Selection intensity against deleterious mutations in RNA secondary structures and rate of compensatory nucleotide substitutions

A two-locus model of reversible mutations with compensatory fitness interactions is presented; single mutations are assumed to be deleterious but neutral in appropriate combinations. The expectation of the time of compensatory nucleotide substitutions is calculated analytically for the case of tight linkage between sites. It is shown that selection increases the substitution time dramatically when selection intensity Ns > 1, where N is the diploid population size and s the selection coefficient. Computer simulations demonstrate that recombination increases the substitution time, but the effect of recombination is small when selection is weak. The amount of linkage disequilibrium generated in the process of compensatory substitution is also investigated. It is shown that significant linkage disequilibrium is expected to be rare in natural populations. The model is applied to the mRNA secondary structure of the bicoid 3' untranslated region of Drosophila. It is concluded that average selection intensity Ns against single deleterious mutations is not likely to be much larger than 1.     

Modeling of selection processes with age-dependent birth and death rates

Two simple models for the competition and selection in age-dependent populations are developed and analyzed mathematically. Following Eigen, competition is introduced by the condition of constant overall-number of the population. In the first model this condition is satisfied by regulation of a dilution flux and in the second case by regulation of a food density. The calculation of maximal fitness is given explicitly for both situations. It is shown that fitness depends in a complicated way on the age-dependence of the birth and death rates. Therefore species have to develop special aging strategies in order to survive in a population under selection pressure. In general, early reproduction is of advantage and increases fitness.     

On the study of two interacting populations one of which acts independently of the other

Many ecological and biological systems can be studied in terms of a bivariate stochastic branching process, {X ₁ (t), X ₂ (t)}, each of whose components (or populations) varies in magnitude according to the laws of a generalized birth-death process. Of particular interest is such a model in which the birth and death rates of the first population,X ₁, are constant while those of the second population,X ₂, exhibit a functional dependence upon the magnitude of the first. It is shown, first, that the existence of the stochastic mean of a birth death process implies the existence of all higher moments. The values of all the factorial moments of such a process are then determined. The moments of the dependent population of the bivariate process are given in terms of its expectation and the joint probability density function of the process is determined. It is possible, therefore, to use Bayesian techniques to infer conclusions about the independent population, given information about the variation of the dependent one.     

Equilibria in systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

First evidence for adoption in California sea lions

Demographic parameters such as birth and death rates determine the persistence of populations. Understanding the mechanisms that influence these rates is essential to developing effective management strategies. Alloparental behavior, or the care of non-filial young, has been documented in many species and has been shown to influence offspring survival. However, the role of alloparental behavior in maintaining population viability has not been previously studied. Here, we provide the first evidence for adoption in California sea lions and show that adoption potentially works to maintain a high survival rate of young and may ultimately contribute to population persistence. Alloparental behavior should have a positive effect on the population growth rate when the sum of the effects on fitness for the alloparent and beneficiary is positive.     

Rate of Gene Silencing at Duplicate Loci: A Theoretical Study and Interpretation of Data from Tetraploid Fishes

A large-scale simulation has been conducted on the rate of gene loss at duplicate loci under irreversible mutation. It is found that tight linkage does not provide a strong sheltering effect, as thought by previous authors; indeed, the mean loss time for the case of tight linkage is of the same order of magnitude as that for no linkage, as long as Nu is not much larger than 1, where N is the effective population size and u the mutation rate. When Nu is 0.01 or less, the two loci behave almost as neutral loci, regardless of linkage, and the mean loss time is about only half the mean extinction time for a neutral allele under irreversible mutation. However, the former becomes two or more times larger than the latter when Nu ≥ 1.——In the simulation, the sojourn times in the frequency intervals (0, 0.01) and (0.99, 1) and the time for the frequency of the null allele to reach 0.99 at one of the two loci have also been recorded. The results show that the population is monomorphic for the normal allele most of the time if Nu ≤ 0.01, but polymorphic for the null and the normal alleles most of the time if Nu ≥ 0.1.——The distribution of the frequency of the null allele in an equilibrium tetraploid population has been studied analytically. The present results have been applied to interpret data from some fish groups that are of tetraploid origin, and a model for explaining the slow rate of gene loss in these fishes is proposed.     

Multilocus dynamics under haploid selection

 A general haploid selection model with arbitrary number of multiallelic loci and arbitrary linkage distribution is considered. The population is supposed to be panmictic. A dynamically equivalent diploid selection model is introduced. There is a position effect in this model if the original haploid selection is not multiplicative. If haploid selection is additive then the fundamental theorem is established even with an estimate for the change in the mean fitness. On this basis exponential convergence to an equilibrium is proved. As rule, the limit states are single-gamete ones. If, moreover, linkage is tight, then the single-gamete state with maximal fitness attracts the population for almost all initial states. Received 27 November 1995; received in revised form 17 January 1996     

The Evolution of Multilocus Systems under Weak Selection

The evolution of multilocus systems under weak selection is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. The genotypic fitnesses may depend on the gametic frequencies and time. The results hold for s << c(min), where s and c(min) denote the selection intensity and the smallest two-locus recombination frequency, respectively. After an evolutionarily short time of t(1) ~ (ln s)/ln(1 - c(min)) generations, all the multilocus linkage disequilibria are of the order of s [i.e., O(s) as s -> 0], and then the population evolves approximately as if it were in linkage equilibrium, the error in the gametic frequencies being O(s). Suppose the explicit time dependence (if any) of the genotypic fitnesses is O(s(2)). Then after a time t(2) ~ 2t(1), the linkage disequilibria are nearly constant, their rate of change being O(s(2)). Furthermore, with an error of O(s(2)), each linkage disequilibrium is proportional to the corresponding epistatic deviation for the interaction of additive effects on fitness. If the genotypic fitnesses change no faster than at the rate O(s(3)), then the single-generation change in the mean fitness is ΔW = W(-1)V(g) + O(s(3)), where V(g) designates the genic (or additive genetic) variance in fitness. The mean of a character with genotypic values whose single-generation change does not exceed O(s(2)) evolves at the rate ΔZ = W(-1)C(g) + O(s(2)), where C(g) represents the genic covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character). Thus, after a short time t(2), the absolute error in the fundamental and secondary theorems of natural selection is small, though the relative error may be large.     

Interaction characteristics as evolutionary features for the spatial Prisoner’s Dilemma in a population modeled by continuous probabilistic cellular automata and evolutionary algorithm

Evolutionary games usually take into consideration individuals’ strategies as the transformative characteristic which leads to the evolution of the population. Here, besides the strategies, interaction aspects are also considered as evolutionary attributes which can change over time as the replacement dynamic renovates the population choosing locally better individuals to reproduce. The population is modeled by cellular automata, interactions by the Prisoner’s Dilemma game and the replacement process is ruled by two versions of death-birth dynamic. Although the average payoff per game is considered as the fitness for choosing better individuals, the number of games per time step and a maximum radius of interaction with neighbours are also present in the individual’s chromosome which is passed to the next generation. Numerical simulations show that individual interaction properties and cooperation level are linked to the version of death-birth dynamic used and the game payoff. For instance, when the fitness bias is on the death event, individuals have more interactions in a larger radius, and the cooperation level is usually lower than the case where the fitness bias is on the birth event. Also, the individuals’ interaction profiles are heterogeneous, and cooperative individuals form clusters in the lattice to protect themselves.     

The Alternative Fitness Sets Which Preserve Allele Trajectories: A General Treatment

A general solution is presented of the problem of specifying all alternative, generally frequency-dependent, (absolute) fitness sets which give rise to the same allele frequency changes and population dynamics as a given fitness set. The one- and two-locus cases are analyzed in detail and the method is then extended to the n-locus case. It is shown that if biological constraints can be used to specify the mean fitness of the population and the relative fitnesses of the heterozygotes, then the allele frequency trajectories determine a unique fitness set.     

Host mating system and the prevalence of disease in a plant population

A modified susceptible-infected-recovered (SIR) host-pathogen model is used to determine the influence of plant mating system on the outcome of a host-pathogen interaction. Unlike previous models describing how interactions between mating system and pathogen infection affect individual fitness, this model considers the potential consequences of varying mating systems on the prevalence of resistance alleles and disease within the population. If a single allele for disease resistance is sufficient to confer complete resistance in an individual and if both homozygote and heterozygote resistant individuals have the same mean birth and death rates, then, for any parameter set, the selfing rate does not affect the proportions of resistant, susceptible or infected individuals at equilibrium. If homozygote and heterozygote individual birth rates differ, however, the mating system can make a difference in these proportions. In that case, depending on other parameters, increased selfing can either increase or decrease the rate of infection in the population. Results from this model also predict higher frequencies of resistance alleles in predominantly selfing compared to predominantly outcrossing populations for most model conditions. In populations that have higher selfing rates, the resistance alleles are concentrated in homozygotes, whereas in more outcrossing populations, there are more resistant heterozygotes.     

A three-stage discrete-time population model: continuous versus seasonal reproduction

We consider a three-stage discrete-time population model with density-dependent survivorship and time-dependent reproduction. We provide stability analysis for two types of birth mechanisms: continuous and seasonal. We show that when birth is continuous there exists a unique globally stable interior equilibrium provided that the inherent net reproductive number is greater than unity. If it is less than unity, then extinction is the population's fate. We then analyze the case when birth is a function of period two and show that the unique two-cycle is globally attracting when the inherent net reproductive number is greater than unity, while if it is less than unity the population goes to extinction. The two birth types are then compared. It is shown that for low birth rates the adult average number over a one-year period is always higher when reproduction is continuous. Numerical simulations suggest that this remains true for high birth rates. Thus periodic birth rates of period two are deleterious for the three-stage population model. This is different from the results obtained for a two-stage model discussed by Ackleh and Jang (J. Diff. Equ. Appl., 13, 261-274, 2007), where it was shown that for low birth rates seasonal breeding results in higher adult averages.     

Classification of large-time behaviors in a reaction-diffusion system modeling diallelic selection

We study a mathematical model from population genetics, describing a single-locus diallelic (A/a) selection-migration process. The model consists of a coupled system of three reaction-diffusion equations, one for the density of each genotype, posed in a bounded domain or in the whole space R(n). The genotype AA is advantageous, due to a smaller death rate, and the main concern is to determine whether or not the disadvantageous gene a is eliminated in the large time limit. This model was studied in the celebrated work of Aronson and Weinberger (1975,1977), where they derived a simplified scalar model as an approximation of the full system and studied the asymptotic behavior for the scalar model. In particular they showed that, in the fully recessive case (same death rate for the heterozygote and inferior homozygote), the behavior crucially depends on the space dimension. In a previous paper, we were able to prove that their results concerning the scalar model in the fully recessive case remain valid in a certain sense for the full system. In this paper, we reconsider the general case (all possible values of the death and birth rates). We succeed to give a complete picture of whether or not the disadvantageous gene a can survive as t→∞, according to the values of the death and birth rates and of the space dimension. We find distinctive behaviors according to whether the homozygote is superior, intermediate, or inferior and, in the latter case, to whether the common birth rate is smaller or higher than the difference of the death rates of the two heterozygotes. In cases when the disadvantageous gene disappears, the decay rate of its frequency is estimated as well.     

Evolution of recombination due to random drift

In finite populations subject to selection, genetic drift generates negative linkage disequilibrium, on average, even if selection acts independently (i.e., multiplicatively) upon all loci. Negative disequilibrium reduces the variance in fitness and hence, by Fisher's (1930) fundamental theorem, slows the rate of increase in mean fitness. Modifiers that increase recombination eliminate the negative disequilibria that impede selection and consequently increase in frequency by "hitchhiking." Thus, stochastic fluctuations in linkage disequilibrium in finite populations favor the evolution of increased rates of recombination, even in the absence of epistatic interactions among loci and even when disequilibrium is initially absent. The method developed within this article allows us to quantify the strength of selection acting on a modifier allele that increases recombination in a finite population. The analysis indicates that stochastically generated linkage disequilibria do select for increased recombination, a result that is confirmed by Monte Carlo simulations. Selection for a modifier that increases recombination is highest when linkage among loci is tight, when beneficial alleles rise from low to high frequency, and when the population size is small.     

Towards a theory of the evolution of modifier genes

The main findings of a study of the evolution of modifier gene frequencies in models of deterministic population genetics are presented. A wide variety of random mating systems are subject to selection with modifiers operating, in different cases, on mutation rates, migration between subpopulations, and linkage between other loci. In all these instances, the modifier frequencies evolve in such a way as to maximize the mean fitness of the population at equilibrium. This is remarkable since, the modifier genes are selectively neutral in the sense that they do not affect the fitness of their individual carriers. In nonrandom mating systems, the mean fitness concept is not well-defined, and there does not appear to be such a simple principle governing the evolution of modifier frequencies. In assortative mating systems, modifiers favoring reduced assortment propensities tend to increase. In contrast, for selfing-outcrossing systems, modifiers favoring increased selfing tend to increase.     

A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence

The iterated birth and death Markov process is defined as an n-fold iteration of a birth and death Markov process describing kinetics of certain population combined with random killing of individuals in the population at moments tau 1,...,tau n with given survival probabilities s1,...,sn. A long-standing problem of computing the distribution of the number of clonogenic tumor cells surviving an arbitrary fractionated radiation schedule is solved within the framework of iterated birth and death Markov process. It is shown that, for any initial population size iota, the distribution of the size N of the population at moment t > or = tau n is generalized negative binomial, and an explicit computationally feasible formula for the latter is found. It is shown that if i --> infinity and sn --> 0 so that the product iota s1...sn tends to a finite positive limit, the distribution of random variable N converges to a probability distribution, which for t = tau n turns out to be Poisson. In the latter case, an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained.     

A mechanistic model based analysis of cotton aphid population dynamics data

1 A 2‐year field study was conducted to generate data on seasonal abundance patterns of cotton aphids Aphis gossypii Glover and to develop a mechanistic model based on cumulative population size. The treatments consisted of three irrigation levels (Low, Medium and High) with 65%, 75% and 85% evapotranspiration replacement and three nitrogen fertility treatments (blanket‐rate‐N, variable‐rate‐N and no nitrogen). 2 A nonlinear regression equation, the analytical solution of a cumulative size mechanistic model, was fitted to each of the 27 individual data sets collected in 2003 and in 2004. The size and time of the peak, the cumulative aphid density, and the birth and death rates were estimated for each population, and each of these five variables was analyzed as a response variable in the analysis of variance. 3 For 2003 (a dry year), the Water (irrigation) main effect was found to be significant for the time of peak, the death rate and the cumulative density. The lower aphid death rate at low water levels might be due to the water stress in plants. 4 For 2004 (a year with moderate precipitation), the Nitrogen main effect was significant for both the birth and death rates. As nitrogen applications were increased, the decrease in both the aphid birth and death rates translates into a decrease in crowding and an increase in aphid survival. 5 The fact that treatment effects may be manifested through birth and death rate parameters in the new mechanistic model opens up new avenues for analyzing population size data of this kind.     

Population genetics of modifiers of meiotic drive III. Equilibrium analysis of a general model for the genetic control of segregation distortion

Prout, Bungaard and Bryant (1973, Theor. Popul. Biol. 4, 446–465) presented the first formal treatment of a model of meiotic drive involving a modifier locus which controls the intensity of drive. They studied the equilibrium behavior in the simplest model where it is assumed that drive is maximal when not suppressed. In that case there is one polymorphic equilibrium at which there is linkage disequilibrium. The equilibrium solutions in the general model of meiotic drive proposed by Prout, et al. are given in this paper together with a stability analysis. It is shown that up to three polymorphic equilibria may exist, two of which are in linkage disequilibrium and one in linkage equilibrium. These equilibria exhibit behavior qualitatively opposite to what is widely accepted as the usual for two locus systems and which is not seem in the simple case originally treated. The polymorphic equilibria with linkage disequilibrium may be stable for loose linkage and not for tight while that with linkage equilibrium is stable in an interval of relatively tight linkage values.     

Incorporating environmental stochasticity within a biological population model

The birth and death transition rates for a population are modelled as functions of both the population size and the environmental condition. An assortment of important theoretical results and techniques that can be utilized to analyze such a population’s behaviour is covered. Consequently, these results and techniques are used to study two examples. Firstly, we study a population with a stable equilibrium state, whose per capita birth and death rates are linearly related to the environmental condition. (The environmental condition in turn is modelled as an Ornstein–Uhlenbeck process.) Secondly, we study a population affected by two interdependent environmental factors.     

The ideal free distribution and bacterial growth on two substrates

A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.