Kavita Jain

Deterministic and Stochastic Regimes of Asexual Evolution on Rugged Fitness Landscapes

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 deff up to which the population can spread in genotype space. For deff = L, the deterministic quasi-species theory operates while for deff < 1, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasi-species theory below a crossover time T× while above T× the population gets trapped at a local fitness peak and manages to find a better peak via either stochastic tunneling or double mutations. In the stochastic regime deff < 1, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interpretation of evolution experiments with microbial populations.

EVOLUTIONARY ADVANTAGE OF SMALL POPULATIONS ON COMPLEX FITNESS LANDSCAPES

Recent experimental and theoretical studies have shown that small asexual populations evolving on complex fitness landscapes may achieve a higher fitness than large ones due to the increased heterogeneity of adaptive trajectories. Here, we introduce a class of haploid three‐locus fitness landscapes that allow the investigation of this scenario in a precise and quantitative way. Our main result derived analytically shows how the probability of choosing the path of the largest initial fitness increase grows with the population size. This makes large populations more likely to get trapped at local fitness peaks and implies an advantage of small populations at intermediate time scales. The range of population sizes where this effect is operative coincides with the onset of clonal interference. Additional studies using ensembles of random fitness landscapes show that the results achieved for a particular choice of three‐locus landscape parameters are robust and also persist as the number of loci increases. Our study indicates that an advantage for small populations is likely whenever the fitness landscape contains local maxima. The advantage appears at intermediate time scales, which are long enough for trapping at local fitness maxima to have occurred but too short for peak escape by the creation of multiple mutants.

Adaptation in Simple and Complex Fitness Landscapes

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 organisms, semiconservative replication and time-dependent fitness peaks are briefly described. A further section is devoted to randomly rugged fitness landscapes, which may display fitness correlations of various degree as well as extended neutral networks. The final two sections address evolutionary dynamics in both rugged and smooth fitness landscapes, and provide a brief overview of pertinent experiments.

Loss of Least-Loaded Class in Asexual Populations Due to Drift and Epistasis

We consider the dynamics of a nonrecombining haploid population of finite size that accumulates deleterious mutations irreversibly. This ratchet-like process occurs at a finite speed in the absence of epistasis, but it has been suggested that synergistic epistasis can halt the ratchet. Using a diffusion theory, we find explicit analytical expressions for the typical time between successive clicks of the ratchet for both nonepistatic and epistatic fitness functions. Our calculations show that the interclick time is of a scaling form that in the absence of epistasis gives a speed that is determined by size of the least-loaded class and the selection coefficient. With synergistic interactions, the ratchet speed is found to approach zero rapidly for arbitrary epistasis. Our analytical results are in good agreement with the numerical simulations.

Breaking records in the evolutionary race

We explore some aspects of the relationship between biological evolution processes and the mathematical theory of records. For Eigens 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 independent, identically distributed random variables. Second, fitness records can be bypassed, which strongly reduces the number of genotypes that take part in an evolutionary trajectory. For exponential and Gaussian fitness distributions, this number scales with sequence length N as N, and it is of order unity for distributions with a power law tail. This is in strong contrast to the number of records, which is of order N for any fitness distribution.

Evolutionary trajectories in rugged fitness landscapes

We consider the evolutionary trajectories traced out by an infinite population undergoing mutation–selection dynamics in static, uncorrelated random fitness landscapes. Starting from the population that consists of a single genotype, the most populated genotype jumps from one local fitness maximum to another and eventually reaches the global maximum. We use a strong selection limit, which reduces the dynamics beyond the first time step to the competition between independent mutant subpopulations, to study the dynamics of this model and of a simpler one-dimensional model which ignores the geometry of the sequence space. We find that the fit genotypes that appear along a trajectory are a subset of suitably defined fitness records, and exploit several results from the record theory for non-identically distributed random variables. The genotypes that contribute to the trajectory are those records that are not bypassed by superior records arising further away from the initial population. Several conjectures concerning the statistics of bypassing are extracted from numerical simulations. In particular, for the one-dimensional model, we propose a simple relation between the bypassing probability and the dynamic exponent which describes the scaling of the typical evolution time with genome size. The latter can be determined exactly in terms of the extremal properties of the fitness distribution.

Purifying Selection, Drift and Reversible Mutation with Arbitrarily High Mutation Rates

Some species exhibit very high levels of DNA sequence variability; there is also evidence for the existence of heritable epigenetic variants that experience state changes at a much higher rate than sequence variants. In both cases, the resulting high diversity levels within a population (hyperdiversity) mean that standard population genetics methods are not trustworthy. We analyze a population genetics model that incorporates purifying selection, reversible mutations, and genetic drift, assuming a stationary population size. We derive analytical results for both population parameters and sample statistics and discuss their implications for studies of natural genetic and epigenetic variation. In particular, we find that (1) many more intermediate-frequency variants are expected than under standard models, even with moderately strong purifying selection, and (2) rates of evolution under purifying selection may be close to, or even exceed, neutral rates. These findings are related to empirical studies of sequence and epigenetic variation.

Relaxation times of unstable states in systems with long range interactions

We consider several models with long range interactions evolving via Hamiltonian dynamics. The microcanonical dynamics of the basic Hamiltonian mean field (HMF) model and perturbed HMF models with either global anisotropy or an on-site potential are studied both analytically and numerically. We find that, in the magnetic phase, the initial zero magnetization state remains stable above a critical energy and is unstable below it. In the dynamically stable state, these models exhibit relaxation timescales that increase algebraically with the number N of particles, indicating the robustness of the quasistationary state seen in previous studies. In the unstable state, the corresponding timescale increases logarithmically in N.

Multiple Adaptive Substitutions During Evolution in Novel Environments

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 among genotypic fitnesses increase. When the fitnesses are exponentially or uniformly distributed, using an evolution equation for the distribution of population fitness, we analytically calculate the fitness distribution of fixed beneficial mutations and the walk length distribution.

Dynamics of a disordered, driven zero-range process in one dimension.

We study a disordered, driven zero range process which models a closed system of attractive particles that hop with site-dependent rates and whose steady state shows a condensation transition with increasing density. We characterize the dynamical properties of the mass fluctuations in the steady state in one dimension both analytically and numerically and show that there is a dynamic phase transition in the density-disorder plane. We also determine the form of the scaling function which describes the growth of the condensate as a function of time, starting from a uniform density distribution.