Theoretical Population Biology

The height of ancestral trees, in other words the time to the most recent common ancestor in a genealogical tree, is an informative statistic in population genomics and used in contexts of genealogical dating, demographic history inference, and assessing signals of selection. Exploiting a general pathwise identity between the height of a tree and its branch lengths, we revisit Zeng et al.s mutation rate estimator to develop a model-free estimator for the product of the expected height of genealogical trees with the scaled mutation rate, both for single loci and genomic regions. This estimator is a linear function of the site frequency spectrum of a sample, i.e., of the distribution of allele counts at all segregating sites across the locus or genomic region and thus both observable and computationally cheap. We show that, under the infinite sites model of mutation, our estimator is unbiased and, for two genome-wide models of ancestries (sequential Markovian coalescents and common-pedigree ancestries), consistent. Furthermore, we show via simulation that our estimator performs smaller errors than averaging over reconstructed ancestral tree heights (extracted from reconstructed ancestral recombination graphs) when considering genome properties similar to human genomes across large genomic regions. We then revisit a publicly available genomic dataset of dogs and wolves and compare our method with extracting TMRCAs from ancestral recombination graph reconstructions.

For decades, population geneticists have relied upon a formula derived by Malecot and by Kimura to estimate the fixation probability of mutant alleles ({phi}). Among other things, this formula leads to the conclusion that in sufficiently large populations{phi} asymptotically approaches 2s(Ne/N ) (for small s), where s is the relative selective advantage of the mutant allele, and Ne and N respectively denote the effective and absolute (haploid) population sizes. In contrast, in sufficiently small populations,{phi} = 1/N, with the two domains of behavior being separated at the point where s [bsime] 1/(2Ne). These results hold when s, N, and Ne remain constant during the fixation process, but require modification when populations are changing in size. Here, we show that if there are ecological effects associated with competing alleles, such that the genetic composition of the population influences the total population size, the fixation probability of a beneficial allele can substantially deviate from the levels suggested above, even in the domain of effective neutrality (i.e., | s | [lsim]1/(2Ne)). We obtain analytical results for a variety of frequency-dependent functions for the overall population size, and show that the overall effect is largely a function of the response at low mutant-allele frequency. We also derive expressions for times to fixation and for fixation probabilities of deleterious alleles, and show how our results relate to prior work for the situation in which population sizes change in frequency-independent manners.

Conventional LD measures such as r2 perform poorly in the rare common regime, particularly in asymmetric configurations such as nested haplotype structure. Because r2 is symmetric and quadratic, it removes directional structure in two ways: squaring discards the sign, or phase, retained by the signed LD coefficient D, while symmetric normalization hides the asymmetry between the conditional probabilities P(A|B) and P(B|A). Although D recovers the phase, it is locus symmetric and unnormalized; its magnitude is hard to compare across frequency regimes and it does not by itself express which way the asymmetry runs. We therefore analyze the conditional-probability asymmetry {Delta} = P(A|B) - P(B|A), together with r2 and D, as distinct scalar functions on the haplotype simplex under the Fisher information metric. The conditional probabilities P(A|B) and P(B|A) are bounded in [0, 1], directly express carrier-set inclusion, and are more readily visualized than D. Moreover, their difference admits the exact decomposition {Delta} = M + C into a marginal frequency term M and an LD-coupled term C. Prior work has characterized either the mathematical behavior of LD normalizations across allele-frequency space or the Fisher geometry of the haplotype simplex, but not their connection. We bridge this gap by showing that the geometric structure of the simplex explains why LD measures disagree in the rare common regime and why symmetric normalizations such as r2 lose directional information. We show that the fixed-frequency leaf is intrinsically anisotropic, positively curved, and frequency-dependent under the Fisher metric. These geometric predictions are tested empirically , in phased 1000 Genomes data1 and a two locus Wright Fisher model, in a companion paper (Ichikawa, preprint); the present note develops the geometry itself. Keywords: linkage disequilibrium; Fisher information metric; haplotype simplex; rare variant; conditional-probability asymmetry; nested haplotype structure

Aedes aegypti is a major vector of arboviruses, and belongs to subfamily Culicinae, a diverse group of mosquitoes with homomorphic sex-determining chromosomes. Males are the heterogametic sex with a dominant male-determining locus (M locus). The M locus and its counterpart m locus are embedded in a region of suppressed recombination, with a large portion of this recombination desert showing significant molecular differentiation despite homomorphy. We developed a mathematical framework to examine M-linked genome editors that specifically target the m-chromosome during spermatogenesis, mimicking the naturally occurring sex ratio distorters (SRDs) in Culicinae that produce male-biased meiotic drives. Unlike previous models for species with heteromorphic sex chromosomes (e.g., X and Y), we incorporate features stemming from the homomorphic nature of the Ae. aegypti sex chromosomes such as varied linkage to the M locus, making the degree of super-Mendelian inheritance readily tunable. We evaluated in silico SRDs with a range of M-linkage and editing efficiencies and established the theoretical foundation for developing highly efficient SRDs that outperform several methods of population suppression. These SRDs can be tuned to mitigate impact on a neighboring population. The framework developed here is suitable for exploring SRD-mediated genetic biocontrol of pests with homomorphic sex chromosomes.

Eusocial Hymenoptera present the highest known recombination rates among metazoans, which evolved several times independently among bees, ants and wasps. Several hypotheses have been proposed to explain this observation, including stronger selection for recombination caused by coevolving parasites and pathogens, and strong sexual selection among haploid males due to male-biased sex ratios among reproductive individuals. In this article, we explore the effects of haplodiploidy and differential selection between sexes on the evolution of recombination, by analyzing a three-locus model in which selection for recombination stems from negative epistasis between selected loci. Our analytical predictions are compared with the results of individual-based simulations in which deleterious mutations occur along a linear chromosome. Our results show that, at mutation-selection balance for deleterious alleles, increasing the strength of selection against deleterious alleles (due to the effect of male haploidy and/or sexual selection) tends to reduce selection for recombination. However, an increase in the overall magnitude of negative epistasis (which may also be due to male haploidy and/or sexual selection) combined with the fact that recombination only occurs in females may increase selection for recombination substantially. Our model also shows that, in conditions favoring recombination, increasing recombination in meioses leading to parthenogenetic ovules (and male offspring) may yield stronger benefits than in meioses leading to fertilized ovules (and female offspring).

Internal evolutionary conflicts arise when elements within an organism have diverging fitness interests. Examples range from meiotic drive and cytoplasmic male sterility to transposable elements and supernumerary B chromosomes. While once treated as genetic curiosities, they are now seen as widespread and major drivers of eukaryotic genome evolution. Yet their study remains fragmented, with no clear entry point not only for those who wish to gain an overview of theoretical advances, or those who wish to construct models of their own. Here, we discuss ways in which internal evolutionary conflicts have been modelled and develop a common population genetic framework for building such models. The framework provides explicit criteria for what counts as conflict, distinguishing it from fitness trade-offs, and formalises how and when internal conflicts arise. By treating different cases within the same structure, it shows that these diverse phenomena share a common logic.

The global potential of a chemical reaction network has many applications and is closely related to the stochastic detailed balance. However, many fundamental questions concerning stochastic detailed balance remain unresolved, such as whether it depends on the system volume and how to construct new systems that satisfy it. In this paper, we show that stochastic detailed balance may depend on the system volume. We therefore introduce four types of stochastic detailed balance according to their dependence on volume and rate constants, and systematically investigate the relationships among them. Our results distinguish detailed balance arising from particular choices of volume and parameters from that enforced by network structure, and identify conditions under which detailed balance at one volume extends to all volumes. We further obtain a class of networks satisfying stochastic detailed balance for every volume and every positive choice of rate constants, and construct new systems whose global potentials exhibit double-well structures.

Many endosymbioses in eukaryotes superficially appear to be beneficial to both participants. However, there is little direct evidence for this, and symbioses naturally set up conditions in which each member of the pair is under selection to extract resources from the other. Ultimately, the endosymbiont either evolves to be in conflict with the interests of the host or to act cooperatively with the host contrary to its own best interests. Focusing on obligate symbioses, we develop theory to clarify the population-genetic conditions favoring the alternative outcomes. The balance is usually tipped in favor of exploitation by the symbiont, particularly when the number of symbionts within host cells is high, selection is strong on symbionts relative to hosts, there is horizontal transfer of symbionts, and/or the symbionts have accelerated mutation rates or turnover times. If the symbiont conditions the host-cell biology to enhance within-host population sizes, selection for selfish symbionts will be further enhanced by the diminished level of within-host drift. Although the host evolves in parallel to exploit resources from the endosymbiont, the net result is often a stalemate in which the host is no better off than prior to host-symbiont coevolution. Strict vertical inheritance can result in an evolutionary alignment of interests of the endosymbiont and the host, as this minimizes the possibility of within-host selection, but even here there is a critical host population size below which the symbiont evolves to exploit the host. These results suggest that the evolutionary enslavement of a symbiont to benefit a host species requires a narrow mix of population-biological features of both participants.

Many proposed mechanisms for the evolution of cooperation among unrelated individuals rely on relatively demanding cognitive abilities that are not widespread across taxa. In contrast, individual heterogeneity is a pervasive feature of animal groups, encompassing differences in personality as well as physical and cognitive traits. Such heterogeneity can promote the evolution of cooperation, yet its role has received comparatively little attention, particularly as a source of variation giving rise to social organization such as leadership. A specific form of leadership can emerge under unstable environmental conditions, when some individuals become better suited than others to initiate action and influence the behavior of their peers. Unlike fixed dominance hierarchies, emergent leadership can rapidly adjust to changing environmental conditions, thereby reshaping group organization. Because it does not require the maintenance of stable hierarchies, this form of leadership can arise even in species that do not have the cognitive capabilities to sustain complex social structures. In this work, we investigate the combined effects of individual heterogeneity and emergent leadership on the evolution of cooperation using an evolutionary game-theoretic model in which individuals may assume the roles of leaders or followers according to their strength, representing individual differences in suitability to prevailing environmental conditions. We examine different levels of population heterogeneity together with increasingly complex strategy sets requiring progressively greater informational requirements, allowing individuals to condition cooperation on their own strength, leadership role, or both. Our results show that the interplay between leadership and heterogeneity promotes the evolution of cooperation, particularly when only a small fraction of individuals act as leaders. Under these circumstances, cooperation evolves even when individuals employ the simplest possible strategies. Under harsher ecological conditions, cooperation can be sustained by more sophisticated strategies, specifically by conditional strategies that prescribe cooperation when individuals are strong or leading and defect when acting independently. Author summaryIn this study, we propose that emergent leadership mediated by individual diversity can boost the evolution of cooperation in animal groups. Building on growing evidence on the heterogeneity of animal capabilities and personalities, we focus on the fleeting leadership that emerges in animal groups when facing rapidly changing environmental conditions. We suggest that this type of leadership that emerges from individual differences in strength--a generic quality encompassing those characteristics that make an individual more fit to lead in a given situation--does not require complex cognitive capabilities from the animals and represents a valid alternative to more demanding strategies proposed in the past to explain the evolution of cooperation. Using an evolutionary game theory model, we show that if a population includes a few strong players, these can become influential leaders and guide the actions of their peers to achieve cooperation. Although the naive strategy of always cooperating is sufficient for cooperation to evolve, the introduction of more complex strategies leads players to cooperate only when they are more likely to be recognized as influential leaders. These strategies are more effective in promoting cooperation under unfavorable ecological conditions and are also more robust against exploitation by defectors.

The general random walk model (GRW) of Hunt (2006) is used to infer directional evolution in mean trait values from sparse fossil data by modeling phenotypic change as the accumulated result of small steps with mean step sizes and step variances. Using simulations and real data cases, Ergon (2026) showed that the step variances can be estimated reasonably well only when the mean trait values have small measurement errors, while for fossil data with realistic measurement errors they appear to be extremely difficult to find, and they are often found to be negative. In the simulations Ergon (2026) assumed that the true phenotypic mean values were known. Here, I essentially repeat these simulations under the assumption that only mean trait values with large measurement errors are known, and based on weighted mean squared error (WMSE) comparisons the conclusion is that weighted least squares (WLS) is a better method than GRW. A second conclusion is that WLS is a better method also in the possibly rare cases with large measurement errors where the GRW parameters are estimated well. The GRW method is simply not flexible enough to handle such cases. A third conclusion is that Akaike Information Criterion (AIC) results for GRW models with large measurement errors relative to the step variance may be overly optimistic.

Multi-environment trials are central to cultivar evaluation because they reveal how candidate cultivars perform across locations, years, management conditions, and stress environments. The resulting yield matrix is a rich source of data on genotype-by-environment interaction, and a wide literature on estimation, decomposition, visualisation, and prediction of yield potential and stability has flourished. However the ultimate question of which cultivar to recommend on the basis of such a matrix is often left implicit. The question is far from trivial, and in this paper we formulate cultivar recommendation as an axiomatic ranking problem. This framework is rich enough to encompass the existing literature on stability indices, as well as any other deterministic ranking procedure. We show that many commonly used stability-based procedures can violate minimal criteria of efficiency or consistency. The result of such violations is that a cultivar with uniformly high yield could be ranked below a cultivar with uniformly low yield, or the relative ranks of two cultivars could depend on whether or not a third cultivar is present in the matrix. Our results prove that under a small number of such criteria the space of admissible rules collapses to the family of power means and their limiting cases. If we further wish to allow multiplication normalisation of yield, we are left with the geometric mean as the unique solution.

In many animal species, individuals acquire knowledge from others that enhances their survival and reproduction. However, among the many available cultural exemplars, not all provide reliable information. Consequently, individuals tend to choose their exemplars selectively. One widespread pattern is a preference for older individuals, who may have accumulated valuable knowledge through life. Yet empirical studies also show that individuals frequently learn from age peers, suggesting that copying elders is not universally optimal. The ecological and social conditions that favor learning from elders rather than peers, therefore, remain unclear. Here, we investigate the evolutionary drivers of age-biased exemplar choice in age-structured populations where individuals accumulate knowledge over their lifespan. We develop a model that captures the coevolution of exemplar age choice and age-specific investments in social learning, individual learning, and the use of acquired knowledge for energy extraction. We show that selection promotes a progressive shift from social to individual learning and from learning to energy extraction with age. Exemplar age choice, in turn, evolves through a trade-off between targeting knowledgeable individuals and accessible ones. This trade-off leads young learners to learn preferentially from relatively young exemplars, who are common and still able to provide substantial amounts of novel knowledge, given learners limited knowledge at early ages. As individuals age, encountering exemplars with substantially novel knowledge becomes increasingly difficult. Consequently, as they age, individuals are expected to shift toward learning from older individuals, who possess more knowledge. Population, environment, and knowledge characteristics can shift this balance, generating a wide range of strategies from learning primarily from peers to consistently targeting the oldest individuals. In particular, learning from age peers is favored in populations with strong within-cohort interaction structure, high mortality, or high encounter rates, in unstable environments with rapid knowledge loss, and when knowledge is easily acquired or transmitted.

In studies of assortative mating, similarity between variables measured in parents is often quantified using correlation. The order of the parents within any given pair can be arbitrary in these applications, but common correlation estimators are not robust to reordering within pairs. These unordered variable pairs are exchangeable, since the joint distributions of both orders are equal, and a given order is biased if the one variable has a lower expectation than the other. In this work, we characterize the effect of order bias on Pearson correlation estimates assuming exchangeable variables, and develop a new unbiased estimator, CorSym, that does not depend on order within each pair. Exchangeable variables have equal marginal distributions for both variables, a property accounted for by CorSym. In contrast, standard correlation estimators assume the two variables have different distributions, so biased orders skew the underlying mean, variance and covariance estimates. We show, through theory and simulations, how order bias often results in upwardly biased Pearson correlation estimates. Simulations confirm CorSym is unbiased, and validate its estimated confidence intervals. Using real admixed trios (parents and a child) from 1000 Genomes, we first demonstrate that the global ancestry of fathers and mothers are consistent with exchangeability, using both Kolmogorov-Smirnov tests and a Binomial test for order bias. However, ANCESTOR, which estimates parental global ancestry from a child's local ancestry, produces significant order biases in its output that result in substantial Pearson biases, which CorSym overcomes. Compared to ancestry proportions calculated directly on the parents, ANCESTOR also overestimates parent ancestry divergence and experiences another estimation artifact. Overall, CorSym solves an important estimation bias likely to be encountered in the study of assortative mating, providing unbiased and deterministic estimates that do not depend on the arbitrary order of the data.

Many questions in population genetics are concerned with reconstructing evolutionary history through time, such as inferring how population structure has changed throughout the past. Yet, many existing approaches have only an implicit temporal component, using quantities such as allele frequency or haplotype length as rough proxies for age. Recent advances in the inference of Ancestral Recombination Graphs (ARGs) have made it possible to estimate the entire sequence of local genealogies along the genome. These genealogies explicitly encode how samples are related to each other at different time points in the past, enabling the inference of how population structure has changed over time. To this end, recent work has used ARGs to define time-stratified versions of widely-used population genetics summary statistics in an attempt to capture the population structure present within a particular time window. Here, we show that naive approaches result in statistics that cannot be interpreted solely in terms of the population structure present within the time window they are targeting. To address this problem, we introduce a framework of coalescent-based time-stratified statistics, which use coalescence probabilities to partition classical summary statistics into interval-specific contributions. Using coalescent simulations, we demonstrate that these statistics accurately isolate population structure at different temporal depths and avoid spurious signals. Our results highlight the necessity of integrating coalescent theory into ARG-based temporal analyses and provide a principled and practical foundation for studying the dynamics of population structure through time.

Mathematical models, e.g. differential equations and stochastic processes, have gained considerable attention for understanding evolution of antibiotic resistance. However, most existing models assume standing genetic variation and do not consider the possibility of random or drug-induced mutation of reference bacterial strains. Therefore, we propose a pharmacokinetics/pharmacodynamics (PK/PD)-based continuous-time Markov chain considering the competition and mutation between sensitive and resistant bacterial within an infected host during treatment. The proposed model is approximated as a generalized birth-death process with immigration, allowing for explicit derivation of the probability resistant population establishes during treatment. Besides capturing the stochasticity of de novo emergence of a resistant bacterial strain, we explore the effects of different antibiotic modes of action, horizontal gene transfer, nutrient availability and drug pharmacokinetics on antibiotic resistance. We find that replication-targeting (biostatic) drugs suppress resistance more than death-targeting (biocidal) drugs. Like prior works, we obtain maximized resistance at intermediate drug concentrations, however the consideration of de novo mutation magnifies the superiority of higher doses in preventing resistance emergence.

Observed group sizes rarely match the size that would maximize what each member gets from belonging. We propose a two-part theory in which group size is regulated by two related conflicts: insider-outsider conflict over admission, and within-group conflict as crowding, competition, and social tensions intensify with size. Three strategies are available: admission, exclusion, and fission. The first part shows that even when exclusion is unavailable, fission dynamics alone drive group size away from the optimum in both directions, with the pattern set by how prospective joiners encounter groups and by the geometry of fission. When joiners compare groups across a shared landscape and fission is asymmetric, the standing distribution is bimodal: supra-optimal large groups coexisting with a sub-optimal mode of small groups, the pattern characteristic of fission-fusion societies. The second part promotes exclusion and fission to active decisions: incumbents weigh the per-capita cost of accommodating entry ({beta}) against the costs of coordinated exclusion (c +{gamma} N*) and fissioning (F). A single inequality, {beta} > c +{gamma} N*, partitions populations into two regimes: where it holds, exclusion is viable and groups lock at the optimum size; where it fails, groups grow past the optimum and cycle through recurrent fission. Modal group size, fission frequency, and exclusion behavior together identify which regime governs a population -- a set of predictions applicable across fishes, social insects, birds, and mammals including primates and human foragers.

Some organisms exhibit declining mortality and increasing fecundity following sexual maturity, a demographic pattern known as negative senescence. According to life history theory, ageing occurs because resources are preferentially allocated to reproduction over somatic maintenance. Models connecting indeterminate growth to negative senescence exist, but none integrate somatic maintenance as a competing allocation decision alongside growth and reproduction. We formulate a life history model in which an individual allocates energy among reproduction, somatic growth, and somatic maintenance and mortality rate depends on both body size and somatic damage. We show that negative actuarial senescence, whereby mortality declines with age, occurs when the proportional change in reproductive value exceeds the proportional change in fitness returns from current investments into reproduction and soma. We derive the necessary conditions for an uninvadable allocation strategy using invasion analysis and Pontryagin's maximum principle, and examine biologically relevant cases numerically. We show that both negative senescence and indeterminate growth arise together as uninvadable outcomes even when maintenance competes for the same resources as growth and reproduction. We show that both diminishing returns to reproduction and diminishing returns to growth can give rise to negative senescence. These results extend the disposable soma theory to organisms with indeterminate growth, in which mortality decreases with size, and identify key mechanisms for the empirically observed association between indeterminate growth and non-senescent demographic trajectories.

The evolution of recombination is thought to be influenced by many factors, including the mating system. Here, we provide an experimental test of how self-fertilization (selfing) affects the evolution of a recombination modifier. We used experimental populations of Caenorhabditis elegans segregating for the recombination modifier rec-1, a mutant that redistributes crossovers from the genetically diverse chromosome arms toward the less diverse central regions. By evolving populations under varying selfing rates, we show that increasing selfing reverses selection acting on the rec-1 mutant, from positive to negative. Simulations show that this reversal can be explained by an expansion of the genomic region over which the modifier remains associated with the genetic combinations it creates. These results demonstrate that selfing can fundamentally alter the evolutionary fate of recombination modifiers and reveal a mechanism not predicted by previous theoretical models of recombination evolution under different mating systems, which assumed uniform recombination landscapes.

The Gene Version Iteration Hypothesis (GVIH) proposes that mutant genes may originate from the Y chromosome, traverse through the X chromosome to autosomes, undergo interchromosomal transfer, and potentially return to the Y chromosome via the X chromosome. This hypothetical closed transmission loop may facilitate the storage, screening, and elimination of different versions of mutant genes. The hypothesis comprises five core propositions: (1) Mutation reservoir: The Y chromosome may serve as a specialized carrier for generating mutant genes, characterized by elevated mutation rates, reduced gene density, and accelerated evolutionary dynamics; (2) Closed-loop transmission: Mutant genes may follow a unidirectional pathway Y[->]X[->]autosomes[->]X[->]Y, forming a complete transmission circuit; (3) Coexistence of multiple versions: A single functional gene may exist in multiple versions across different chromosomes, constituting a dynamic gene version library; (4) Reproductive screening: Environmentally adaptive gene versions may persist across generations and potentially migrate to upstream chromosomes, while maladaptive versions may be eliminated; (5) Terminal elimination: Gene versions reaching the Y chromosome may undergo elimination processes, potentially preventing version monopolization and maintaining evolutionary dynamics. This hypothesis provides a novel framework for understanding adaptive evolution at the genetic level. If empirically validated, it may offer new insights into the molecular mechanisms underlying certain genetic phenomena and evolutionary processes.

Despite the impacts of within-host disease dynamics on disease outcomes in individual hosts and disease spread among-hosts, generic models of within-host population dynamics have received far less attention than their among-host counterparts. While a number of models have been proposed to explore theoretical eco-evolutionary dynamics, they have yet to be evaluated for estimability, raising questions about their ability to provide reliable inference when confronted with data. We evaluated the estimability of two generic within-host population dynamics models by assessing: (1) parameter estimation, our ability to recover correct values of model parameters from data, (2) the consequences of mis-assigning the underlying mechanistic model on parameter estimation, and (3) the reproduction of qualitative dynamics, or, our ability to use parameter estimates to reproduce observed dynamical behaviours. In some cases, fitting a mis-matched mechanistic model to time series data produced reasonable parameter estimates that were able to reproduce system dynamics, and that when provided the data-generating model, parameter uncertainty can produce substantial behavioural uncertainty. Our findings highlight the impacts of structural, parametric, and behavioural uncertainty on inference, and demonstrate the value of improving system-specific knowledge to prevent the use of incorrect functional forms and of measuring consequential parameters to improve estimability.