A minimum on the mean number of steps taken in adaptive walks

I consider the adaptation of a DNA sequence when mutant fitnesses are drawn randomly from a probability distribution. I focus on "gradient" adaptation in which the population jumps to the best mutant sequence available at each substitution. Given a random starting point, I derive the distribution of the number of substitutions that occur during adaptive walks to a locally optimal sequence. I show that the mean walk length is a constant:L = e-1, where e approximately 2.72. I argue that this result represents a limit on what is possible under any form of adaptation. No adaptive algorithm on any fitness landscape can arrive at a local optimum in fewer than a mean of L = e-1 steps when starting from a random sequence. Put differently, evolution must try out at least e wild-type sequences during an average bout of adaptation.     

Towards a general theory of adaptive walks on rugged landscapes

Adaptive evolution, to a large extent, is a complex combinatorial optimization process. In this article we take beginning steps towards developing a general theory of adaptive "walks" via fitter variants in such optimization processes. We introduce the basic idea of a space of entities, each a 1-mutant neighbor of many other entities in the space, and the idea of a fitness ascribed to each entity. Adaptive walks proceed from an initial entity, via fitter neighbors, to locally or globally optimal entities that are fitter than their neighbors. We develop a general theory for the number of local optima, lengths of adaptive walks, and the number of alternative local optima accessible from any given initial entity, for the baseline case of an uncorrelated fitness landscape. Most fitness landscapes are correlated, however. Therefore we develop parts of a universal theory of adaptation on correlated landscapes by adaptive processes that have sufficient numbers of mutations per individual to "jump beyond" the correlation lengths in the underlying landscape. In addition, we explore the statistical character of adaptive walks in two independent complex combinatorial optimization problems, that of evolving a specific cell type in model genetic networks, and that of finding good solutions to the traveling salesman problem. Surprisingly, both show similar statistical features, encouraging the hope that a general theory for adaptive walks on correlated and uncorrelated landscapes can be found. In the final section we explore two limits to the efficacy of selection. The first is new, and surprising: for a wide class of systems, as the complexity of the entities under selection increases, the local optima that are attainable fall progressively closer to the mean properties of the underlying space of entities. This may imply that complex biological systems, such as genetic regulatory systems, are "close" to the mean properties of the ensemble of genomic regulatory systems explored by evolution. The second limit shows that with increasing complexity and a fixed mutation rate, selection often becomes unable to pull an adapting population to those local optima to which connected adaptive walks via fitter variants exist. These beginning steps in theory development are applied to maturation of the immune response, and to the problem of radiation and stasis. Despite the limitations of the adaptive landscape metaphor, we believe that further development along the lines begun here will prove useful.     

The NK model of rugged fitness landscapes and its application to maturation of the immune response

Adaptive evolution is, to a large extent, a complex combinatorial optimization process. Such processes can be characterized as "uphill walks on rugged fitness landscapes". Concrete examples of fitness landscapes include the distribution of any specific functional property such as the capacity to catalyze a specific reaction, or bind a specific ligand, in "protein space". In particular, the property might be the affinity of all possible antibody molecules for a specific antigenic determinant. That affinity landscape presumably plays a critical role in maturation of the immune response. In this process, hypermutation and clonal selection act to select antibody V region mutant variants with successively higher affinity for the immunizing antigen. The actual statistical structure of affinity landscapes, although knowable, is currently unknown. Here, we analyze a class of mathematical models we call NK models. We show that these models capture significant features of the maturation of the immune response, which is currently thought to share features with general protein evolution. The NK models have the important property that, as the parameter K increases, the "ruggedness" of the NK landscape varies from a single peaked "Fujiyama" landscape to a multi-peaked "badlands" landscape. Walks to local optima on such landscapes become shorter as K increases. This fact allows us to choose a value of K that corresponds to the experimentally observed number of mutational "steps", 6-8, taken as an antibody sequence matures. If the mature antibody is taken to correspond to a local optimum in the model, tuning the model requires that K be about 40, implying that the functional contribution of each amino acid in the V region is affected by about 40 others. Given this value of K, the model then predicts several features of "antibody space" that are in qualitative agreement with experiment: (1) The fraction of fitter variants of an initial "roughed in" germ line antibody amplified by clonal selection is about 1-2%. (2) Mutations at some sites of the mature antibody hardly affect antibody function at all, but mutations at other sites dramatically decrease function. (3) The same "roughed in" antibody sequence can "walk" to many mature antibody sequences. (4) Many adaptive walks can end on the same local optimum. (5) Comparison of different mature sequences derived from the same initial V region shows evolutionary hot spots and parallel mutations. All these predictions are open to detailed testing by obtaining monoclonal antibodies early in the immune response and carrying out in vitro mutagenesis and adaptive hill climbing with respect to affinity for the immunizing antigen.     

Adaptive evolutionary walks require neutral intermediates in RNA fitness landscapes

In RNA fitness landscapes with interconnected networks of neutral mutations, neutral precursor mutations can play an important role in facilitating the accessibility of epistatic adaptive mutant combinations. I use an exhaustively surveyed fitness landscape model based on short sequence RNA genotypes (and their secondary structure phenotypes) to calculate the minimum rate at which mutants initially appearing as neutral are incorporated into an adaptive evolutionary walk. I show first, that incorporating neutral mutations significantly increases the number of point mutations in a given evolutionary walk when compared to estimates from previous adaptive walk models. Second, that incorporating neutral mutants into such a walk significantly increases the final fitness encountered on that walk - indeed evolutionary walks including neutral steps often reach the global optimum in this model. Third, and perhaps most importantly, evolutionary paths of this kind are often extremely winding in their nature and have the potential to undergo multiple mutations at a given sequence position within a single walk; the potential of these winding paths to mislead phylogenetic reconstruction is briefly considered.     

The population genetics of adaptation on correlated fitness landscapes: the block model

Several recent theoretical studies of the genetics of adaptation have focused on the mutational landscape model, which considers evolution on rugged fitness landscapes (i.e., ones having many local optima). Adaptation in this model is characterized by several simple results. Here I ask whether these results also hold on correlated fitness landscapes, which are smoother than those considered in the mutational landscape model. In particular, I study the genetics of adaptation in the block model, a tunably rugged model of fitness landscapes. Considering the scenario in which adaptation begins from a high fitness wild-type DNA sequence, I use extreme value theory and computer simulations to study both single adaptive steps and entire adaptive walks. I show that all previous results characterizing single steps in adaptation in the mutational landscape model hold at least approximately on correlated landscapes in the block model; many entire-walk results, however, do not.     

Properties of selected mutations and genotypic landscapes under Fisher's geometric model

The fitness landscape—the mapping between genotypes and fitness—determines properties of the process of adaptation. Several small genotypic 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 small genotypic landscapes under Fisher's geometric model of adaptation. When the ancestral strain is far from the fitness optimum, we analytically compute the fitness effect of selected mutations and their epistatic interactions. Epistasis may be negative or positive on average depending on the distance of the ancestral genotype to the optimum and whether mutations were independently selected, or coselected in an adaptive walk. Simulations show that genotypic landscapes built from Fisher's model are very close to an additive landscape when the ancestral strain is far from the optimum. However, when it is close to the optimum, a large diversity of landscape with substantial roughness and sign epistasis emerged. Strikingly, small genotypic landscapes built from several replicate adaptive walks on the same underlying landscape were highly variable, suggesting that several realizations of small genotypic landscapes are needed to gain information about the underlying architecture of the fitness landscape.     

Properties of adaptive walks on uncorrelated landscapes under strong selection and weak mutation

We examine properties of adaptive walks on uncorrelated (i.e. random) fitness landscapes starting from moderately fit genotypes under strong selection weak mutation. As an extension of Orr's model for a single step in an adaptive walk under these conditions, we show that the fitness rank of the dominant genotype in a population after the fixation of a beneficial mutation is, on average, (i+6)/4, where i is the fitness rank of the starting genotype. This accounts for the change in rank due to acquiring a new set of single-mutation neighbors after fixing a new allele through natural selection. Under this scenario, adaptive walks can be modeled as a simple Markov chain on the space of possible fitness ranks with an absorbing state at i = 1, from which no beneficial mutations are accessible. We find that these walks are typically short and are often completed in a single step when starting from a moderately fit genotype. As in Orr's original model, these results are insensitive to both the distribution of fitness effects and most biological details of the system under consideration.     

The Properties of Adaptive Walks in Evolving Populations of Fungus

The rarity of beneficial mutations has frustrated efforts to develop a quantitative theory of adaptation. Recent models of adaptive walks, the sequential substitution of beneficial mutations by selection, make two compelling predictions: adaptive walks should be short, and fitness increases should become exponentially smaller as successive mutations fix. We estimated the number and fitness effects of beneficial mutations in each of 118 replicate lineages of Aspergillus nidulans evolving for approximately 800 generations at two population sizes using a novel maximum likelihood framework, the results of which were confirmed experimentally using sexual crosses. We find that adaptive walks do indeed tend to be short, and fitness increases become smaller as successive mutations fix. Moreover, we show that these patterns are associated with a decreasing supply of beneficial mutations as the population adapts. We also provide empirical distributions of fitness effects among mutations fixed at each step. Our results provide a first glimpse into the properties of multiple steps in an adaptive walk in asexual populations and lend empirical support to models of adaptation involving selection towards a single optimum phenotype. In practical terms, our results suggest that the bulk of adaptation is likely to be accomplished within the first few steps.     

Adaptive walks through fitness landscapes for early vascular land plants

Hypothetical adaptive walks (i. e., morphological transformation series gaining increasing relative fitness) were simulated through a computer-generated domain for early vascular land plant morphologies to examine the relationship between the dynamics of adaptive walks and the topologies of fitness landscapes. A total of 15 hypothetical adaptive walks were simulated, assuming that relative fitness was based on performing one or more of four biological tasks: maximizing light interception, mechanical stability, and reproductive success, and minimizing total surface area. Morphologies occupying fitness peaks typically were similar to some early vascular land plant remains. The most stringent task (the minimization of total surface area) resulted in a few, comparatively small Y-shaped morphologies. Based on the 15 walks, the number of fitness peaks increased and their heights decreased as the number of tasks simultaneously performed increased. These results (which are consistent with prior computer-simulated walks treating light interception, mechanical stability, and reproductive success) suggest that the biological requirement to conserve water reduced the number of phenotypic options available to the earliest land plants, and that, once this adaptive hurtle was overcome, the simultaneous performance of two or more tasks, increased the number of phenotypic options with equivalent relative fitnesses that could be rapidly reached due to the comparatively small fitness differential between derived and ancestral morphologies.     

Random field models for fitness landscapes

 In many cases fitness landscapes are obtained as particular instances of random fields by randomly assigning a large number of parameters. Models of this type are often characterized reasonably well by their covariance matrices. We characterize isotropic random fields on finite graphs in terms of their Fourier series expansions and investigate the relation between the covariance matrix of the random field model and the correlation structure of the individual landscapes constructed from this random field. Correlation measures are a good characteristic of “rugged landscapes” models as they are closely related to quantities like the number of local optima or the length of adaptive walks. Our formalism suggests to approximate landscape with known autocorrelation function by a random field model that has the same correlation structure. Received: 10 November 1995 / Revised version: 19 February 1996     

Thermodynamical interpretation of an adaptive walk on a Mt. Fuji-type fitness landscape: Einstein relation-like formula holds in a stochastic evolution

We have theoretically studied the statistical properties of adaptive walks (or hill-climbing) on a Mt. Fuji-type fitness landscape in the multi-dimensional sequence space through mathematical analysis and computer simulation. The adaptive walk is characterized by the "mutation distance" d as the step-width of the walker and the "population size" N as the number of randomly generated d-fold point mutants to be screened. In addition to the fitness W, we introduced the following quantities analogous to thermodynamical concepts: "free fitness" G(W) is identical with W+T x S(W), where T is the "evolutionary temperature" T infinity square root of d/lnN and S(W) is the entropy as a function of W, and the "evolutionary force" X is identical with d(G(W)/T)/dW, that is caused by the mutation and selection pressure. It is known that a single adaptive walker rapidly climbs on the fitness landscape up to the stationary state where a "mutation-selection-random drift balance" is kept. In our interpretation, the walker tends to the maximal free fitness state, driven by the evolutionary force X. Our major findings are as follows: First, near the stationary point W*, the "climbing rate" J as the expected fitness change per generation is described by J approximately L x X with L approximately V/2, where V is the variance of fitness distribution on a local landscape. This simple relationship is analogous to the well-known Einstein relation in Brownian motion. Second, the "biological information gain" (DeltaG/T) through adaptive walk can be described by combining the Shannon's information gain (DeltaS) and the "fitness information gain" (DeltaW/T).     

The length of adaptive walks is insensitive to starting fitness in Aspergillus nidulans

Adaptation involves the successive substitution of beneficial mutations by selection, a process known as an adaptive walk. Gradualist models of adaptation, which assume that all mutations are small relative to the distance to a fitness optimum, predict that adaptive walks should be longer when the founding genotype is less well adapted. More recent work modeling adaptation as a sequence of moves in phenotype or genotype space predicts, by contrast, much shorter adaptive walks irrespective of the fitness of the founding genotype. Here, we provide what is, to the best of our knowledge, the first direct test of these alternative models, measuring the length of adaptive walks in evolving lineages of fungus that differ initially in fitness. Contrary to the gradualist view, we show that the length of adaptive walks in the fungus Aspergillus nidulans is insensitive to starting fitness and involves just two mutations on average. This arises because poorly adapted populations tend to fix mutations of larger average effect than those of better-adapted populations. Our results suggest that the length of adaptive walks may be independent of the fitness of the founding genotype and, moreover, that poorly adapted populations can quickly adapt to novel environments.     

Evolutionary optimization in multimodal search space

Optimization by a simple evolution strategy based on a mutation and selection scheme without recombination was tested for its efficiency in multimodal search space. A modified Rastrigin function served as an objective function providing fitness landscapes with many local optima. It turned out that the evolutionary algorithm including adaptive stepsize control is wellsuited for optimization. The process is able to efficiently surmount local energy barriers and converge to the global optimum. The relation between the optimization time available and the optimal number of offspring was investigated and a simple rule proposed. Several numbers of offspring are nearly equally suited in a smooth search space, whereas in rough fitness landscapes an optimum is observed. In either case both very large and very small numbers of offspring turned out to be unfavourable for optimization.     

Impact of genetic architecture on the relative rates of X versus autosomal adaptive substitution

Molecular evolutionary theory predicts that the ratio of autosomal to X-linked adaptive substitution (K(A)/K(x)) is primarily determined by the average dominance coefficient of beneficial mutations. Although this theory has profoundly influenced analysis and interpretation of comparative genomic data, its predictions are based upon two unverified assumptions about the genetic basis of adaptation. The theory assumes that 1) the rate of adaptively driven molecular evolution is limited by the availability of beneficial mutations, and 2) the scaling of evolutionary parameters between the X and the autosomes (e.g., the beneficial mutation rate, and the fitness effect distribution of beneficial alleles, per X-linked versus autosomal locus) is constant across molecular evolutionary timescales. Here, we show that the genetic architecture underlying bouts of adaptive substitution can influence both assumptions, and consequently, the theoretical relationship between K(A)/K(x) and mean dominance. Quantitative predictions of prior theory apply when 1) many genomically dispersed genes potentially contribute beneficial substitutions during individual steps of adaptive walks, and 2) the population beneficial mutation rate, summed across the set of potentially contributing genes, is sufficiently small to ensure that adaptive substitutions are drawn from new mutations rather than standing genetic variation. Current research into the genetic basis of adaptation suggests that both assumptions are plausibly violated. We find that the qualitative positive relationship between mean dominance and K(A)/K(x) is relatively robust to the specific conditions underlying adaptive substitution, yet the quantitative relationship between dominance and K(A)/K(x) is quite flexible and context dependent. This flexibility may partially account for the puzzlingly variable X versus autosome substitution patterns reported in the empirical evolutionary genomics literature. The new theory unites the previously separate analysis of adaptation using new mutations versus standing genetic variation and makes several useful predictions about the interaction between genetic architecture, evolutionary genetic constraints, and effective population size in determining the ratio of adaptive substitution between autosomal and X-linked genes.     

Cover-Encodings of Fitness Landscapes

The traditional way of tackling discrete optimization problems is by using local search on suitably defined cost or fitness landscapes. Such approaches are however limited by the slowing down that occurs when the local minima that are a feature of the typically rugged landscapes encountered arrest the progress of the search process. Another way of tackling optimization problems is by the use of heuristic approximations to estimate a global cost minimum. Here, we present a combination of these two approaches by using cover-encoding maps which map processes from a larger search space to subsets of the original search space. The key idea is to construct cover-encoding maps with the help of suitable heuristics that single out near-optimal solutions and result in landscapes on the larger search space that no longer exhibit trapping local minima. We present cover-encoding maps for the problems of the traveling salesman, number partitioning, maximum matching and maximum clique; the practical feasibility of our method is demonstrated by simulations of adaptive walks on the corresponding encoded landscapes which find the global minima for these problems.     

Adaptation in Tunably Rugged Fitness Landscapes: The Rough Mount Fuji Model

Much of the current theory of adaptation is based on Gillespie’s mutational landscape model (MLM), which assumes that the fitness values of genotypes linked by single mutational steps are independent random variables. On the other hand, a growing body of empirical evidence shows that real fitness landscapes, while possessing a considerable amount of ruggedness, are smoother than predicted by the MLM. In the present article we propose and analyze a simple fitness landscape model with tunable ruggedness based on the rough Mount Fuji (RMF) model originally introduced by Aita et al. in the context of protein evolution. We provide a comprehensive collection of results pertaining to the topographical structure of RMF landscapes, including explicit formulas for the expected number of local fitness maxima, the location of the global peak, and the fitness correlation function. The statistics of single and multiple adaptive steps on the RMF landscape are explored mainly through simulations, and the results are compared to the known behavior in the MLM model. Finally, we show that the RMF model can explain the large number of second-step mutations observed on a highly fit first-step background in a recent evolution experiment with a microvirid bacteriophage.     

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.     

Adaptive walks on behavioural landscapes and the evolution of optimal behaviour by natural selection

Summary One of the main challenges to the adaptationist programme in general and to the use of optimality models in behavioural and evolutionary ecology in particular is that natural selection need not optimise fitness. This challenge is addressed by considering the evolution of optimal patch choice by natural selection. The behavioural model is based on a state variable approach in which a strategy consists of a sequence denoting the patch to be visited as a function of the organism's state and time. The optimal strategy maximises expected terminal reproduction. The fitnesses of alternative strategies are computed by iteration of the associated equations for fitness; this characterises the adaptive behavioural landscape. There may be enormous numbers of strategies that have near optimal fitnesses. A population model is used to connect frequencies of behavioural types from one generation to the next. Theories on adaptive walks on fitness landscapes are considered in the context of behaviour. The main result is that within the context of optimality arguments at selective equilibrium, sub-optimal behaviours can persist. General implications for research in behavioural ecology, including tests of behavioural theories, are discussed.     

Development of simple fitness landscapes for peptides by artificial neural filter systems

The applicability of artificial neural filter systems as fitness functions for sequence-oriented peptide design was evaluated. Two example applications were selected: classification of dipeptides according to their hydrophobicity and classification of proteolytic cleavage-sites of protein precursor sequences according to their mean hydrophobicities and mean side-chain volumes. The cleavage-sites covered 12 residues. In the dipeptide experiments the objective was to separate a selected set of molecules from all other possible dipeptide sequences. Perceptrons, feedforward networks with one hidden layer, and a hybrid network were applied. The filters were trained by a (1,) evolution strategy. Two types of network units employing either a sigmoidal or a unimodal transfer function were used in the feedforward filters, and their influence on classification was investigated. The two-layer hybrid network employed gaussian activation functions. To analyze classification of the different filter systems, their output was plotted in the two-dimensional sequence space. The diagrams were interpreted as fitness landscapes qualifying the markedness of a characteristic peptide feature which can be used as a guide through sequence space for rational peptide design. It is demonstrated that the applicability of neural filter systems as a heuristic method for sequence optimization depends on both the appropriate network architecture and selection of representative sequence data. The networks with unimodal activation functions and the hybrid networks both led to a number of local optima. However, the hybrid networks produced the best prediction results. In contrast, the filters with sigmoidal activation produced good reclassification results leading to fitness landscapes lacking unreasonable local optima. Similar results were obtained for classification of both dipeptides and cleavage-site sequences.     

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.