A General Approximation for the Dynamics of Quantitative Traits

Abstract

Selection, mutation, and random drift affect the dynamics of allele frequencies and consequently of quantitative traits. While the macroscopic dynamics of quantitative traits can be measured, the underlying allele frequencies are typically unobserved. Can we understand how the macroscopic observables evolve without following these microscopic processes? This problem has been studied previously by analogy with statistical mechanics: the allele frequency distribution at each time point is approximated by the stationary form, which maximizes entropy. We explore the limitations of this method when mutation is small (4Nm , 1) so that populations are typically close to fixation, and we extend the theory in this regime to account for changes in mutation strength. We consider a single diallelic locus either under directional selection or with overdominance and then generalize to multiple unlinked biallelic loci with unequal effects. We find that the maximum-entropy approximation is remarkably accurate, even when mutation and selection change rapidly.KEYWORDS quantitative genetics; maximum entropy; diffusion approximation, quasi-stationarity M OST traits of interest have a complex genetic basis depending on very many loci. Quantitative genetics gives a sophisticated statistical description of the components of trait variance that can predict the immediate change due to selection. The present abundance of genetic markers allows us to find some of the loci that affect traits, but such QTL typically account for only a small fraction of the genetic variance (Hill and Kirkpatrick 2010;Yang et al. 2010). While we may be able to predict breeding values and estimate the distribution of effects, it does not seem possible-even in principle-to identify the individual alleles responsible for the bulk of heritable variance. Thus, we cannot hope to predict the evolution of quantitative traits by using a direct population genetics approach based on the frequencies of each individual allele.Here we develop a general method that allows us to closely approximate the evolution of quantitative traits knowing only the distribution of allelic effects and mutation rates but without requiring knowledge of individual allele frequencies. This can be seen as an extension of the classical infinitesimal model to include arbitrary gene interactions and the effects of selection, mutation, and drift on the genetic variance. It also can be viewed as a generalized version of the quasi-steadystate assumption (QSSA) that is often made in dynamical reaction systems (Segel and Slemrod 1989;Goeke and Walcher 2013) to noisy systems described by partial differential equations (PDEs), where the dynamics are approximated using a quasi-stationary distribution assumption (QSDA); here we use the maximum-entropy (MaxEnt) principle to define that distribution.In physics, the MaxEnt principle has a long history, starting with the seminal work of Jaynes (1957), who interpreted the Boltzmann distribution of statistical physics as the most random distribution sub...

Modes of Rapid Polygenic Adaptation

“…The derived population may thereby undergo a population size reduction (bottleneck). It is therefore important to study polygenic selection for populations that are finite in size (Bod'ová et al, 2016;Franssen et al, 2017) and may undergo size changes in time.…”

Well posedness and maximum entropy approximation for the dynamics of quantitative traits

“…The surprisingly good approximation properties of the DynMaxEnt method, as documented by the numerical results in [6] and Section 6 of this paper, suggest that the infinitely-dimensional dynamics of the Fokker-Planck equation (1.1) can be well approximated by suitable finitely-dimensional dynamical systems. This is reminiscent of the recent series of works of E. Titi and collaborators [18,12,2,17,1] where a data assimilation (downscaling) approach to fluid flow problems is developed, inspired by ideas applied for designing finite-parameters feedback control for dissipative systems.…”

“…It has been applied, e.g., to modeling of cosmic ray transport [14], general Fokker-Planck equation [21], analysis of genetic algorithms [22], and population genetics [23,4,6]. In [6] it is observed that the "classical" DynMaxEnt method cannot be applied in the regime of small mutations, and the theory is extended for this regime to account for changes in mutation strength. Surprisingly, systematic numerical simulations document superb approximation properties of the method even far from the quasi-stationary regime.…”

“…where u α is the normalized stationary solution of the Fokker-Planck equation (1.1), given by formula (2.1). Note that this is a different approach compared with [6], where the logarithmic entropy is taken relative to the neutral distribution of allele frequencies in the absence of mutation or selection, L i=1 ξ −1 i , and the variational problem is complemented with normalization and moment constraints.…”

“…The diffusion term captures the stochasticity of the allele frequencies arising from random sampling. Here we assume that linkage disequilibria are negligible, otherwise this term would be of cross-diffusion type, reflecting correlations between loci [6]. The drift term captures deterministic effects on allele frequencies that are described by a vector of coefficients α and a vector of complementary quantities A.…”