Abstract
A general theory is developed for the eigenvalue effective size (N_{eE}) of structured populations in which a gene with two alleles segregates in discrete time. Generalizing results of Ewens (Theor Popul Biol 21:373–378, 1982), we characterize N_{eE} in terms of the largest non-unit eigenvalue of the transition matrix of a Markov chain of allele frequencies. We use Perron–Frobenius Theorem to prove that the same eigenvalue appears in a linear recursion of predicted gene diversities between all pairs of subpopulations. Coalescence theory is employed in order to characterize this recursion, so that explicit novel expressions for N_{eE} can be derived. We then study N_{eE} asymptotically, when either the inverse size and/or the overall migration rate between subpopulations tend to zero. It is demonstrated that several previously known results can be deduced as special cases. In particular when the coalescence effective size N_{eC} exists, it is an asymptotic version of N_{eE} in the limit of large populations.
Highlights
The effective size Ne was introduced by Wright (1931, 1938) as the size of an ideal homogeneous population with the same rate of loss of heterozygosity per generation as the studied population
It is equivalent to the eigenvalue effective size NeE, defined in terms of the largest non-unit eigenvalue of a Markov chain of allele frequencies (Ewens 1982, 2004)
The nucleotide diversity or mutation effective size Neπ is essentially the expected coalescence time of a pair of haploid individuals (Ewens 1989; Durrett 2008), whereas the coalescent effective size NeC is defined for populations such that the ancestral tree of any finite number of individuals converges to a Kingman coalescent in the limit of large populations (Kingman 1982; Nordborg and Krone 2002; Sjödin et al 2005; Wakeley and Sargsyan 2009; Hössjer 2011)
Summary
The effective size Ne was introduced by Wright (1931, 1938) as the size of an ideal homogeneous population with the same rate of loss of heterozygosity per generation as the studied population. It is equivalent to the eigenvalue effective size NeE , defined in terms of the largest non-unit eigenvalue of a Markov chain of allele frequencies (Ewens 1982, 2004). Whitlock and Barton (1997) showed that these linear recursions are closely related They argued briefly that the transition matrix of the Markov chain of allele frequencies has its largest non-unit eigenvalue equal to λ, and all effective sizes of the previous paragraph agree with NeE. 2 we introduce the population genetic model and Ewens’ definition of NeE in terms of the rate at which the Markov process of allele frequencies in all subpopulations reaches an absorbing state, quantified by the largest non-unit eigenvalue λ of its transition matrix. As the size of a WF population for which the largest non-unit eigenvalue in (20) is the same as for the studied population
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