Clonal interference, the competition between lineages arising from different beneficial mutations in an asexually reproducing population, is an important factor determining the tempo and mode of microbial adaptation. The standard theory of this phenomenon neglects the occurrence of multiple mutations as well as the correlation between loss by genetic drift and clonal competition, which is questionable in large populations. Working within the Wright-Fisher model with multiplicative fitness (no epistasis), we determine the rate of adaptation asymptotically for very large population sizes and show that the standard theory fails in this regime. Our study also explains the success of the standard theory in predicting the rate of adaptation for moderately large populations. Furthermore, we show that the nature of the substitution process changes qualitatively when multiple mutations are allowed for, because several mutations can be fixed in a single fixation event. As a consequence, the index of dispersion for counts of the fixation process displays a minimum as a function of population size, whereas the origination process of fixed mutations becomes completely regular for very large populations. We find that the number of mutations fixed in a single event is geometrically distributed as in the neutral case. These conclusions are based on extensive simulations combined with analytic results for the limit of infinite population size. microbial adaptation ͉ substitution process ͉ rate of adaptation ͉ index of dispersion ͉ Wright-Fisher model

The nature of epistasis has important consequences for the evolutionary significance of sex and recombination. Recent efforts to find negative epistasis as a source of negative linkage disequilibrium and associated long-term advantage to sex have yielded little support. Sign epistasis, where the sign of the fitness effects of alleles varies across genetic backgrounds, is responsible for the ruggedness of the fitness landscape, with several unexplored implications for the evolution of sex. Here, we describe fitness landscapes for two sets of strains of the asexual fungus Aspergillus niger involving all combinations of five mutations. We find that ∼30% of the single-mutation fitness effects are positive despite their negative effect in the wildtype strain and that several local fitness maxima and minima are present. We then compare adaptation of sexual and asexual populations on these empirical fitness landscapes by using simulations. The results show a general disadvantage of sex on these rugged landscapes, caused by the breakdown by recombination of genotypes on fitness peaks. Sex facilitates movement to the global peak only for some parameter values on one landscape, indicating its dependence on the landscape's topography. We discuss possible reasons for the discrepancy between our results and the reports of faster adaptation of sexual populations.

The pair contact process with diffusion (PCPD) has been recently investigated extensively, but its critical behavior is not yet clearly established. By introducing biased diffusion, we show that the external driving is relevant and the driven PCPD exhibits a mean-field-type critical behavior even in one dimension. In systems which can be described by a single-species bosonic field theory, the Galilean invariance guarantees that the driving is irrelevant. The well-established directed percolation (DP) and parity conserving (PC) classes are such examples. This leads us to conclude that the PCPD universality class should be distinct from the DP or PC class. Moreover, it implies that the PCPD is generically a multi-species model and a field theory of two species is suitable for proper description.PACS numbers: 64.60. Ht,05.70.Ln,89.75.Da The steady state of an equilibrium system is characterized by its Hamiltonian and Gibbs measure. There is no systematic generalization to the stationary state of nonequilibrium systems so far. Since nonequilibrium systems encompass all kinds of many body systems without a constraint of detailed balance, it may be hopeless to find a universal formalism applied to general nonequilibrium systems. At this point, model studies or case-by-case studies are rather useful to accumulate our knowledge on nonequilibrium systems.Our experience on equilibrium systems illustrates the scale-free fluctuation or power law behavior at the critical point where the continuous phase change occurs. The scale-free nature is worth while to be studied not only because of its theoretical attraction, but also because of ubiquity in nature -the clustering of galaxies [1], 1/f noise [2], percolation structure [3], to name only a few. This scale-free nature is also expected at criticality under nonequilibrium circumstances. As a prototype of nonequilibrium critical phenomena, absorbing phase transitions (APTs) have been studied extensively [4]. APT is a transition from an active phase to an absorbing phase in nonequilibrium steady states. The absorbing states are defined as the configurations where the system cannot escape by the prescribed dynamic rules. As in equilibrium systems, this transition is possible only at the thermodynamic limit because the finite systems eventually fall into the absorbing states.In epidemiology, for example, the virus extinct state is an absorbing state. Actually, the disease spreading is modeled and dubbed the contact process (CP) by Harris [5]. CP shows a phase transition from the virus infested state (active state), to the quiescent state (absorbing state). This transition is known to belong to the directed percolation (DP) universality class. Actually, many types of models belong to the DP class and it is conjectured that a phase transition occurred in a system with a single absorbing state should share the critical behavior with the DP [6,7].As in equilibrium critical phenomena, a symmetry or conservation may play an important role in determining the universality class. Accor...

We consider an asexual biological population of constant size N evolving in discrete time under the influence of selection and mutation. Beneficial mutations appear at rate U and their selective effects s are drawn from a distribution g(s). After introducing the required models and concepts of mathematical population genetics, we review different approaches to computing the speed of logarithmic fitness increase as a function of N, U and g(s). We present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid. We then discuss approximate approaches to the finite population problem, distinguishing between the case of a single selection coefficient, g(s) = δ (s − s b ), and a continuous distribution of selection coefficients. Analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10 300 .

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.

We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution g(w). This is the finite population version of Kingman's house of cards model [J.F.C. Kingman, J. Appl. Probab. 15, 1 (1978)]. In contrast to Kingman's work, the focus here is on unbounded distributions g(w) which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size N → ∞ and simulated numerically for finite N . When the genomewide mutation probability U is small, the long time behavior of the model reduces to a point process of fixation events, which is referred to as a diluted record process (DRP). The DRP is similar to the standard record process except that a new record candidate (a number that exceeds all previous entries in the sequence) is accepted only with a certain probability that depends on the values of the current record and the candidate. We develop a systematic analytic approximation scheme for the DRP. At finite U the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of 1 − U compared to the U → 0 limit.

Quantifying the dynamics of intrahost HIV-1 sequence evolution is one means of uncovering information about the interaction between HIV-1 and the host immune system. In the chronic phase of infection, common dynamics of sequence divergence and diversity have been reported. We developed an HIV-1 sequence evolution model that simulated the effects of mutation and fitness of sequence variants. The amount of evolution was described by the distance from the founder strain, and fitness was described by the number of offspring a parent sequence produces. Analysis of the model suggested that the previously observed saturation of divergence and decrease of diversity in later stages of infection can be explained by a decrease in the proportion of offspring that are mutants as the distance from the founder strain increases rather than due to an increase of viral fitness. The prediction of the model was examined by performing phylogenetic analysis to estimate the change in the rate of evolution during infection. In agreement with our modeling, in 13 out of 15 patients (followed for 3–12 years) we found that the rate of intrahost HIV-1 evolution was not constant but rather slowed down at a rate correlated with the rate of CD4+ T-cell decline. The correlation between the dynamics of the evolutionary rate and the rate of CD4+ T-cell decline, coupled with our HIV-1 sequence evolution model, explains previously conflicting observations of the relationships between the rate of HIV-1 quasispecies evolution and disease progression.

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