Maximization principles for frequency-dependent selection I: the one-locus two-allele case

In this article we study the one-locus two-allele version of the pairwise-interaction model of frequency-dependent selection in discrete and continuous time. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles. We provide a systematic approach that covers all possible facets of the dynamical behavior of the model, and we illustrate our results by concrete examples. We show that the mean fitness of the population is nondecreasing if the interaction coefficients are symmetric and positive. Moreover, monotonic convergence to the set of equilibria always occurs, which is not true if we also consider negative interaction coefficients. For asymmetric interaction, we provide necessary conditions when the mean fitness is nondecreasing and sufficient conditions when it is not. Furthermore, in discrete time, we show that limit cycles cannot occur, unless some interaction coefficients are negative.     

The donation game with roles played between relatives

We consider a social game with two choices, played between two relatives, where roles are assigned to individuals so that the interaction is asymmetric. Behaviour in each of the two roles is determined by a separate genetic locus. Such asymmetric interactions between relatives, in which individuals occupy different behavioural contexts, may occur in nature, for example between adult parents and juvenile offspring. The social game considered is known to be equivalent to a donation game with non-additive payoffs, and has previously been analysed for the single locus case, both for discrete and continuous strategy traits. We present an inclusive fitness analysis of the discrete trait game with roles and recover equilibrium conditions including fixation of selfish or altruistic behaviour under both behavioural contexts, or fixation of selfish behaviour under one context and altruistic behaviour under the other context. These equilibrium solutions assume that the payoff matrices under each behavioural context are identical. The equilibria possible do depend crucially, however, on the deviation from payoff additivity that occurs when both interacting individuals act altruistically.     

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.     

Sex differences in linkage between autosomal loci

In the case of conventional selection theory with multiplicative gene action between loci and no sex differences except in crossover frequencies, it is shown that the usual conditions for stability hold when the mean of the recombination frequencies for the two sexes is used. For additive gene action between loci, it is shown that, after one generation of random mating, the gene frequencies of male and female origin are the same. This equality implies the nondecreasing property of the mean fitness function. Some attention is also given to neutral loci.     

Asymptotic behaviour of solutions of the diffusive Lotka-Volterra equations

A spatially discrete version of the diffusive Lotka-Volterra equations is considered. Asymptotical spatial homogeneity of solutions of the equations with equilibrium, periodic or zero flux boundary conditions is proved without regard to crowding effects. The proof does not require the assumption of equal diffusion coefficients and the restrictions on the dimension of space and on the initial data, which are necessary in the spatially continuous model.     

时标上一捕食二食饵系统的周期解（英文）

研究时标上一捕食二食饵系统.运用时标上Gaines和Mawhin的连续拓扑度定理,得到了系统存在周期解的新的充分条件.其研究方法可以广泛地运用来研究微分或者差分方程的周期解存在性问题.     

Host-parasite interaction as a factor in the evolution of genetic recombination

A model of the rec-system evolution determined by species interactions of the host-parasite type has been studied. In contrast to known formalizations, the genetic structure of both populations is clearly represented in our model, which makes it possible to set the mode of their interrelations in a more natural and biologically consistent way. The numerical analysis has revealed the situations, where nondecreasing oscillations of the linkage disequilibrium coefficient and, consequently, the selection favourable for higher recombination occur in a system. The evolutionary advantage of recombination has been demonstrated both in terms of population mean fitness and in the models with locus modifier of recombination.     

Chromosome Interactions in DROSOPHILA MELANOGASTER. II. Total Fitness

In a large experiment, using nearly 200 population cages, we have measured the fitness of Drosophila melanogaster homozygous (1) for the second chromosome, (2) for the third chromosome, and (3) for both chromosomes. Twentyfour second chromosomes and 24 third chromosomes sampled from a natural population were tested. The mean fitness of the homozygous flies is 0.081 ± 0.014 for the second chromosome, 0.080 ± 0.017 for the third chromosome, and 0.079 ± 0.024 for both chromosomes simultaneously. Assuming that fitnesses are multiplicative (the additive fitness model makes no sense in the present case because of the large selection coefficients involved), the expected mean fitness of the homozygotes for both chromosomes is 0.0066; their observed fitness is more than ten times greater. Thus, it appears that synergistic interactions between loci are considerable; and that, consequently, the fitness function substantially departs from linearity. Two models are tentatively suggested for the fitness function: a "threshold" model and a "synergistic" model.—The experiments reported here confirm previous results showing that the concealed genetic load present in natural populations of Drosophila is sufficient to account for the selective maintenance of numerous polymorphisms (of the order of 1000).     

Measuring selection coefficients below 10(-3): method, questions, and prospects

Measuring fitness with precision is a key issue in evolutionary biology, particularly in studying mutations of small effects. It is usually thought that sampling error and drift prevent precise measurement of very small fitness effects. We circumvented these limits by using a new combined approach to measuring and analyzing fitness. We estimated the mutational fitness effect (MFE) of three independent mini-Tn10 transposon insertion mutations by conducting competition experiments in large populations of Escherichia coli under controlled laboratory conditions. Using flow cytometry to assess genotype frequencies from very large samples alleviated the problem of sampling error, while the effect of drift was controlled by using large populations and massive replication of fitness measures. Furthermore, with a set of four competition experiments between ancestral and mutant genotypes, we were able to decompose fitness measures into four estimated parameters that account for fitness effects of our fluorescent marker (α), the mutation (β), epistasis between the mutation and the marker (γ), and departure from transitivity (τ). Our method allowed us to estimate mean selection coefficients to a precision of 2 × 10(-4). We also found small, but significant, epistatic interactions between the allelic effects of mutations and markers and confirmed that fitness effects were transitive in most cases. Unexpectedly, we also detected variation in measures of s that were significantly bigger than expected due to drift alone, indicating the existence of cryptic variation, even in fully controlled experiments. Overall our results indicate that selection coefficients are best understood as being distributed, representing a limit on the precision with which selection can be measured, even under controlled laboratory conditions.     

Models of Frequency-Dependent Selection with Mutation from Parental Alleles

Frequency-dependent selection (FDS) remains a common heuristic explanation for the maintenance of genetic variation in natural populations. The pairwise-interaction model (PIM) is a well-studied general model of frequency-dependent selection, which assumes that a genotype’s fitness is a function of within-population intergenotypic interactions. Previous theoretical work indicated that this type of model is able to sustain large numbers of alleles at a single locus when it incorporates recurrent mutation. These studies, however, have ignored the impact of the distribution of fitness effects of new mutations on the dynamics and end results of polymorphism construction. We suggest that a natural way to model mutation would be to assume mutant fitness is related to the fitness of the parental allele, i.e., the existing allele from which the mutant arose. Here we examine the numbers and distributions of fitnesses and alleles produced by construction under the PIM with mutation from parental alleles and the impacts on such measures due to different methods of generating mutant fitnesses. We find that, in comparison with previous results, generating mutants from existing alleles lowers the average number of alleles likely to be observed in a system subject to FDS, but produces polymorphisms that are highly stable and have realistic allele-frequency distributions.     

A mathematical model for storage and recall functions in plants

In plantlets of Bidens pilosa L., under severely limiting environmental conditions the growth of the buds at the axil of the cotyledons (cotyledonary buds) is asymmetric (i.e. one of the buds starts growing before the other one), this asymmetry being oriented by the pricking of one of the cotyledons (i.e. pricking one cotyledon increases the probability that the bud at the axil of the other cotyledon be the first to start to grow). As long as the plant apex (i.e. the terminal bud) is present, the growth of the cotyledonary buds is inhibited (apical dominance), but the souvenir of the asymmetric message caused by sub-optimal environmental conditions and the orientation given by the cotyledon pricking is always present in the plant and can be revealed by removing the apex. Depending on the conditions for removing the plant apex and/or on the application of a variety of symmetrical treatments (e.g. thermal treatment, symmetrical pricking treatments, etc.) the stored asymmetry will either take effect (the bud at the axil of the non-pricked cotyledon will be the first to start to grow more often than the other one) or not (both buds will have equal chance to be the first to start to grow). This has been termed 'recalling' the stored asymmetry. By combining several successive symmetrical treatments, it is possible to reversibly switch on and off the recall function several times. This recall of the stored plant-asymmetry is analogous to the evocation function of a memory system. In this paper, we will present first a discrete logical version of the observed interaction structure between the main components of the bud growth system, then a continuous differential version, taking into account the main features of the observed experimental reality and trying to explain this phenomenology. The interaction structure of both the discrete and the continuous models presents similar positive and negative feedback circuits, necessary condition for observing multistationarity and stability.     

A comparison of sexual and asexual replication strategies in a simplified model based on the yeast life cycle

This paper develops simplified mathematical models describing the mutation-selection balance for the asexual and sexual replication pathways in Saccharomyces cerevisiae, or Baker’s yeast. The simplified models are based on the single-fitness-peak approximation in quasispecies theory. We assume diploid genomes consisting of two chromosomes, and we assume that each chromosome is functional if and only if its base sequence is identical to some master sequence. The growth and replication of the yeast cells is modeled as a first-order process, with first-order growth rate constants that are determined by whether a given genome consists of zero, one, or two functional chromosomes. In the asexual pathway, we assume that a given diploid cell divides into two diploids. For the sake of generality, our model allows for mitotic recombination and asymmetric chromosome segregation. In the sexual pathway, we assume that a given diploid cell divides into two diploids, each of which then divide into two haploids. The resulting four haploids enter a haploid pool, where they grow and replicate until they meet another haploid with which to fuse. In the sexual pathway, we consider two mating strategies: (1) a selective strategy, where only haploids with functional chromosomes can fuse with one another; (2) a random strategy, where haploids randomly fuse with one another. When the cost for sex is low, we find that the selective mating strategy leads to the highest mean fitness of the population, when compared to all of the other strategies. When the cost for sex is low, sexual replication with random mating also has a higher mean fitness than asexual replication without mitotic recombination or asymmetric chromosome segregation. We also show that, at low replication fidelities, sexual replication with random mating has a higher mean fitness than asexual replication, as long as the cost for sex is low. If the fitness penalty for having a defective chromosome is sufficiently high and the cost for sex sufficiently low, then at low replication fidelities the random mating strategy has a mean fitness that is a factor of larger than the asexual mean fitness. We argue that for yeast, the selective mating strategy is the one that is closer to reality, which if true suggests that sex may provide a selective advantage under considerably more relaxed conditions than previous research has indicated. The results of this paper also suggest that S. cerevisiae switches from asexual to sexual replication when stressed, because stressful growth conditions provide an opportunity for the yeast to clear out deleterious mutations from their genomes. That being said, our model does not contradict theories for the evolution of sex that argue that sex evolved because it allows a population to more easily adapt to changing conditions.     

EVOLUTIONARY EPIDEMIOLOGY AND THE DYNAMICS OF ADAPTATION

The mean fitness of a population, often equal to its growth rate, measures its level of adaptation to particular environmental conditions. A better understanding of the evolution of mean fitness could thus provide a natural link between evolution and demography. Yet, after the seminal work of Fisher and its renowned "fundamental theorem of natural selection," the dynamics of mean fitness has attracted little attention, and mostly from theoretical population geneticists. Here we analyze the dynamics of mean fitness in the context of host-parasite interactions. We illustrate the potential relevance of this analysis under different scenarios ranging from a simple situation in which a parasite evolves in a homogeneous host population to a more complex one with host-parasite coevolution. In each case, we contrast the effects of natural selection, recurrent mutations, and the change of the biotic environment, on the dynamics of adaptation. Decoupling these three components helps elucidate the interplay between evolutionary and ecological dynamics. In particular, it offers new perspectives on situations leading to evolutionary suicide. As mean fitness is an easily measurable quantity in microbial systems, this analysis provides new ways to track the dynamics of adaptation in experimental evolution and coevolution studies.     

The Arabidopsis thaliana flower organ specification gene regulatory network determines a robust differentiation process

The Arabidopsis thaliana flower organ specification gene regulatory network (FOS-GRN) has been modeled previously as a discrete dynamical system, recovering as steady states configurations that match the genetic profiles described in primordial cells of inflorescence, sepals, petals, stamens and carpels during early flower development. In this study, we first update the FOS-GRN by adding interactions and modifying some rules according to new experimental data. A discrete model of this updated version of the network has a dynamical behavior identical to previous versions, under both wild type and mutant conditions, thus confirming its robustness. Then, we develop a continuous version of the FOS-GRN using a new methodology that builds upon previous proposals. The fixed point attractors of the discrete system are all observed in the continuous model, but the latter also contains new steady states that might correspond to genetic activation states present briefly during the early phases of flower development. We show that both the discrete and the continuous models recover the observed stable gene configurations observed in the inflorescence meristem, as well as the primordial cells of sepals, petals, stamens and carpels. Additionally, both models are subjected to perturbations in order to establish the nature of additional signals that may suffice to determine the experimentally observed order of appearance of floral organs. Our results thus describe a possible mechanism by which the network canalizes molecular signals and/or noise, thus conferring robustness to the differentiation process.     

A continuous optimization approach for inferring parameters in mathematical models of regulatory networks

Background

The advances of systems biology have raised a large number of sophisticated mathematical models for describing the dynamic property of complex biological systems. One of the major steps in developing mathematical models is to estimate unknown parameters of the model based on experimentally measured quantities. However, experimental conditions limit the amount of data that is available for mathematical modelling. The number of unknown parameters in mathematical models may be larger than the number of observation data. The imbalance between the number of experimental data and number of unknown parameters makes reverse-engineering problems particularly challenging.

Results

To address the issue of inadequate experimental data, we propose a continuous optimization approach for making reliable inference of model parameters. This approach first uses a spline interpolation to generate continuous functions of system dynamics as well as the first and second order derivatives of continuous functions. The expanded dataset is the basis to infer unknown model parameters using various continuous optimization criteria, including the error of simulation only, error of both simulation and the first derivative, or error of simulation as well as the first and second derivatives. We use three case studies to demonstrate the accuracy and reliability of the proposed new approach. Compared with the corresponding discrete criteria using experimental data at the measurement time points only, numerical results of the ERK kinase activation module show that the continuous absolute-error criteria using both function and high order derivatives generate estimates with better accuracy. This result is also supported by the second and third case studies for the G1/S transition network and the MAP kinase pathway, respectively. This suggests that the continuous absolute-error criteria lead to more accurate estimates than the corresponding discrete criteria. We also study the robustness property of these three models to examine the reliability of estimates. Simulation results show that the models with estimated parameters using continuous fitness functions have better robustness properties than those using the corresponding discrete fitness functions.

Conclusions

The inference studies and robustness analysis suggest that the proposed continuous optimization criteria are effective and robust for estimating unknown parameters in mathematical models.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-256) contains supplementary material, which is available to authorized users.     

Evolution under Fertility and Viability Selection

Evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability is investigated. Discrete, nonoverlapping generations are posited for autosomal and X-linked loci in dioecious populations, but monoecious populations are studied in both discrete and continuous time. Mating is random. The results hold after several generations have elapsed. With an error of order s [i.e., O(s)], where s represents the selection intensity, the population evolves in Hardy-Weinberg proportions. Provided the change per generation of the fertilities and viabilities due to their explicit time dependence (if any) is O(s2), the rate of change of the deviation from Hardy-Weinberg proportions is O(s2). If the change per generation of the viabilities and genotypic fertilities is smaller than second order [i.e., o(s2)], then to O(s2) the rate of change of the mean fitness is equal to the genic variance. The mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. Hence, as long as there is significant gene frequency change, the mean fitness increases. If it is the fertilities of matings that change slowly [at rate o(s2)], the above conclusions apply to a modified mean fitness, defined as the product of the mean viability and the square root of the mean fertility.     

Variability in a single compartment system: A note on S. R. Bernard's model

In this note we examine a continuous time version of a compartmental model introduced in a discrete time setting by S. R. Bernard. The model allows for more than one particle to leave the system at any time. This introduces additional randomness into the system, over the pure death system and this is reflected in the variance function.     

An epizootic model of an insect-fungal pathogen system

A system of integro-differential equations is derived to describe epizootics of a fungal pathogen in an insect population. Because of piecewise continuous behavior under some parametric conditions, it is concluded that standard phase orbits can be misleading. Using a different analytic approach yields a simple system of finite difference equations. Both the continuous and discrete versions are compared to classical forms. The continuous version differs from a classical one in possessing a second derivative dependent on population density. The discrete version differs in maintaining positive, non-zero populations of both infectives and susceptibles in finite time. Scientific paper No. 81-7-129 of the Kentucky Agricultural Experiment Station, Lexington. This research has been financed (in part) with Federal funds from the Environmental Protection Agency under grant number CR-806277-020. The contents do not necessarily reflect the views and policies of the Environmental Protection Agency, nor does mention of trade name or commercial products constitute endorsement or recommendation for use.     

Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite‐site fitness landscapes. First, we analyze the mean fitness trajectory—that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non‐epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics—the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions—can detect certain forms of epistasis in the underlying fitness landscape.     

Stochastic hybrid delay population dynamics: well-posed models and extinction

Nonlinear differential equations have been used for decades for studying fluctuations in the populations of species, interactions of species with the environment, and competition and symbiosis between species. Over the years, the original non-linear models have been embellished with delay terms, stochastic terms and more recently discrete dynamics. In this paper, we investigate stochastic hybrid delay population dynamics (SHDPD), a very general class of population dynamics that comprises all of these phenomena. For this class of systems, we provide sufficient conditions to ensure that SHDPD have global positive, ultimately bounded solutions, a minimum requirement for a realistic, well-posed model. We then study the question of extinction and establish conditions under which an ecosystem modelled by SHDPD is doomed.