Consider a mathematical model of evolutionary adaptation of fitness landscape and mutation matrix as a reaction to population changes. As a basis, we use an open quasispecies model, which is modified to include explicit death flow. We assume that evolutionary parameters of mutation and selection processes vary in a way to maximize the mean fitness of the system. From this standpoint, Fisher's theorem of natural selection is being rethought and discussed. Another assumption is that system dynamics has two significant timescales. According to our central hypothesis, major evolutionary transitions happen in the steady-state of the corresponding dynamical system, so the evolutionary time is much slower than the one of internal dynamics. For the specific cases of quasispecies systems, we show how our premises form the fitness landscape adaptation process.
We discuss a population of sequences subject to mutations and frequency-dependent selection, where the fitness of a sequence depends on the composition of the entire population. This type of dynamics is crucial to understand the evolution of genomic regulation. Mathematically, it takes the form of a reaction-diffusion problem that is nonlinear in the population state. In our model system, the fitness is determined by a simple mathematical game, the hawk-dove game. The stationary population distribution is found to be a quasispecies with properties different from those which hold in fixed fitne…
The quasispecies model introduced by Eigen in 1971 has close connections with the isometry group of the space of binary sequences relative to the Hamming distance metric. Generalizing this observation we introduce an abstract quasispecies model on a finite metric space $X$ together with a group of isometries $Γ$ acting transitively on $X$. We show that if the domain of the fitness function has a natural decomposition into the union of $t$ $G$-orbits, $G$ being a subgroup of $Γ$, then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most $t\cdot {\…
We study the adaptation dynamics of an initially maladapted population evolving via the elementary processes of mutation and selection. The evolution occurs on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local peaks separated by low fitness valleys. We mainly focus on the Eigen's model that describes the deterministic dynamics of an infinite number of self-replicating molecules. In the stationary state, for small mutation rates such a population forms a {\it quasispecies} which consists of the fittest genotype and its closely related mutan…
We investigate the evolutionary dynamics of a finite population of RNA sequences adapting to a neutral fitness landscape. Despite the lack of differential fitness between viable sequences, we observe typical properties of adaptive evolution, such as increase of mean fitness over time and punctuated equilibrium transitions. We discuss the implications of these results for understanding evolution at high mutation rates, and extend the relevance of the quasispecies concept to finite populations and time scales. Our results imply that the quasispecies concept and neutral drift are not complementar…
A quasispecies is a set of interrelated genotypes that have reached a situation of equilibrium while evolving according to the usual Darwinian principles of selection and mutation. Quasispecies studies invariably assume that it is possible for any genotype to mutate into any other, but recent finds indicate that this assumption is not necessarily true. Here we revisit the traditional quasispecies theory by adopting a network structure to constrain the occurrence of mutations. Such structure is governed by a random-graph model, whose single parameter (a probability p) controls both the graph's …
Quasispecies theory provides the conceptual and theoretical bases for describing the dynamics of biological information of replicators subject to large mutation rates. This theory, initially conceived within the framework of prebiotic evolution, is also being used to investigate the evolutionary dynamics of RNA viruses and heterogeneous cancer cells populations. In this sense, efforts to approximate the initial quasispecies theory to more realistic scenarios have been made in recent decades. Despite this, how time lags in RNA synthesis and periodic fluctuations impact quasispecies dynamics rem…
A non-local rare mutations model for quasispecies and Prisoner's dilemma: numerical assessment of qualitative behaviour
Università di Napoli “Parthenope”（那不勒斯帕尔特诺佩大学）；Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche（意大利国家研究委员会计算应用研究所）
Co-evolution of two coupled quasispecies is studied, motivated by the competition between viral evolution and adapting immune response. In this co-adaptive model, besides the classical error catastrophe for high virus mutation rates, a second ``adaptation-'' catastrophe occurs, when virus mutation rates are too small to escape immune attack. Maximizing both regimes of viral error catastrophes is a possible strategy for an optimal immune response, reducing the range of allowed viral mutation rates to a minimum. From this requirement one obtains constraints on B-cell mutation rates and receptor …
We consider an asexual population evolving on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local optima. We track the most populated genotype as it changes when the population jumps from a fitness peak to a better one during the process of adaptation. This is done using the dynamics of the shell model which is a simplified version of the quasispecies model for infinite populations and standard Wright-Fisher dynamics for large finite populations. We show that the population fraction of a genotype obtained within the quasispecies model and th…
In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species.
In spite of their many facets, the phenomena of autoimmunity and immunodeficiency seem to be related to each other through the subtle links connecting retroviral mutation and action to immune response and adaptation. In a previous work, we introduced a network model of how a set of interrelated genotypes (called a quasispecies, in the stationary state) and a set of interrelated idiotypes (an idiotypic network) interact. That model, which does not cover the case of a retroviral quasispecies, was instrumental for the study of quasispecies survival when confronting the immune system and led to th…
Shared quasispecies architecture in experimental and natural RNA virus populations
Museo Nacional de Ciencias Naturales (CSIC)（国家自然科学博物馆（西班牙科学理事会））；Grupo Interdisciplinar de Sistemas Complejos (GISC)（复杂系统跨学科小组）；Centro de Astrobiología (CAB), CSIC-INTA（天体生物学中心（西班牙科学理事会-国家航空航天技术研究所））；Centro de Biología Molecular “Severo Ochoa” (CSIC-UAM)（“塞韦罗·奥乔亚”分子生物学中心（西班牙科学理事会-马德里自治大学））；Instituto de Investigación Sanitaria-Fundación Jiménez Díaz University Hospital (UAM)（卫生研究基金会-希门尼斯·迪亚斯大学医院附属研究所（马德里自治大学））；Institut de Biologia Evolutiva (UPF-CSIC)（进化生物学研究所（庞培法布拉大学-西班牙科学理事会））
A simplified form of the time dependent evolutionary dynamics of a quasispecies model with a rugged fitness landscape is solved via a mapping onto a random flux model whose asymptotic behavior can be described in terms of a random walk. The statistics of the number of changes of the dominant genotype from a finite set of genotypes are exactly obtained confirming existing conjectures based on numerics.
This study focuses on open quasispecies systems with competition and death flow, described by modified Eigen and Crow-Kimura models. We examine the evolutionary adaptation process as a reaction to changes in rates. One of the fundamental assumptions, which forms the basis of our mathematical model, is the existence of two different timescales: internal dynamics time and evolutionary time. The latter is much slower and exhibits significant adaptation events. These conditions allow us to represent the whole evolutionary process through a series of steady-state equations, where all the elements c…
During the HIV infection several quasispecies of the virus arise, which are able to use different coreceptors, in particular the CCR5 and CXCR4 coreceptors (R5 and X4 phenotypes, respectively). The switch in coreceptor usage has been correlated with a faster progression of the disease to the AIDS phase. As several pharmaceutical companies are starting large phase III trials for R5 and X4 drugs, models are needed to predict the co-evolutionary and competitive dynamics of virus strains. We present a model of HIV early infection which describes the dynamics of R5 quasispecies and a model of HIV l…
George Williams defined an evolutionary unit as hereditary information for which the selection bias between competing units dominates the informational decay caused by imperfect transmission. In this article, I extend Williams' approach to show that the ratio of selection bias to transmission bias provides a unifying framework for diverse biological problems. Specific examples include Haldane and Lande's mutation-selection balance, Eigen's error threshold and quasispecies, Van Valen's clade selection, Price's multilevel formulation of group selection, Szathmary and Demeter's evolutionary origi…
Many important real-world networks manifest "small-world" properties such as scale-free degree distributions, small diameters, and clustering. The most common model of growth for these networks is "preferential attachment", where nodes acquire new links with probability proportional to the number of links they already have. We show that preferential attachment is a special case of the process of molecular evolution. We present a new single-parameter model of network growth that unifies varieties of preferential attachment with the quasispecies equation (which models molecular evolution), and a…
Growing efforts to measure fitness landscapes in molecular and microbial systems are premised on a tight relationship between landscape topography and evolutionary trajectories. This relationship, however, is far from being straightforward: depending on their mutation rate, Darwinian populations can climb the closest fitness peak (survival of the fittest), settle in lower regions with higher mutational robustness (survival of the flattest), or fail to adapt altogether (error catastrophes). These bifurcations highlight that evolution does not necessarily drive populations "from lower peak to hi…
We explore some aspects of the relationship between biological evolution processes and the mathematical theory of records. For Eigen's quasispecies model with an uncorrelated fitness landscape, we show that the evolutionary trajectories traced out by a population initially localized at a randomly chosen point in sequence space can be described in close analogy to record dynamics, with two complications. First, the increasing number of genotypes that become available with increasing distance from the starting point implies that fitness records are more frequent than for the standard case of ind…
The idea of incompetence as a learning or adaptation function was introduced in the context of evolutionary games as a fixed parameter. However, live organisms usually perform different nonlinear adaptation functions such as a power law or exponential fitness growth. Here, we examine how the functional form of the learning process may affect the social competition between different behavioral types. Further, we extend our results for the evolutionary games where fluctuations in the environment affect the behavioral adaptation of competing species and demonstrate importance of the starting leve…
We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length $L$. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance $d_{\mathrm{eff}}$ upto which the population can spread in genotype space. For $d_{\mathrm{eff}}=L$, the deterministic quasispecies theory operates while for $d_{\mathrm{eff}} < 1$, the evolution is completely stochastic. Between thes…
Genetic inactivation of essential genes creates an evolutionary scenario distinct from escape from drug inhibition, but the mechanisms of microbe adaptations in such cases remain unknown. Here we inactivate E. coli dihydrofolate reductase (DHFR) by introducing D27G,N,F chromosomal mutations in a key catalytic residue with subsequent adaptation by serial dilutions. The partial reversal G27->C occurred in three evolutionary trajectories. Conversely, in one trajectory for D27G and in all trajectories for D27F,N strains adapted to grow at very low supplement folAmix concentrations but did not esca…
This is an introductory review of deterministic mutation-selection models for asexual populations (i.e., quasispecies theory) and related topics. First, the basic concepts of fitness, mutations, and sequence space are introduced. Different types of mutation-selection dynamics are defined and their relation to problems of statistical physics are outlined. Then the stationary population distribution in simple, single peak fitness landscapes is discussed at length, with particular emphasis on the error threshold phenomenon. Extensions of the theory covering e.g. epistatic interactions, diploid or…
A principle of evolutionary adaptation is applied to the Lotka--Volterra models, in particular to the food webs. We present a relatively simple computational algorithm of optimization with respect to a given criterion. This algorithm boils down to a sequence of easy to solve linear programming problems. As a criterion for the optimization we use the total weighted population size of the given community and an ecological fitness, which is an analogue of the potential energy in physics. We show by computational experiments that it is almost always possible to substantially increase the total wei…
In this paper, we discuss the fitness landscape evolution of permanent replicator systems using a hypothesis that the specific time of evolutionary adaptation of the system parameters is much slower than the time of internal evolutionary dynamics. In other words, we suppose that the extreme principle of Darwinian evolution based the Fisher's fundamental theorem of natural selection is valid for the steady-states. Various cases of the evolutionary adaptation for permanent replicator system are considered.
A key challenge to make effective use of evolutionary algorithms is to choose appropriate settings for their parameters. However, the appropriate parameter setting generally depends on the structure of the optimisation problem, which is often unknown to the user. Non-deterministic parameter control mechanisms adjust parameters using information obtained from the evolutionary process. Self-adaptation -- where parameter settings are encoded in the chromosomes of individuals and evolve through mutation and crossover -- is a popular parameter control mechanism in evolutionary strategies. However, …
Game dynamics in which three or more strategies are cyclically competitive, as represented by the rock-scissors-paper game, have attracted practical and theoretical interests. In evolutionary dynamics, cyclic competition results in oscillatory dynamics of densities of individual strategists. In finite-size populations, it is known that oscillations blow up until all but one strategies are eradicated if without mutation. In the present paper, we formalize replicator dynamics with players that have different adaptation rates. We show analytically and numerically that the heterogeneous adaptation…
This Letter studies the quasispecies dynamics of a population capable of genetic repair evolving on a time-dependent fitness landscape. We develop a model that considers an asexual population of single-stranded, conservatively replicating genomes, whose only source of genetic variation is due to copying errors during replication. We consider a time-dependent, single-fitness-peak landscape where the master sequence changes by a single point mutation every time $ τ$. We are able to analytically solve for the evolutionary dynamics of the population in the point-mutation limit. In particular, our …
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…
The enormous size and complexity of genotypic sequence space frequently requires consideration of coarse-grained sequences in empirical models. We develop scaling relations to quantify the effect of this coarse-graining on properties of fitness landscapes and evolutionary paths. We first consider evolution on a simple Mount Fuji fitness landscape, focusing on how the length and predictability of evolutionary paths scale with the coarse-grained sequence length and alphabet. We obtain simple scaling relations for both the weak- and strong-selection limits, with a non-trivial crossover regime at …
Evolutionary adaptation through bet-hedging in finite populations under fluctuating environments
Grupo Interdisciplinar de Sistemas Complejos (GISC) and Departamento de Estructura de la Materia, Física Térmica y Electrónica, Universidad Complutense de Madrid（马德里康普顿斯大学复杂系统跨学科小组及物质结构、热学与电子学系）
Evolutionary adaptation is often likened to climbing a hill or peak. While this process is simple for fitness landscapes where mutations are independent, the interaction between mutations (epistasis) as well as mutations at loci that affect more than one trait (pleiotropy) are crucial in complex and realistic fitness landscapes. We investigate the impact of epistasis and pleiotropy on adaptive evolution by studying the evolution of a population of asexual haploid organisms (haplotypes) in a model of N interacting loci, where each locus interacts with K other loci. We use a quantitative measure…
Mathematical Models of Evolution and Replicator Systems Dynamics
Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University（莫斯科国立大学基础与应用数学莫斯科中心）；A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation（阿·伊·阿利哈尼扬国家科学实验室（耶烈万物理研究所）基金会）
This paper develops a point-mutation model describing the evolutionary dynamics of a population of adult stem cells. Such a model may prove useful for quantitative studies of tissue aging and the emergence of cancer. We consider two modes of chromosome segregation: (1) Random segregation, where the daughter chromosomes of a given parent chromosome segregate randomly into the stem cell and its differentiating sister cell. (2) ``Immortal DNA strand'' co-segregation, for which the stem cell retains the daughter chromosomes with the oldest parent strands. Immortal strand co-segregation is a mechan…
Because mutations are mostly deleterious, mutation rates should be reduced by natural selection. However, mutations also provide the raw material for adaptation. Therefore, evolutionary theory suggests that the mutation rate must balance between adaptability - the ability to adapt - and adaptedness - the ability to remain adapted. We model an asexual population crossing a fitness valley and analyze the rate of complex adaptation with and without stress-induced mutagenesis - the increase of mutation rates in response to stress or maladaptation. We show that stress-induced mutagenesis increases …
We perform individual-based Monte Carlo simulations in a community consisting of two predator species competing for a single prey species, with the purpose of studying biodiversity stabilization in this simple model system. Predators are characterized with predation efficiency and death rates, to which Darwinian evolutionary adaptation is introduced. Competition for limited prey abundance drives the populations' optimization with respect to predation efficiency and death rates. We study the influence of various ecological elements on the final state, finding that both indirect competition and …
Risk aversion is a common behavior universal to humans and animals alike. Economists have traditionally defined risk preferences by the curvature of the utility function. Psychologists and behavioral economists also make use of concepts such as loss aversion and probability weighting to model risk aversion. Neurophysiological evidence suggests that loss aversion has its origins in relatively ancient neural circuitries (e.g., ventral striatum). Could there thus be an evolutionary origin to risk avoidance? We study this question by evolving strategies that adapt to play the equivalent mean payof…
Understanding why we age is a long-lived open problem in evolutionary biology. Aging is prejudicial to the individual and evolutionary forces should prevent it, but many species show signs of senescence as individuals age. Here, I will propose a model for aging based on assumptions that are compatible with evolutionary theory: i) competition is between individuals; ii) there is some degree of locality, so quite often competition will between parents and their progeny; iii) optimal conditions are not stationary, mutation helps each species to keep competitive. When conditions change, a senescen…
In this work we employ various methods of analysis (unfolding simulations and comparative analysis of structures and sequences of proteomes of thermophilic organisms) to show that organisms can follow two major strategies of thermophilic adaptation: (i) General, non-specific, structure-based, when proteomes of certain thermophilic organisms show significant structural bias toward proteins of higher compactness. In this case thermostability is achieved by greater overall number of stabilizing contacts, none of which may be especially strong, and (ii) Specific, sequence-based, whereby sequence v…
The challenge of scale in molecular adaptation: Local searches in astronomical genotype networks
Museo Nacional de Ciencias Naturales (CSIC)（国家自然科学博物馆）；Institut de Biologia Evolutiva (UPF-CSIC)（进化生物学研究所）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
The adaptation process of a species to a new environment is a significant area of study in biology. As part of natural selection, adaptation is a mutation process which improves survival skills and reproductive functions of species. Here, we investigate this process by combining the idea of incompetence with evolutionary game theory. In the sense of evolution, incompetence and training can be interpreted as a special learning process. With focus on the social side of the problem, we analyze the influence of incompetence on behavior of species. We introduce an incompetence parameter into a lear…
Adaptation is a central topic in theoretical biology, of practical importance for analyzing drug resistance mutations. Several authors have used arguments based on extreme value theory in their work on adaptation. There are complications with these approaches if fitness is additive (meaning that fitness effects of mutations sum), or whenever there is more additivity than what one would expect in an uncorrelated fitness landscape. However, the approaches have been used in published work, even in situations with substantial amounts of additivity. In particular, extreme value theory has been used…
We combine incentive, adaptive, and time-scale dynamics to study multipopulation dynamics on the simplex equipped with a large class of Riemmanian metrics, simultaneously generalizing and extending many dynamics commonly studied in dynamic game theory and evolutionary dynamics. Each population has its own geometry, method of adaptation (incentive), and time-scale (discrete, continuous, and others). Using an information-theoretic measure of distance we give a widely-applicable Lyapunov result for the dynamic. We include a wealth of examples leading up to and beyond the main results.
On rugged fitness landscapes where sign epistasis is common, adaptation can often involve either individually beneficial "uphill" mutations or more complex mutational trajectories involving fitness valleys or plateaus. The dynamics of the evolutionary process determine the probability that evolution will take any specific path among a variety of competing possible trajectories. Understanding this evolutionary choice is essential if we are to understand the outcomes and predictability of adaptation on rugged landscapes. We present a simple model to analyze the probability that evolution will es…
In this paper, we inspect well-known population genetics and social dynamics models. In these models, interacting individuals, while participating in a self-organizing process, give rise to the emergence of complex behaviors and patterns. While one main focus in population genetics is on the adaptive behavior of a population, social dynamics is more often concerned with the splitting of a connected array of individuals into a state of global polarization, that is, the emergence of speciation. Applying computational and mathematical tools we show that the way the mechanisms of selection, intera…
Population genetics struggles to model extinction; standard models track the relative rather than absolute fitness of genotypes, while the exceptions describe only the short-term transition from imminent doom to evolutionary rescue. But extinction can result from failure to adapt not only to catastrophes, but also to a backlog of environmental challenges. We model long-term evolution to long series of small challenges, where fitter populations reach higher population sizes. The population's long-term fitness dynamic is well approximated by a simple stochastic Markov chain model. Long-term pers…
We present a model for evolving agents using both genetic and cultural inheritance mechanisms. Within each agent our model maintains two distinct information stores we call the genome and the memome. Processes of adaptation are modeled as evolutionary processes at each level of adaptation (phylogenetic, ontogenetic, sociogenetic). We review relevant competing models and we show how our model improves on previous attempts to model genetic and cultural evolutionary processes. In particular we argue our model can achieve divergent gene-culture co-evolution.
Genetic interactions can strongly influence the fitness effects of individual mutations, yet the impact of these epistatic interactions on evolutionary dynamics remains poorly understood. Here we investigate the evolutionary role of epistasis over 50,000 generations in a well-studied laboratory evolution experiment in E. coli. The extensive duration of this experiment provides a unique window into the effects of epistasis during long-term adaptation to a constant environment. Guided by analytical results in the weak-mutation limit, we develop a computational framework to assess the compatibili…
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.
Background: Acceleration of adaptation dynamics by stress-induced hypermutation has been found experimentally. Evolved evolvability is a prominent explanation. We investigate a more generally applicable explanation by a physical constraint. Methods and Results: A generic thermodynamical analysis of genetic information storage obviates physical constraints on the integrity of genetic information. The capability to employ metabolic resources is found as a major determinant of mutation probability in stored genetic information. Incorporation into a non-recombinant, asexual adaptation toy model pr…
In classical evolutionary theory, genetic variation provides the source of heritable phenotypic variation on which natural selection acts. Against this classical view, several theories have emphasized that developmental variability and learning enhance nonheritable phenotypic variation, which in turn can accelerate evolutionary response. In this paper, I show how developmental variability alters evolutionary dynamics by smoothing the landscape that relates genotype to fitness. In a fitness landscape with multiple peaks and valleys, developmental variability can smooth the landscape to provide …
We consider a fixed size population that undergoes an evolutionary adaptation in the weak mutuation rate limit, which we model as a biased Langevin process in the genotype space. We show analytically and numerically that, if the fitness landscape has a small highly epistatic (rough) and time-varying component, then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability. We argue that our principal finding that even very small time-dependent fluctuations of fitness can substantially speed up evolution…
As organisms are faced with intense rapidly changing selective pressures, new genetic material is required to facilitate adaptation. Among sources of genetic novelty, gene duplications and transposable elements (TEs) offer new genes or new regulatory patterns that can facilitate evolutionary change. With advances in genome sequencing it is possible to gain a broader view of how gene family proliferation and TE content evolve in non-model species when populations become threatened. Freshwater bivalves (Unionidae) currently face severe anthropogenic challenges. Over 70% of species in the United …
A previously established multiscale population genetics model states that fitness can be inferred from the physical properties of proteins under the physiological assumption that a loss of stability by any protein confers the lethal phenotype to an organism. Here we develop this model further by positing that replication rate (fitness) of a bacterial or viral strain directly depends on the copy number of folded proteins which determine its replication rate. Using this model, and both numerical and analytical approaches, we studied the adaptation process of bacteria and viruses at varied enviro…
In a complex community, species continuously adapt to each other. On rare occasions, the adaptation of a species can lead to the extinction of others, and even its own. "Adaptive dynamics" is the standard mathematical framework to describe evolutionary changes in community interactions, and in particular, predict adaptation driven extinction. Unfortunately, most authors implement the equations of adaptive dynamics through computer simulations, that require assuming a large number of questionable parameters and fitness functions. In this study we present analytical solutions to adaptive dynamic…
