Dynamics of neutral and selected alleles when the offspring distribution is skewed

We analyze the dynamics of two alternative alleles in a simple model of a population that allows for large family sizes in the distribution of offspring number. This population model was first introduced by Eldon and Wakeley, who described the backward-time genealogical relationships among sampled individuals, assuming neutrality. We study the corresponding forward-time dynamics of allele frequencies, with or without selection. We derive a continuum approximation, analogous to Kimura's diffusion approximation, and we describe three distinct regimes of behavior that correspond to distinct regimes in the coalescent processes of Eldon and Wakeley. We demonstrate that the effect of selection is strongly amplified in the Eldon-Wakeley model, compared to the Wright-Fisher model with the same variance effective population size. Remarkably, an advantageous allele can even be guaranteed to fix in the Eldon-Wakeley model, despite the presence of genetic drift. We compute the selection coefficient required for such behavior in populations of Pacific oysters, based on estimates of their family sizes. Our analysis underscores that populations with the same effective population size may nevertheless experience radically different forms of genetic drift, depending on the reproductive mechanism, with significant consequences for the resulting allele dynamics.     

Dynamics of neutral biodiversity

Hubbell's neutral model has become a major paradigm in ecology. Whereas the steady-state structure is well understood, results about the dynamical aspects of the model are scarce. Here we derive dynamical equations for the Simpson diversity index. Both mean and variance of the diversity are proven to satisfy stable linear system dynamics. We show that in the stationary limit we indeed recover previous results, and we supplement this with numerical simulations to validate the dynamical part of our analytical computations. These findings are especially relevant for experiments in microbial ecology, where the Simpson diversity index can be accurately measured as a function of time.     

The onion model, a simple neutral model for the evolution of diversity in bacterial biofilms

Bacterial biofilms are particularly resistant to a wide variety of antimicrobial compounds. Their persistence in the face of antibiotic therapies causes significant problems in the treatment of infectious diseases. Seldom have evolutionary processes like genetic drift and mutation been invoked to explain how resistance to antibiotics emerges in biofilms, and we lack a simple and tractable model for the genetic and phenotypic diversification that occurs in bacterial biofilms. Here, we introduce the 'onion model', a simple neutral evolutionary model for phenotypic diversification in biofilms. We explore its properties and show that the model produces patterns of diversity that are qualitatively similar to observed patterns of phenotypic diversity in biofilms. We suggest that models like our onion model, which explicitly invoke evolutionary process, are key to understanding biofilm resistance to bactericidal and bacteriostatic agents. Elevated phenotypic variance provides an insurance effect that increases the likelihood that some proportion of the population will be resistant to imposed selective agents and may thus enhance persistence of the biofilm. Accounting for evolutionary change in biofilms will improve our ability to understand and counter diseases that are caused by biofilm persistence.     

Relaxing the zero-sum assumption in neutral biodiversity theory

The zero-sum assumption is one of the ingredients of the standard neutral model of biodiversity by Hubbell. It states that the community is saturated all the time, which in this model means that the total number of individuals in the community is constant over time, and therefore introduces a coupling between species abundances. It was shown recently that a neutral model with independent species, and thus without any coupling between species abundances, has the same sampling formula (given a fixed number of individuals in the sample) as the standard model [Etienne, R.S., Alonso, D., McKane, A.J., 2007. The zero-sum assumption in neutral biodiversity theory. J. Theor. Biol. 248, 522-536]. The equilibria of both models are therefore equivalent from a practical point of view. Here we show that this equivalence can be extended to a class of neutral models with density-dependence on the community-level. This result can be interpreted as robustness of the model, i.e. insensitivity of the model to the precise interaction of the species in a neutral community. It can also be interpreted as a lack of resolution, as different mechanisms of interactions between neutral species cannot be distinguished using only a single snapshot of species abundance data.     

A neutral metapopulation model of biodiversity in river networks

In this paper, we develop a stochastic, discrete, structured metapopulation model to explore the dynamics and patterns of biodiversity of riparian vegetation. In the model, individual plants spread along a branched network via directional dispersal and undergo neutral ecological drift. Simulation results suggest that in comparison to 2-D landscapes with non-directional dispersal, river networks with directional dispersal have lower local (alpha) and overall (gamma) diversities, but higher between-community (beta) diversity, implying that riparian species are distributed in a more localized pattern and more vulnerable to local extinction. The relative abundance patterns also change, such that higher percentages of species are in low-abundance, or rare, classes, accompanied by concave rank-abundance curves. In contrast to existing theories, the results suggest that in river networks, increased directional dispersal reduces alpha diversity. These altered patterns and trends result from the combined effects of directionality of dispersal and river network structure, whose relative importance is in need of continuing study. In addition, riparian communities obeying neutral dynamics seem to exhibit abrupt changes where large tributaries confluence; this pattern may provide a signature to identify types of interspecific dynamics in river networks.     

A spatially explicit model for tropical tree diversity patterns

A simple two-parameter model resembling the classical voter model is introduced to describe macroecological properties of tropical tree communities. The parameters of the model characterize the speciation- and global-dispersion rates. Monte Carlo type computer simulations are performed on the model, investigating species abundances and the spatial distribution of individuals and species. Simulation results are critically compared with the experimental data obtained from a tree census on a 50 hectare area of the Barro Colorado Island (BCI), Panama. Fitting to only two observable quantities from the BCI data (total species number and the slope of the log-log species-area curve at the maximal area), it is possible to reproduce the full species-area curve, the relative species abundance distribution, and a more realistic spatial distribution of species.     

Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory

The allele frequency spectrum has attracted considerable interest for the simultaneous inference of the demographic and adaptive history of populations. In a recent study, Evans et al. (2007) developed a forward diffusion equation describing the allele frequency spectrum, when the population is subject to size changes, selection and mutation. From the diffusion equation, the authors derived a system of ordinary differential equations (ODEs) for the moments in a Wright–Fisher diffusion with varying population size and constant selection. Here, we present an explicit solution for this system of ODEs with variable population size, but without selection, and apply this result to derive the expected spectrum of a sample for time-varying population size. We use this forward-in-time-solution of the allele frequency spectrum to obtain the backward-in-time-solution previously derived via coalescent theory by Griffiths and Tavaré (1998). Finally, we discuss the applicability of the theoretical results to the analysis of nucleotide polymorphism data.     

Integrating genealogy and epidemiology: the ancestral infection and selection graph as a model for reconstructing host virus histories

We model the genealogies of coupled haploid host-virus populations. Hosts reproduce and replace other hosts as in the Moran model. The virus can be transmitted between individuals of the same and succeeding generations. The epidemic model allows a selective advantage for susceptible over infected hosts. The coupled host-virus ancestry of a sample of hosts is embedded in a branching and coalescing structure that we call the Ancestral Infection and Selection Graph, a direct analogue to the Ancestral Selection Graph of Krone and Neuhauser [1997. Theoret. Population Biol. 51, 210-237]. We prove this and discuss various special cases. We show that the inter-host viral genealogy is a scaled coalescent. Using simulations, we compare the viral genealogy under this model to earlier published models and investigate the estimatability of the selection and infectious contact rates. We use simulations to compare the persistence of the disease with the time to the ultimate ancestor.     

Eigenanalysis for the Kolmogorov backward equation for the neutral multi-allelic model

In a previous paper [9], we have given an algorithm for obtaining the time dependent solution of all polynomial moments of gene frequencies in neutral models. The recurrence formula for the moment generating functions obtained in [9] gives information about the eigenvalues and the eigenfunctions in neutral models. In this article, we solve the eigenvalue problem for the Kolmogorov backward equation for the case of neutral alleles under mutation pressure.     

Existence and nonexistence of steady-state solutions for a selection-migration model in population genetics

We discuss a selection-migration model in population genetics, involving two alleles A₁ and A₂ such that A₁ is at an advantage over A₂ in certain subregions and at a disadvantage in others. It is shown that if A₁ is at an overall disadvantage to A₂ and the rate of gene flow is sufficiently large than A₁ must die out; on the other hand, if the two alleles are in some sense equally advantaged overall, then A₁ and A₂ can coexist no matter how great the rate of gene flow.     

Genealogical-tree probabilities in the infinitely-many-site model

This paper considers the distribution of the genealogical tree of a sample of genes in the infinitely-many-site model where the relative age ordering of the mutations (nodes in the tree) is known. A computer implementation of a recursion for the probability of such trees is discussed when the nodes are age-labeled, or not.     

The effect of recombination on the neutral evolution of genetic robustness

Conventional population genetics considers the evolution of a limited number of genotypes corresponding to phenotypes with different fitness. As model phenotypes, in particular RNA secondary structure, have become computationally tractable, however, it has become apparent that the context dependent effect of mutations and the many-to-one nature inherent in these genotype-phenotype maps can have fundamental evolutionary consequences. It has previously been demonstrated that populations of genotypes evolving on the neutral networks corresponding to all genotypes with the same secondary structure only through neutral mutations can evolve mutational robustness [E. van Nimwegen, J.P. Crutchfield, M. Huynen, Neutral evolution of mutational robustness, Proc. Natl. Acad. Sci. USA 96(17), 9716-9720 (1999)], by concentrating the population on regions of high neutrality. Introducing recombination we demonstrate, through numerically calculating the stationary distribution of an infinite population on ensembles of random neutral networks that mutational robustness is significantly enhanced and further that the magnitude of this enhancement is sensitive to details of the neutral network topology. Through the simulation of finite populations of genotypes evolving on random neutral networks and a scaled down microRNA neutral network, we show that even in finite populations recombination will still act to focus the population on regions of locally high neutrality.     

The number of distinguishable alleles according to the Ohta-Kimura model of neutral mutation

We calculate how many alleles one can expect to distinguish in a large sample from a large population which develops according to the Ohta-Kimura model. This number tends to infinity with the sample size, but so slowly that it is bounded for all practical purposes.Research supported by the NSF through a grant to Cornell University     

A neutral model of edge effects

In this paper a spatially implicit neutral model for explaining the edge effects between habitats is proposed. To analyze this model we use two different approaches: a discrete approach that is based on the Master equation for a one step jump process and a continuous approach based on the approximation of the discrete jump process with the Kolmogorov-Fokker-Planck forward and backward equations. The discrete and continuous approaches are applied to analyze the species abundance distributions and the time to species extinction. Moreover, with the aid of the continuous approach a realistic classification of the behavior of species in local communities is developed. The species abundance dynamics at the edge between two distinct habitats is compared with those located in the homogeneous interior habitats using species abundance distributions and the first time to species extinction. We show that the structure of the links between local community and the metacommunity plays an important role on species persistence. Specifically, species at the edge between two distinct metacommunities have higher extinction rate than those in the interior habitats connected only to one metacommunity. Moreover, the same species might be persistent in the homogeneous interior habitat, but its probability of extinction from the edge local community could be very high.     

Research of population with impulsive perturbations based on dynamics of a neutral delay equation and ecological quality system

This paper studies the global behaviors of a nonlinear autonomous neutral delay differential population model with impulsive perturbation. This model may be suitable for describing the dynamics of population with long larval and short adult phases. It is shown that the system may have global stability of the extinction and positive equilibria, or grow without being bounded under some conditions.     

Slight differences among individuals and the unified neutral theory of biodiversity

The unified neutral theory of biodiversity provides a very simple and counterintuitive explanation of species diversity patterns. By specifying speciation, community size and dispersal, and completely ignoring differences among individual organisms and species, it generates biodiversity patterns that remarkably resemble natural ones. Here I show that adding even slight differences among organisms generates very different patterns and predictions. In large communities with widespread dispersal, heritable differences in viability among individual organisms lead to biodiversity patterns characterised by the overdominance of a single species comprising organisms with relatively high fitness. In communities with local dispersal, the same differences produce rapid community extinction. I conclude that the unified neutral theory is not robust to slight deviations from its most controversial assumption.     

A compact result for the time-dependent probability of fixation at a neutral locus

A result is derived, in the form of a sum, for the time-dependent probability of fixation of an unlinked neutral locus. The result captures many of the key features of the probability of fixation in a highly compact form. For ‘small’ times (t?4N_e) a single term of the sum accurately determines the time-dependent probability of fixation. This is in contrast to the well-known result of Kimura, which requires the contribution of many terms in a different sum, for ‘small’ times. Going beyond small times, an approximation is derived for the time-dependent probability of fixation which applies for all times when the initial relative allele frequency is small.     

Population growth enhances the mean fixation time of neutral mutations and the persistence of neutral variation

A fundamental result of population genetics states that a new mutation, at an unlinked neutral locus in a randomly mating diploid population, has a mean time of fixation of ～4N(e) generations, where N(e) is the effective population size. This result is based on an assumption of fixed population size, which does not universally hold in natural populations. Here, we analyze such neutral fixations in populations of changing size within the framework of the diffusion approximation. General expressions are derived for the mean and variance of the fixation time in changing populations. Some explicit results are given for two cases: (i) the effective population size undergoes a sudden change, representing a sudden population expansion or a sudden bottleneck; (ii) the effective population changes linearly for a limited period of time and then remains constant. Additionally, a lower bound for the mean time of fixation is obtained for an effective population size that increases with time, and this is applied to exponentially growing populations. The results obtained in this work show, among other things, that for populations that increase in size, the mean time of fixation can be enhanced, sometimes substantially so, over 4N(e,0) generations, where N(e,0) is the effective population size at the time the mutation arises. Such an enhancement is associated with (i) an increased probability of neutral polymorphism in a population and (ii) an enhanced persistence of high-frequency neutral variation, which is the variation most likely to be observed.     

Nitrogen deposition mediates the effects and importance of chance in changing biodiversity

Nitrogen deposition is changing biodiversity on Earth. We need to understand the underlying mechanisms to conserve biodiversity better. Both selection and chance are potential mechanisms, and they may operate concurrently. Then, what are the respective effects of selection and chance, what is their relative importance and how do they change with increasing nitrogen deposition rate? Here, we performed a 6‐year nitrogen addition experiment (0–28 g N/m²/year) in a typical steppe ecosystem of Inner Mongolia to investigate the community structure of plants, bacteria and ammonia‐oxidizing Archaea (AOA). We developed an experimentally based calculation method to first separate the structural variations between plots into the effects of selection (S) and chance (C), and then calculate their relative importance. Our results showed that as nitrogen addition rate increased, S for both plants and bacteria increased, but C for plants first increased and then decreased, and C for bacteria also increased; meanwhile, both S and C for AOA changed nonlinearly. As nitrogen addition rate increased, the importance of chance decreased on the whole for all these communities, but it decreased nonlinearly for plants and bacteria, with a local increase at certain intermediate rates. At all treatments, the importance of chance was <0.5 for plants, but >0.5 for AOA. These results demonstrated that nitrogen deposition changed biodiversity by mediating the effects and importance of chance, implicating different strategies should be adopted in conserving biodiversity according to nitrogen deposition rate and community properties.     

On realizing shapes in the theory of RNA neutral networks

It is known (Reidys et al., 1997b. Bull. Math. Biol. 59(2), 339-397) that for any two secondary structures S,S' there exists an RNA sequence compatible with both, and that this result does not extend to more than two secondary structures. Indeed, a simple formula for the number of RNA sequences compatible with secondary structures S,S' plays a role in the algorithms of Flamm et al. (2001. RNA 7, 254-265) and of Abfalter et al. (2003. Proceedings of the German Conference on Bioinformatics, ) to design an RNA switch. Here we show that a natural extension of this problem is NP-complete. Unless P=NP, there is no polynomial time algorithm, which when given secondary structures S1,...,S(k), for k4, determines the least number of positions, such that after removal of all base pairs incident to these positions there exists an RNA nucleotide sequence compatible with the given secondary structures. We also consider a restricted version of this problem with a "fixed maximum" number of possible stars and show that it has a simple polynomial time solution.