The foldability landscape of model proteins

Molecular evolution may be considered as a walk in a multidimensional fitness landscape, where the fitness at each point is associated with features such as the function, stability, and survivability of these molecules. We present a simple model for the evolution of protein sequences on a landscape with a precisely defined fitness function. We use simple lattice models to represent protein structures, with the ability of a protein sequence to fold into the structure with lowest energy, quantified as the foldability, representing the fitness of the sequence. The foldability of the sequence is characterized based on the spin glass model of protein folding. We consider evolution as a walk in this foldability landscape and study the nature of the landscape and the resulting dynamics. Selective pressure is explicitly included in this model in the form of a minimum foldability requirement. We find that different native structures are not evenly distributed in interaction space, with similar structures and structures with similar optimal foldabilities clustered together. Evolving proteins marginally fulfill the selective criteria of foldability. As the selective pressure is increased, evolutionary trajectories become increasingly confined to “neutral networks,” where the sequence and the interactions can be significantly changed while a constant structure is maintained. © 1997 John Wiley & Sons, Inc. Biopoly 42: 427–438, 1997     

Adaptive evolutionary walks require neutral intermediates in RNA fitness landscapes

In RNA fitness landscapes with interconnected networks of neutral mutations, neutral precursor mutations can play an important role in facilitating the accessibility of epistatic adaptive mutant combinations. I use an exhaustively surveyed fitness landscape model based on short sequence RNA genotypes (and their secondary structure phenotypes) to calculate the minimum rate at which mutants initially appearing as neutral are incorporated into an adaptive evolutionary walk. I show first, that incorporating neutral mutations significantly increases the number of point mutations in a given evolutionary walk when compared to estimates from previous adaptive walk models. Second, that incorporating neutral mutants into such a walk significantly increases the final fitness encountered on that walk - indeed evolutionary walks including neutral steps often reach the global optimum in this model. Third, and perhaps most importantly, evolutionary paths of this kind are often extremely winding in their nature and have the potential to undergo multiple mutations at a given sequence position within a single walk; the potential of these winding paths to mislead phylogenetic reconstruction is briefly considered.     

Metastable evolutionary dynamics: Crossing fitness barriers or escaping via neutral paths?

We analytically study the dynamics of evolving populations that exhibit metastability on the level of phenotype or fitness. In constant selective environments, such metastable behavior is caused by two qualitatively different mechanisms. On the one hand, populations may become pinned at a local fitness optimum, being separated from higher-fitness genotypes by a fitness barrier of low-fitness genotypes. On the other hand, the population may only be metastable on the level of phenotype or fitness while, at the same time, diffusing over neutral networks of selectively neutral genotypes. Metastability occurs in this case because the population is separated from higher-fitness genotypes by an entropy barrier: the population must explore large portions of these neutral networks before it discovers a rare connection to fitter phenotypes. We derive analytical expressions for the barrier crossing times in both the fitness barrier and entropy barrier regime. In contrast with ‘landscape’ evolutionary models, we show that the waiting times to reach higher fitness depend strongly on the width of a fitness barrier and much less on its height. The analysis further shows that crossing entropy barriers is faster by orders of magnitude than fitness barrier crossing. Thus, when populations are trapped in a metastable phenotypic state, they are most likely to escape by crossing an entropy barrier, along a neutral path in genotype space. If no such escape route along a neutral path exists, a population is most likely to cross a fitness barrier where the barrier is narrowest, rather than where the barrier is shallowest.     

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.     

Selective neutrality and enzyme kinetics

This article appeals to a recent theory of enzyme evolution to show that the properties, neutral or adaptive, which characterize the observed allelic variation in natural populations can be inferred from the functional parameters, substrate specificity, and reaction rate. This study delineates the following relations between activity variables, and the forces—adaptive or neutral—determining allelic variation: (1) Enzymes with broad substrate specificity: The observed polymorphism is adaptive; mutations in this class of enzymes can result in increased fitness of the organism and hence be relevant for positive selection. (2) Enzymes with absolute substrate specificity and diffusion-controlled rates: Observed allelic variation will be absolutely neutral; mutations in this class of enzymes will be either deleterious or have no effect on fitness. (3) Enzymes with absolute or group specificity and nondiffusion-controlled rates: Observed variation will be partially neutral; mutants which are selectively neutral may become advantageous under an appropriate environmental condition or different genetic background. We illustrate each of the relations between kinetic properties and evolutionary states with examples drawn from enzymes whose evolutionary dynamics have been intensively studied. Received: 12 December 1996 / Accepted: 22 April 1997     

Exploring phenotype space through neutral evolution

RNA secondary-structure folding algorithms predict the existence of connected networks of RNA sequences with identical secondary structures. Fitness landscapes that are based on the mapping between RNA sequence and RNA secondary structure hence have many neutral paths. A neutral walk on these fitness landscapes gives access to a virtually unlimited number of secondary structures that are a single point mutation from the neutral path. This shows that neutral evolution explores phenotype space and can play a role in adaptation. Received: 23 December 1995 / Accepted: 17 March 1996     

Local evolvability of statistically neutral GasNet robot controllers

In this paper we introduce and apply the concept of local evolvability to investigate the behaviour of populations during evolutionary search. We focus on the evolution of GasNet neural network controllers for a robotic visual discrimination problem, showing that the evolutionary process undergoes long neutral fitness epochs. We show that the local evolvability properties of the search space surrounding a group of statistically neutral solutions do vary across the course of an evolutionary run, especially during periods of population takeover. However, once takeover is complete there is no evidence for further increase in local evolvability across fitness epochs. We also see no evidence for the neutral evolution of increased solution robustness, but show that this may be due to the ability of evolutionary algorithms to focus search on volumes of the fitness landscape with above average robustness.     

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.     

Exploring protein sequence space using knowledge-based potentials

Knowledge-based potentials can be used to decide whether an amino acid sequence is likely to fold into a prescribed native protein structure. We use this idea to survey the sequence-structure relations in protein space. In particular, we test the following two propositions which were found to be important for efficient evolution: the sequences folding into a particular native fold form extensive neutral networks that percolate through sequence space. The neutral networks of any two native folds approach each other to within a few point mutations. Computer simulations using two very different potential functions, M. Sippl's PROSA pair potential and a neural network based potential, are used to verify these claims.     

Origin of the fittest: link between emergent variation and evolutionary change as a critical question in evolutionary biology

In complex organisms, neutral evolution of genomic architecture, associated compensatory interactions in protein networks and emergent developmental processes can delineate the directions of evolutionary change, including the opportunity for natural selection. These effects are reflected in the evolution of developmental programmes that link genomic architecture with a corresponding functioning phenotype. Two recent findings call for closer examination of the rules by which these links are constructed. First is the realization that high dimensionality of genotypes and emergent properties of autonomous developmental processes (such as capacity for self-organization) result in the vast areas of fitness neutrality at both the phenotypic and genetic levels. Second is the ubiquity of context- and taxa-specific regulation of deeply conserved gene networks, such that exceptional phenotypic diversification coexists with remarkably conserved generative processes. Establishing the causal reciprocal links between ongoing neutral expansion of genomic architecture, emergent features of organisms' functionality, and often precisely adaptive phenotypic diversification therefore becomes an important goal of evolutionary biology and is the latest reincarnation of the search for a framework that links development, functioning and evolution of phenotypes. Here I examine, in the light of recent empirical advances, two evolutionary concepts that are central to this framework-natural selection and inheritance-the general rules by which they become associated with emergent developmental and homeostatic processes and the role that they play in descent with modification.     

Evolutionary genetics of plant adaptation

Plants provide unique opportunities to study the mechanistic basis and evolutionary processes of adaptation to diverse environmental conditions. Complementary laboratory and field experiments are important for testing hypotheses reflecting long-term ecological and evolutionary history. For example, these approaches can infer whether local adaptation results from genetic tradeoffs (antagonistic pleiotropy), where native alleles are best adapted to local conditions, or if local adaptation is caused by conditional neutrality at many loci, where alleles show fitness differences in one environment, but not in a contrasting environment. Ecological genetics in natural populations of perennial or outcrossing plants can also differ substantially from model systems. In this review of the evolutionary genetics of plant adaptation, we emphasize the importance of field studies for understanding the evolutionary dynamics of model and nonmodel systems, highlight a key life history trait (flowering time) and discuss emerging conservation issues.     

An in silico exploration of the neutral network in protein sequence space

Designating amino-acid sequences that fold into a common main-chain structure as "neutral sequences" for the structure, regardless of their function or stability, we investigated the distribution of neutral sequences in protein sequence space. For four distinct target structures (alpha, beta,alpha/beta and alpha+beta types) with the same chain length of 108, we generated the respective neutral sequences by using the inverse folding technique with a knowledge-based potential function. We assumed that neutral sequences for a protein structure have Z scores higher than or equal to fixed thresholds, where thresholds are defined as the Z score for the corresponding native sequence (case 1) or much greater Z score (case 2). An exploring walk simulation suggested that the neutral sequences mapped into the sequence space were connected with each other through straight neutral paths and formed an inherent neutral network over the sequence space. Through another exploring walk simulation, we investigated contiguous regions between or among the neutral networks for the distinct protein structures and obtained the following results. The closest approach distance between the two neutral networks ranged from 5 to 29 on the Hamming distance scale, showing a linear increase against the threshold values. The sequences located at the "interchange" regions between the two neutral networks have intermediate sequence-profile-scores for both corresponding structures. Introducing a "ball" in the sequence space that contains at least one neutral sequence for each of the four structures, we found that the minimal radius of the ball that is centered at an arbitrary position ranged from 35 to 50, while the minimal radius of the ball that is centered at a certain special position ranged from 20 to 30, in the Hamming distance scale. The relatively small Hamming distances (5-30) may support an evolution mechanism by transferring from a network for a structure to another network for a more beneficial structure via the interchange regions.     

Decay of unused characters by selection and drift

The reduction and loss of redundant phenotypic characters is a common feature of evolution. However, the mechanisms that drive deterioration of unused characters remain unclear. Here, we outline a simple framework where the relative importance of selective and neutral processes varies with environmental factors, because of variation in the fitness costs associated with unused traits. We tested our hypotheses using experimental evolution of the bacterium Pseudomonas fluorescens in spatially uniform environments. Results show that an unused character, swimming motility, decayed over evolutionary time and the rate of this decay varied among selection environments with different levels of resource availability. This is explained in the context of an environment-specific genetic correlation between motility and fitness, which is negative when resources are limited but neutral at higher resource levels. Thus, selection against an unused character was most effective in environments where the fitness cost was the greatest. This suggests that the same character can decay by different mechanisms depending upon environmental factors and supports previous evidence to show that resource availability can critically affect the outcomes of evolution.     

Evolutionary Dynamics on Protein Bi-stability Landscapes can Potentially Resolve Adaptive Conflicts

Experimental studies have shown that some proteins exist in two alternative native-state conformations. It has been proposed that such bi-stable proteins can potentially function as evolutionary bridges at the interface between two neutral networks of protein sequences that fold uniquely into the two different native conformations. Under adaptive conflict scenarios, bi-stable proteins may be of particular advantage if they simultaneously provide two beneficial biological functions. However, computational models that simulate protein structure evolution do not yet recognize the importance of bi-stability. Here we use a biophysical model to analyze sequence space to identify bi-stable or multi-stable proteins with two or more equally stable native-state structures. The inclusion of such proteins enhances phenotype connectivity between neutral networks in sequence space. Consideration of the sequence space neighborhood of bridge proteins revealed that bi-stability decreases gradually with each mutation that takes the sequence further away from an exactly bi-stable protein. With relaxed selection pressures, we found that bi-stable proteins in our model are highly successful under simulated adaptive conflict. Inspired by these model predictions, we developed a method to identify real proteins in the PDB with bridge-like properties, and have verified a clear bi-stability gradient for a series of mutants studied by Alexander et al. (Proc Nat Acad Sci USA 2009, 106:21149–21154) that connect two sequences that fold uniquely into two different native structures via a bridge-like intermediate mutant sequence. Based on these findings, new testable predictions for future studies on protein bi-stability and evolution are discussed.     

Protein Folding and Evolution are Driven by the Maxwell Demon Activity of Proteins

In this paper we propose a theoretical model of protein folding and protein evolution in which a polypeptide (sequence/structure) is assumed to behave as a Maxwell Demon or Information Gathering and Using System (IGUS) that performs measurements aiming at the construction of the native structure. Our model proposes that a physical meaning to Shannon information (H) and Chaitin's algorithmic information (K) parameters can be both defined and referred from the IGUS standpoint. Our hypothesis accounts for the interdependence of protein folding and protein evolution through mutual influencing relationships mediated by the IGUS. In brief, IGUS activity in protein folding determines long term tendencies that emerge at the evolutionary time-scale.Thus, protein evolution is a consequence of measurements executed by proteins at the cellular level, where the IGUS imposes a tendency to attain a highly unique stable native form that promotes the updating of the information content. The folding kinetics observed is, thus, the outcome of an evolutionary process where the polypeptide-IGUS drives the evolution of its linear sequence. Finally, we describe protein evolution as an entropic process that tends to increase the content of mutual algorithmic information between the sequence and the structure. This model enables one: 1. To comprehend that full determination of the three-dimensional structure by the linear sequence is a tendency where satisfaction is only possible at thermodynamic equilibrium.2. To account for the observed randomness of the amino acid sequences. 3. To predict an alternation of periods of selection and neutral diffusion during protein evolutionary time.     

A niche for neutrality

Ecologists now recognize that controversy over the relative importance of niches and neutrality cannot be resolved by analyzing species abundance patterns. Here, we use classical coexistence theory to reframe the debate in terms of stabilizing mechanisms (niches) and fitness equivalence (neutrality). The neutral model is a special case where stabilizing mechanisms are absent and species have equivalent fitness. Instead of asking whether niches or neutral processes structure communities, we advocate determining the degree to which observed diversity reflects strong stabilizing mechanisms overcoming large fitness differences or weak stabilization operating on species of similar fitness. To answer this question, we propose combining data on per capita growth rates with models to: (i) quantify the strength of stabilizing processes; (ii) quantify fitness inequality and compare it with stabilization; and (iii) manipulate frequency dependence in growth to test the consequences of stabilization and fitness equivalence for coexistence.     

Molecular evolution, mutation size and gene pleiotropy: a geometric reexamination

The influence of phenotypic effects of genetic mutations on molecular evolution is not well understood. Neutral and nearly neutral theories of molecular evolution predict a negative relationship between the evolutionary rate of proteins and their functional importance; nevertheless empirical studies seeking relationships between evolutionary rate and the phenotypic role of proteins have not produced conclusive results. In particular, previous studies have not found the expected negative correlation between evolutionary rate and gene pleiotropy. Here, we studied the effect of gene pleiotropy and the phenotypic size of mutations on the evolutionary rate of genes in a geometrical model, in which gene pleiotropy was characterized by n molecular phenotypes that affect organismal fitness. For a nearly neutral process, we found a negative relationship between evolutionary rate and mutation size but pleiotropy did not affect the evolutionary rate. Further, for a selection model, where most of the substitutions were fixed by natural selection in a randomly fluctuating environment, we also found a negative relationship between evolutionary rate and mutation size, but interestingly, gene pleiotropy increased the evolutionary rate as √n. These findings may explain part of the disagreement between empirical data and traditional expectations.     

Neutral mutation as the source of genetic variation in life history traits

The mechanism underlying the maintenance of adaptive genetic variation is a long-standing question in evolutionary genetics. There are two concepts (mutation-selection balance and balancing selection) which are based on the phenotypic differences between alleles. Mutation - selection balance and balancing selection cannot properly explain the process of gene substitution, i.e. the molecular evolution of quantitative trait loci affecting fitness. I assume that such loci have non-essential functions (small effects on fitness), and that they have the potential to evolve into new functions and acquire new adaptations. Here I show that a high amount of neutral polymorphism at these loci can exist in real populations. Consistent with this, I propose a hypothesis for the maintenance of genetic variation in life history traits which can be efficient for the fixation of alleles with very small selective advantage. The hypothesis is based on neutral polymorphism at quantitative trait loci and both neutral and adaptive gene substitutions. The model of neutral - adaptive conversion (NAC) assumes that neutral alleles are not neutral indefinitely, and that in specific and very rare situations phenotypic (relative fitness) differences between them can appear. In this paper I focus on NAC due to phenotypic plasticity of neutral alleles. The important evolutionary consequence of NAC could be the increased adaptive potential of a population. Loci responsible for adaptation should be fast evolving genes with minimally discernible phenotypic effects, and the recent discovery of genes with such characteristics implicates them as suitable candidates for loci involved in adaptation.