Coevolution: the geographic mosaic of coevolutionary arms races

Coevolutionary arms races between species can favor exaggeration of traits for attack and defense, but relentless escalation of these arms races does not necessarily occur in all populations.     

Coevolution across landscapes: a spatially explicit model of parasitoid-host coevolution

The geographic mosaic theory of coevolution suggests that population spatial structure may have a strong impact on coevolutionary dynamics. Therefore, coevolution must be studied across geographic scales, not just in single populations. To examine the impact of movement rate on coevolutionary dynamics, we developed a spatially explicit model of host–parasitoid coevolution. We described space as a coupled-map lattice and assumed that resistance (defined as the ability of a host to encapsulate a parasitoid egg) and virulence (defined as the successful parasitization of a host) traits were graded and costly. The model explicitly detailed population and evolutionary dynamics. When holding all parameters constant and varying only the movement rate of the host and parasitoid, profoundly different dynamics were observed. We found that fluctuations in the mean levels of resistance and virulence in the global population were greatest when the movement rate of the host and parasitoid was high. In addition, we found that the variation in resistance and virulence levels among neighboring patches was greatest when the movement rates of the host and parasitoid was low. However, as the distance among patches increased, so did the variation in resistance and virulence levels regardless of movement rate. These generalizations did not hold when spatial patterns in the distribution of resistance and virulence traits, such as spirals, were observed. Finally, we found that the evolution of resistance and virulence caused the abundance of hosts to increase and the abundance of parasitoids to decrease. As a result, the spatial distribution of hosts and parasitoids was influenced.     

Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities

We study how correlations in the random fitness assignment may affect the structure of fitness landscapes, in three classes of fitness models. The first is a phenotype space in which individuals are characterized by a large number n of continuously varying traits. In a simple model of random fitness assignment, viable phenotypes are likely to form a giant connected cluster percolating throughout the phenotype space provided the viability probability is larger than 1/2(n). The second model explicitly describes genotype-to-phenotype and phenotype-to-fitness maps, allows for neutrality at both phenotype and fitness levels, and results in a fitness landscape with tunable correlation length. Here, phenotypic neutrality and correlation between fitnesses can reduce the percolation threshold, and correlations at the point of phase transition between local and global are most conducive to the formation of the giant cluster. In the third class of models, particular combinations of alleles or values of phenotypic characters are "incompatible" in the sense that the resulting genotypes or phenotypes have zero fitness. This setting can be viewed as a generalization of the canonical Bateson-Dobzhansky-Muller model of speciation and is related to K-SAT problems, prominent in computer science. We analyze the conditions for the existence of viable genotypes, their number, as well as the structure and the number of connected clusters of viable genotypes. We show that analysis based on expected values can easily lead to wrong conclusions, especially when fitness correlations are strong. We focus on pairwise incompatibilities between diallelic loci, but we also address multiple alleles, complex incompatibilities, and continuous phenotype spaces. In the case of diallelic loci, the number of clusters is stochastically bounded and each cluster contains a very large sub-cube. Finally, we demonstrate that the discrete NK model shares some signature properties of models with high correlations.     

Correlated and uncorrelated fitness landscapes and how to tell the difference

The properties of multi-peaked fitness landscapes have attracted attention in a wide variety of fields, including evolutionary biology. However, relaively little attention has been paid to the properties of the landscapes themselves. Herein, we suggest a framework for the mathematical treatment of such landscapes, including an explicit mathematical model. A central role in this discussion is played by the autocorrelation of fitnesses obtained from a random walk on the landscape. Our ideas about average autocorrelations allow us to formulate a condition (satisfied by a wide class of landscapes we call AR(1) landscapes) under which the average autocorrelation approximates a decaying exponential. We then show how our mathematical model can be used to estimate both the globally optimal fitnesses of AR(1) landscapes and their local structure. We illustrate some aspects of our method with computer experiments based on a single family of landscapes (Kauffman's N-k model), that is shown to be a generic AR(1) landscape. We close by discussing how these ideas might be useful in the tuning of combinatorial optimization algorithms, and in modelling in the experimental sciences.     

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

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Why do evolutionary systems stick to the edge of chaos

Summary  The long-term behaviour of dynamic systems can be classified in two different regimes, regular or chaotic, depending on the values of the control parameters, which are kept constant during the time evolution. Starting from slightly different initial conditions, a regular system converges to the same final trajectory, whereas a chaotic system follows two distinct trajectories exponentially diverging from each other. In spite of these differences, regular and chaotic systems share a common property: both arrive exponentially fast to their final destiny, becoming trapped there. In both cases one has finite transient times. This is not a profitable property in what concerns evolutionary strategies, where the eternal search for new forms, better than the current one, is imperative. That is why evolutionary dynamic systems tend to tune themselves in very particular situations in between regular and chaotic regimes. These particular situations present eternal transients, and the system actually never reaches its final destiny, preserving diversity. This feature allows the system to visit other regions of the space of possibilities, not only the tiny region covered by its final attractor.     

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.     

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.     

The role of coevolutionary signatures in protein interaction dynamics,complex inference,molecular recognition,and mutational landscapes

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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...     

Ecosystem engineering,environmental decay and environmental states of landscapes

Although environmental modification by ecosystem engineers influences species distributions and abundances and ecological process rates, general determinants of the environmental states of engineered landscapes are not well understood. Here we develop a general, spatially implicit model of engineered landscapes that includes parameters driving engineer populations (demographics, environmental modification) and environmental decay. We show that average environmental states and heterogeneities of landscapes are the result of a balance between parameters determining engineering rates and decay rates that can be expressed as a net engineering ratio (NER). This ratio highlights the need to include environmental decay in ecosystem engineering studies. Moreover, it defines a significant engineer as one that can alter the environment despite decay and generates expectations for different kinds of effects on the engineer, other species and ecological processes depending on ratio values. Finally, it suggests that, in general, decay places limits as to what can be inferred about engineer population dynamics from environmental dynamics and vice versa.     

M.V. Volkenstein, evolutionary thinking and the structure of fitness landscapes.

High dimensional fitness landscapes are robustly dominated by saddle points, not isolated peaks. We present an argument to this effect that is reminiscent of May's complexity stability analysis and trace out the significance for the dynamics of speciation, the connection between the neutral and punctuated aspects of evolution and evolution on moving landscapes. The paper is written in honor of M.V. Volkenstein (October 23rd, 1912-February 18th, 1992), who devoted his last papers to uniting dynamics with evolutionary thinking.     

The population genetics of adaptation on correlated fitness landscapes: the block model

Several recent theoretical studies of the genetics of adaptation have focused on the mutational landscape model, which considers evolution on rugged fitness landscapes (i.e., ones having many local optima). Adaptation in this model is characterized by several simple results. Here I ask whether these results also hold on correlated fitness landscapes, which are smoother than those considered in the mutational landscape model. In particular, I study the genetics of adaptation in the block model, a tunably rugged model of fitness landscapes. Considering the scenario in which adaptation begins from a high fitness wild-type DNA sequence, I use extreme value theory and computer simulations to study both single adaptive steps and entire adaptive walks. I show that all previous results characterizing single steps in adaptation in the mutational landscape model hold at least approximately on correlated landscapes in the block model; many entire-walk results, however, do not.     

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 of codon usage and tRNA genes leads to alternative stable states of biased codon usage

The typical number of tRNA genes in bacterial genomes is around 50, but this number varies from under 30 to over 120. We argue that tRNA gene copy numbers evolve in response to translational selection. In rapidly multiplying organisms, the time spent in translation is a limiting factor in cell division; hence, it pays to duplicate tRNA genes, thereby increasing the concentration of tRNA molecules in the cell and speeding up translation. In slowly multiplying organisms, translation time is not a limiting factor, so the overall translational cost is minimized by reducing the tRNAs to only one copy of each required gene. Translational selection also causes a preference for codons that are most rapidly translated by the current tRNAs; hence, codon usage and tRNA gene content will coevolve to a state where each is adapted to the other. We show that there is often more than one stable coevolved state. This explains why different combinations of tRNAs and codon bias can exist for different amino acids in the same organism. We analyze a set of 80 complete bacterial genomes and show that the theory predicts many of the trends that are seen in these data.     

Adding a third dimension to the edge of a species' range: altitude and genetic structuring in mountainous landscapes

In addition to the topographical and ecological barriers, other landscape features may also subtly influence the patterns of gene flow and spatial genetic structuring at species' borders. This paper focuses on the role played by altitudinal gradients that characterize mountainous landscapes. We formulate and test the hypothesis that when the distribution boundaries of plant species intersect mountainous landscapes, altitudinal gradients in ecological conditions may considerably enhance population subdivision and genetic structuring at the regional level. Using amplified fragment length polymorphism markers, we studied genetic diversity and differentiation in a set of 21 peripheral populations of the evergreen shrub Lavandula latifolia Med. (Labiatae) at its southernmost distribution limit in the Betic mountain ranges of southern Spain. Population size and abundance, and within-population genetic diversity, varied predictably with altitude, being highest at middle elevations and declining steadily towards both the upper and lower altitudinal distribution margins. Genetic differentiation tended to follow the opposite trend. These altitudinal patterns result from variation with elevation in the relative influence of gene flow and drift on the distribution of genetic variation. Genetic drift prevails around the upper and lower altitudinal limits, whereas a situation closer to a drift-gene flow equilibrium exists at the center of the altitudinal distribution. Altitudinal variation in the relative influences of gene flow and drift appears as an essential element in the interpretation of regional genetic structuring of L. latifolia at its mountainous distribution edge, and a factor which may influence the evolutionary potential of peripheral populations and the likelihood of local adaptation.     

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.