Self-incompatibility (SI) is a genetically based recognition system that functions to prevent self-fertilization and mating among related plants. An enduring puzzle in SI is how the high diversity observed in nature arises and is maintained. Based on the underlying recognition mechanism, SI can be classified into two main groups: self-recognition (SR) and nonself-recognition (NSR). Most work has focused on diversification within SR systems despite expected differences between the two groups in the evolutionary pathways and outcomes of diversification. Here, we use a deterministic population genetic model and stochastic simulations to investigate how novel S-haplotypes evolve in a gametophytic NSR [SRNase/S Locus F-box (SLF)] SI system. For this model, the pathways for diversification involve either the maintenance or breakdown of SI and can vary in the order of mutations of the female (SRNase) and male (SLF) components. We show analytically that diversification can occur with high inbreeding depression and self-pollination, but this varies with evolutionary pathway and level of completeness (which determines the number of potential mating partners in the population), and, in general, is more likely for lower haplotype number. The conditions for diversification are broader in stochastic simulations of finite population size. However, the number of haplotypes observed under high inbreeding and moderate-to-high self-pollination is less than that commonly observed in nature. Diversification was observed through pathways that maintain SI as well as through self-compatible intermediates. Yet the lifespan of diversified haplotypes was sensitive to their level of completeness. By examining diversification in a NSR SI system, this model extends our understanding of the evolution and maintenance of haplotype diversity observed in a recognition system common in flowering plants.

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...

Recent developments in automated tracking allow uninterrupted, high-resolution recording of animal trajectories, sometimes coupled with the identification of stereotyped changes of body pose or other behaviors of interest. Analysis and interpretation of such data represents a challenge: the timing of animal behaviors may be stochastic and modulated by kinematic variables, by the interaction with the environment or with the conspecifics within the animal group, and dependent on internal cognitive or behavioral state of the individual. Existing models for collective motion typically fail to incorporate the discrete, stochastic, and internal-state-dependent aspects of behavior, while models focusing on individual animal behavior typically ignore the spatial aspects of the problem. Here we propose a probabilistic modeling framework to address this gap. Each animal can switch stochastically between different behavioral states, with each state resulting in a possibly different law of motion through space. Switching rates for behavioral transitions can depend in a very general way, which we seek to identify from data, on the effects of the environment as well as the interaction between the animals. We represent the switching dynamics as a Generalized Linear Model and show that: (i) forward simulation of multiple interacting animals is possible using a variant of the Gillespie’s Stochastic Simulation Algorithm; (ii) formulated properly, the maximum likelihood inference of switching rate functions is tractably solvable by gradient descent; (iii) model selection can be used to identify factors that modulate behavioral state switching and to appropriately adjust model complexity to data. To illustrate our framework, we apply it to two synthetic models of animal motion and to real zebrafish tracking data.