THEWEIGHT: A simple and flexible algorithm for simulating non-ideal, age-structured populations

1. The Wright-Fisher model, which directs how matings occur and how genes are transmitted across generations, has long been a lynchpin of evolutionary biology. This model is elegantly simple, analytically tractable, and easy to implement, but it has one serious limitation: essentially no real species satisfies its many assumptions. With growing awareness of the importance of jointly considering both ecology and evolution in eco-evolutionary models, this limitation has become more apparent, causing many researchers to search for more realistic simulation models. 2. A recently described variation retains most of the Wright-Fisher simplicity but provides greater flexibility to accommodate departures from model assumptions. This generalized Wright-Fisher model relaxes the assumption that all individuals have identical expected reproductive success by introducing a vector of parental weights w that specifies relative probabilities different individuals have of producing offspring. With parental weights specified this way, expectations of key demographic parameters are simple functions of w. This allows researchers to quantitatively predict the consequences of non-Wright-Fisher features incorporated into their models. 3. An important limitation of the Wright-Fisher model is that it assumes discrete generations, whereas most real species are age-structured. Here I show how an algorithm (THEWEIGHT) that implements the generalized Wright-Fisher model can be used to model evolution in age-structured populations with overlapping generations. Worked examples illustrate simulation of seasonal and lifetime reproductive success and show how the user can pick vectors of weights expected to produce a desired level of reproductive skew or a desired Ne/N ratio. Alternatively, weights can be associated with heritable traits to provide a simple, quantitative way to model natural selection. Using THEWEIGHT, it is easy to generate positive or negative correlations of individual reproductive success over time, thus allowing explicit modeling of common biological processes like skip breeding and persistent individual differences. 4. R code is provided to implement basic features of THEWEIGHT and applications described here. However, required coding changes to the Wright-Fisher model are modest, so the real value of the new algorithm is to encourage users to adopt its features into their own or others models.

Weak approximations of Wright-Fisher equation

We construct weak approximations of the Wright-Fisher model and illustrate their accuracy by simulation examples.

Diffusion approximations in population genetics and the rate of Muller's ratchet

Diffusion theory is a central tool of modern population genetics, yielding simple expressions for fixation probabilities and other quantities that are not easily derived from the underlying Wright-Fisher model. Unfortunately, the textbook derivation of diffusion equations as scaling limits requires evolutionary parameters (selection coefficients, mutation rates) to scale like the inverse population size---a severe restriction that does not always reflect biological reality. Here we note that the Wright-Fisher model can be approximated by diffusion equations under more general conditions, including in regimes where selection and/or mutation are strong compared to genetic drift. As an illustration, we use a diffusion approximation of the Wright-Fisher model to improve estimates for the expected time to fixation of a strongly deleterious allele, i.e. the rate of Muller's ratchet.

AdmixSim 2: a forward-time simulator for modeling complex population admixture

Background Computer simulations have been widely applied in population genetics and evolutionary studies. A great deal of effort has been made over the past two decades in developing simulation tools. However, there are not many simulation tools suitable for studying population admixture. Results We here developed a forward-time simulator, AdmixSim 2, an individual-based tool that can flexibly and efficiently simulate population genomics data under complex evolutionary scenarios. Unlike its previous version, AdmixSim 2 is based on the extended Wright-Fisher model, and it implements many common evolutionary parameters to involve gene flow, natural selection, recombination, and mutation, which allow users to freely design and simulate any complex scenario involving population admixture. AdmixSim 2 can be used to simulate data of dioecious or monoecious populations, autosomes, or sex chromosomes. To our best knowledge, there are no similar tools available for the purpose of simulation of complex population admixture. Using empirical or previously simulated genomic data as input, AdmixSim 2 provides phased haplotype data for the convenience of further admixture-related analyses such as local ancestry inference, association studies, and other applications. We here evaluate the performance of AdmixSim 2 based on simulated data and validated functions via comparative analysis of simulated data and empirical data of African American, Mexican, and Uyghur populations. Conclusions AdmixSim 2 is a flexible simulation tool expected to facilitate the study of complex population admixture in various situations.

Boundary conformal field theory at large charge

We study operators with large internal charge in boundary conformal field theories (BCFTs) with internal symmetries. Using the state-operator correspondence and the existence of a macroscopic limit, we find a non-trivial relation between the scaling dimension of the lowest dimensional CFT and BCFT charged operators to leading order in the charge. We also construct the superfluid effective field theory for theories with boundaries and use it to systematically calculate the BCFT spectrum in a systematic expansion. We verify explicitly many of the predictions from the EFT analysis in concrete examples including the classical conformal scalar field with a |ϕ|6 interaction in three dimensions and the O(2) Wilson-Fisher model near four dimensions in the presence of boundaries. In the appendices we additionally discuss a systematic background field approach towards Ward identities in general boundary and defect conformal field theories, and clarify its relation with Noether’s theorem in perturbative theories.

Two Applications of the Analytic Conformal Bootstrap: A Quick Tour Guide

We reviewed the recent developments in the study of conformal field theories in generic space time dimensions using the methods of the conformal bootstrap, in its analytic aspect. These techniques are solely based on symmetries, particularly on the analytic structure and in the associativity of the operator product expansion. We focused on two applications of the analytic conformal bootstrap: the study of the ϵ expansion of the Wilson–Fisher model via the introduction of a dispersion relation and the large N expansion of the maximally supersymmetric Super Yang–Mills theory in four dimensions.

Applications of dispersive sum rules: $ε$-expansion and holography

We use Mellin space dispersion relations together with Polyakov conditions to derive a family of sum rules for Conformal Field Theories (CFTs). The defining property of these sum rules is suppression of the contribution of the double twist operators. Firstly, we apply these sum rules to the Wilson-Fisher model in d=4-\epsilond=4−ϵ dimensions. We re-derive many of the known results to order \epsilon^4ϵ4 and we make new predictions. No assumption of analyticity down to spin 00 was made. Secondly, we study holographic CFTs. We use dispersive sum rules to obtain tree-level and one-loop anomalous dimensions. Finally, we briefly discuss the contribution of heavy operators to the sum rules in UV complete holographic theories.

Accessing large global charge via the ϵ-expansion

We compute the lowest operator dimension ∆(J; D) at large global charge J in the O(2) Wilson-Fisher model in D = 4 − ϵ dimensions, to leading order in both 1/J and ϵ. While the effective field theory approach of [1] could only determine ∆(J; 3) as a series expansion in 1/J up to an undetermined constant in front of each term, this time we try to determine the coefficient in front of J3/2 in the ϵ-expansion. The final result for ∆(J; D) in the (resummed) ϵ-expansion, valid when J ≫ 1/ϵ ≫ 1, turns out to be$$ \Delta \left(J;D\right)=\left[\frac{2\left(D-1\right)}{3\left(D-2\right)}{\left(\frac{9\left(D-2\right)\pi }{5D}\right)}^{\frac{D}{2\left(D-1\right)}}{\left[\frac{5\Gamma \left(\frac{D}{2}\right)}{24{\pi}^2}\right]}^{\frac{1}{D-1}}{\epsilon}^{\frac{D-1}{2\left(D-1\right)}}\right]\times {J}^{\frac{D}{D-1}}+O\left({J}^{\frac{D-2}{D-1}}\right) $$ Δ J D = 2 D − 1 3 D − 2 9 D − 2 π 5 D D 2 D − 1 5 Γ D 2 24 π 2 1 D − 1 ϵ D − 1 2 D − 1 × J D D − 1 + O J D − 2 D − 1 where next-to-leading order onwards were not computed here due to technical cumbersomeness, despite there are no fundamental difficulties. We also compare the result at ϵ = 1,$$ \Delta (J)=0.293\times {J}^{3/2}+\cdots $$ Δ J = 0.293 × J 3 / 2 + ⋯ to the actual data from the Monte-Carlo simulation in three dimensions [2], and the discrepancy of the coefficient 0.293 from the numerics turned out to be 13%. Additionally, we also find a crossover of ∆(J; D) from ∆(J) ∝ $$ {J}^{\frac{D}{D-1}} $$ J D D − 1 to ∆(J) ∝ J, at around J ∼ 1/ϵ, as one decreases J while fixing ϵ (or vice versa), reflecting the fact that there are no interacting fixed-point at ϵ = 0. Based on this behaviour, we propose an interesting double-scaling limit which fixes λ ≡ Jϵ, suitable for probing the region of the crossover. I will give ∆(J; D) to next-to-leading order in perturbation theory, either in 1/λ or in λ, valid when λ ≫ 1 and λ ≪ 1, respectively.

Moment-based approximations for the Wright-Fisher model of population dynamics under natural selection at two linked loci

Properly modelling genetic recombination and local linkage has been shown to bring a significant improvement to the inference of natural selection from time series genetic data under a Wright-Fisher model. Existing approaches that can take genetic recombination effect and local linkage information into account are built upon either the diffusion approximation or the moment-based approximation of the Wright-Fisher model. However, such approximations are either limited to the increased computational cost like the diffusion approximation or suffer from the distribution support issue like the normal approximation, which can seriously affect computational efficiency and accuracy. In this work, we propose two novel moment-based approximations for the Wright-Fisher model of population dynamics subject to natural selection at a pair of linked loci. Our key innovation is that we extend existing approaches to calculate the mean and (co)variance of the two-locus Wright-Fisher model with selection and suggest a logistic normal distribution or a hierarchical beta distribution as a parametric continuous probability distribution to approximate the Wright-Fisher model by matching its first two moments to those of the Wright-Fisher model. Compared with the diffusion approximation, our approximations enable the computation of the transition probability distribution of the Wright-Fisher model at a far smaller computational cost and also allow us to avoid the distribution support issue found in the normal approximation.