The decay of genetic variability in geographically structured populations. II

The ultimate rate of approach to equilibrium in the infinite stepping-stone model is calculated. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. Let f(t, x) be the probability that two homologous genes separated by the vector x in generation t are the same allele. It is supposed that $\text{f(}\text{0, x}\text{) = O(x}{}^{\mathrm{?2?\eta }}\text{), η > 0,}\text{as}\text{x ≡ ¦}\text{x}\text{¦ → ∞}$. In the absence of mutation, f(t, x) tends to unity at the rate $\text{t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ in one dimension and (ln t)^?1 in two dimensions. Thus, the loss of genetic variability in two dimensions is so slow that evolutionary forces not considered in this model would supervene long before a two-dimensional natural population became completely homogeneous. If the mutation rate, u, is not zero f(t, x) asymptotically approaches equilibrium at the rate $\text{(1 ? u)}{}^{\mathrm{2t}}\text{t}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ in one dimension and (1 ? u)^2tt^?1(lnt)^?2 in two dimensions. Integral formulas are presented for the spatial dependence of the deviation of f(t, x) from its stationary value as t → ∞, and for large separations this dependence is shown to be (const + x) in one dimension and (const + ln x) in two dimensions. All the results are the same for the Malécot model of a continuously distributed population provided the number of individuals per colony is replaced by the population density. The relatively slow algebraic and logarithmic rates of convergence for the infinite habitat contrast sharply with the exponential one for a finite habitat.     

On the evolution of behavioral reproductive isolation: The Wallace effect

A model of speciation is constructed in which two equally fit but cross-sterile interbreeding species W and C meet in a zone of overlap Z. Within the type W population, which is held at a constant proportion γ of Z by migration from the outside, a mutation arises which causes a partial aversion to individuals of type C. Formulas for the expected frequency of the various possible mating pairs in Z are derived, and an approximation for small aversion is found. The increase in frequency of the mutant gene in the type W population in Z is the found to be approximately pq(1 ? γ)[r₁(1 ? 2p) + r₂p], where p is the relative proportion of the mutant gene in the type W population, q = 1 ? p, and r₁ and r₂ are the aversion probabilities for mutant heterozygotes and homozygotes, respectively. The resulting selective pressure towards behavioral reproductive isolation is discussed.     

The Distribution of Mutant Alleles in a Subdivided Population

The results are presented from a simulation study of the spatial distribution of mutant alleles in a subdivided population. Statistical measures of the spatial pattern are defined in such a way that the same quantities could be measured in a geographic survey of allele frequencies in natural populations. Two types of quantities are discussed in this paper: (1) the occupancy distribution provides information on the presence or absence of the mutant in different numbers of demes; and (2) the conditional frequency distribution provides information about the extent of local differentiation when the mutant is present in different numbers of demes. Properties of these distributions are found for different types of natural selection acting on the mutant. Some results are presented for the same statistical measures based on samples of individuals from a fraction of the total number of demes. The simulation results for intermediate levels of the migration rates are compared with analytic results obtained on the limits of high and low migration rates. The main conclusion is that these measures of the spatial distribution of mutants in a subdivided population have simple properties that could provide a new perspective on data from natural populations.     

Natural selection and fitness entropy in a density-regulated population

The entropy H(p_o,p*) of a population with the initial allele frequency p_o given the equilibrium polymorphic frequency p* has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium, N(t), with the size trajectories of populations not initially at p*,N(t), but which do eventually converge to a common equilibrium allele frequency and equilibrium density, N*. The following experimentally testable hyopthesis was established. The total area defined by the difference between the trajectories of N(t) and N(t) as they converge to N* is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle Tribolium castaneum.     

A model of the complex response of Staphylococcus aureus to methicillin

It is widely accepted that β-lactam antimicrobials cause cell death through a mechanism that interferes with cell wall synthesis. Later studies have also revealed that β-lactams modify the autolysis function (the natural process of self-exfoliation of the cell wall) of cells. The dynamic equilibrium between growth and autolysis is perturbed by the presence of the antimicrobial. Studies with Staphylococcus aureus to determine the minimum inhibitory concentration (MIC) have revealed complex responses to methicillin exposure. The organism exhibits four qualitatively different responses: homogeneous sensitivity, homogeneous resistance, heterogeneous resistance and the so-called ‘Eagle-effect’. A mathematical model is presented that links antimicrobial action on the molecular level with the overall response of the cell population to antimicrobial exposure. The cell population is modeled as a probability density function F(x,t) that depends on cell wall thickness x and time t. The function F(x,t) is the solution to a Fokker-Planck equation. The fixed point solutions are perturbed by the antimicrobial load and the advection of F(x,t) depends on the rates of cell wall synthesis, autolysis and the antimicrobial concentration. Solutions of the Fokker-Planck model are presented for all four qualitative responses of S. aureus to methicillin exposure.     

Probabilistic Model of Microbial Cell Growth,Division, and Mortality

After a short time interval of length δt during microbial growth, an individual cell can be found to be divided with probability P_d(t)δt, dead with probability P_m(t)δt, or alive but undivided with the probability 1 − [P_d(t) + P_m(t)]δt, where t is time, P_d(t) expresses the probability of division for an individual cell per unit of time, and P_m(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat''s properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the population''s size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a “shoulder,” and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With P_d(t) and P_m(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.     

Effects of the Mode of Competition on Stationary Size Distribution in Plant Populations

The effects of the mode of between-individual competition onthe stationary size distribution in uneven-aged and continuallyreproducing tree populations were investigated theoretically,based on the diffusion and continuity equation models. The B(t,x) function, which gives a measure proportional to total leafarea of individual trees larger than dbh (trunk diameter atbreast height) x at time t, expresses the effect of one-sidedcompetition on the subject individual of dbh x. The effect oftwo-sided competition on individuals of any size is equallygiven by B(t, x₀), i.e. total leaf area of the stand (or LAI),where x₀ is the minimum dbh. The following two points, whichwere shown by Kohyama (1991) from results of model simulation,were theoretically derived: (1) the effect of the diffusionterm [D(t, x) function] of the diffusion model on the stationarysize distribution is small for the range of parameter valuesobtained empirically for the warm-temperate and tropical rainforests; (2) the shape of the stationary size distribution underone-sided competition is less changed by a change in the seedlingsupply rate than under two-sided competition. The followingtwo points were further shown theoretically: (3) one-sided competitionis likely to lead to a monotonically decreasing size distributionof dbh with the peak at x = x₀, which is a characteristic ofthe rain forest; (4) the slope of size distribution of dbh steepensas one-sided competition intensifies. Relationships betweenthe mode of competition and the stability of population structure,and general functional forms of one-sided and two-sided competitionare discussed. Diffusion model, one-sided competition, stability of population structure, stationary size distribution, two-sided competition     

Index measures for assessing the mode of inheritance of continuously distributed traits: I, theory and justifications

A class of indices that may be applied to quantitative data on nuclear families and that can help to assess degrees of mode of inheritance is developed. Given phenotype values of spouses x⁽¹⁾ and x⁽²⁾ and offspring y, the deviation of an offspring value from the midparent is $\text{¦y ?}\text{1}\text{2}\text{(x}{}^{\mathrm{(1)}}\text{+ x}{}^{\mathrm{(2)}}\text{)¦}$, and those from the separate parents are $\text{¦y ? x}{}^{\mathrm{(1)}}\text{¦}$ and $\text{¦y ? x}{}^{\mathrm{(2)}}\text{¦}$. The indices called major-gene indices (MGI) investigated are functions of the deviations from midparental values compared to corresponding symmetric functions of the deviations from separate parents. Major-gene indices exceeding 1 may indicate some extent of major-gene inheritance, whereas an MGI less than 1 is suggestive of relatively more polygenic inheritance. Superposition of assortative mating and environmental effects will tend not to shift the MGI greater than 1 for polygenic inheritance, nor will they shift the MGI less than 1 for major-gene factors. The reliance on the proposed indices is reinforced on the basis of a hierarchy of representative models of monogenic and multifactorial inheritance. Extensions of the method to deal with multigenerational pedigrees are briefly discussed.     

Bounds on the total population for species governed by reaction-diffusion equations in arbitrary two-dimensional regions

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ _R θ(x ₁,x ₂,t) dx ₁ dx ₂ of a biological species with an area-density distribution function θ=θ(x ₁,x ₂,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇²θ +fθ −gθ ⁿ⁺¹ whereD (>0),n (>0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(_{x 1},x ₂, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn>1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ _R θ(x ₁,x ₂, 0)² dx ₁ dx ₂ sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.     

The rate of allelism of lethal genes in a geographically structured population

The expected rate of allelism, E[I(x)], of lethal genes between two colonies with distance x in a structured population is studied by using one- and two-dimensional stepping-stone models. It is shown that E[I(x)] depends on the magnitude of selection in heterozygous condition (h), the rate of migration among adjacent colonies (m), the number of loci which produce lethal mutations (n) and the effective population size of each colony (N).——E[I(x)] always decreases with distance x. The rate of decrease is affected strongly by the magnitude of m. The rate of decrease is faster when m is small. E[I(x)] also decreases with increasing N and n. The effect of h on E[I(x)] is somewhat complicated. However, E[I(0)] is always smaller when h is small than when it is large.——For large x, the following approximate formulae may be obtained: (see PDF) where q and Var (q) are the mean and the variance of gene frequencies in each colony, t is approximated as t=h, (see PDF), -h for the partially recessive, completely recessive, and overdominant lethals, respectively, and C₀ is a function of m and t. It is clear that E[I(x)] declines exponentially with x in a one-dimensional habitat. The decrease E[I(x)] is faster in a two-dimensional habitat than in a one-dimensional habitat. The present result is applied to some of the existing data and the estimation of population parameters is also discussed.     

Fixation probabilities and effective population numbers in diploid populations with overlapping generations

A finite diploid population, observed at times t = 0, 1, 2,…, is studied. An individual is said to be in age group i at time t if its age is between i and i + 1 units at that time, where i ? 1. It is assumed that the number of individuals in a particular age-sex class is the same for every t and that the probability that a male offspring was produced by a mating of a male in age group i and a female in age group j is p_ij^m (with a corresponding probability p^f_ij for a female offspring), regardless of when the individual is born. The probability of ultimate fixation of an allele A₁ and the inbreeding effective number, for large populations, is calculated under the further assumptions that A₁ is neutral and that mating is random, given the ages of the mates.     

New approach to statistical description of fluctuating particle fluxes

The probability density functions (PDFs) of the increments of fluctuating particle fluxes are investigated. It is found that the PDFs have heavy power-law tails decreasing as x ^{?α ? 1} at x → ∞. This makes it possible to describe these PDFs in terms of fractionally stable distributions (FSDs) q(x; α, β, θ, λ). The parameters α, β, γ, and λ were estimated statistically using as an example the time samples of fluctuating particle fluxes measured in the edge plasma of the L-2M stellarator. Two series of fluctuating fluxes measured before and after boronization of the vacuum chamber were processed. It is shown that the increments of fluctuating fluxes are well described by DSDs. The effect of boronization on the parameters of FSDs is analyzed. An algorithm for statistically estimating the FSD parameters and a procedure for processing experimental data are described.     

Chaotic Dynamics of Neuromuscular System Parameters and the Problems of the Evolution of Complexity

The evolution rate v(t) varies among diverse biosystems, but a general theory can be formulated when the dynamics of the biosystem stater x = x(t) = (x₁, x₂, x _m ) ^T is considered in the m-dimensional space of states. A mathematical approach is proposed for evaluating such processes and describes the processes in terms of particular chaos of the statistical distribution functions f(x). In the case of complex multicomponent systems with a high dimension number m (m ?1) of the phase space of states, we propose using pairwise comparison matrices of samples x(t) when homeostasis is constant and calculating the parameters of quasiattractors. The Glensdorff–Prigogine thermodynamic approach to estimating evolution is inefficient in assessing the third-type systems, while it is applicable and the Prigogine theorem works at the level of molecular systems. Alterations in the state of the human neuromuscular system were found to lead to chaotic changes in the statistical functions f(x) in tremor recording samples, while quasiattractor parameters demonstrate a certain regularity.     

A Markov Process of Gene Frequency Change in a Geographically Structured Population

A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.     

Estimating weight�Clength relationships without individual weight data: an application to the American lobster (Homarus americanus) fishery of Long Island Sound

The estimation of weight?Clength relationship of fish species requires having data on individual weight and length. However, individual weight data are often not available because they are too expensive or not feasible to gather and the relationship cannot be explicitly estimated. Yet, in this paper I develop a simple methodology that allows me to estimate a weight?Clength relationship when only aggregate weight data are available. To show its usefulness, the methodology is applied to the American lobster (Homarus americanus) population of Long Island Sound. Results indicate the existence of isometric growth for American lobsters in this geographical location: W = 0.000924L ^2.9619. The estimated relationship is used to predict individual weight of lobsters which are then used to construct biomass indexes for three size classes of lobsters for the time period 1987?C2006. This analysis suggests that is not necessary to invest efforts in collecting individual weight data to be able to construct meaningful indicators of fish population.     

Chromosome numbers,characterization of chromosomal pairing during meiosis,origin and natural propagation in polyploid cytotypes (4x, 5x and 6x) of Agrimonia eupatoria L. (Rosaceae) in northwest Himalayas (India)

Despite the presence of intraspecific polyploidy (2x, 4x, 5x and 6x) in Agrimonia eupatoria, origin of these cytotypes has never been addressed adequately. The aim of the present study was to record the original chromosome counts and characterize chromosomal pairing during meiosis and microsporogenesis in the 5x cytotype, and discussing the hypothesis regarding the possible origin of polyploid cytotypes (4x, 5x and 6x) in the species. The geographical distribution pattern of cytotypes in the Indian Himalayas and elsewhere has also been analyzed. The present meiotic analysis revealed three chromosomes counts, the tetraploid (2n?=?4x?=?56), the pentaploid (2n?=?5x?=?70) and the hexaploid (2n?=?6x?=?84) cytotypes based on x?=?14. Meiotic course was perfectly normal in the 4x and 6x cytotypes resulting into high pollen fertility (94–100 %). Meiotic course in the imbalanced 5x cytotype has been found to be irregular characterized by the presence of high frequency of univalents at diakinesis and metaphase-I. Abnormal meiotic course contributed towards high pollen sterility (74–88 %). Even the apparently fertile/stained pollen grains were of irregular shape and of heterogeneous sizes. Meiotic behaviour of the 5x cytotype is like typical of allopolyploid. Individuals of 5x cytotype did not produce seeds and propagate vegetatively (root suckers) while 4x and 6x cytotypes exploited sexual (seeds) as well as vegetative means for propagation. Chromosomal pairing in pentaploid cytotype is like typical of an allopolyploid and we assume that it might have originated owing to natural inter-cytotype hybridization between 4x and 6x cytotypes in a mixed population. Analysis of geographical distribution pattern of cytotypes shows that Indian Himalayas represent the most cytotype-diverse region for A. eupatoria with the existence of all the four cytotypes (2x, 4x, 5x, 6x). This shows the dynamic nature of the species at chromosomal level in this part of the world.     

Global mutations and local mutations have very different effects on evolution, illustrated by mixed strategies of asymmetric binary games

We study the evolutionary effect of rare mutations causing global changes in traits. We consider asymmetric binary games between two players. The first player takes two alternative options with probability x and 1−x; and the second player takes options with probability y and 1−y. Due to natural selection and recurrent mutation, the population evolves to have broad distributions of x and y. We analyze three cases showing qualitatively different dynamics, exemplified by (1) vigilance-intrusion game, (2) asymmetric hawk-dove game and (3) cleaner-client game. We found that the evolutionary outcome is strongly dependent upon the distribution of mutants’ traits, more than the mutation rates. For example in the vigilance-intrusion game, the evolutionary dynamics show a perpetual stable oscillation if mutants are always close to the parent (local-mutation mode), whilst the population converges to a stable equilibrium distribution if mutants can be quite different from the parent (global-mutation mode), even for extremely low mutation rate. When common local mutations and rare global mutations occur simultaneously, the evolutionary outcome is controlled by the latter.