Elsevier

Theoretical Population Biology

Volume 79, Issues 1–2, February–March 2011, Pages 19-38
Theoretical Population Biology

The influence of partial panmixia on neutral models of spatial variation

https://doi.org/10.1016/j.tpb.2010.08.006Get rights and content

Abstract

Partial panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into neutral models of geographical variation is investigated. The monoecious, diploid population is subdivided into randomly mating colonies that exchange gametes independently of genotype. The gametes fuse wholly at random, including self-fertilization. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus; every allele mutates to new alleles at the same rate. Introducing some panmixia intensifies sufficiently weak migration. A general formula is derived for the migration effective population number, Ne, and Ne is evaluated explicitly in a number of models with nonconservative migration. Usually, Ne increases as the panmictic rate, b, increases; in particular, this result holds for two demes, and generically if the underlying migration is either sufficiently weak or panmixia is sufficiently strong. However, in an analytic model, there exists an open set of parameters for which Ne decreases as b increases. Migration is conservative in the island and circular-habitat models, which are studied in detail. In the former, including some panmixia simply alters the underlying migration rate, increasing (decreasing) it if it is less (greater) than the panmictic value. For the circular habitat, the probability of identity in allelic state at equilibrium is calculated in a nonlocal, continuous-space, continuous-time approximation. In both models, by an efficient, general method, the expected homozygosity, effective number of alleles, and differentiation of gene frequencies are evaluated and discussed; their monotonicity properties with respect to all the parameters are determined; and in the model of infinitely many sites, the mean coalescence times and nucleotide diversities are studied similarly. For the probability of identity at equilibrium in the unbounded stepping-stone model in arbitrarily many dimensions, introducing some panmixia merely replaces the mutation rate by a larger parameter. If the average probability of identity is initially zero, as for identity by descent, then the same conclusion holds for all time.

Introduction

In many species, dispersion includes both local migration and a small proportion of long-distance migration (Petrovskii and Morozov, 2009, and the references therein). In particular, this observation applies to man (Cavalli-Sforza et al., 1994), Arabidopsis thaliana (Nordborg et al., 2005, Platt et al., 2010), and Caenorhabditis elegans (Rockman and Kruglyak, 2009).

The analytic study of the genetic effect of arbitrary long-distance migration in discrete-space models would be quite difficult, and migration is intrinsically local in the diffusion approximation. Therefore, we seek insight by exploring the influence of incorporating the extreme case of long-distance migration, partial panmixia, into neutral models of geographical variation. The corresponding generalization of migration-selection models, especially clines, is equally important and is under investigation.

We use the classic Malécot (1951) model for migration, mutation, and random genetic drift. This model has been extensively analyzed: consult Nagylaki (1989), Charlesworth et al. (2003), and Rousset (2004) for reviews; see Barton et al. (2002), Matsen and Wakeley (2006), Notohara and Umeda (2006), Wakeley and Lessard (2006), Eldon and Wakeley (2009), and the references therein for recent studies.

In Section 2, we formulate our model and develop an efficient, general method for examining the expected homozygosity, effective number of alleles, differentiation of gene frequencies, and (in the model of infinitely many sites) nucleotide diversities. The Malécot (1951) model holds for all forms of genotype-independent gametic dispersion (Sawyer, 1976, Nagylaki, 1983, Nagylaki, 1986), and at equilibrium, it is usually a good approximation for genotype-independent diploid migration (Nagylaki, 1983). We generalize many underlying migration patterns by including partial panmixia.

In Section 3, we derive a general formula for the migration effective population number, calculate it explicitly in a number of models with nonconservative migration (Nagylaki, 1980), and examine its dependence on the proportion of panmixia.

The island model (Moran, 1959, Maruyama, 1970a, Maynard Smith, 1970, Latter, 1973, Nei, 1975, pp. 121–122; Nagylaki, 1983, Nagylaki, 1986) is the subject of Section 4. We discuss the variables introduced in Section 2 and deduce their monotonicity properties with respect to every parameter.

In Section 5, we establish for a circular habitat a nonlocal, continuous-space, continuous-time approximation for the probability of identity in allelic state and solve it at equilibrium. This enables us to investigate the same variables as in the island model and to compare their behavior. This investigation and comparison are important because the island model has population subdivision and a homing tendency, but no isolation by distance, whereas the circular habitat has isolation by distance. Furthermore, although the formulation of partial panmixia requires finiteness of the habitat, it will be possible to obtain the infinite line as the limit of the circle.

The probability of identity in state in the unbounded stepping-stone model in arbitrarily many dimensions (Malécot, 1949, Malécot, 1951, Kimura, 1953) is the topic of Section 6. We prove that at equilibrium, introducing some panmixia merely replaces the mutation rate by a larger parameter. Furthermore, if the average probability of identity is initially zero, as for identity by descent, then the same conclusion holds for all time.

In Section 7, we summarize and briefly discuss our results.

Section snippets

Formulation, assessment, and partial panmixia

In Section 2.1, we briefly review the formulation of the Malécot (1951) model. In Section 2.2, we introduce the parameters for efficiently investigating the expected homozygosity, effective number of alleles, differentiation of gene frequencies, and (in the model of infinitely many sites) nucleotide diversities. In Section 2.3, we incorporate partial panmixia.

The migration effective population number

Since the well-studied migration models are conservative, the migration effective population number, Ne, has not been evaluated in any nonconservative models. Here, we posit (2.44) and derive a general formula for Ne. Specifying the underlying backward migration matrix, M̌, defines a particular model, and we calculate Ne for a number of cases. Since we do not study fˆij in this section, therefore u does not appear and we do not need the approximation u1.

Since increasing the rate of partial

The island model

Here, we incorporate partial panmixia into the classical island model (Moran, 1959, Maruyama, 1970a, Maynard Smith, 1970, Nagylaki, 1983, Nagylaki, 1986). We demonstrate that this extension simply alters the rate of migration; therefore, the entire stochastic process is still an island model. With respect to every parameter in our model, we derive the monotonicity properties of the expected heterozygosity, genetic diversity, and each measure of genetic differentiation. In Section 5, we compare

The circular habitat

Here, we investigate the influence of partial panmixia on the simplest model of isolation by distance: a chain of n colonies, each of N individuals, which form a closed loop (Malécot, 1951). This arrangement might apply to an atoll; demes around a mountain, lake, or shore of an island; and colonies of amphibians or shallow-water organisms in a large, deep lake or around an island. In Sections 5.1 Formulation and approximation, 5.2 Equilibrium, 5.3 Assessment, we formulate and approximate our

The unbounded stepping-stone model

In (5.53) we found that if b=O(u), then in the diffusion limit of the equilibrium probability of identity in state in the unbounded, unidimensional stepping-stone model, the mutation rate is simply augmented by the panmictic rate. Here, we generalize this result to d(d1) dimensions. However, since the diffusion approximation fails when d>1 (Nagylaki, 1978b), we must investigate the discrete model (Malécot, 1949, Malécot, 1951, Kimura, 1953).

We start with a toroidal model in d dimensions (

Discussion

In this section, we summarize our principal results and comment on some unsolved problems.

To explore the biological consequences of the main idea of this paper, the incorporation of some panmixia as the extreme limit of long-distance migration, in Section 2.2 we developed an efficient, general method for evaluating at equilibrium the expected heterozygosity (h̄ˆ0), genetic diversity (measured by the effective number of alleles, ne), and genetic differentiation (D and D̃ for high and low

Acknowledgments

The author is most grateful to Prof. Ethan Akin for a helpful discussion concerning Remark 4.7. It is a pleasure to thank two conscientious referees for finding some misprints and requesting some clarifications and additional explanations.

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