This paper develops a mathematical model describing the influence that conjugation-mediated Horizontal Gene Transfer (HGT) has on the mutation-selection balance in an asexually reproducing population of unicellular, prokaryotic organisms. It is assumed that mutation-selection balance is reached in the presence of a fixed background concentration of antibiotic, to which the population must become resistant in order to survive. We analyze the behavior of the model in the limit of low and high antibiotic-induced first-order death rate constants, and find that the highest mean fitness is obtained at low rates of bacterial conjugation. As the rate of conjugation crosses a threshold, the mean fitness decreases to a minimum, and then rises asymptotically to a limiting value as the rate of conjugation becomes infinitely large. However, this limiting value is smaller than the mean fitness obtained in the limit of low conjugation rate. This dependence of the mean fitness on the conjugation rate is fairly small for the parameter ranges we have considered, and disappears as the first-order death rate constant due to the presence of antibiotic approaches zero. For large values of the antibiotic death rate constant, we have obtained an analytical solution for the behavior of the mean fitness that agrees well with the results of simulations. The results of this paper suggest that conjugation-mediated HGT has a slightly deleterious effect on the mean fitness of a population at mutation-selection balance. Therefore, we argue that HGT confers a selective advantage by allowing for faster adaptation to a new or changing environment. The results of this paper are consistent with the observation that HGT can be promoted by environmental stresses on a population.
This paper develops a quasispecies model that incorporates the SOS response. We consider a unicellular, asexually replicating population of organisms, whose genomes consist of a single, double-stranded DNA molecule, i.e. one chromosome. We assume that repair of post-replication mismatched base-pairs occurs with probability $ λ$, and that the SOS response is triggered when the total number of mismatched base-pairs exceeds $ l_S $. We further assume that the per-mismatch SOS elimination rate is characterized by a first-order rate constant $ κ_{SOS} $. For a single fitness peak landscape where th…
This paper develops mathematical models describing the evolutionary dynamics of both asexually and sexually reproducing populations of diploid unicellular organisms. We consider two forms of genome organization. In one case, we assume that the genome consists of two multi-gene chromosomes, while in the second case we assume that each gene defines a separate chromosome. If the organism has $ l $ homologous pairs that lack a functional copy of the given gene, then the fitness of the organism is $ κ_l $. The $ κ_l $ are assumed to be monotonically decreasing, so that $ κ_0 = 1 > κ_1 > κ_2 > ... >…
Understanding the emergence and evolution of multicellularity and cellular differentiation is a core problem in biology. We develop a quantitative model that shows that a multicellular form emerges from genetically identical unicellular ancestors when the compartmentalization of poorly compatible physiological processes into component cells of an aggregate produces a fitness advantage. This division of labour between the cells in the aggregate occurs spontaneously at the regulatory level due to mechanisms present in unicellular ancestors and does not require any genetic pre-disposition for a p…
The growth rate of organisms depends both on external conditions and on internal states, such as the expression levels of various genes. We show that to achieve a criterion mean growth rate over an ensemble of conditions, the internal variables must carry a minimum number of bits of information about those conditions. Evolutionary competition thus can select for cellular mechanisms that are more efficient in an abstract, information theoretic sense. Estimates based on recent experiments suggest that the minimum information required for reasonable growth rates is close to the maximum informatio…
Which factors govern the evolution of mutation rates and emergence of species? Here, we address this question using a first principles model of life where population dynamics of asexual organisms is coupled to molecular properties and interactions of proteins encoded in their genomes. Simulating evolution of populations, we found that fitness increases in punctuated steps via epistatic events, leading to formation of stable and functionally interacting proteins. At low mutation rates, species - populations of organisms with identical genotypes - form, while at higher mutation rates, species ar…
Asymmetric segregation of key proteins at cell division -- be it a beneficial or deleterious protein -- is ubiquitous in unicellular organisms and often considered as an evolved trait to increase fitness in a stressed environment. Here, we provide a general framework to describe the evolutionary origin of this asymmetric segregation. We compute the population fitness as a function of the protein segregation asymmetry $a$, and show that the value of $a$ which optimizes the population growth manifests a phase transition between symmetric and asymmetric partitioning phases. Surprisingly, the natu…
Deleterious genetic variants can be evaluated as quantitative traits using information theory-based sequence analysis of recognition sites. To assess the effect of such variants, fitness and genetic load of SNPs which alter binding site affinity are derived from changes in individual information and allele frequencies. Human SNPs that alter mRNA splicing are partitioned according to their genetic load. SNPs with high genetic loads (>0.5) are common in the genome and, in many instances, predicted effects are supported by gene expression studies.
In unicellular organisms such as bacteria and in most viruses, mutations mainly occur during reproduction. Thus, genotypes with a high birth rate should have a higher mutation rate. However, standard models of asexual adaptation such as the 'replicator-mutator equation' often neglect this effect. In this study, we investigate the emergence of a positive dependence between the birth rate and the mutation rate in models of asexual adaptation and the consequences of this dependence. We show that it emerges naturally at the population scale, based on a large population limit of a stochastic timeco…
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 …
We consider evolution of a large population, where fitness of each organism is defined by many phenotypical traits. These traits result from expression of many genes. We propose a new model of gene regulation, where gene expression is controlled by a gene network with a threshold mechanism and there is a feedback between that threshold and gene expression. We show that this regulation is very powerful: depending on parameters we can obtain any functional connection between thresholds and genes. Under general assumptions on fitness we prove that such model organisms are capable, to some extent,…
Epistasis occurs when the effect of a mutation depends on its carrier's genetic background. Despite increasing evidence that epistasis for fitness is common, its role during evolution is contentious. Fitness landscapes, mappings of genotype or phenotype to fitness, capture the full extent and complexity of epistasis. Fitness landscape theory has shown how epistasis affects the course and the outcome of evolution. Moreover, by measuring the competitive fitness of sets of tens to thousands of connected genotypes, empirical fitness landscapes have shown that epistasis is frequent and depends on t…
It has recently been noted that the relative prevalence of the various kinds of epistasis varies along an adaptive walk. This has been explained as a result of mean regression in NK model fitness landscapes. Here we show that this phenomenon occurs quite generally in fitness landscapes. We propose a simple and general explanation for this phenomemon, confirming the role of mean regression. We provide support for this explanation with simulations, and discuss the empirical relevance of our findings.
Fitness landscapes are central in analyzing evolution, in particular for drug resistance mutations for bacteria and virus. We show that the fitness landscapes associated with antibiotic resistance are not compatible with any of the classical models; additive, uncorrelated and block fitness landscapes. The NK model is also discussed. It is frequently stated that virtually nothing is known about fitness landscapes in nature. We demonstrate that available records of antimicrobial drug mutations can reveal interesting properties of fitness landscapes in general. We apply the methods to analyze the…
Evolutionary dynamics and patterns of molecular evolution are strongly influenced by selection on linked regions of the genome, but our quantitative understanding of these effects remains incomplete. Recent work has focused on predicting the distribution of fitness within an evolving population, and this forms the basis for several methods that leverage the fitness distribution to predict the patterns of genetic diversity when selection is strong. However, in weakly selected populations random fluctuations due to genetic drift are more severe, and neither the distribution of fitness nor the se…
One essential ingredient of evolutionary theory is the concept of fitness as a measure for a species' success in its living conditions. Here, we quantify the effect of environmental fluctuations onto fitness by analytical calculations on a general evolutionary model and by studying corresponding individual-based microscopic models. We demonstrate that not only larger growth rates and viabilities, but also reduced sensitivity to environmental variability substantially increases the fitness. Even for neutral evolution, variability in the growth rates plays the crucial role of strongly reducing t…
The genetic composition of a naturally developing population is considered as due to mutation, selection, genetic drift and recombination. Selection is modeled as single-locus terms (additive fitness) and two-loci terms (pairwise epistatic fitness). The problem is posed to infer epistatic fitness from population-wide whole-genome data from a time series of a developing population. We generate such data in silico, and show that in the Quasi-Linkage Equilibrium (QLE) phase of Kimura, Neher and Shraiman, that pertains at high enough recombination rates and low enough mutation rates, epistatic fit…
How to define and use the concept of inclusive fitness is a contentious topic in evolutionary theory. Inclusive fitness can be used to calculate selection on a focal gene, but it is also applied to whole organisms. Individuals are then predicted to appear designed as if to maximise their inclusive fitness, provided that certain conditions are met (formally when interactions between individuals are 'additive'). Here we argue that applying the concept of inclusive fitness to organisms is justified under far broader conditions than previously shown, but only if it is appropriately defined. Specif…
Phenotypic fluctuations and plasticity can generally affect the course of evolution, a process known as the Baldwin effect. Several studies have recast this effect and claimed that phenotypic plasticity acceler- ates evolutionary rate (the Baldwin expediting effect); however, the validity of this claim is still controversial. In this study, we investi- gate the evolutionary population dynamics of a quantitative genetic model under a multi-peaked fitness landscape, in order to evaluate the validity of the effect. We provide analytical expressions for the evolutionary rate and average population…
Mounting evidence suggests that natural populations can harbor extensive fitness diversity with numerous genomic loci under selection. It is also known that genealogical trees for populations under selection are quantifiably different from those expected under neutral evolution and described statistically by Kingman's coalescent. While differences in the statistical structure of genealogies have long been used as a test for the presence of selection, the full extent of the information that they contain has not been exploited. Here we shall demonstrate that the shape of the reconstructed geneal…
Using the Hamilton-Jacobi equation approach to study genomes of length $L$, we obtain 1/L corrections for the steady state population distributions and mean fitness functions for horizontal gene transfer model, as well as for the diploid evolution model with general fitness landscapes. Our numerical solutions confirm the obtained analytic equations. Our method could be applied to the general case of nonlinear Markov models.
The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the two-locus case. Gene interactions among multiple loci are usually reduced to two-way interactions. We present a geometric theory of shapes of fitness landscapes for multiple loci. A central concept is the genotope, which is the convex hull of all possible allele frequencies in populations. Triangulations of the genotope correspond to different shapes of fitness landscapes and reveal all the gene interactions. The theory is applied to fitness data fro…
Biological systems are modular, and this modularity affects the evolution of biological systems over time and in different environments. We here develop a theory for the dynamics of evolution in a rugged, modular fitness landscape. We show analytically how horizontal gene transfer couples to the modularity in the system and leads to more rapid rates of evolution at short times. The model, in general, analytically demonstrates a selective pressure for the prevalence of modularity in biology. We use this model to show how the evolution of the influenza virus is affected by the modularity of the …
If deleterious mutations near a fitness maximum in a spatially distributed population are sufficiently frequent or detrimental, the population can undergo a fitness collapse, similarly to the Muller's ratchet effect in well-mixed populations. Recent studies of one-dimensional habitats (e.g., the frontier of a two-dimensional range expansion) have shown that the onset of the fitness collapse is described by a directed percolation phase transition with its associated critical exponents. We consider population fitness collapse in three-dimensional range expansions with both inflating and fixed-si…
Epistatic interactions between mutations add substantial complexity to adaptive landscapes, and are often thought of as detrimental to our ability to predict evolution. Yet, patterns of global epistasis, in which the fitness effect of a mutation is well-predicted by the fitness of its genetic background, may actually be of help in our efforts to reconstruct fitness landscapes and infer adaptive trajectories. Microscopic interactions between mutations, or inherent nonlinearities in the fitness landscape, may cause global epistasis patterns to emerge. In this brief review, we provide a succinct …
This work presents a population genetic model of evolution, which includes haploid selection, mutation, recombination, and drift. The mutation-selection equilibrium can be expressed exactly in closed form for arbitrary fitness functions without resorting to diffusion approximations. Tractability is achieved by generating new offspring using n-parent rather than 2-parent recombination. While this enforces linkage equilibrium among offspring, it allows analysis of the whole population under linkage disequilibrium. We derive a general and exact relationship between fitness fluctuations and respon…
We introduce a generalization of the parallel, or Crow-Kimura, and Eigen models of molecular evolution to represent the exchange of genetic information between individuals in a population. We study the effect of different schemes of genetic recombination on the steady-state mean fitness and distribution of individuals in the population, through an analytic field theoretic mapping. We investigate both horizontal gene transfer from a population and recombination between pairs of individuals. Somewhat surprisingly, these nonlinear generalizations of quasi-species theory to modern biology are anal…
Mutation rates and fitness costs of deleterious mutations are difficult to measure in vivo but essential for a quantitative understanding of evolution. Using whole genome deep sequencing data from longitudinal samples during untreated HIV-1 infection, we estimated mutation rates and fitness costs in HIV-1 from the temporal dynamics of genetic variation. At approximately neutral sites, mutations accumulate with a rate of 1.2 x 10^-5 per site per day, in agreement with the rate measured in cell cultures. The rate from G to A is largest, followed by the other transitions C to T, T to C, and A to …
Evolutionary trajectories and phenotypic states available to cell populations are ultimately dictated by intermolecular interactions between DNA, RNA, proteins, and other molecular species. Here we study how evolution of gene regulation in a single-cell eukaryote S. cerevisiae is affected by the interactions between transcription factors (TFs) and their cognate genomic sites. Our study is informed by high-throughput in vitro measurements of TF-DNA binding interactions and by a comprehensive collection of genomic binding sites. Using an evolutionary model for monomorphic populations evolving on…
We consider an asexual population under strong selection-weak mutation conditions evolving on rugged fitness landscapes with many local fitness peaks. Unlike the previous studies in which the initial fitness of the population is assumed to be high, here we start the adaptation process with a low fitness corresponding to a population in a stressful novel environment. For generic fitness distributions, using an analytic argument we find that the average number of steps to a local optimum varies logarithmically with the genotype sequence length and increases as the correlations amongst genotypic …
We adapt a fitness function from evolutionary game theory as a mechanism for aggregation and dispersal in a partial differential equation (PDE) model of two interacting populations, described by density functions $u$ and $v$. We consider a spatial model where individuals migrate up local fitness gradients, seeking out locations where their given traits are more advantageous. The resulting system of fitness gradient equations is a degenerate system having spatially structured, smooth, steady state solutions characterized by constant fitness throughout the domain. When populations are viewed as …
Darwinian evolution can be modeled in general terms as a flow in the space of fitness (i.e. reproductive rate) distributions. In the diffusion approximation, Tsimring et al. have showed that this flow admits "fitness wave" solutions: Gaussian-shape fitness distributions moving towards higher fitness values at constant speed. Here we show more generally that evolving fitness distributions are attracted to a one-parameter family of distributions with a fixed parabolic relationship between skewness and kurtosis. Unlike fitness waves, this statistical pattern encompasses both positive and negative…
How self-incompatibility systems are maintained in plant populations is still a debated issue. Theoretical models predict that self-incompatibility systems break down according to the intensity of inbreeding depression and number of S-alleles. Other studies have explored the function of asexual reproduction in the maintenance of self-incompatibility. However, the population genetics of partially asexual, self-incompatible populations are poorly understood and previous studies have failed to consider all possible effects of asexual reproduction or could only speculate on those effects. In this …
The accumulation of deleterious mutations is driven by rare fluctuations which lead to the loss of all mutation free individuals, a process known as Muller's ratchet. Even though Muller's ratchet is a paradigmatic process in population genetics, a quantitative understanding of its rate is still lacking. The difficulty lies in the nontrivial nature of fluctuations in the fitness distribution which control the rate of extinction of the fittest genotype. We address this problem using the simple but classic model of mutation selection balance with deleterious mutations all having the same effect o…
Fitness inference tested by in silico population genetics
Nanjing University of Posts and Telecommunications（南京邮电大学）；Nanjing University（南京大学）；University of Pittsburgh School of Medicine（匹兹堡大学医学院）；AlbaNova University Center（阿尔巴诺瓦大学中心）
The fitness landscape - the mapping between genotypes and fitness - determines properties of the process of adaptation. Several small genetic fitness landscapes have recently been built by selecting a handful of beneficial mutations and measuring fitness of all combinations of these mutations. Here we generate several testable predictions for the properties of these landscapes under Fisher's geometric model of adaptation (FGMA). When far from the fitness optimum, we analytically compute the fitness effect of beneficial mutations and their epistatic interactions. We show that epistasis may be n…
Compared to a neutral model, purifying selection distorts the structure of genealogies and hence alters the patterns of sampled genetic variation. Although these distortions may be common in nature, our understanding of how we expect purifying selection to affect patterns of molecular variation remains incomplete. Genealogical approaches such as coalescent theory have proven difficult to generalize to situations involving selection at many linked sites, unless selection pressures are extremely strong. Here, we introduce an effective coalescent theory (a "fitness-class coalescent") to describe …
Biological fitness is not an observable quantity and must be inferred from population dynamics. Bayesian inference applied to the Moran process and variants yields a robust inference method that can infer fitness in populations evolving via a Moran dynamic and generalizations. Information about fitness is derived solely from birth-events in birth-death and death-birth processes in which selection acts proportionally to fitness, which allows the method to be applied to populations on a network where the network itself may be changing in time. Populations may also be allowed to change size while…
Simple sign epistasis and evolutionary detours in fitness landscapes
UK Health Security Agency（英国卫生安全局）；Biomathematics and Statistics Scotland, The James Hutton Institute（苏格兰生物数学与统计学，詹姆斯·赫顿研究所）；Group of Complex Systems and Statistical Mechanics, Physics Faculty, University of Havana（哈瓦那大学物理学院复杂系统与统计力学小组）；System Modeling Group, Institute for Veterinary Epidemiology and Biostatistics, Freie Universitaet Berlin（柏林自由大学兽医流行病学和生物统计学研究所系统建模小组）；Accenture（埃森哲）；National University of General Sarmiento（圣萨尔米恩托国立大学）；Department of Biology, University of Oxford（牛津大学生物学系）；Institute for Biological Physics, University of Cologne（科隆大学生物物理学研究所）；Pandemic Sciences Institute and Big Data Institute, Nuffield Department of Medicine, University of Oxford（牛津大学纳菲尔德医学系大流行病科学与大数据研究所）
A fitness landscape is a genetic space -- with two genotypes adjacent if they differ in a single locus -- and a fitness function. Evolutionary dynamics produce a flow on this landscape from lower fitness to higher; reaching equilibrium only if a local fitness peak is found. I use computational complexity to question the common assumption that evolution on static fitness landscapes can quickly reach a local fitness peak. I do this by showing that the popular NK model of rugged fitness landscapes is PLS-complete for K >= 2; the reduction from Weighted 2SAT is a bijection on adaptive walks, so th…
Darwinian evolution can be illustrated as an uphill walk in a landscape, where the surface consists of genotypes, the height coordinates represent fitness, and each step corresponds to a point mutation. Epistasis, roughly defined as the dependence between the fitness effects of mutations, is a key concept in the theory of adaptation. Important recent approaches depend on graphs and polytopes. Fitness graphs are useful for describing coarse properties of a landscape, such as mutational trajectories and the number of peaks. The graphs have been used for relating global and local properties of fi…
The adaptive evolution of a population under the influence of mutation and selection is strongly influenced by the structure of the underlying fitness landscape, which encodes the interactions between mutations at different genetic loci. Theoretical studies of such landscapes have been carried out for several decades, but only recently experimental fitness measurements encompassing all possible combinations of small sets of mutations have become available. The empirical studies have spawned new questions about the accessibility of optimal genotypes under natural selection. Depending on populat…
Fitness effects of mutations fall on a continuum ranging from lethal to deleterious to beneficial. The distribution of fitness effects (DFE) among random mutations is an essential component of every evolutionary model and a mathematical portrait of robustness. Recent experiments on five viral species all revealed a characteristic bimodal shaped DFE, featuring peaks at neutrality and lethality. However, the phenotypic causes underlying observed fitness effects are still unknown, and presumably thought to vary unpredictably from one mutation to another. By combining population genetics simulatio…
We consider the optimal dynamics in the infinite population evolution models with general symmetric fitness landscape. The search of optimal evolution trajectories are complicated due to sharp transitions (like shock waves) in evolution dynamics with smooth fitness landscapes, which exist even in case of popular quadratic fitness. We found exact analytical solutions for discontinuous dynamics at the large genome length limit. We found the optimal mutation rates for the fixed fitness landscape. The single peak fitness landscape gives the fastest dynamics to send the vast majority of the populat…
Evolutionary adaptation is the process that increases the fit of a population to the fitness landscape it inhabits. As a consequence, evolutionary dynamics is shaped, constrained, and channeled, by that fitness landscape. Much work has been expended to understand the evolutionary dynamics of adapting populations, but much less is known about the structure of the landscapes. Here, we study the global and local structure of complex fitness landscapes of interacting loci that describe protein folds or sets of interacting genes forming pathways or modules. We find that in these landscapes, high pe…
We construct a pathwise formulation of a growing population of cells, based on two different samplings of lineages within the population, namely the forward and backward samplings. We show that a general symmetry relation, called fluctuation relation relates these two samplings, independently of the model used to generate divisions and growth in the cell population. These relations lead to estimators of the population growth rate, which can be very efficient as we demonstrate by an analysis of a set of mother machine data. These fluctuation relations lead to general and important inequalities …
We study the evolution of asexual microorganisms with small mutation rate in fluctuating environments, and develop techniques that allow us to expand the formal solution of the evolution equations to first order in the mutation rate. Our method can be applied to both discrete time and continuous time systems. While the behavior of continuous time systems is dominated by the average fitness landscape for small mutation rates, in discrete time systems it is instead the geometric mean fitness that determines the system's properties. In both cases, we find that in situations in which the arithmeti…
