The influence of partial panmixia on neutral models of spatial variation
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, , has not been evaluated in any nonconservative models. Here, we posit (2.44) and derive a general formula for . Specifying the underlying backward migration matrix, , defines a particular model, and we calculate for a number of cases. Since we do not study in this section, therefore does not appear and we do not need the approximation .
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 colonies, each of 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 , 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 dimensions. However, since the diffusion approximation fails when (Nagylaki, 1978b), we must investigate the discrete model (Malécot, 1949, Malécot, 1951, Kimura, 1953).
We start with a toroidal model in 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 , genetic diversity (measured by the effective number of alleles, ), and genetic differentiation ( and 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.
References (70)
- et al.
Neutral evolution in spatially continuous populations
Theor. Popul. Biol.
(2002) The coalescent
Stochastic Process. Appl.
(1982)Distribution of nucleotide differences between two randomly chosen cistrons in a subdivided population: the finite island model
Theor. Popul. Biol.
(1976)Heterozygosity and relationship in regularly subdivided populations
Theor. Popul. Biol.
(1975)Effective number of alleles in a subdivided population
Theor. Popul. Biol.
(1970)The decay of genetic variability in geographically structured populations. II
Theor. Popul. Biol.
(1976)Geographical invariance in population genetics
J. Theoret. Biol.
(1982)The robustness of neutral models of geographical variation
Theor. Popul. Biol.
(1983)- et al.
The coalescence time of sampled genes in the structured coalescent model
Theor. Popul. Biol.
(2006) - et al.
Corridors for migration between large subdivided populations, and the structured coalescent
Theor. Popul. Biol.
(2006)