Evolution in silico and in vitro: the RNA model.

Theoretical concepts and experiments dealing with the evolution of molecules in vitro reached a state that allows for direct applications to the design of biomolecules with predefined properties. RNA evolution in vitro represents a basis for the development of a new and comprehensive model of evolution, focusing on the phenotype and its fitness relevant properties. Relations between genotypes and phenotypes are described by mappings from genotype space onto a space of phenotypes, which are many-to-one and thus give ample room for neutrality as expressed by the existence of extended neutral networks in genotype space. The RNA model reduces genotype-phenotype relations to mappings from sequences into secondary structures of minimal free energies and allows for derivation of otherwise inaccessible quantitative results. Continuity and discontinuity in evolution are defined through a new notion of accessibility in phenotype space that provides a basis for straight forward interpretation of computer simulations on RNA optimization; furthermore, it reveals the constructive role of random genomic drift in the search for phenotypes of higher fitness. The effects of population size on the course of evolutionary optimization can be predicted quantitatively by means of a simple stochastic model based on a birth-anddeath process with immigration.     

Quasi-independence,homology and the unity of type: a topological theory of characters

In this paper Lewontin's notion of "quasi-independence" of characters is formalized as the assumption that a region of the phenotype space can be represented by a product space of orthogonal factors. In this picture each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a "type", i.e. as a set of phenotypes that share the same set of phenotypic characters. Using the notion of local factorizations we develop a theory of character identity based on the continuation of common factors among different regions of the phenotype space. We also consider the topological constraints on evolutionary transitions among regions with different regional factorizations, i.e. for the evolution of new types or body plans. It is shown that direct transition between different "types" is only possible if the transitional forms have all the characters that the ancestral and the derived types have and are thus compatible with the factorization of both types. Transitional forms thus have to go over a "complexity hump" where they have more quasi-independent characters than either the ancestral as well as the derived type. The only logical, but biologically unlikely, alternative is a "hopeful monster" that transforms in a single step from the ancestral type to the derived type. Topological considerations also suggest a new factor that may contribute to the evolutionary stability of "types". It is shown that if the type is decomposable into factors which are vertex irregular (i.e. have states that are more or less preferred in a random walk), the region of phenotypes representing the type contains islands of strongly preferred states. In other words types have a statistical tendency of retaining evolutionary trajectories within their interior and thus add to the evolutionary persistence of types.     

A model of developmental evolution: selection, pleiotropy and compensation

Development and physiology translate genetic variation into phenotypic variation and determine the genotype-phenotype map, such as which gene affects which character (pleiotropy). Any genetic change in this mapping reflects a change in development. Here, we discuss evidence for variation in pleiotropy and propose the selection, pleiotropy and compensation model (SPC) for adaptive evolution. It predicts that adaptive change in one character is associated with deleterious pleiotropy in others and subsequent selection to compensate for these pleiotropic effects. The SPC model provides a unifying perspective for a variety of puzzling phenomena, including developmental systems drift and character homogenization. The model suggests that most adaptive signatures detected in genome scans could be the result of compensatory changes, rather than of progressive character adaptations.     

Plasticity, evolvability, and modularity in RNA

RNA folding from sequences into secondary structures is a simple yet powerful, biophysically grounded model of a genotype-phenotype map in which concepts like plasticity, evolvability, epistasis, and modularity can not only be precisely defined and statistically measured but also reveal simultaneous and profoundly non-independent effects of natural selection. Molecular plasticity is viewed here as the capacity of an RNA sequence to assume a variety of energetically favorable shapes by equilibrating among them at constant temperature. Through simulations based on experimental designs, we study the dynamics of a population of RNA molecules that evolve toward a predefined target shape in a constant environment. Each shape in the plastic repertoire of a sequence contributes to the overall fitness of the sequence in proportion to the time the sequence spends in that shape. Plasticity is costly, since the more shapes a sequence can assume, the less time it spends in any one of them. Unsurprisingly, selection leads to a reduction of plasticity (environmental canalization). The most striking observation, however, is the simultaneous slow-down and eventual halting of the evolutionary process. The reduction of plasticity entails genetic canalization, that is, a dramatic loss of variability (and hence a loss of evolvability) to the point of lock-in. The causal bridge between environmental canalization and genetic canalization is provided by a correlation between the set of shapes in the plastic repertoire of a sequence and the set of dominant (minimum free energy) shapes in its genetic neighborhood. This statistical property of the RNA genotype-phenotype map, which we call plastogenetic congruence, traps populations in regions where most genetic variation is phenotypically neutral. We call this phenomenon neutral confinement. Analytical models of neutral confinement, made tractable by the assumption of perfect plastogenetic congruence, formally connect mutation rate, the topography of phenotype space, and evolvability. These models identify three mutational regimes: that corresponding to neutral confinement, an exploration threshold corresponding to a breakdown of neutral confinement with the simultaneous persistence of the dominant phenotype, and a classic error threshold corresponding to the loss of the dominant phenotype. In a final step, we analyze the structural properties of canalized phenotypes. The reduction of plasticity leads to extreme modularity, which we analyze from several perspectives: thermophysical (melting--the RNA version of a norm of reaction), kinetic (folding pathways--the RNA version of development), and genetic (transposability--the insensitivity to genetic context). The model thereby suggests a possible evolutionary origin of modularity as a side effect of environmental canalization.     

Topological structure of the space of phenotypes: the case of RNA neutral networks

The evolution and adaptation of molecular populations is constrained by the diversity accessible through mutational processes. RNA is a paradigmatic example of biopolymer where genotype (sequence) and phenotype (approximated by the secondary structure fold) are identified in a single molecule. The extreme redundancy of the genotype-phenotype map leads to large ensembles of RNA sequences that fold into the same secondary structure and can be connected through single-point mutations. These ensembles define neutral networks of phenotypes in sequence space. Here we analyze the topological properties of neutral networks formed by 12-nucleotides RNA sequences, obtained through the exhaustive folding of sequence space. A total of 4(12) sequences fragments into 645 subnetworks that correspond to 57 different secondary structures. The topological analysis reveals that each subnetwork is far from being random: it has a degree distribution with a well-defined average and a small dispersion, a high clustering coefficient, and an average shortest path between nodes close to its minimum possible value, i.e. the Hamming distance between sequences. RNA neutral networks are assortative due to the correlation in the composition of neighboring sequences, a feature that together with the symmetries inherent to the folding process explains the existence of communities. Several topological relationships can be analytically derived attending to structural restrictions and generic properties of the folding process. The average degree of these phenotypic networks grows logarithmically with their size, such that abundant phenotypes have the additional advantage of being more robust to mutations. This property prevents fragmentation of neutral networks and thus enhances the navigability of sequence space. In summary, RNA neutral networks show unique topological properties, unknown to other networks previously described.     

Evolution of phenotypic fluctuation under host-parasite interactions

Robustness and plasticity are essential features that allow biological systems to cope with complex and variable environments. In a constant environment, robustness, i.e., insensitivity of phenotypes, is expected to increase, whereas plasticity, i.e., the changeability of phenotypes, tends to diminish. Under a variable environment, existence of plasticity will be relevant. The robustness and plasticity, on the other hand, are related to phenotypic variances. As phenotypic variances decrease with the increase in robustness to perturbations, they are expected to decrease through the evolution. However, in nature, phenotypic fluctuation is preserved to a certain degree. One possible cause for this is environmental variation, where one of the most important “environmental” factors will be inter-species interactions. As a first step toward investigating phenotypic fluctuation in response to an inter-species interaction, we present the study of a simple two-species system that comprises hosts and parasites. Hosts are expected to evolve to achieve a phenotype that optimizes fitness. Then, the robustness of the corresponding phenotype will be increased by reducing phenotypic fluctuations. Conversely, plasticity tends to evolve to avoid certain phenotypes that are attacked by parasites. By using a dynamic model of gene expression for the host, we investigate the evolution of the genotype-phenotype map and of phenotypic variances. If the host–parasite interaction is weak, the fittest phenotype of the host evolves to reduce phenotypic variances. In contrast, if there exists a sufficient degree of interaction, the phenotypic variances of hosts increase to escape parasite attacks. For the latter case, we found two strategies: if the noise in the stochastic gene expression is below a certain threshold, the phenotypic variance increases via genetic diversification, whereas above this threshold, it is increased mediated by noise-induced phenotypic fluctuation. We examine how the increase in the phenotypic variances caused by parasite interactions influences the growth rate of a single host, and observed a trade-off between the two. Our results help elucidate the roles played by noise and genetic mutations in the evolution of phenotypic fluctuation and robustness in response to host–parasite interactions.     

Phenotypic noise and the cost of complexity

Experimental studies demonstrate the existence of phenotypic diversity despite constant genotype and environment. Theoretical models based on a single phenotypic character predict that during an adaptation event, phenotypic noise should be positively selected far from the fitness optimum because it increases the fitness of the genotype, and then be selected against when the population reaches the optimum. It is suggested that because of this fitness gain, phenotypic noise should promote adaptive evolution. However, it is unclear how the selective advantage of phenotypic noise is linked to the rate of evolution, and whether any advantage would hold for more realistic, multidimensional phenotypes. Indeed, complex organisms suffer a cost of complexity, where beneficial mutations become rarer as the number of phenotypic characters increases. Using a quantitative genetics approach, we first show that for a one-dimensional phenotype, phenotypic noise promotes adaptive evolution on plateaus of positive fitness, independently from the direct selective advantage on fitness. Second, we show that for multidimensional phenotypes, phenotypic noise evolves to a low-dimensional configuration, with elevated noise in the direction of the fitness optimum. Such a dimensionality reduction of the phenotypic noise promotes adaptive evolution and numerical simulations show that it reduces the cost of complexity.     

The evolutionary dynamics of direct phenotypic overdominance: emergence possible, loss probable

An evolutionary dynamical system with explicit diploid genetics is used to investigate the likelihood of observing phenotypically overdominant heterozygotes versus heterozygous phenotypes that are intermediate between the homozygotes. In this model, body size evolves in a population with discrete demographic episodes and with competition limiting reproduction. A genotype-phenotype map for body size is used that can generate the two qualitative types of dominance interactions (overdominance versus intermediate dominance). It is written as a single-locus model with one focal locus and parameters summarizing the effects of alleles at other loci. Two types of evolutionarily stable strategy (ESS; continuously stable strategy, CSS) occur. The ESS is generated either (1) by the population ecology; or (2) by a local maximum of the genotype-phenotype map. Overdominant heterozygotes are expected to arise if the population evolves toward the second type of ESS, where nearly maximum body sizes are found. When other loci with partially dominant inheritance also evolve, the location of the maximum in the genotype-phenotype map repeatedly changes. It is unlikely that an evolving population will track these changes; ESSs of the second type now are at best quasi-stationary states of the evolutionary dynamics. Considering the restrictions on its probability, a pattern of phenotypic overdominance is expected to be rare.     

Is the genotype-phenotype map modular? A statistical approach using mouse quantitative trait loci data.

Various theories about the evolution of complex characters make predictions about the statistical distribution of genetic effects on phenotypic characters, also called the genotype-phenotype map. With the advent of QTL technology, data about these distributions are becoming available. In this article, we propose simple tests for the prediction that functionally integrated characters have a modular genotype-phenotype map. The test is applied to QTL data on the mouse mandible. The results provide statistical support for the notion that the ascending ramus region of the mandible is modularized. A data set comprising the effects of QTL on a more extensive portion of the phenotype is required to determine if the alveolar region of the mandible is also modularized.     

Visualizing fitness landscapes

Fitness landscapes are a classical concept for thinking about the relationship between genotype and fitness. However, because the space of genotypes is typically high-dimensional, the structure of fitness landscapes can be difficult to understand and the heuristic approach of thinking about fitness landscapes as low-dimensional, continuous surfaces may be misleading. Here, I present a rigorous method for creating low-dimensional representations of fitness landscapes. The basic idea is to plot the genotypes in a manner that reflects the ease or difficulty of evolving from one genotype to another. Such a layout can be constructed using the eigenvectors of the transition matrix describing the evolution of a population on the fitness landscape when mutation is weak. In addition, the eigendecomposition of this transition matrix provides a new, high-level view of evolution on a fitness landscape. I demonstrate these techniques by visualizing the fitness landscape for selection for the amino acid serine and by visualizing a neutral network derived from the RNA secondary structure genotype-phenotype map.     

A comparison of genotype-phenotype maps for RNA and proteins

The relationship between the genotype (sequence) and the phenotype (structure) of macromolecules affects their ability to evolve new structures and functions. We here compare the genotype space organization of proteins and RNA molecules to identify differences that may affect this ability. To this end, we computationally study the genotype-phenotype relationship for short RNA and lattice proteins of a reduced monomer alphabet size, to make exhaustive analysis and direct comparison of their genotype spaces feasible. We find that many fewer protein molecules than RNA molecules fold, but they fold into many more structures than RNA. In consequence, protein phenotypes have smaller genotype networks whose member genotypes tend to be more similar than for RNA phenotypes. Neighborhoods in sequence space of a given radius around an RNA molecule contain more novel structures than for protein molecules. We compare this property to evidence from natural RNA and protein molecules, and conclude that RNA genotype space may be more conducive to the evolution of new structure phenotypes.     

The effect of recombination on the neutral evolution of genetic robustness

Conventional population genetics considers the evolution of a limited number of genotypes corresponding to phenotypes with different fitness. As model phenotypes, in particular RNA secondary structure, have become computationally tractable, however, it has become apparent that the context dependent effect of mutations and the many-to-one nature inherent in these genotype-phenotype maps can have fundamental evolutionary consequences. It has previously been demonstrated that populations of genotypes evolving on the neutral networks corresponding to all genotypes with the same secondary structure only through neutral mutations can evolve mutational robustness [E. van Nimwegen, J.P. Crutchfield, M. Huynen, Neutral evolution of mutational robustness, Proc. Natl. Acad. Sci. USA 96(17), 9716-9720 (1999)], by concentrating the population on regions of high neutrality. Introducing recombination we demonstrate, through numerically calculating the stationary distribution of an infinite population on ensembles of random neutral networks that mutational robustness is significantly enhanced and further that the magnitude of this enhancement is sensitive to details of the neutral network topology. Through the simulation of finite populations of genotypes evolving on random neutral networks and a scaled down microRNA neutral network, we show that even in finite populations recombination will still act to focus the population on regions of locally high neutrality.     

Sequential search and the influence of male quality on female mating decisions

 The patterns of phenotypic association between mated males and females depend on the decision rules that individuals employ during search for a mate. We generalize the sequential search rule and examine how the shape of the function that relates a male character to the benefit of a mating decision influences the threshold value of the male trait that induces females to terminate search. If the fitness function is linear the optimal threshold value of a male character increases with the slope of the function. The phenotypic threshold criterion declines, all else being equal, if the fitness function is made more concave (or less convex) by an increase of the risk of the function. The expression of the trait in females has no effect on the optimal threshold value of a male character if the fitness function is linear and phenotypic values combine additively to influence the benefit of a mating decision; the phenotypic threshold criterion is ubiquitous among females. A convex fitness function induces females with high trait values to adopt a relatively high phenotypic threshold criterion, whereas a concave fitness function induces such females to adopt a low threshold value for the male trait. Thus, linear, convex and concave fitness functions effect random, assortative and disassortative combinations of phenotypes among mated individuals, respectively. Changes of female search behavior induced by changes of the distribution of a male character similarly depend on the shape of the fitness function. A variance-preserving increase of male trait values produces a relatively small increase of the threshold criterion for the male character if the fitness function is concave, relative to conditions in which the fitness function is either linear or convex. Our results suggest that a sequential search rule can in principle induce the kinds of mating patterns observed in nature and that the phenotypic association between mated individuals is likely to depend on how a male character translates into fitness, the distribution of the trait among males and attributes of searching females. Received: 20 September 1997 / Revised version: 13 August 1998     

Modelling 'evo-devo' with RNA

The folding of RNA sequences into secondary structures is a simple yet biophysically grounded model of a genotype-phenotype map. Its computational and mathematical analysis has uncovered a surprisingly rich statistical structure characterized by shape space covering, neutral networks and plastogenetic congruence. I review these concepts and discuss their evolutionary implications.     

On the Accessibility of Adaptive Phenotypes of a Bacterial Metabolic Network

The mechanisms by which adaptive phenotypes spread within an evolving population after their emergence are understood fairly well. Much less is known about the factors that influence the evolutionary accessibility of such phenotypes, a pre-requisite for their emergence in a population. Here, we investigate the influence of environmental quality on the accessibility of adaptive phenotypes of Escherichia coli''s central metabolic network. We used an established flux-balance model of metabolism as the basis for a genotype-phenotype map (GPM). We quantified the effects of seven qualitatively different environments (corresponding to both carbohydrate and gluconeogenic metabolic substrates) on the structure of this GPM. We found that the GPM has a more rugged structure in qualitatively poorer environments, suggesting that adaptive phenotypes could be intrinsically less accessible in such environments. Nevertheless, on average ∼74% of the genotype can be altered by neutral drift, in the environment where the GPM is most rugged; this could allow evolving populations to circumvent such ruggedness. Furthermore, we found that the normalized mutual information (NMI) of genotype differences relative to phenotype differences, which measures the GPM''s capacity to transmit information about phenotype differences, is positively correlated with (simulation-based) estimates of the accessibility of adaptive phenotypes in different environments. These results are consistent with the predictions of a simple analytic theory that makes explicit the relationship between the NMI and the speed of adaptation. The results suggest an intuitive information-theoretic principle for evolutionary adaptation; adaptation could be faster in environments where the GPM has a greater capacity to transmit information about phenotype differences. More generally, our results provide insight into fundamental environment-specific differences in the accessibility of adaptive phenotypes, and they suggest opportunities for research at the interface between information theory and evolutionary biology.     

Replication and mutation on neutral networks

Folding of RNA sequences into secondary structures is viewed as a map that assigns a uniquely defined base pairing pattern to every sequence. The mapping is non-invertible since many sequences fold into the same minimum free energy (secondary) structure or shape. The pre-images of this map, called neutral networks, are uniquely associated with the shapes and vice versa. Random graph theory is used to construct networks in sequence space which are suitable models for neutral networks. The theory of molecular quasispecies has been applied to replication and mutation on single-peak fitness landscapes. This concept is extended by considering evolution on degenerate multi-peak landscapes which originate from neutral networks by assuming that one particular shape is fitter than all the others. On such a single-shape landscape the superior fitness value is assigned to all sequences belonging to the master shape. All other shapes are lumped together and their fitness values are averaged in a way that is reminiscent of mean field theory. Replication and mutation on neutral networks are modeled by phenomenological rate equations as well as by a stochastic birth-and-death model. In analogy to the error threshold in sequence space the phenotypic error threshold separates two scenarios: (i) a stationary (fittest) master shape surrounded by closely related shapes and (ii) populations drifting through shape space by a diffusion-like process. The error classes of the quasispecies model are replaced by distance classes between the master shape and the other structures. Analytical results are derived for single-shape landscapes, in particular, simple expressions are obtained for the mean fraction of master shapes in a population and for phenotypic error thresholds. The analytical results are complemented by data obtained from computer simulation of the underlying stochastic processes. The predictions of the phenomenological approach on the single-shape landscape are very well reproduced by replication and mutation kinetics of tRNA(phe). Simulation of the stochastic process at a resolution of individual distance classes yields data which are in excellent agreement with the results derived from the birth-and-death model.     

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.     

Evolutionary aspects and sensitivity studies of some major gene models

Extensive biometrical and statistically oriented studies in segregation and pedigree analyses reflect current efforts to demonstrate major gene factors playing a significant role for a whole hierarchy of multifactorial diseases and related risk factors exhibiting continuous variation. The evolutionary aspects of the changes in gene frequencies of some major gene one locus models admitting a broad range of genotype-phenotype associations and different forms of selection functions are investigated. The flexibility of differences among the genotypic-phenotypic distribution can take account of variable penetrance expressivity, complex multifarious heterogeneous background effects, or partial dominance concepts. The phenotype distribution and selection function are assumed to be time invariant such that the environments with which the population interacts do not depend on either the phenotypes or the genotypes present in the population of any particular generation. Viability selection optimizing or directional acts on the phenotypic level. We consider random mating, and concentrate mostly on evaluating the nature of the equilibrium structure for the cases of “strong” and “weak” selection. For weak stabilizing selection the determinants of superior genotypic fitness in the class of phenotypic symmetric distributions reside in minimizing a combination of the phenotypic variance and the deviation of the phenotypic mean from the optimal phenotype. With equal means of central phenotype values, a canalizing selection effect signifying fitness superiority for the genotype with minimal variance is in force. For strong stabilizing selection the genotype-phenotype density at the optimal value determines the relative genotype fitness value. For directional selection the determinants of the selection realizations depend on a “standardized” deviation of the mean phenotype distributional value relative to its total variance. The effects of symmetry as against asymmetry in the genotype distributions with prescribed means and variances were investigated by numerical computations.     

What is wrong with absolute individual fitness?

One of the most basic facts about evolution is that fitness is a relative concept. It does not matter how well an organism survives and reproduces, only that it does so better than other organisms bearing alternative traits. Nevertheless, many evolutionary arguments are framed in terms of absolute individual fitness. The absolute fitness criterion (AFC) can be justified in terms of relative fitness only given certain assumptions that are frequently violated in nature. In particular, interactions must occur in groups that are randomly formed and phenotypic variation among groups must be tightly coupled to genetic variation. Complicating the genotype-phenotype relationship can cause phenotypic variation among groups to become nonrandom, even when the groups are randomly formed, favoring traits that do not maximize absolute individual fitness. Complex genotype-phenotype relationships and complex population structures require explicit models of evolutionary change based on relative fitness differences within and among groups.