Predictions of single-nucleotide polymorphism differentiation between two populations in terms of mutual information

Abstract

Mutual information (I) provides a robust measure of genetic differentiation for the purposes of estimating dispersal between populations. At present, however, there is little predictive theory for I. The growing importance in population biology of analyses of single-nucleotide and other single-feature polymorphisms (SFPs) is a potent reason for developing an analytic theory for I with respect to a single locus. This study represents a first step towards such a theory. We present theoretical predictions of I between two populations with respect to a single haploid biallelic locus. Dynamical and steady-state forecasts of I are derived from a Wright-Fisher model with symmetrical mutation between alleles and symmetrical dispersal between populations. Analytical predictions of a simple Taylor approximation to I are in good agreement with numerical simulations of I and with data on I from SFP analyses of dispersal experiments on Drosophila fly populations. The theory presented here also provides a basis for the future inclusion of selection effects and extension to multiallelic loci.

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model

“…Shannon entropy H and its monotonic transformations, such as exp( H ), connect directly to the rich mathematics of information theory initiated by Shannon [11], singularly appropriate for DNA information [12,13,14]. Unlike heterozygosity, information measures weigh alleles in proportion to their population fraction.…”

“…Shannon entropy and mutual information have more recently been used to analyze a wide variety of genetic processes and patterns [12,13]. Examples cover a range of taxa, including viruses [21], bacteria [22], protist parasites [23], mosses [24], higher plants [25–31], invertebrates [14,32] and vertebrates including humans [33,34,35]. Many concentrate on microsatellites [12], but they have also assessed AFLPs [29], and single-nucleotide polymorphisms [14].…”

“…Examples cover a range of taxa, including viruses [21], bacteria [22], protist parasites [23], mosses [24], higher plants [25–31], invertebrates [14,32] and vertebrates including humans [33,34,35]. Many concentrate on microsatellites [12], but they have also assessed AFLPs [29], and single-nucleotide polymorphisms [14]. Recent theoretical uses of Shannon entropy and mutual information in genetics also include: dynamics of populations of genetically variable individuals in landscapes [36]; dynamics of molecules in gene expression networks [37,38,39]; analysis of gene-environment interactions, including genome wide association studies [40–44]; phylogenetic reconstruction [45,46,47]; mapping genes [48,49]; and derivations of classical population genetic results regarding drift and selection [50].…”

“…Previous attempts [12,14,55] fell short of general analytic expressions. For a single isolated population, Sherwin et al [12] used the diffusion approximation to predict equilibrium Shannon entropy under the infinite allele model (IAM) or stepwise mutation model (SMM).…”

“…For two populations connected by dispersal, with SMM, simulation results provided an empirical equation for mutual information at equilibrium, but no analytical equation was obtained [12]. Dewar et al [14] derived a Taylor approximation to mutual information for bi-allelic genes only. Even with this incomplete armory of methods, Sherwin et al [12] and Sherwin [13] showed firstly that for analysis of geographic subdivision and genetic exchange between sub-populations, mutual information readily yields an estimate of the dispersal rate per generation, and secondly that compared to all other approaches for analyzing such data, this method is robust to an extraordinarily wide range of dispersal rates and population sizes.…”