on measures to protect not only the transfusion recipient but also the blood donor. Donor selection criteria have been voluntarily adopted or enforced through regulation in different countries, but review of practices in different blood centers reveals wide disparity in the current approaches. Such variability in practice suggests that the criteria for the protection of donor are often arbitrary or reflect deeply engrained precautionary practices and exposes the inherent uncertainty about the best way to minimize risk to the donor. Certain selection criteria introduced years ago have become dogma in some countries but were never subjected to systematic study and persist despite available evidence that the measures do not measurably improve donor safety. Current efforts to define a rational, evidence-based approach are crucial to eliminate practices that lead to the unnecessary deferral of large numbers of blood donors without improving the safety of the donation process. Future prospects to improve the safety of the donation process rest with hemovigilance initiatives to monitor the effectiveness of interventions to minimize the risks to blood donors. © 2009 Elsevier Inc. All rights reserved. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Wright Fisher model for a finite population of diploid sexual organisms where selection acts at a locus with multiple alleles. The mathematical description of a such a model requires vectors and matrices of a multidimensional nature, and hence has a considerable level of complexity. In the present work we avoid this complexity by introducing a simple mathematical transformation. This yields a description of the model in terms of ordinary vectors and ordinary matrices, thereby allowing standard linear algebra techniques to be directly employed. The new description yields a common mathematical representation of the Wright Fisher model that applies for arbitrary numbers of alleles. Within this framework, it is shown how the dynamics decomposes into component parts that are responsible for the different possible transitions of segregating and fixed populations, thereby allowing a clearer understanding of the population dynamics. This decomposition allows expressions to be directly derived for the mean time of fixation, the mean time of segregation (i.e., the sojourn time) and the probability of fixation. Numerical methods are discussed for the evaluation of these quantities. [Copyright &y& Elsevier]

Abstract: We investigate a genetic model of a large population of sexual organisms in a changing environment. The organisms are subject to stabilising selection on a quantitative trait, with environmental change causing the fitness optimum to move. When the fitness optimum moves slowly, adaptation to the changing environment occurs by means of reasonably well-separated substitutions at the loci controlling the trait. In this way, the trait generally tracks the moving optimum, but in such a case, the population may exhibit periods of time where the mean trait value overshoots the moving optimal trait value, thereby exhibiting an apparent anticipation of selection. The mechanism underlying this phenomenon is determined from consideration of a simpler model that correctly captures the observed dynamical behaviour. We note that very slow rates of changes of traits are seen in the fossil record and the present work may be relevant to this topic. [Copyright &y& Elsevier]

Abstract: The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the “weight” associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability. [Copyright &y& Elsevier]