Parsimony and the Fisher–Wright debate

In the past five years, there have been a series of papers in the journal Evolution debating the relative significance of two theories of evolution, a neo-Fisherian and a neo-Wrightian theory, where the neo-Fisherians make explicit appeal to parsimony. My aim in this paper is to determine how we can make sense of such an appeal. One interpretation of parsimony takes it that a theory that contains fewer entities or processes, (however we demarcate these) is more parsimonious. On the account that I defend here, parsimony is a ‘local’ virtue. Scientists’ appeals to parsimony are not necessarily an appeal to a theory’s simplicity in the sense of it’s positing fewer mechanisms. Rather, parsimony may be proxy for greater probability or likelihood. I argue that the neo-Fisherians appeal is best understood on this interpretation. And indeed, if we interpret parsimony as either prior probability or likelihood, then we can make better sense of Coyne et al. argument that Wright’s three phase process operates relatively infrequently.     

The free energy method and the Wright–Fisher model with 2 alleles

We systematically investigate the Wright–Fisher model of population genetics with the free energy functional formalism of statistical mechanics and in the light of recent mathematical work on the connection between Fokker–Planck equations and free energy functionals. In statistical physics, entropy increases, or equivalently, free energy decreases, and the asymptotic state is given by a Gibbs-type distribution. This also works for the Wright–Fisher model when rewritten in divergence to identify the correct free energy functional. We not only recover the known results about the stationary distribution, that is, the asymptotic equilibrium state of the model, in the presence of positive mutation rates and possibly also selection, but can also provide detailed formulae for the rate of convergence towards that stationary distribution. In the present paper, the method is illustrated for the simplest case only, that of two alleles.     

The Quasispecies for the Wright–Fisher Model

We consider the classical Wright–Fisher model of population genetics. We prove the existence of an error threshold for the mutation probability per nucleotide, below which a quasispecies is formed. We show a new phenomenon, specific to a finite population model, namely the existence of a population threshold: to ensure the stability of the quasispecies, the population size has to be at least of the same order as the genome length. We derive an explicit formula describing the quasispecies.     

A mathematical model to predict the tissue response to parthenin — An allelochemical

Parthenin — a sesquiterpene lactone fromParthenium hysterophorus L. is an allelochemical that prevents the germination ofPhaseolus aureus Roxb. cv. ML-5 seeds. The response of the seed has been attributed to the inhibition of the respiratory electron transport ability of its embryo. It has been shown to depend directly not only upon concentration of parthenin, but simultaneously on the duration of exposure of the seeds to the chemical as well. A strong correlation exists between the quantum of the response and the product of the period of exposure and the concentration of parthenin. In order to predict the maximum possible germination ability of the seed exposed for a given period to a given concentration of parthenin, an expression X = 10000 Y³ / [e^Y/0.31-1] was formulated from the equation X = AY³/[e^Y/Y0-1], where X represents values of maximum respiratory activity, Y represents the product of concentration and time in units mg cn^-3 h, A represents a dimensional constant. The trend and nature of response that is calculated on the basis of formulation coincides with that of measured response through spectrophotometry.     

The Variance of Identity-by-Descent Sharing in the Wright–Fisher Model

Widespread sharing of long, identical-by-descent (IBD) genetic segments is a hallmark of populations that have experienced recent genetic drift. Detection of these IBD segments has recently become feasible, enabling a wide range of applications from phasing and imputation to demographic inference. Here, we study the distribution of IBD sharing in the Wright–Fisher model. Specifically, using coalescent theory, we calculate the variance of the total sharing between random pairs of individuals. We then investigate the cohort-averaged sharing: the average total sharing between one individual and the rest of the cohort. We find that for large cohorts, the cohort-averaged sharing is distributed approximately normally. Surprisingly, the variance of this distribution does not vanish even for large cohorts, implying the existence of “hypersharing” individuals. The presence of such individuals has consequences for the design of sequencing studies, since, if they are selected for whole-genome sequencing, a larger fraction of the cohort can be subsequently imputed. We calculate the expected gain in power of imputation by IBD and subsequently in power to detect an association, when individuals are either randomly selected or specifically chosen to be the hypersharing individuals. Using our framework, we also compute the variance of an estimator of the population size that is based on the mean IBD sharing and the variance in the sharing between inbred siblings. Finally, we study IBD sharing in an admixture pulse model and show that in the Ashkenazi Jewish population the admixture fraction is correlated with the cohort-averaged sharing.IN isolated populations, even purported unrelated individuals often share genetic material that is identical-by-descent (IBD). Traditionally, the term IBD sharing referred to coancestry at a single site (or autozygosity, in the case of a diploid individual) and was widely investigated as a measure of the degree of inbreeding in a population (Hartl and Clark 2006). Recent years have brought dramatic increases in the quantity and density of available genetic data and, together with new computational tools, these data have enabled the detection of IBD sharing of entire genomic segments (see, e.g., Purcell et al. 2007; Kong et al. 2008; Albrechtsen et al. 2009; Gusev et al. 2009; Browning and Browning 2011; Carr et al. 2011; Brown et al. 2012). The availability of IBD detection tools that are efficient enough to detect shared segments in large cohorts has resulted in numerous applications, from demographic inference (Davison et al. 2009; Palamara et al. 2012) and characterization of populations (Gusev et al. 2012a) to selection detection (Albrechtsen et al. 2010), relatedness detection and pedigree reconstruction (Huff et al. 2011; Kirkpatrick et al. 2011; Stevens et al. 2011; Henn et al. 2012), prioritization of individuals for sequencing (Gusev et al. 2012b), inference of HLA type (Setty et al. 2011), detection of haplotypes associated with a disease or a trait (Akula et al. 2011; Gusev et al. 2011; Browning and Thompson 2012), imputation (Uricchio et al. 2012), and phasing (Palin et al. 2011).Recently, some of us used coalescent theory to calculate several theoretical quantities of IBD sharing under a number of demographic histories. Then, shared segments were detected in real populations, and their demographic histories were inferred (Palamara et al. 2012). Here, we expand upon Palamara et al. (2012) to investigate additional aspects of the stochastic variation in IBD sharing. Specifically, we provide a precise calculation for the variance of the total sharing in the Wright–Fisher model, either between a random pair of individuals or between one individual and all others in the cohort.Understanding the variation in IBD sharing is an important theoretical characterization of the Wright–Fisher model, and additionally, it has several practical applications. For example, it can be used to calculate the variance of an estimator of the population size that is based on the sharing between random pairs. In a different domain, the variance in IBD sharing is needed to accurately assess strategies for sequencing study design, specifically, in prioritization of individuals to be sequenced. This is because imputation strategies use IBD sharing between sequenced individuals and genotyped, not-sequenced individuals to increase the number of effective sequences analyzed in the association study (Palin et al. 2011; Gusev et al. 2012b; Uricchio et al. 2012).In the remainder of this article, we first review the derivation of the mean fraction of the genome shared between two individuals (Palamara et al. 2012). We then calculate the variance of this quantity, using coalescent theory with recombination. We provide a number of approximations, one of which results in a surprisingly simple expression, which is then generalized to a variable population size and to the sharing of segments in a length range. We also numerically investigate the pairwise sharing distribution and provide an approximate fit. We then turn to the average total sharing between each individual and the entire cohort. We show that this quantity, which we term the cohort-averaged sharing, is approximately normally distributed, but is much wider than naively expected, implying the existence of hypersharing individuals. We consider several applications: the number of individuals needed to be sequenced to achieve a certain imputation power and the implications to disease mapping, inference of the population size based on the total sharing, and the variance of the sharing between siblings. We finally calculate the mean and the variance of the sharing in an admixture pulse model and show numerically that admixture results in a broader than expected cohort-averaged sharing. Therefore, large variance of the cohort-averaged sharing can indicate admixture. In the Ashkenazi Jewish population, we show that the cohort-averaged sharing is strongly anticorrelated with the fraction of European ancestry.     

How to calculate the non-synonymous to synonymous rate ratio of protein-coding genes under the Fisher–Wright mutation–selection framework

First principles of population genetics are used to obtain formulae relating the non-synonymous to synonymous substitution rate ratio to the selection coefficients acting at codon sites in protein-coding genes. Two theoretical cases are discussed and two examples from real data (a chloroplast gene and a virus polymerase) are given. The formulae give much insight into the dynamics of non-synonymous substitutions and may inform the development of methods to detect adaptive evolution.     

The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright–Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection–diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.     

Investigations on the genetics and population genetics of the β2 polymorphism

Summary The results of studies on 49 families with 107 children and various populations of Caucasoid, Negroid and Mongoloid origin concerning the genetics and population genetics of the ₂-glycoprotein I polymorphism are reported. In general the genetical model proposed by Cleve (1968) is confirmed: two autosomal alleles Bg^N and Bg^D controlling the phenotypes Bg N-N, Bg N-D and Bg D-D. However, divergences from this model were found in two families. They indicate the assumption of non-genetic factors influencing the phenotype expression rather than more complicated genetical control mechanisms. Within Caucasoid populations phenotype and gene frequencies show almost a homogeneous distribution. This racial stock is striking due to a significant higher ₂-glycoprotein I concentration in serum as compared to Negroids and Mongoloids. In connection with this, these racial stocks differ obviously in the gene frequencies: Caucasoids Bg^N=0.937, Negroids=0.742, Mongoloids =0.780; resp. Bg^D=0.063, 0.258, 0.220.Supported by the Deutsche Forschungsgemeinschaft.D 77.     

The stationary distribution of a sample from the Wright–Fisher diffusion model with general small mutation rates

Journal of Mathematical Biology - The stationary distribution of a sample taken from a Wright–Fisher diffusion with general small mutation rates is found using a coalescent approach. The...     

Quantitative approximation of the discrete Moran process by a Wright–Fisher diffusion

Journal of Mathematical Biology - The Moran discrete process and the Wright–Fisher model are the most popular models in population genetics. The Wright–Fisher diffusion is commonly used...     

A dual process for the coupled Wright–Fisher diffusion

Journal of Mathematical Biology - Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is...     

The basic structure and dynamics of cell membranes: An update of the Singer–Nicolson model

The fluid mosaic model of Singer and Nicolson (1972) is a commonly used representation of the cell membrane structure and dynamics. However a number of features, the result of four decades of research, must be incorporated to obtain a valid, contemporary version of the model. Among the novel aspects to be considered are: (i) the high density of proteins in the bilayer, that makes the bilayer a molecularly “crowded” space, with important physiological consequences; (ii) the proteins that bind the membranes on a temporary basis, thus establishing a continuum between the purely soluble proteins, never in contact with membranes, and those who cannot exist unless bilayer-bound; (iii) the progress in our knowledge of lipid phases, the putative presence of non-lamellar intermediates in membranes, and the role of membrane curvature and its relation to lipid geometry, (iv) the existence of lateral heterogeneity (domain formation) in cell membranes, including the transient microdomains known as rafts, and (v) the possibility of transient and localized transbilayer (flip-flop) lipid motion. This article is part of a Special Issue entitled: Membrane Structure and Function: Relevance in the Cell's Physiology, Pathology and Therapy.     

Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics

An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.     

Hunter–Gatherer population structure and the evolution of contingent cooperation

Unlike other vertebrates, humans cooperate in large groups with unrelated individuals. Many authors have argued that the evolution of such cooperation has resulted from reciprocity and other forms of contingent cooperation. This argument is not well supported by existing theory. The theory of contingent cooperation in pairs is well developed: reciprocating strategies are stable when common, and can increase when rare as long as population structure leads to modest levels of relatedness. In larger groups, however, it is not clear whether contingent cooperation can increase when rare. Existing work suggests that contingent strategies cannot increase unless relatedness is high, but depends on unrealistic assumptions about the effects of population structure. Here we develop and analyze a model incorporating a two level population structure that captures important features of human hunter–gatherer societies. This model suggests that previous work underestimates the range of conditions under which contingent cooperation can evolve, but also predicts that cooperation will not evolve unless (1) social groups are small, and (2) the relatedness within ethnolinguistic groups is at the high end of the range of empirical estimates.     

On the Relation Between the Eigen Model and the Asexual Wright–Fisher Model

We show that the Eigen model and the asexual Wright–Fisher model can be obtained as two different limit cases of a single stochastic model. This result is used to enlighten the mathematical similarities and differences among these two models.     

Plague dynamics and population genetics of the desert locust: can turnover during recession maintain population genetic structure?

The desert locust (Schistocerca gregaria) undergoes crowding-induced phase transformation from solitary form to gregarious form. The transformations involves changes in behaviour, colour, development, morphometry, fecundity and endocrine physiology. Recession populations of the desert locust exist primarily in the solitary phase as small populations in patchy environments and are prone to extinction because of climatic events. Significant genetic differentiation among recession populations along the Red Sea coast of Eritrea was previously reported. It was hypothesized that despite the mixing effect of recurrent swarms, metapopulation dynamics could have produced genetic divergence among these highly scattered recession populations. A Monte Carlo simulation of the population dynamics of the desert locust in a metapopulation setting, with a realistic range of parameter values clearly demonstrated that this is possible. Population growth was represented by a discrete-time logistic equation. The duration of recessions and swarms was sampled from normal distributions whose means and standard deviations were varied based on reported estimates. An average recession duration of 10 +/- 3 generations and swarm periods half as long but almost twice as variable produced a partitioning of the total genetic variance most similar to that in the empirical study. In conventional metapopulation analysis, whether turnover leads to increased or reduced divergence is dependent on the number of colonists relative to the number of recurrent migrants, and on whether the colonists arise from a single patch or many patches. In the case of locusts, the stochastic boom and bust cycle is the overriding factor. Divergence between patches during recession due to founder effect and recurrent drift is balanced by the high rate of mixing during plagues.