Small-world networks decrease the speed of Muller's ratchet

Nature, 2005

Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially-extended populations 1−3. Here we generalize population structure by arranging individuals on a graph. Each vertex represents an individual. The weighted edges denote reproductive rates which govern how often individuals place offspring into adjacent vertices. The homogeneous population, described by the Moran process 4 , is the special case of a fully connected graph with evenly-weighted edges. Spatial structures are described by graphs where vertices are connected with their nearest neighbors. We also explore evolution on random and scalefree networks 5−6. We determine the fixation probability of mutants, and characterize those graphs whose fixation behavior is identical to that of a homogeneous population 7. Furthermore, some graphs act as suppressors and others as amplifiers of selection. It is even possible to find graphs that guarantee the fixation of any advantageous mutant. We also study frequency dependent selection and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph. Evolutionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization, and economics.

Physical Review E, 2011

In population genetics, the Moran model describes the neutral evolution of a biallelic gene in a population of haploid individuals subjected to mutations. We show in this paper that this model can be mapped into an influence dynamical process on networks subjected to external influences. The panmictic case considered by Moran corresponds to fully connected networks and can be completely solved in terms of hypergeometric functions. Other types of networks correspond to structured populations, for which approximate solutions are also available. This approach to the classic Moran model leads to a relation between regular networks based on spatial grids and the mechanism of isolation by distance. We discuss the consequences of this connection for topopatric speciation and the theory of neutral speciation and biodiversity. We show that the effect of mutations in structured populations, where individuals can mate only with neighbors, is greatly enhanced with respect to the panmictic case. If mating is further constrained by genetic proximity between individuals, a balance of opposing tendencies takes place: increasing diversity promoted by enhanced effective mutations versus decreasing diversity promoted by similarity between mates. Resolution of large enough opposing tendencies occurs through speciation via pattern formation. We derive an explicit expression that indicates when speciation is possible involving the parameters characterizing the population. We also show that the time to speciation is greatly reduced in comparison with the panmictic case.

Genetics, 2012

The accumulation of deleterious mutations is driven by rare fluctuations that lead to the loss of all mutation free individuals, a process known as Muller’s ratchet. Even though Muller’s ratchet is a paradigmatic process in population genetics, a quantitative understanding of its rate is still lacking. The difficulty lies in the nontrivial nature of fluctuations in the fitness distribution, which control the rate of extinction of the fittest genotype. We address this problem using the simple but classic model of mutation selection balance with deleterious mutations all having the same effect on fitness. We show analytically how fluctuations among the fittest individuals propagate to individuals of lower fitness and have dramatically amplified effects on the bulk of the population at a later time. If a reduction in the size of the fittest class reduces the mean fitness only after a delay, selection opposing this reduction is also delayed. This delayed restoring force speeds up Muller...

Advances in Physics, 2002

We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short-a feature known as the "smallworld" effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc. CONTENTS 42 IX L. Eigenvalue spectrum of the adjacency matrix 42 IX M. Scale-free trees 43 X. Non-scale-free networks with preferential linking 43 XI. Percolation on networks 44 XI A. Theory of percolation on undirected equilibrium networks 44 XI B. Percolation on directed equilibrium networks 48 XI C. Failures and attacks 49 XI D. Resilience against random breakdowns 50 XI E. Intentional damage 52 XI F. Disease spread within networks 54 XI G. Anomalous percolation on growing networks 55 XII. Growth of networks and self-organized criticality 57 XII A. Linking with sand-pile problems 57 XII B. Preferential linking and the Simon model 57 XII C. Multiplicative stochastic models and the generalized Lotka-Volterra equation 58 XIII. Concluding remarks 58 Acknowledgements 59 References 59

2011

In an evolving population, network structure can have striking effects on the survival probability of a mutant allele and on the rate at which it spreads. In networks with ‘hubs’ (representing geographic or other constraints), the heightened probability of an initially rare mutant has led to the prediction that such networks act to amplify the effects of selection over drift. But selection and mutation interplay in a subtle way in such populations: hubs also slow the mutant’s rate of invasion, so that if multiple mutants are allowed to spread at the same time, more of them may be present. In other words it might be misleading to consider only the fixation probability, because new mutants spread at different rates in these networks. Instead of following a single mutation to fixation, we give a very simple model that allows for a stream of mutations, leading to a dynamic equilibrium. In this way we take account of ongoing evolution rather than simply following a single mutant to fixat...

Physical Review E, 2009

Networks of selectively neutral genotypes underlie the evolution of populations of replicators in constant environments. Previous theoretical analysis predicted that such populations will evolve toward highly connected regions of the genome space. We first study the evolution of populations of replicators on simple networks and quantify how the transient time to equilibrium depends on the initial distribution of sequences on the neutral network, on the topological properties of the latter, and on the mutation rate. Second, network neutrality is broken through the introduction of an energy for each sequence. This allows to study the competition between two features ͑neutrality and energetic stability͒ relevant for survival and subjected to different selective pressures. In cases where the two features are negatively correlated, the population experiences sudden migrations in the genome space for values of the relevant parameters that we calculate. The numerical study of larger networks indicates that the qualitative behavior to be expected in more realistic cases is already seen in representative examples of small networks.

We present a numerical study of a reaction-diffusion model on a small-world network. We characterize the model's steady state average activity F ss and the transition from a collective (global) extinct state to an active state in parameter space, and provide an explicit relation between the parameters of our model at the frontier between these states. A collective active state can be associated to a global epidemic spread, or to a persistent neuronal activity. We found that F ss does not depends on disorder in the network if the transmission rate r or the average coordination number K are large enough. The collective extinct-active transition can be induced by changing two parameters associated to the network: K and the disorder parameter p (which controls the variance of K). We can also induce the transition by changing r, which controls the threshold size in the dynamics. We find that, in order to stay at the transition, to increase disorder in the network is equivalent to increase the critical threshold size. Our results are relevant for systems that operate at the transition in order to increase its dynamic range and/or to operate under optimal information-processing conditions. Many problems in Science can be cast in terms of dynamics on networks: social phenomena [1, 2, 3], epidemic spread [4], food webs [5] and ecosystem's diversity [6], brain activity , granular materials [13, 14, 15] and, in general, complex systems [16]. Among the most studied models, the small-world network model of Watts and Strogatz (WS) can be tuned to interpolate between a regular and a random network, a very atractive property that allows us to explore the consequences of network disorder on dynamics. In this work we consider a stochastic reaction-difusion cellular automata model on a small-world network and study its average activity in stationary state and its collective extinct-active transition. We provide a explicit relation between the parameters of the model for the system to operate at the transition. A collective active state can be associated to a global epidemic spread, or to a persistent neuronal activity.

virtualknowledgestudio.nl

Computing Research Repository, 2011

Evolutionary dynamics have been traditionally studied in the context of homogeneous populations, mainly described my the Moran process. Recently, this approach has been generalized in \cite{LHN} by arranging individuals on the nodes of a network. Undirected networks seem to have a smoother behavior than directed ones, and thus it is more challenging to find suppressors/amplifiers of selection. In this paper

The European Physical Journal B - Condensed Matter, 2004

The behavior of spatially inhomogeneous populations in networks of habitats provides examples of dynamical systems on random graphs with structure. A particular example is a butterfly species inhabiting theÅland archipelago. A metapopulation description of the patch occupancies is here mapped to a quenched graph, using the empirical ecology-based incidence function description as a starting point. Such graphs are shown to have interesting features that both reflect the probably "self-organized" nature of a metapopulation that can survive and the geographical details of the landscape. Simulations of the Susceptible-Infected-Susceptible model, to mimick the time-dependent population dynamics relate to the graph features: lack of a typical scale, large connectivity per vertex, and the existence of independent subgraphs. Finally, ideas related to the application and extension of scale-free graphs to metapopulations are discussed.

2007

Adaptation of populations takes place with the occurrence and subsequent fixation of mutations that confer some selective advantage to the individuals which acquire it. For this reason, the study of the process of fixation of advantageous mutations has a long history in the population genetics literature. Particularly, the previous investigations aimed to find out the main evolutionary forces affecting the strength of natural selection in the populations. In the current work, we investigate the dynamics of fixation of beneficial mutations in a subdivided population. The subpopulations (demes) can exchange migrants among their neighbors, in a migration network which is assumed to have either a random graph or a scale-free topology.

Physical Review E, 2001

For asexual organisms point mutations correspond to local displacements in the genotypic space, while other genotypic rearrangements represent longrange jumps. We investigate the spreading properties of an initially homogeneous population in a flat fitness landscape, and the equilibrium properties on a smooth fitness landscape. We show that a small-world effect is present: even a small fraction of quenched long-range jumps makes the results indistinguishable from those obtained by assuming all mutations equiprobable. Moreover, we find that the equilibrium distribution is a Boltzmann one, in which the fitness plays the role of an energy, and mutations that of a temperature.

The 17th Annual Colloquium of the Spatial …, 2005

This paper considers spatially-structured populations described as a network, and examines the prop-erties of these networks in terms of their affect on fixation of neutral alleles due solely to genetic drift. Individuals are modelled as two allele, one locus haploid, diploid and ...

BioSystems, 2006

Molecular ecology, 2017

In populations occupying discrete habitat patches, gene flow between habitat patches may form an intricate population structure. In such structures, the evolutionary dynamics resulting from interaction of gene-flow patterns with other evolutionary forces may be exceedingly complex. Several models describing gene flow between discrete habitat patches have been presented in the population-genetics literature; however, these models have usually addressed relatively simple settings of habitable patches and have stopped short of providing general methodologies for addressing nontrivial gene-flow patterns. In the last decades, network theory - a branch of discrete mathematics concerned with complex interactions between discrete elements - has been applied to address several problems in population genetics by modelling gene flow between habitat patches using networks. Here, we present the idea and concepts of modelling complex gene flows in discrete habitats using networks. Our goal is to ...