Neutral networks of genotypes: Evolution behind the curtain

Our understanding of the evolutionary process has gone a long way since the publication, 150 years ago, of "On the origin of species" by Charles R. Darwin. The XXth Century witnessed great efforts to embrace replication, mutation, and selection within the framework of a formal theory, able eventually to predict the dynamics and fate of evolving populations. However, a large body of empirical evidence collected over the last decades strongly suggests that some of the assumptions of those classical models necessitate a deep revision. The viability of organisms is not dependent on a unique and optimal genotype. The discovery of huge sets of genotypes (or neutral networks) yielding the same phenotype --in the last term the same organism--, reveals that, most likely, very different functional solutions can be found, accessed and fixed in a population through a low-cost exploration of the space of genomes. The 'evolution behind the curtain' may be the answer to some of the current puzzles that evolutionary theory faces, like the fast speciation process that is observed in the fossil record after very long stasis periods.Comment: 7 pages, 7 color figures, uses a modification of pnastwo.cls called pnastwo-modified.cls (included

Stochastic multiplicative processes with reset events

We study a stochastic multiplicative process with reset events. It is shown that the model develops a stationary power-law probability distribution for the relevant variable, whose exponent depends on the model parameters. Two qualitatively different regimes are observed, corresponding to intermittent and regular behaviour. In the boundary between them, the mean value of the relevant variable is time-independent, and the exponent of the stationary distribution equals -2. The addition of diffusion to the system modifies in a non-trivial way the profile of the stationary distribution. Numerical and analytical results are presented.Comment: 8 pages, 3 figures. To appear in Phys. Rev.

Small-world behavior in a system of mobile elements

We analyze the propagation of activity in a system of mobile automata. A number r L^d of elements move as random walkers on a lattice of dimension d, while with a small probability p they can jump to any empty site in the system. We show that this system behaves as a Dynamic Small-World (DSW) and present analytic and numerical results for several quantities. Our analysis shows that the persistence time T* (equivalent to the persistence size L* of small-world networks) scales as T* ~ (r p)^(-t), with t = 1/(d+1).Comment: To appear in Europhysics Letter

Replica-symmetry breaking in dynamical glasses

Systems of globally coupled logistic maps (GCLM) can display complex collective behaviour characterized by the formation of synchronous clusters. In the dynamical clustering regime, such systems possess a large number of coexisting attractors and might be viewed as dynamical glasses. Glass properties of GCLM in the thermodynamical limit of large system sizes $N$ are investigated. Replicas, representing orbits that start from various initial conditions, are introduced and distributions of their overlaps are numerically determined. We show that for fixed-field ensembles of initial conditions, as used in previous numerical studies, all attractors of the system become identical in the thermodynamical limit up to variations of order $1/\sqrt{N}$ because the initial value of the coupling field is characterized by vanishing fluctuations, and thus replica symmetry is recovered for $N\to \infty$. In contrast to this, when random-field ensembles of initial conditions are chosen, replica symmetry remains broken in the thermodynamical limit.Comment: 19 pages, 18 figure

Biodiversity in model ecosystems, II: Species assembly and food web structure

This is the second of two papers dedicated to the relationship between population models of competition and biodiversity. Here we consider species assembly models where the population dynamics is kept far from fixed points through the continuous introduction of new species, and generalize to such models thecoexistence condition derived for systems at the fixed point. The ecological overlap between species with shared preys, that we define here, provides a quantitative measure of the effective interspecies competition and of the trophic network topology. We obtain distributions of the overlap from simulations of a new model based both on immigration and speciation, and show that they are in good agreement with those measured for three large natural food webs. As discussed in the first paper, rapid environmental fluctuations, interacting with the condition for coexistence of competing species, limit the maximal biodiversity that a trophic level can host. This horizontal limitation to biodiversity is here combined with either dissipation of energy or growth of fluctuations, which in our model limit the length of food webs in the vertical direction. These ingredients yield an effective model of food webs that produce a biodiversity profile with a maximum at an intermediate trophic level, in agreement with field studies

Biodiversity in model ecosystems, I: Coexistence conditions for competing species

This is the first of two papers where we discuss the limits imposed by competition to the biodiversity of species communities. In this first paper we study the coexistence of competing species at the fixed point of population dynamic equations. For many simple models, this imposes a limit on the width of the productivity distribution, which is more severe the more diverse the ecosystem is (Chesson, 1994). Here we review and generalize this analysis, beyond the ``mean-field''-like approximation of the competition matrix used in previous works, and extend it to structured food webs. In all cases analysed, we obtain qualitatively similar relations between biodiversity and competition: the narrower the productivity distribution is, the more species can stably coexist. We discuss how this result, considered together with environmental fluctuations, limits the maximal biodiversity that a trophic level can host

Evolution on neutral networks accelerates the ticking rate of the molecular clock

Large sets of genotypes give rise to the same phenotype because phenotypic expression is highly redundant. Accordingly, a population can accept mutations without altering its phenotype, as long as they transform its genotype into another one on the same set. By linking every pair of genotypes that are mutually accessible through mutation, genotypes organize themselves into genotype networks (GN). These networks are known to be heterogeneous and assortative. As these features condition the probability that mutations keep the phenotype unchanged---hence becoming blind to natural selection---it follows that the topology of the GN will influence the evolutionary dynamics of the population. In this letter we analyze this effect by studying the dynamics of random walks (RW) on assortative networks with arbitrary topology. We find that the probability that a RW leaves the network is smaller the longer the time spent in it---i.e., the process is not Markovian. From the biological viewpoint, this "phenotypic entrapment" entails an acceleration in the fixation of neutral mutations, thus implying a non-uniform increase in the ticking rate of the molecular clock with the age of branches in phylogenetic trees. We also show that this effect is stronger the larger the fitness of the current phenotype relative to that of neighboring phenotypes.The authors acknowledge the Fnancial support of the Spanish Ministerio de Ciencia e Innovación under projects FIS2011-27569, PRODIEVO (FIS2011-22449), and Complexity-NET RESINEE, and of Comunidad de Madrid under project MODELICO-CM (S2009/ESP-1691)

Phenotypic effect of mutations in evolving populations of RNA molecules

Abstract Background The secondary structure of folded RNA sequences is a good model to map phenotype onto genotype, as represented by the RNA sequence. Computational studies of the evolution of ensembles of RNA molecules towards target secondary structures yield valuable clues to the mechanisms behind adaptation of complex populations. The relationship between the space of sequences and structures, the organization of RNA ensembles at mutation-selection equilibrium, the time of adaptation as a function of the population parameters, the presence of collective effects in quasispecies, or the optimal mutation rates to promote adaptation all are issues that can be explored within this framework. Results We investigate the effect of microscopic mutations on the phenotype of RNA molecules during their in silico evolution and adaptation. We calculate the distribution of the effects of mutations on fitness, the relative fractions of beneficial and deleterious mutations and the corresponding selection coefficients for populations evolving under different mutation rates. Three different situations are explored: the mutation-selection equilibrium (optimized population) in three different fitness landscapes, the dynamics during adaptation towards a goal structure (adapting population), and the behavior under periodic population bottlenecks (perturbed population). Conclusions The ratio between the number of beneficial and deleterious mutations experienced by a population of RNA sequences increases with the value of the mutation rate μ at which evolution proceeds. In contrast, the selective value of mutations remains almost constant, independent of μ, indicating that adaptation occurs through an increase in the amount of beneficial mutations, with little variations in the average effect they have on fitness. Statistical analyses of the distribution of fitness effects reveal that small effects, either beneficial or deleterious, are well described by a Pareto distribution. These results are robust under changes in the fitness landscape, remarkably when, in addition to selecting a target secondary structure, specific subsequences or low-energy folds are required. A population perturbed by bottlenecks behaves similarly to an adapting population, struggling to return to the optimized state. Whether it can survive in the long run or whether it goes extinct depends critically on the length of the time interval between bottlenecks.Support from the Spanish MICINN through research project FIS2008-05273 is gratefully acknowledged.Peer Reviewe