Nonlinear Dynamics of the Rock-Paper-Scissors Game with Mutations

We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game.Comment: 6 pages, 5 figure

The genetic equidistance result of molecular evolution is independent of mutation rates

The well-established genetic equidistance result shows that sister species are approximately equidistant to a simpler outgroup as measured by DNA or protein dissimilarity. The equidistance result is the most direct evidence, and remains the only evidence, for the constant mutation rate interpretation of this result, known as the molecular clock. However, data independent of the equidistance result have steadily accumulated in recent years that often violate a constant mutation rate. Many have automatically inferred non-equidistance whenever a non-constant mutation rate was observed, based on the unproven assumption that the equidistance result is an outcome of constant mutation rate. Here it is shown that the equidistance result remains valid even when different species can be independently shown to have different mutation rates. A random sampling of 50 proteins shows that nearly all proteins display the equidistance result despite the fact that many proteins have non-constant mutation rates. Therefore, the genetic equidistance result does not necessarily mean a constant mutation rate. Observations of different mutation rates do not invalidate the genetic equidistance result. New ideas are needed to explain the genetic equidistance result that must grant different mutation rates to different species and must be independently testable

Whole genome sequencing of Mycobacterium tuberculosis reveals slow growth and low mutation rates during latent infections in humans

Very little is known about the growth and mutation rates of Mycobacterium tuberculosis during latent infection in humans. However, studies in rhesus macaques have suggested that latent infections have mutation rates that are higher than that observed during active tuberculosis disease. Elevated mutation rates are presumed risk factors for the development of drug resistance. Therefore, the investigation of mutation rates during human latency is of high importance. We performed whole genome mutation analysis of M. tuberculosis isolates from a multi-decade tuberculosis outbreak of the New Zealand Rangipo strain. We used epidemiological and phylogenetic analysis to identify four cases of tuberculosis acquired from the same index case. Two of the tuberculosis cases occurred within two years of exposure and were classified as recently transmitted tuberculosis. Two other cases occurred more than 20 years after exposure and were classified as reactivation of latent M. tuberculosis infections. Mutation rates were compared between the two recently transmitted pairs versus the two latent pairs. Mean mutation rates assuming 20 hour generation times were 5.5X10⁻¹⁰ mutations/bp/generation for recently transmitted tuberculosis and 7.3X10⁻¹¹ mutations/bp/generation for latent tuberculosis. Generation time versus mutation rate curves were also significantly higher for recently transmitted tuberculosis across all replication rates (p = 0.006). Assuming identical replication and mutation rates among all isolates in the final two years before disease reactivation, the u20hr mutation rate attributable to the remaining latent period was 1.6×10⁻¹¹ mutations/bp/generation, or approximately 30 fold less than that calculated during the two years immediately before disease. Mutations attributable to oxidative stress as might be caused by bacterial exposure to the host immune system were not increased in latent infections. In conclusion, we did not find any evidence to suggest elevated mutation rates during tuberculosis latency in humans, unlike the situation in rhesus macaques

Silting mutation in triangulated categories

In representation theory of algebras the notion of `mutation' often plays important roles, and two cases are well known, i.e. `cluster tilting mutation' and `exceptional mutation'. In this paper we focus on `tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing `silting mutation' for silting objects as a generalization of `tilting mutation'. We shall develope a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with `silting mutation' by generalizing the theory of Riedtmann-Schofield and Happel-Unger. We show that iterated silting mutation act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally we give a bijection between silting subcategories and certain t-structures.Comment: 29 page

An Evolutionary Reduction Principle for Mutation Rates at Multiple Loci

A model of mutation rate evolution for multiple loci under arbitrary selection is analyzed. Results are obtained using techniques from Karlin (1982) that overcome the weak selection constraints needed for tractability in prior studies of multilocus event models. A multivariate form of the reduction principle is found: reduction results at individual loci combine topologically to produce a surface of mutation rate alterations that are neutral for a new modifier allele. New mutation rates survive if and only if they fall below this surface - a generalization of the hyperplane found by Zhivotovsky et al. (1994) for a multilocus recombination modifier. Increases in mutation rates at some loci may evolve if compensated for by decreases at other loci. The strength of selection on the modifier scales in proportion to the number of germline cell divisions, and increases with the number of loci affected. Loci that do not make a difference to marginal fitnesses at equilibrium are not subject to the reduction principle, and under fine tuning of mutation rates would be expected to have higher mutation rates than loci in mutation-selection balance. Other results include the nonexistence of 'viability analogous, Hardy-Weinberg' modifier polymorphisms under multiplicative mutation, and the sufficiency of average transmission rates to encapsulate the effect of modifier polymorphisms on the transmission of loci under selection. A conjecture is offered regarding situations, like recombination in the presence of mutation, that exhibit departures from the reduction principle. Constraints for tractability are: tight linkage of all loci, initial fixation at the modifier locus, and mutation distributions comprising transition probabilities of reversible Markov chains.Comment: v3: Final corrections. v2: Revised title, reworked and expanded introductory and discussion sections, added corollaries, new results on modifier polymorphisms, minor corrections. 49 pages, 64 reference

Generalized Hybrid Evolutionary Algorithm Framework with a Mutation Operator Requiring no Adaptation

This paper presents a generalized hybrid evolutionary optimization structure that not only combines both nondeterministic and deterministic algorithms on their individual merits and distinct advantages, but also offers behaviors of the three originating classes of evolutionary algorithms (EAs). In addition, a robust mutation operator is developed in place of the necessity of mutation adaptation, based on the mutation properties of binary-coded individuals in a genetic algorithm. The behaviour of this mutation operator is examined in full and its performance is compared with adaptive mutations. The results show that the new mutation operator outperforms adaptive mutation operators while reducing complications of extra adaptive parameters in an EA representation

Fast Genetic Algorithms

For genetic algorithms using a bit-string representation of length~$n$, the general recommendation is to take $1/n$ as mutation rate. In this work, we discuss whether this is really justified for multimodal functions. Taking jump functions and the $(1+1)$ evolutionary algorithm as the simplest example, we observe that larger mutation rates give significantly better runtimes. For the \jump_{m,n} function, any mutation rate between $2/n$ and $m/n$ leads to a speed-up at least exponential in $m$ compared to the standard choice. The asymptotically best runtime, obtained from using the mutation rate $m/n$ and leading to a speed-up super-exponential in $m$, is very sensitive to small changes of the mutation rate. Any deviation by a small (1 \pm \eps) factor leads to a slow-down exponential in $m$. Consequently, any fixed mutation rate gives strongly sub-optimal results for most jump functions. Building on this observation, we propose to use a random mutation rate $\alpha/n$, where $\alpha$ is chosen from a power-law distribution. We prove that the $(1+1)$ EA with this heavy-tailed mutation rate optimizes any \jump_{m,n} function in a time that is only a small polynomial (in~$m$) factor above the one stemming from the optimal rate for this $m$. Our heavy-tailed mutation operator yields similar speed-ups (over the best known performance guarantees) for the vertex cover problem in bipartite graphs and the matching problem in general graphs. Following the example of fast simulated annealing, fast evolution strategies, and fast evolutionary programming, we propose to call genetic algorithms using a heavy-tailed mutation operator \emph{fast genetic algorithms}

Use of the q-Gaussian mutation in evolutionary algorithms

Copyright @ Springer-Verlag 2010.This paper proposes the use of the q-Gaussian mutation with self-adaptation of the shape of the mutation distribution in evolutionary algorithms. The shape of the q-Gaussian mutation distribution is controlled by a real parameter q. In the proposed method, the real parameter q of the q-Gaussian mutation is encoded in the chromosome of individuals and hence is allowed to evolve during the evolutionary process. In order to test the new mutation operator, evolution strategy and evolutionary programming algorithms with self-adapted q-Gaussian mutation generated from anisotropic and isotropic distributions are presented. The theoretical analysis of the q-Gaussian mutation is also provided. In the experimental study, the q-Gaussian mutation is compared to Gaussian and Cauchy mutations in the optimization of a set of test functions. Experimental results show the efficiency of the proposed method of self-adapting the mutation distribution in evolutionary algorithms.This work was supported in part by FAPESP and CNPq in Brazil and in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant EP/E060722/1 and Grant EP/E060722/2