Simon-Ando decomposability and fitness landscapes

In this paper, we investigate fitness landscapes (under point mutation and recombination) from the standpoint of whether the induced evolutionary dynamics have a “fast-slow” time scale associated with the differences in relaxation time between local quasi-equilibria and the global equilibrium. This dynamical hevavior has been formally described in the econometrics literature in terms of the spectral properties of the appropriate operator matrices by Simon and Ando (Econometrica 29 (1961) 111), and we use the relations they derive to ask which fitness functions and mutation/recombination operators satisfy these properties. It turns out that quite a wide range of landscapes satisfy the condition (at least trivially) under point mutation given a sufficiently low mutation rate, while the property appears to be difficult to satisfy under genetic recombination. In spite of the fact that Simon-Ando decomposability can be realized over fairly wide range of parameters, it imposes a number of restriction on which landscape partitionings are possible. For these reasons, the Simon-Ando formalism does not appear to be applicable to other forms of decomposition and aggregation of variables that are important in evolutionary systems.     

Fourier and Taylor series on fitness landscapes

Holland's hyperplane transform of a fitness landscape, a random, real valued function of the verticies of a regular finite graph, is shown to be a special case of the Fourier transform of a function of a finite group. It follows that essentially all of the powerful Fourier theory, which assumes a simple form for commutative groups, can be used to characterize such landscapes. In particular, an analogue of the KarhunenLoève expansion can be used to prove that the Fourier coefficients of landscapes on commutative groups are uncorrelated and to infer their variance from the autocorrelation function of a random walk on the landscape. There is also a close relationship between the Fourier coefficients and Taylor coefficients, which provide information about the landscape's local properties. Special attention is paid to a particularly simple, but ubiquitous class of landscapes, so-called AR(1) landscapes.     

Experimental fitness landscapes to understand the molecular evolution of RNA-based life

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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.     

Multidimensional epistasis and fitness landscapes in enzyme evolution

The conventional analysis of enzyme evolution is to regard one single salient feature as a measure of fitness, expressed in a milieu exposing the possible selective advantage at a given time and location. Given that a single protein may serve more than one function, fitness should be assessed in several dimensions. In the present study we have explored individual mutational steps leading to a triple-point-mutated human GST (glutathione transferase) A2-2 displaying enhanced activity with azathioprine. A total of eight alternative substrates were used to monitor the diverse evolutionary trajectories. The epistatic effects of the mutations on catalytic activity were variable in sign and magnitude and depended on the substrate used, showing that epistasis is a multidimensional quality. Evidently, the multidimensional fitness landscape can lead to alternative trajectories resulting in enzymes optimized for features other than the selectable markers relevant at the origin of the evolutionary process. In this manner the evolutionary response is robust and can adapt to changing environmental conditions.     

Combinatorial vector fields and the valley structure of fitness landscapes

Adaptive (downhill) walks are a computationally convenient way of analyzing the geometric structure of fitness landscapes. Their inherently stochastic nature has limited their mathematical analysis, however. Here we develop a framework that interprets adaptive walks as deterministic trajectories in combinatorial vector fields and in return associate these combinatorial vector fields with weights that measure their steepness across the landscape. We show that the combinatorial vector fields and their weights have a product structure that is governed by the neutrality of the landscape. This product structure makes practical computations feasible. The framework presented here also provides an alternative, and mathematically more convenient, way of defining notions of valleys, saddle points, and barriers in landscape. As an application, we propose a refined approximation for transition rates between macrostates that are associated with the valleys of the landscape.     

Cluster partitions and fitness landscapes of the Drosophila fly microbiome

Journal of Mathematical Biology - The concept of genetic epistasis defines an interaction between two genetic loci as the degree of non-additivity in their phenotypes. A fitness landscape describes...     

Male competition fitness landscapes predict both forward and reverse speciation

Speciation is facilitated when selection generates a rugged fitness landscape such that populations occupy different peaks separated by valleys. Competition for food resources is a strong ecological force that can generate such divergent selection. However, it is unclear whether intrasexual competition over resources that provide mating opportunities can generate rugged fitness landscapes that foster speciation. Here we use highly variable male F2 hybrids of benthic and limnetic threespine sticklebacks, Gasterosteus aculeatus Linnaeus, 1758, to quantify the male competition fitness landscape. We find that disruptive sexual selection generates two fitness peaks corresponding closely to the male phenotypes of the two parental species, favouring divergence. Most surprisingly, an additional region of high fitness favours novel hybrid phenotypes that correspond to those observed in a recent case of reverse speciation after anthropogenic disturbance. Our results reveal that sexual selection through male competition plays an integral role in both forward and reverse speciation.     

The NK model of rugged fitness landscapes and its application to maturation of the immune response

Adaptive evolution is, to a large extent, a complex combinatorial optimization process. Such processes can be characterized as "uphill walks on rugged fitness landscapes". Concrete examples of fitness landscapes include the distribution of any specific functional property such as the capacity to catalyze a specific reaction, or bind a specific ligand, in "protein space". In particular, the property might be the affinity of all possible antibody molecules for a specific antigenic determinant. That affinity landscape presumably plays a critical role in maturation of the immune response. In this process, hypermutation and clonal selection act to select antibody V region mutant variants with successively higher affinity for the immunizing antigen. The actual statistical structure of affinity landscapes, although knowable, is currently unknown. Here, we analyze a class of mathematical models we call NK models. We show that these models capture significant features of the maturation of the immune response, which is currently thought to share features with general protein evolution. The NK models have the important property that, as the parameter K increases, the "ruggedness" of the NK landscape varies from a single peaked "Fujiyama" landscape to a multi-peaked "badlands" landscape. Walks to local optima on such landscapes become shorter as K increases. This fact allows us to choose a value of K that corresponds to the experimentally observed number of mutational "steps", 6-8, taken as an antibody sequence matures. If the mature antibody is taken to correspond to a local optimum in the model, tuning the model requires that K be about 40, implying that the functional contribution of each amino acid in the V region is affected by about 40 others. Given this value of K, the model then predicts several features of "antibody space" that are in qualitative agreement with experiment: (1) The fraction of fitter variants of an initial "roughed in" germ line antibody amplified by clonal selection is about 1-2%. (2) Mutations at some sites of the mature antibody hardly affect antibody function at all, but mutations at other sites dramatically decrease function. (3) The same "roughed in" antibody sequence can "walk" to many mature antibody sequences. (4) Many adaptive walks can end on the same local optimum. (5) Comparison of different mature sequences derived from the same initial V region shows evolutionary hot spots and parallel mutations. All these predictions are open to detailed testing by obtaining monoclonal antibodies early in the immune response and carrying out in vitro mutagenesis and adaptive hill climbing with respect to affinity for the immunizing antigen.     

Coevolution to the edge of chaos: coupled fitness landscapes, poised states, and coevolutionary avalanches

We introduce a broadened framework to study aspects of coevolution based on the NK class of statistical models of rugged fitness landscapes. In these models the fitness contribution of each of N genes in a genotype depends epistatically on K other genes. Increasing epistatic interactions increases the rugged multipeaked character of the fitness landscape. Coevolution is thought of, at the lowest level, as a coupling of landscapes such that adaptive moves by one player deform the landscapes of its immediate partners. In these models we are able to tune the ruggedness of landscapes, how richly intercoupled any two landscapes are, and how many other players interact with each player. All these properties profoundly alter the character of the coevolutionary dynamics. In particular, these parameters govern how readily coevolving ecosystems achieve Nash equilibria, how stable to perturbations such equilibria are, and the sustained mean fitness of coevolving partners. In turn, this raises the possibility that an evolutionary metadynamics due to natural selection may sculpt landscapes and their couplings to achieve coevolutionary systems able to coadapt well. The results suggest that sustained fitness is optimized when landscape ruggedness relative to couplings between landscapes is tuned such that Nash equilibria just tenuously form across the ecosystem. In this poised state, coevolutionary avalanches appear to propagate on all length scales in a power law distribution. Such avalanches may be related to the distribution of small and large extinction events in the record.     

Tell me how and where you play football and I’ll tell you how much you have to run

The aim of this study was to describe the team accumulated physical response per minute considering only the effective playing time (EPT) attending to different contextual and strategic variables in the Spanish Football First Division (LaLiga Santander). A sample of 2,959 performances was included in the analysis, considering a 4-year period of analysis (from 2016–17 to 2019–20). The physical variables were: total distance covered with (DTminPOS) and without (DTminNOpos) possession of the ball, and distance covered at > 21 km·h^-1 with possession (DT21minPOS) and without (DT21minNOpos). Two contextual variables, i.e. Place (Home/Away) and Score (Lost/Draw/Win), and two strategic variables, i.e. level of effective playing time (LevelETP) and level of possession of the ball (LevelPOS), were analysed. The teams ran more without possession of the ball than with possession; nevertheless, the teams that had less possession of the ball had higher values in the distance covered at > 21 km·h^-1 with possession of the ball and vice versa. Furthermore, the strategic variables also had influence on the physical response (DT and DT21) of the teams,LevelETP and LevelPOS, although with interactive effects: longer playing time, less accumulated distance, and greater possession, greater accumulated distance in the defensive phase, both per min. The findings of this study may offer important practical implications to practitioners in order to assess physical performances of the players in matches, because it is crucial to integrate in the analysis the different contextual and strategic variables where the match has taken place to assess performances of the teams.     

Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length L. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance d(eff) up to which the population can spread in genotype space. For d(eff) = L, the deterministic quasi-species theory operates while for d(eff) < 1, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasi-species theory below a crossover time T(x) while above T(x) the population gets trapped at a local fitness peak and manages to find a better peak via either stochastic tunneling or double mutations. In the stochastic regime d(eff) < 1, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interpretation of evolution experiments with microbial populations.     

Cambrian explosion and Permian quiescence: Implications of rugged fitness landscapes

Summary The transition from the late Precambrian to Cambrian coincided with a massive increase in diversity of multicellular organisms and the rapid establishment of a large number of phyla. In contrast, the Permian extinction 200 million years ago was followed by as rapid an increase at the family level, but no new phyla or classes emerged. This asymmetry has suggested alternative theories based on greater ecological opportunity in the Cambrian, or special developmental canalization locking in development by the Permian. I suggest instead that the asymmetry reflects generic features of adaptive evolution on rugged fitness landscapes.     

Puddling: from natural history to understanding how it affects fitness

The term ‘puddling’ includes feeding on (dried) mud and various excrements and secretions of vertebrates, and carrion. It is thought to be a form of supplementary feeding, not targeted at obtaining energy. Although the natural history of the puddling phenomenon in herbivorous arthropods becomes better known, it is still largely unclear how puddling (in particular for sodium) affects fitness despite the growing knowledge of insect physiology at the cellular level. If we follow the definition used for puddling in Lepidoptera, representatives of a wide range of herbivorous and detrivorous terrestrial arthropods (Lepidoptera, Orthoptera, Blattodea, Hymenoptera, Hemiptera, Diptera, and Diplopoda) have been observed to puddle. It appears that those species with diets low in sodium (e.g., folivorous larvae) puddle for sodium whereas those with diets low in nitrogen (e.g., detritivores) puddle for nitrogen. Sex differentials in puddling behavior can usually be explained by transfers of nutrients from males to females during mating. Puddling is rare or absent in immature stages and there is some evidence that nutrients from puddles increase female reproductive success. Strong evidence for the widely cited hypothesis that sodium from puddles is used to enhance neuromuscular activity is still lacking. High mobility and long life spans could be associated with puddling behavior, whereas insects that are concealed or well defended are less likely to puddle (e.g., beetles). The role that risks of pathogen and parasite infection as well as predation at puddling substrates may play in the evolution of puddling remains virtually unexplored.