本文关注生物进化能够发生的最大速度。我推导了若干不等式，用以限制由自然选择、突变或遗传漂变驱动的进化过程的速率。这些“速率限制”将种群中的变异性与进化速率联系起来。具体而言，种群适合度以及某一数量性状的高方差使该性状的平均值能够快速变化。相比之下，低变异性会使性状较不易受到由遗传漂变引起的随机变化影响。本文的结果通过约束任意性状进化速率的权衡关系，将 Fisher 的自然选择基本定理推广到允许突变和遗传漂变的动力学。这些速率限制可用于探究各种进化生物学和生态学情境中的问题。例如，它们适用于物种内部或跨物种的性状动态，也适用于细菌菌株的进化。它们适用于任何数量性状，例如从物种体重到 DNA 链长度。
英文摘要：
This paper focuses on the maximum speed at which biological evolution can occur. I derive inequalities that limit the rate of evolutionary processes driven by natural selection, mutations, or genetic drift. These \emph{rate limits} link the variability in a population to evolutionary rates. In particular, high variances in the fitness of a population and of a quantitative trait allow for fast changes in the trait's average. In contrast, low variability makes a trait less susceptible to random changes due to genetic drift. The results in this article generalize Fisher's fundamental theorem of natural selection to dynamics that allow for mutations and genetic drift, via trade-off relations that constrain the evolutionary rates of arbitrary traits. The rate limits can be used to probe questions in various evolutionary biology and ecology settings. They apply, for instance, to trait dynamics within or across species or to the evolution of bacteria strains. They apply to any quantitative trait, e.g., from species' weights to the lengths of DNA strands.
Molecular traits, such as gene expression levels or protein binding affinities, are increasingly accessible to quantitative measurement by modern high-throughput techniques. Such traits measure molecular functions and, from an evolutionary point of view, are important as targets of natural selection. We review recent developments in evolutionary theory and experiments that are expected to become building blocks of a quantitative genetics of molecular traits. We focus on universal evolutionary characteristics: these are largely independent of a trait's genetic basis, which is often at least par…
Many biological characteristics of evolutionary interest are not scalar variables but continuous functions. Given a dataset of function-valued traits generated by evolution, we develop a practical statistical approach to infer ancestral function-valued traits, and estimate the generative evolutionary process. We do this by combining dimension reduction and phylogenetic Gaussian process regression, a nonparametric procedure which explicitly accounts for known phylogenetic relationships. We test the methods' performance on simulated function-valued data generated from a stochastic evolutionary m…
How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. Population dynamics are driven by births and deaths, mutations at birth, and competition between individuals. Trait values influence ecological processes (demographic events, competition), and competition generates selection on trait variation, thus closing the eco-evolutionary feedback loop. The demographic effec…
The branching structure of biological evolution confers statistical dependencies on phenotypic trait values in related organisms. For this reason, comparative macroevolutionary studies usually begin with an inferred phylogeny that describes the evolutionary relationships of the organisms of interest. The probability of the observed trait data can be computed by assuming a model for trait evolution, such as Brownian motion, over the branches of this fixed tree. However, the phylogenetic tree itself contributes statistical uncertainty to estimates of other evolutionary quantities, and many compa…
When studying the dynamics of trait distribution of populations in a heterogeneous environment, classical models from quantitative genetics choose to look at its system of moments, specifically the first two ones. Additionally, in order to close the resulting system of equations, they often assume that the trait distribution is Gaussian (see for instance Ronce and Kirkpatrick 2001). The aim of this paper is to introduce a mathematical framework that follows the whole trait distribution (without prior assumption) to study evolutionary dynamics of sexually reproducing populations. Specifically, …
Natural Averaging May Complement Known Biological Constraints in Bi-parental Reproduction's Advantages Over Mono-parental in Conserving Species Quantitative Traits
Evaluating Phylogenetic Comparative Methods under Reticulate Evolutionary Scenarios
Texas A&M University（德克萨斯农工大学）；Lakehead University（湖首大学）；IISD Experimental Lakes Area（国际可持续发展研究所实验湖区）；The Institute for Computational and Experimental Research in Mathematics, Brown University（布朗大学计算与实验数学研究所）；RIKEN iTHEMS（理化学研究所创新智能数学科学中心）
Inferring dependencies between complex biological traits while accounting for evolutionary relationships between specimens is of great scientific interest yet remains infeasible when trait and specimen counts grow large. The state-of-the-art approach uses a phylogenetic multivariate probit model to accommodate binary and continuous traits via a latent variable framework, and utilizes an efficient bouncy particle sampler (BPS) to tackle the computational bottleneck -- integrating many latent variables from a high-dimensional truncated normal distribution. This approach breaks down as the number…
Inferring concerted changes among biological traits along an evolutionary history remains an important yet challenging problem. Besides adjusting for spurious correlation induced from the shared history, the task also requires sufficient flexibility and computational efficiency to incorporate multiple continuous and discrete traits as data size increases. To accomplish this, we jointly model mixed-type traits by assuming latent parameters for binary outcome dimensions at the tips of an unknown tree informed by molecular sequences. This gives rise to a phylogenetic multivariate probit model. Wi…
Molecular phenotypes link genomic information with organismic functions, fitness, and evolution. Quantitative traits are complex phenotypes that depend on multiple genomic loci. In this paper, we study the adaptive evolution of a quantitative trait under time-dependent selection, which arises from environmental changes or through fitness interactions with other co-evolving phenotypes. We analyze a model of trait evolution under mutations and genetic drift in a single-peak fitness seascape. The fitness peak performs a constrained random walk in the trait amplitude, which determines the time-dep…
Many biological characteristics of evolutionary interest are not scalar variables but continuous functions. Here we use phylogenetic Gaussian process regression to model the evolution of simulated function-valued traits. Given function-valued data only from the tips of an evolutionary tree and utilising independent principal component analysis (IPCA) as a method for dimension reduction, we construct distributional estimates of ancestral function-valued traits, and estimate parameters describing their evolutionary dynamics.
We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. …
C$_4$ photosynthesis has independently evolved from the ancestral C$_3$ pathway in at least 60 plant lineages, but, as with other complex traits, how it evolved is unclear. Here we show that the polyphyletic appearance of C$_4$ photosynthesis is associated with diverse and flexible evolutionary paths that group into four major trajectories. We conducted a meta-analysis of 18 lineages containing species that use C$_3$, C$_4$, or intermediate C$_3$-C$_4$ forms of photosynthesis to parameterise a 16-dimensional phenotypic landscape. We then developed and experimentally verified a novel Bayesian a…
We study a general setting of neutral evolution in which the population is of finite, constant size and can have spatial structure. Mutation leads to different genetic types ("traits"), which can be discrete or continuous. Under minimal assumptions, we show that the marginal trait distributions of the evolutionary process, which specify the probability that any given individual has a certain trait, all converge to the stationary distribution of the mutation process. In particular, the stationary frequencies of traits in the population are independent of its size, spatial structure, and evoluti…
Sexual selection theory models evolution of sexual signals and preferences using simple life histories. However, life-history models predict that males benefit from increasing sexual investment approaching old age, producing age-dependent sexual traits. Age-dependent traits require time and energy to grow, and will not fully mature before individuals enter mating competition. Early evolutionary stages pose several problems for these traits. Age-dependent traits suffer from strong viability selection and gain little benefit from mate choice when rare. Few males will grow large traits, and they …
The Tangled Nature Model of biological and cultural evolution features interacting agents which compete for limited resources and reproduce in an error prone fashion and at a rate depending on the `tangle' of interactions they maintain with others. The set of interactions linking a TNM individual to others is key to its reproductive success and arguably constitutes its most important property. Yet, in many studies, the interactions of an individual and those of its mutated off-spring are unrelated, a rather unrealistic feature corresponding to a point mutation turning a giraffe into an elephan…
We consider a simple mathematical model of gradual Darwinian evolution in continuous time and continuous trait space, due to intraspecific competition for common resource in an asexually reproducing population in constant environment, while far from evolutionary stable equilibrium. The model admits exact analytical solution. In particular, Gaussian distribution of the trait emerges from generic initial conditions.
Understanding the processes that give rise to quantitative measurements associated with molecular sequence data remains an important issue in statistical phylogenetics. Examples of such measurements include geographic coordinates in the context of phylogeography and phenotypic traits in the context of comparative studies. A popular approach is to model the evolution of continuously varying traits as a Brownian diffusion process. However, standard Brownian diffusion is quite restrictive and may not accurately characterize certain trait evolutionary processes. Here, we relax one of the major res…
Statistical analysis of alignments of large numbers of protein sequences has revealed "sectors" of collectively coevolving amino acids in several protein families. Here, we show that selection acting on any functional property of a protein, represented by an additive trait, can give rise to such a sector. As an illustration of a selected trait, we consider the elastic energy of an important conformational change within an elastic network model, and we show that selection acting on this energy leads to correlations among residues. For this concrete example and more generally, we demonstrate tha…
We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate L, conditioned on the observed number of particles to be equal n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance s2. In this paper we treat $n$ particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator Rn2 of the evolutionary rate r2=s2/L. The estimator Rn2 is propo…
Horizontal transfer (HT) of heritable information or `traits' (carried by genetic elements, endosymbionts, or culture) is widespread among living organisms. Yet current ecological and evolutionary theory addressing HT is limited. We present a modeling framework for the dynamics of two populations that compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We capture this feedback with a stochastic individual-based model, from whic…
Mortality is an instrument of natural selection. Evolutionary motivated theories imply its irreversibility and life history dependence. This is inconsistent with mortality data for protected populations. Accurate analysis yields mortality law, which is specific for their evolutionary unprecedented conditions, yet universal for species as evolutionary remote as humans and flies. The law is exact, instantaneous, reversible, stepwise, and allows for a rapid (within less than two years for humans) and significant mortality decrease at any (but very old) age.
We present a mathematical simplification for the evolutionary dynamics of a heritable trait within a two-sex population. This trait is assumed to control the timing of sex-specific life-history events, such as the age of sexual maturity and end of female fertility, and each sex has a distinct fitness tradeoff associated with the trait. We provide a formula for the fitness landscape of the population and show a natural extension of the result to an arbitrary number of heritable traits. Our method can be viewed as a dynamical systems generalisation of the Price Equation to include two sexes{, ag…
We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence t…
In evolutionary dynamics, a key measure of a mutant trait's success is the probability that it takes over the population given some initial mutant-appearance distribution. This "fixation probability" is difficult to compute in general, as it depends on the mutation's effect on the organism as well as the population's spatial structure, mating patterns, and other factors. In this study, we consider weak selection, which means that the mutation's effect on the organism is small. We obtain a weak-selection perturbation expansion of a mutant's fixation probability, from an arbitrary initial config…
Geographic ranges of communities of species evolve in response to environmental, ecological, and evolutionary forces. Understanding the effects of these forces on species' range dynamics is a major goal of spatial ecology. Previous mathematical models have jointly captured the dynamic changes in species' population distributions and the selective evolution of fitness-related phenotypic traits in the presence of an environmental gradient. These models inevitably include some unrealistic assumptions, and biologically reasonable ranges of values for their parameters are not easy to specify. As a …
The adaptation of biological species to their environment depends on their traits. When various biological processes occur (survival, reproduction, migration, etc.), the trait distribution may change with respect to time and space. In the context of invasions, when considering the evolution of a heritable trait that encodes the dispersive ability of individuals, the trait distribution develops a particular spatial structure that leads to the acceleration of the front propagation. That phenomenon is known as spatial sorting. Many biological examples can be cited like the bush cricket in Britain…
We study the effects of demographic stochasticity on the long-term dynamics of biological coevolution models of community assembly. The noise is induced in order to check the validity of deterministic population dynamics. While mutualistic communities show little dependence on the stochastic population fluctuations, predator-prey models show strong dependence on the stochasticity, indicating the relevance of the finiteness of the populations. For a predator-prey model, the noise causes drastic decreases in diversity and total population size. The communities that emerge under influence of the …
We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large $n$ the normalized sample mean is approximately normal…
In his highly influential view of evolution, G. G. Simpson hypothesized that clades of species evolve in adaptive zones, defined as collections of niches occupied by species with similar traits and patterns of habitat use. Simpson hypothesized that species enter new adaptive zones in one of three ways: extinction of competitor species, dispersal to a new geographic region, or the evolution of a key trait that allows species to exploit resources in a new way. However, direct tests of Simpson's hypotheses for the entry into new adaptive zones remain elusive. Here we evaluate the fit of a Simpson…
We study fixation probabilities and times as a consequence of neutral genetic drift in subdivided populations, motivated by a model of the cultural evolutionary process of language change that is described by the same mathematics as the biological process. We focus on the growth of fixation times with the number of subpopulations, and variation of fixation probabilities and times with initial distributions of mutants. A general formula for the fixation probability for arbitrary initial condition is derived by extending a duality relation between forwards- and backwards-time properties of the m…
Understanding which phenotypic traits are consistently correlated throughout evolution is a highly pertinent problem in modern evolutionary biology. Here, we propose a multivariate phylogenetic latent liability model for assessing the correlation between multiple types of data, while simultaneously controlling for their unknown shared evolutionary history informed through molecular sequences. The latent formulation enables us to consider in a single model combinations of continuous traits, discrete binary traits and discrete traits with multiple ordered and unordered states. Previous approache…
One of the main properties of biological systems is modularity, which manifests itself at all levels of their organization, starting with the level of molecular genetics, ending with the level of whole organisms and their communities. In a simplified form, these basic principles were transferred from the genetics of populations to the field of evolutionary computations, in order to solve applied optimization problems. Over almost half a century of development in this field of computer science, considerable practical experience has been gained and interesting theoretical results have been obtai…
The phenomenon of preadaptation, or exaptation (wherein a trait that originally evolved to solve one problem is co-opted to solve a new problem) presents a formidable challenge to efforts to describe biological phenomena using a classical (Kolmogorovian) mathematical framework. We develop a quantum framework for exaptation with examples from both biological and cultural evolution. The state of a trait is written as a linear superposition of a set of basis states, or possible forms the trait could evolve into, in a complex Hilbert space. These basis states are represented by mutually orthogonal…
Concomitant with the evolution of biological diversity must have been the evolution of mechanisms that facilitate evolution, due to the essentially infinite complexity of protein sequence space. We describe how evolvability can be an object of Darwinian selection, emphasizing the collective nature of the process. We quantify our theory with computer simulations of protein evolution. These simulations demonstrate that rapid or dramatic environmental change leads to selection for greater evolvability. The selective pressure for large scale genetic moves, such as DNA exchange, becomes increasingl…
Background. Organisms use a variety of mechanisms to protect themselves against perturbations. For example, repair mechanisms fix damage, feedback loops keep homeostatic systems at their setpoints, and biochemical filters distinguish signal from noise. Such buffering mechanisms are often discussed in terms of robustness, which may be measured by reduced sensitivity of performance to perturbations. Methodology/Principal Findings. I use a mathematical model to analyze the evolutionary dynamics of robustness in order to understand aspects of organismal design by natural selection. I focus on two …
In the classic view introduced by R. A. Fisher, a quantitative trait is encoded by many loci with small, additive effects. Recent advances in QTL mapping have begun to elucidate the genetic architectures underlying vast numbers of phenotypes across diverse taxa, producing observations that sometimes contrast with Fisher's blueprint. Despite these considerable empirical efforts to map the genetic determinants of traits, it remains poorly understood how the genetic architecture of a trait should evolve, or how it depends on the selection pressures on the trait. Here we develop a simple, populati…
Biological organisms adapt to changes by processing informations from different sources, most notably from their ancestors and from their environment. We review an approach to quantify these informations by analyzing mathematical models of evolutionary dynamics, and show how explicit results are obtained for a solvable subclass of these models. In several limits, the results coincide with those obtained in studies of information processing for communication, gambling or thermodynamics. In the most general case, however, information processing by biological populations shows unique features tha…
We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability $μ$ of mutation at each birth event. We introduce a renormalization parameter $K$ scaling the size of the population, which leads, when $K\to+\infty$, to a deterministic dynamics for the density of individuals holding a given trait. By combining in a non-standard way the limits of large population ($K\to+\infty$) and of small mutations ($μ\to 0$), we prove that a time scal…
evolSOM: an R Package for evolutionary conservation analysis with SOMs
Universidad Nacional del Litoral（国立滨海大学）；Research Institute for Signals, Systems and Computational Intelligence, sinc(i)（信号、系统与计算智能研究所）；CONICET（国家科学技术研究委员会）
Hierarchical Conditioning of Diffusion Models Using Tree-of-Life for Studying Species Evolution
Virginia Tech（弗吉尼亚理工大学）；Oak Ridge National Laboratory（橡树岭国家实验室）；University of California, Irvine（加利福尼亚大学欧文分校）；Tulane University（杜兰大学）；Battelle（巴特尔公司）；Duke University（杜克大学）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；The Ohio State University（俄亥俄州立大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
Identifying undocumented or potential future interactions among species is a challenge facing modern ecologists. Recent link prediction methods rely on trait data, however large species interaction databases are typically sparse and covariates are limited to only a fraction of species. On the other hand, evolutionary relationships, encoded as phylogenetic trees, can act as proxies for underlying traits and historical patterns of parasite sharing among hosts. We show that using a network-based conditional model, phylogenetic information provides strong predictive power in a recently published g…
University of Birmingham（伯明翰大学）；University of Maryland School of Medicine（马里兰大学医学院）；Baltimore VA Medical Center（巴尔的摩退伍军人事务医疗中心）
Adaptation to changing environments is a hallmark of biological systems. Diversity in traits is necessary for adaptation and can influence the survival of a population faced with novelty. In habitats that remain stable over many generations, stabilizing selection reduces trait differences within populations, thereby appearing to remove the diversity needed for heritable adaptive responses in new environments. Paradoxically, field studies have documented numerous populations under long periods of stabilizing selection and evolutionary stasis that have rapidly evolved under changed environmental…
Anthropogenic changes of the biota and human hyper-dominance are modulating the evolution of life on our planet. Humankind has spread worldwide supported by cultural and technological knowledge, and has already modified uncountable biological interactions. While numerous species have been extinguished by human actions, others are directly favored, such as alien species, hybrids, and genetically modified organisms. These biodiversity shifts have generated new interactions among all living organisms in anthropized or anthropogenic ecosystems, with the consequent establishment of novel evolutiona…
In biology, the evolution of increasingly cooperative groups has shaped the history of life. Genes collaborate in the control of cells; cells efficiently divide tasks to produce cohesive multicellular individuals; individual members of insect colonies cooperate in integrated societies. Biological cooperation provides a foundation on which to understand human behavior. Conceptually, the economics of efficient allocation and the game-like processes of strategy are well understood in biology; we find the same essential processes in many successful theories of human sociality. Historically, the tr…
The evolution of complex molecular traits such as disulphide bridges often requires multiple mutations. The intermediate steps in such evolutionary trajectories are likely to be selectively neutral or deleterious. Therefore, large populations and long times may be required to evolve such traits. We propose that errors in transcription and translation may allow selection for the intermediate mutations if the final trait provides a large enough selective advantage. We test this hypothesis using a population based model of protein evolution. If an individual acquires one of two mutations needed f…
University of Washington（华盛顿大学）；University of Calgary（卡尔加里大学）
We consider the effects of social learning on the individual learning and genetic evolution of a colony of artificial agents capable of genetic, individual and social modes of adaptation. We confirm that there is strong selection pressure to acquire traits of individual learning and social learning when these are adaptive traits. We show that selection pressure for learning of either kind can supress selection pressure for reproduction or greater fitness. We show that social learning differs from individual learning in that it can support a second evolutionary system that is decoupled from the…
Phylogenetic comparative methods for real-valued traits usually make use of stochastic process whose trajectories are continuous. This is despite biological intuition that evolution is rather punctuated than gradual. On the other hand, there has been a number of recent proposals of evolutionary models with jump components. However, as we are only beginning to understand the behaviour of branching Ornstein-Uhlenbeck (OU) processes the asymptotics of branching OU processes with jumps is an even greater unknown. In this work we build up on a previous study concerning OU with jumps evolution on a …
A major aim of evolutionary biology is to explain the respective roles of adaptive versus non-adaptive changes in the evolution of complexity. While selection is certainly responsible for the spread and maintenance of complex phenotypes, this does not automatically imply that strong selection enhances the chance for the emergence of novel traits, that is, the origination of complexity. Population size is one parameter that alters the relative importance of adaptive and non-adaptive processes: as population size decreases, selection weakens and genetic drift grows in importance. Because of this…
