Random Fitness Landscapes Research Articles

Accessibility percolation on Cartesian power graphs

A fitness landscape is a mapping from a space of discrete genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, on which fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph {mathcal {A}^L}, where L is the number of genetic loci and the allele graph mathcal {A} encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in L displays a transition from 0 to a positive value at a threshold beta _text {c} for the fitness difference between the initial and final genotype. We derive a lower bound on beta _text {c} for general mathcal {A} and show that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs.

Gene-influx-driven evolution.

Here we analyze the evolutionary process in the presence of continuous influx of genotypes with submaximum fitness from the outside to the given habitat with finite resources. We show that strong influx from the outside allows the low-fitness genotype to win the competition with the higher fitness genotype, and in a finite population, drive the latter to extinction. We analyze a mathematical model of this phenomenon and obtain the conditions for the transition from the high-fitness to the low-fitness genotype caused by the influx of the latter. We calculate the time to extinction of the high-fitness genotype in a finite population with two alleles and find the exact analytical dynamics of extinction for the case of many genes with epistasis. We solve a related quasispecies model for a single peak (random) fitness landscape as well as for a symmetric fitness landscape. In the symmetric landscape, a nonperturbative effect is observed such that even an extremely low influx of the low-fitness genotype drastically changes the steady state fitness distribution. A similar nonperturbative phenomenon is observed for the allele fixation time as well. The identified regime of influx-driven evolution appears to be relevant for a broad class of biological systems and could be central to the evolution of prokaryotes and viruses.

Success probability for selectively neutral invading species in the line model with a random fitness landscape

AbstractWe consider a spatial (line) model for invasion of a population by a single mutant with a stochastically selectively neutral fitness landscape, independent from the fitness landscape for nonmutants. This model is similar to those considered earlier. We show that the probability of mutant fixation in a population of size , starting from a single mutant, is greater than , which would be the case if there were no variation in fitness whatsoever. In the small variation regime, we recover precise asymptotics for the success probability of the mutant. This demonstrates that the introduction of randomness provides an advantage to minority mutations in this model, and shows that the advantage increases with the system size. We further demonstrate that the mutants have an advantage in this setting only because they are better at exploiting unusually favorable environments when they arise, and not because they are any better at exploiting pockets of favorability in an environment that is selectively neutral overall.

Simulating Evolution in Asexual Populations with Epistasis.

I show how to use OncoSimulR, software for forward-time genetic simulations, to simulate evolution of asexual populations in the presence of epistatic interactions. This chapter emphasizes the specification of fitness and epistasis, both directly (i.e., specifying the effects of individual mutations and their epistatic interactions) and indirectly (using models for random fitness landscapes).

The parabolic Anderson model on the hypercube

We consider the parabolic Anderson model ∂∂tvn=κΔnvn+ξnvn on the n-dimensional hypercube {−1,+1}n with random i.i.d. potential ξn. We study vn at the location of the kth largest potential, xk,2n. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales vn(tn,xk,2n) behaves like a system without diffusion and grows as exp{(ξn(xk,2n)−κ)tn}, whereas, for long time scales the growth is dictated by the principal eigenvalue and the corresponding eigenfunction of the operator κΔn+ξn, for which we give precise asymptotics. Moreover, the transition time depends only on the difference ξn(x1,2n)−ξn(xk,2n). One of our main motivations is to investigate the mutation–selection model of population genetics on a random fitness landscape, which is given by the ratio of vn to its total mass, with ξn corresponding to the fitness landscape. We show that the above mentioned phase transition translates to the mutation–selection model as follows: a population initially concentrated at xk,2n moves completely to x1,2n on time scales where the transition of growth rates occurs. The class of potentials we consider includes the Random Energy Model (REM) which is studied in the statistical physics literature as one of the main examples of a random fitness landscape.

Adaptive walks on high-dimensional fitness landscapes and seascapes with distance-dependent statistics

The dynamics of evolution is intimately shaped by epistasis — interactions between genetic elements which cause the fitness-effect of combinations of mutations to be non-additive. Analyzing evolutionary dynamics that involves large numbers of epistatic mutations is intrinsically difficult. A crucial feature is that the fitness landscape in the vicinity of the current genome depends on the evolutionary history. A key step is thus developing models that enable study of the effects of past evolution on future evolution. In this work, we introduce a broad class of high-dimensional random fitness landscapes for which the correlations between fitnesses of genomes are a general function of genetic distance. Their Gaussian character allows for tractable computational as well as analytic understanding. We study the properties of these landscapes focusing on the simplest evolutionary process: random adaptive (uphill) walks. Conventional measures of “ruggedness” are shown to not much affect such adaptive walks. Instead, the long-distance statistics of epistasis cause all properties to be highly conditional on past evolution, determining the statistics of the local landscape (the distribution of fitness-effects of available mutations and combinations of these), as well as the global geometry of evolutionary trajectories. In order to further explore the effects of conditioning on past evolution, we model the effects of slowly changing environments. At long times, such fitness “seascapes” cause a statistical steady state with highly intermittent evolutionary dynamics: populations undergo bursts of rapid adaptation, interspersed with periods in which adaptive mutations are rare and the population waits for more new directions to be opened up by changes in the environment. Finally, we discuss prospects for studying more complex evolutionary dynamics and on broader classes of high-dimensional landscapes and seascapes.

The rich phase structure of a mutator model.

We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase and non-selective phase. We calculate the phase structure, the mean fitness and the fraction of the mutator allele in the population, which can be applied to describe cancer development and RNA viruses. We find that depending on the genome length, either the normal or the mutator allele dominates in the mixed phase. We analytically solve the model for a general fitness function. We conclude that the random fitness landscape is an appropriate choice for describing the observed mutator phenomenon in the case of a small fraction of mutators. It is shown that the increase in the mutation rates in the regular and the mutator parts of the genome should be set independently; only some combinations of these increases can push the complex biomedical system to the non-selective phase, potentially related to the eradication of tumors.

Phase transition in random adaptive walks on correlated fitness landscapes

We study biological evolution on a random fitness landscape where correlations are introduced through a linear fitness gradient of strength c. When selection is strong and mutations rare the dynamics is a directed uphill walk that terminates at a local fitness maximum. We analytically calculate the dependence of the walk length on the genome size L. When the distribution of the random fitness component has an exponential tail, we find a phase transition of the walk length D between a phase at small c, where walks are short (D∼lnL), and a phase at large c, where walks are long (D∼L). For all other distributions only a single phase exists for any c>0. The considered process is equivalent to a zero temperature Metropolis dynamics for the random energy model in an external magnetic field, thus also providing insight into the aging dynamics of spin glasses.

Evolutionary Accessibility in Tunably Rugged Fitness Landscapes

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 population dynamic parameters such as mutation rate and population size, evolutionary accessibility can be quantified through the statistics of accessible mutational pathways (along which fitness increases monotoncially), or through the study of the basin of attraction of the optimal genotype under greedy (steepest ascent) dynamics. Here we investigate these two measures of accessibility in the framework of Kauffman's $LK$-model, a paradigmatic family of random fitness landscapes with tunable ruggedness. The key parameter governing the strength of genetic interactions is the number $K$ of interaction partners of each of the $L$ sites in the genotype sequence. In general, accessibility increases with increasing genotype dimensionality $L$ and decreases with increasing number of interactions $K$. Remarkably, however, we find that some measures of accessibility behave non-monotonically as a function of $K$, indicating a special role of the most sparsely connected, non-trivial cases K=1 and 2. The relation between models for fitness landscapes and spin glasses is also addressed.

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.

Evolution models with lethal mutations on symmetric or random fitness landscapes

We calculate the mean fitness for evolution models, when the fitness is a function of the Hamming distance from a reference sequence, and there is a probability that this fitness is nullified (Eigen model case) or tends to the negative infinity (Crow-Kimura model case). We calculate the mean fitness of these models. The mean fitness is calculated also for the random fitnesses with logarithmic-normal distribution, reasonably describing sometimes the situation with RNA viruses.

Evolution in random fitness landscapes: the infinite sites model

We consider the evolution of an asexually reproducing population in an uncorrelatedrandom fitness landscape in the limit of infinite genome size, which implies that eachmutation generates a new fitness value drawn from a probability distributiong(w). This is the finite population version of Kingman’s house of cards model (Kingman1978 J. Appl. Probab. 15 1). In contrast to Kingman’s work, the focus here is on unbounded distributionsg(w) which lead to an indefinite growth of the population fitness. The model is solvedanalytically in the limit of infinite population size and simulated numerically for finiteN. When the genome-wide mutation probabilityU is small, the long-time behavior of the model reduces to a point process of fixationevents, which is referred to as a diluted record process (DRP). The DRP is similarto the standard record process except that a new record candidate (a numberthat exceeds all previous entries in the sequence) is accepted only with a certainprobability that depends on the values of the current record and the candidate.We develop a systematic analytic approximation scheme for the DRP. At finiteU the fitness frequency distribution of the population decomposes into astationary part due to mutations and a traveling wave component due toselection, which is shown to imply a reduction of the mean fitness by a factor of1−U compared to the limit.

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.

Embracing Plagiarism: Theoretical, Biological and Empirical Justification for Copy Operators in Genetic Optimisation

A novel genetic operator, the plagiarism operator, is introduced for evolutionary design and optimisation. This operator is analogous in some respects to crossover and to biological transposition. Plagiarism is shown to be theoretically superior to uniform mutation for generalised counting-ones problems, and also to outperform uniform mutation on certain classes of random fitness landscapes. Experimental results are presented showing that plagiarism speeds up the artificial evolution of certain digital logic circuits. The performance of this operator is interpreted in terms of the non-uniform distribution of genetic primitives in good solutions for certain problems.

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

It was recently conjectured by H.A. Orr [2003. A minimum on the mean number of steps taken in adaptive walks. J. Theor. Biol. 220, 241–247] that from a random initial point on a random fitness landscape of alphabetic sequences with one-mutation adjacency, chosen from a larger class of landscapes, no adaptive algorithm can arrive at a local optimum in fewer than on average e - 1 steps. Here, using an example in which the mean number of steps to a local optimum equals ( A - 1 ) / A , where A is the number of distinct “letters” in the “alphabet” from which sequences are constructed, it is shown that as originally stated, the conjecture does not hold. It is also demonstrated that ( A - 1 ) / A is a sharp minimum on the mean number of steps taken in adaptive walks on fitness landscapes of alphabetic sequences with one-mutation adjacency. As the example that achieves the new lower bound has properties that are not often considered as potential attributes for fitness landscapes—non-identically distributed fitnesses and negative fitness correlations for adjacent points—a weaker set of conditions characteristic of more commonly studied fitness landscapes is proposed under which the lower bound on the mean length of adaptive walks is conjectured to equal e - 1 .

Evolutionary trajectories in rugged fitness landscapes

We consider the evolutionary trajectories traced out by an infinite population undergoingmutation–selection dynamics in static, uncorrelated random fitness landscapes. Startingfrom the population that consists of a single genotype, the most populated genotype jumpsfrom 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 stepto the competition between independent mutant subpopulations, to study thedynamics of this model and of a simpler one-dimensional model which ignores thegeometry of the sequence space. We find that the fit genotypes that appear along atrajectory are a subset of suitably defined fitness records, and exploit severalresults from the record theory for non-identically distributed random variables.The genotypes that contribute to the trajectory are those records that are notbypassed by superior records arising further away from the initial population. Severalconjectures concerning the statistics of bypassing are extracted from numericalsimulations. In particular, for the one-dimensional model, we propose a simple relationbetween the bypassing probability and the dynamic exponent which describesthe scaling of the typical evolution time with genome size. The latter can bedetermined exactly in terms of the extremal properties of the fitness distribution.

Self-similar potentials in random media, fractal evolutionary landscapes and Kimura's neutral theory of molecular evolution

A general method for constructing self-similar scalar fractal random fields is suggested based on the assumption that the fields are generated by a broad distribution of punctual sources. The method is illustrated by a problem of population biology, the evolution on a random fitness landscape. A random fitness landscape is constructed based on the following hypotheses. The landscape is defined by the dependence of a fitness variable φ on the state vector x of the individuals: φ = φ( x) ; the corresponding hypersurface has a large number of local maxima characterized by a local probability law with variable parameters. These maxima are uniformly randomly distributed throughout the state space of the individuals. From generation to generation the heights and the shapes of these local maxima can change; this change is described in terms of two probabilities p and α that an individual modification occurs and that the process of variation as a whole stops, respectively. A general method for computing the stochastic properties of the evolutionary landscape is suggested based on the use of characteristic functionals. An explicit computation of the Fourier spectrum of the cumulants of the evolutionary landscape is performed in a limit of the thermodynamic type for which the number of maxima and the volume of the state space of the individuals tend to infinity but the density of maxima remains constant. It is shown that, although a typical realization of the evolutionary landscape is very rough, its average properties expressed by the Fourier spectrum of its cumulants are smooth and characterized by scaling laws of the power law type. The average landscape which is made up of the frozen contributions of the changes corresponding to different generations is flat, a result which is consistent with the Kimura's theory of molecular evolution. Some general implications of the suggested approach for the statistical physics of systems with random ultrametric topology are also investigated.

Evolutionary Walks on Rugged Landscapes

The rapid evolution of antibody molecules during an immune response enables the immune system to respond efficiently to an immense variety of challenges. To investigate this response, a theory for molecular evolution in biological systems is developed and analyzed. A molecule can be represented as a sequence of Nletters, with each letter being chosen from an alphabet of size a; for protein molecules the alphabet consists of the twenty amino acids, while for nucleic acids it consists of the four base pairs. Together, these $a^N $ possibilities form a sequence space ${\bf S}$ It is assumed that a fitness can be assigned to each sequence in ${\bf S}$; for the immune response the fitness is just the chemical affinity of the antibody for the immunizing antigen. Evolution is assumed to occur by random point mutations that change a single letter in the sequence; this defines the set of one-mutant neighbors of a sequence. It is assumed that the original sequence will be replaced by the one-mutant neighbor if and only if the mutant has a higher fitness than the original. Thus, molecular evolution is modeled as being a strictly uphill walk on a fitness landscape, with the landscape being determined by the function that assigns a fitness to each sequence in ${\bf S}$. Here evolution is studied on a completely random fitness landscape; evolution on other landscapes is considered elsewhere. It is shown that the fitness landscape is characterized by a large number of local optima, and that evolutionary walks can be expected to become trapped fairly quickly at local optima, rather than at the global optimum. Various statistics of the trapping process are computed, such as the probability of being trapped on the kth mutational step, and the mean and variance of the number of steps to a local optimum. It is also shown that, on average, the local optimum obtained at the end of an evolutionary walk is closer to the global optimum than a randomly selected local optimum, thus establishing that the evolutionary process is more efficient than random search. Because not all mutations improve fitness, various statistics are examined that characterize the total number of mutations and the number of different mutations attempted during the evolutionary process. Finally, the theory is applied to somatic mutation during an immune response.

Protein evolution on rugged landscapes.

We analyze a mathematical model of protein evolution in which the evolutionary process is viewed as hill-climbing on a random fitness landscape. In studying the structure of such landscapes, we note that a large number of local optima exist, and we calculate the time and number of mutational changes until a protein gets trapped at a local optimum. Such a hill-climbing process may underlie the evolution of antibody molecules by somatic hypermutation.