Simulating protein evolution in sequence and structure space

Naturally occurring proteins comprise a special subset of all plausible sequences and structures selected through evolution. Simulating protein evolution with simplified and all-atom models has shed light on the evolutionary dynamics of protein populations, the nature of evolved sequences and structures, and the extent to which today's proteins are shaped by selection pressures on folding, structure and function. Extensive mapping of the native structure, stability and folding rate in sequence space using lattice proteins has revealed organizational principles of the sequence/structure map important for evolutionary dynamics. Evolutionary simulations with lattice proteins have highlighted the importance of fitness landscapes, evolutionary mechanisms, population dynamics and sequence space entropy in shaping the generic properties of proteins. Finally, evolutionary-like simulations with all-atom models, in particular computational protein design, have helped identify the dominant selection pressures on naturally occurring protein sequences and structures.     

How are model protein structures distributed in sequence space?

The figure-to-structure maps for all uniquely folding sequences of short hydrophobic polar (HP) model proteins on a square lattice is analyzed to investigate aspects considered relevant to evolution. By ranking structures by their frequencies, few very frequent and many rare structures are found. The distribution can be empirically described by a generalized Zipf's law. All structures are relatively compact, yet the most compact ones are rare. Most sequences falling to the same structure belong to "neutral nets." These graphs in sequence space are connected by point mutations and centered around prototype sequences, which tolerate the largest number (up to 55%) of neutral mutations. Profiles have been derived from these homologous sequences. Frequent structures conserve hydrophobic cores only while rare ones are sensitive to surface mutations as well. Shape space covering, i.e., the ability to transform any structure into most others with few point mutations, is very unlikely. It is concluded that many characteristic features of the sequence-to-structure map of real proteins, such as the dominance of few folds, can be explained by the simple HP model. In analogy to protein families, nets are dense and well separated in sequence space. Potential implications in better understanding the evolution of proteins and applications to improving database searches are discussed.     

The evolutionary landscape of functional model proteins.

To study the distinct influences of structure and function on evolution, we propose a minimalist model for proteins with binding pockets, called functional model proteins, based on a shifted-HP model on a two-dimensional square lattice. These model proteins are not maximally compact and contain an empty lattice site surrounded by at least three nearest neighbors, thus providing a binding pocket. Functional model proteins possess a unique native state, cooperative folding and tolerance to mutation. Due to the explicit functionality in these models (by design), we have been able to explore their fitness or evolutionary landscapes, as characterized by the size and distribution of homologous families and by the complexity of the inter-relatedness of the functional model proteins. Mindful that these minimalist models are highly idealized and two-dimensional, functional model proteins should nevertheless provide a useful means for exploring the constraints of maintaining structure and function on the evolution of proteins.     

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     

The role of protein homochirality in shaping the energy landscape of folding

The homochirality, or isotacticity, of the natural amino acids facilitates the formation of regular secondary structures such as alpha-helices and beta-sheets. However, many examples exist in nature where novel polypeptide topologies use both l- and d-amino acids. In this study, we explore how stereochemistry of the polypeptide backbone influences basic properties such as compactness and the size of fold space by simulating both lattice and all-atom polypeptide chains. We formulate a rectangular lattice chain model in both two and three dimensions, where monomers are chiral, having the effect of restricting local conformation. Syndiotactic chains with alternating chirality of adjacent monomers have a very large ensemble of accessible conformations characterized predominantly by extended structures. Isotactic chains on the other hand, have far fewer possible conformations and a significant fraction of these are compact. Syndiotactic chains are often unable to access maximally compact states available to their isotactic counterparts of the same length. Similar features are observed in all-atom models of isotactic versus syndiotactic polyalanine. Our results suggest that protein isotacticity has evolved to increase the enthalpy of chain collapse by facilitating compact helical states and to reduce the entropic cost of folding by restricting the size of the unfolded ensemble of competing states.     

Genotypes with phenotypes: adventures in an RNA toy world

Evolution has created the complexity of the animate world and deciphering the language of evolution is the key towards understanding nature. The dynamics of evolution is simplified by considering it as a superposition of three less sophisticated processes: population dynamics, population support dynamics, and genotype-phenotype mapping. Evolution of molecules in laboratory assays provides a sufficiently simple system for the quantitative analysis of the three phenomena. Coarse-grained notions of structures like RNA secondary structures are used as model phenotypes. They provide an excellent tool for a comprehensive analysis of the entire complex of molecular evolution. The mapping from RNA genotypes into secondary structures is highly redundant. In order to find at least one sequence for every common structures one need only search a (relatively) small part of sequence space. The existence of selectively neutral phenotypes plays an important role for the the success and the efficiency of evolutionary optimization. Molecular evolution found a highly promising technological application in the design of biomolecules with predefined properties.     

Weak selection and protein evolution

The "nearly neutral" theory of molecular evolution proposes that many features of genomes arise from the interaction of three weak evolutionary forces: mutation, genetic drift, and natural selection acting at its limit of efficacy. Such forces generally have little impact on allele frequencies within populations from generation to generation but can have substantial effects on long-term evolution. The evolutionary dynamics of weakly selected mutations are highly sensitive to population size, and near neutrality was initially proposed as an adjustment to the neutral theory to account for general patterns in available protein and DNA variation data. Here, we review the motivation for the nearly neutral theory, discuss the structure of the model and its predictions, and evaluate current empirical support for interactions among weak evolutionary forces in protein evolution. Near neutrality may be a prevalent mode of evolution across a range of functional categories of mutations and taxa. However, multiple evolutionary mechanisms (including adaptive evolution, linked selection, changes in fitness-effect distributions, and weak selection) can often explain the same patterns of genome variation. Strong parameter sensitivity remains a limitation of the nearly neutral model, and we discuss concave fitness functions as a plausible underlying basis for weak selection.     

Social structure modulates the evolutionary consequences of social plasticity: A social network perspective on interacting phenotypes

Organisms express phenotypic plasticity during social interactions. Interacting phenotype theory has explored the consequences of social plasticity for evolution, but it is unclear how this theory applies to complex social structures. We adapt interacting phenotype models to general social structures to explore how the number of social connections between individuals and preference for phenotypically similar social partners affect phenotypic variation and evolution. We derive an analytical model that ignores phenotypic feedback and use simulations to test the predictions of this model. We find that adapting previous models to more general social structures does not alter their general conclusions but generates insights into the effect of social plasticity and social structure on the maintenance of phenotypic variation and evolution. Contribution of indirect genetic effects to phenotypic variance is highest when interactions occur at intermediate densities and decrease at higher densities, when individuals approach interacting with all group members, homogenizing the social environment across individuals. However, evolutionary response to selection tends to increase at greater network densities as the effects of an individual's genes are amplified through increasing effects on other group members. Preferential associations among similar individuals (homophily) increase both phenotypic variance within groups and evolutionary response to selection. Our results represent a first step in relating social network structure to the expression of social plasticity and evolutionary responses to selection.     

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     

The Amino Acid Alphabet and the Architecture of the Protein Sequence-Structure Map. I. Binary Alphabets

The correspondence between protein sequences and structures, or sequence-structure map, relates to fundamental aspects of structural, evolutionary and synthetic biology. The specifics of the mapping, such as the fraction of accessible sequences and structures, or the sequences'' ability to fold fast, are dictated by the type of interactions between the monomers that compose the sequences. The set of possible interactions between monomers is encapsulated by the potential energy function. In this study, I explore the impact of the relative forces of the potential on the architecture of the sequence-structure map. My observations rely on simple exact models of proteins and random samples of the space of potential energy functions of binary alphabets. I adopt a graph perspective and study the distribution of viable sequences and the structures they produce, as networks of sequences connected by point mutations. I observe that the relative proportion of attractive, neutral and repulsive forces defines types of potentials, that induce sequence-structure maps of vastly different architectures. I characterize the properties underlying these differences and relate them to the structure of the potential. Among these properties are the expected number and relative distribution of sequences associated to specific structures and the diversity of structures as a function of sequence divergence. I study the types of binary potentials observed in natural amino acids and show that there is a strong bias towards only some types of potentials, a bias that seems to characterize the folding code of natural proteins. I discuss implications of these observations for the architecture of the sequence-structure map of natural proteins, the construction of random libraries of peptides, and the early evolution of the natural amino acid alphabet.     

Wagner's canalization model

Wagner (1996, Does evolutionary plasticity evolve? Evolution 50, 1008-1023.) and Siegal and Bergman, 2002 and Azevedo et al., 2006 have studied a simple model of the evolution of a network of N genes, in order to explain the observed phenomenon that systems evolve to be robust. These authors primarily considered the case N=10 and used simulations to reach their conclusions. Here we investigate this model in more detail, considering systems of different sizes with and without recombination, and with selection for convergence instead of to a specified limit. For the simpler evolutionary model lacking recombination, we analyze the system as a neutral network. This allows us to describe the equilibrium distribution networks within genotype space. Our results show that, given a sufficiently large population size, the qualitative observation that systems evolve to be robust, is itself robust, as it does not depend on the details of the model. In simple terms, robust systems have more viable offspring, so the evolution of robustness is merely selection for increased fecundity, an observation that is well known in the theory of neutral networks.     

Functional genetics of intraspecific ecological interactions in Arabidopsis thaliana

Studying the genetic basis of traits involved in ecological interactions is a fundamental part of elucidating the connections between evolutionary and ecological processes. Such knowledge allows one to link genetic models of trait evolution with ecological models describing interactions within and between species. Previous work has shown that connections between genetic and ecological processes in Arabidopsis thaliana may be mediated by the fact that quantitative trait loci (QTL) with 'direct' effects on traits of individuals also have pleiotropic 'indirect' effects on traits expressed in neighbouring plants. Here, we further explore these connections by examining functional relationships between traits affected directly and indirectly by the same QTL. We develop a novel approach using structural equation models (SEMs) to determine whether observed pleiotropic effects result from traits directly affected by the QTL in focal individuals causing the changes in the neighbours' phenotypes. This hypothesis was assessed using SEMs to test whether focal plant phenotypes appear to mediate the connection between the focal plants' genotypes and the phenotypes of their neighbours, or alternatively, whether the connection between the focal plants' genotypes and the neighbours' phenotypes is mediated by unmeasured traits. We implement this analysis using a QTL of major effect that maps to the well-characterized flowering locus, FRIGIDA. The SEMs support the hypothesis that the pleiotropic indirect effects of this locus arise from size and developmental timing-related traits in focal plants affecting the expression of developmental traits in their neighbours. Our findings provide empirical insights into the genetics and nature of intraspecific ecological interactions. Our technique holds promise in directing future work into the genetic basis and functional relationship of traits mediating and responding to ecological interactions.     

Quaternary structure constraints on evolutionary sequence divergence

The structurally constrained protein evolution (SCPE) model simulates protein divergence considering protein structure explicitly. The model is based on the observation that protein structure is more conserved during evolution than the sequences encoding for that structure. In the previous work, the SCPE model considered only the tertiary structure. Here we show that the performance of the model is enhanced when the oligomeric structure is taken into account. Our results agree with recent evolutionary studies of oligomeric proteins, which show that conservation of the quaternary structure imposes additional constraints on sequence divergence. The incorporation of protein-protein interactions into protein evolution models may be important in the study of quaternary protein structures and complex protein assemblies.     

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.     

Comparing folding codes for proteins and polymers

Proteins fold to unique compact native structures. Perhaps other polymers could be designed to fold in similar ways. The chemical nature of the monomer “alphabet” determines the “energy matrix” of monomer interactions—which defines the folding code, the relationship between sequence and structure. We study two properties of energy matrices using two-dimensional lattice models: uniqueness, the number of sequences that fold to only one structure, and encodability, the number of folds that are unique lowest-energy structures of certain monomer sequences. For the simplest model folding code, involving binary sequences of H (hydrophobic) and P (polar) monomers, only a small fraction of sequences fold uniquely, and not all structures can be encoded. Adding strong repulsive interactions results in a folding code with more sequences folding uniquely and more designable folds. Some theories suggest that the quality of a folding code depends only on the number of letters in the monomer alphabet, but we find that the energy matrix itself can be at least as important as the size of the alphabet. Certain multi-letter codes, including some with 20 letters, may be less physical or protein-like than codes with smaller numbers of letters because they neglect correlations among inter-residue interactions, treat only maximally compact conformations, or add arbitrary energies to the energy matrix.     

Developmental quantitative genetic models of evolutionary change

Discussions about evolutionary change in developmental processes or morphological structures are predicated on specific quantitative genetic models whose parameters predict whether evolutionary change can occur, its relative rate and direction, and if correlated change will occur in other related and unrelated structures. The appropriate genetic model should reflect the relevant genetical and developmental biology of the organisms, yet be simple enough in its parameters so that deductions can be made and hypotheses tested. As a consequence, the choice of the most appropriate genetic model for polygenically controlled traits is a complex tissue and the eventual choice of model is often a compromise between completeness of the model and computational expediency. Herein, we discuss several developmental quantitative genetic models for the evolution of development and morphology. The models range from the classical direct effects model to complex epigenetic models. Further, we demonstrate the algebraic equivalency of the Cowley and Atchley epigenetic model and Wagner's developmental mapping model. Finally, we propose a new multivariate model for continuous growth trajectories. The relative efficacy of these various models for understanding evolutionary change in developmental and morphological traits is discussed. © 1994 Wiley-Liss, Inc.     

167 Network properties of decoy and CASP predicted models: a comparison with native protein structures

Principles that govern protein folding still remain elusive. Given the huge sequence space, it is reasonable to assume that sequences follow a particular pattern to attain one of the folds already defined in the relatively small structural space. In this study, we have used protein structure networks at different interaction strengths of non-covalent interactions (I_min) (Brinda & Vishveshwara, 2005; Kannan & Vishveshwara, 1999), to identify patterns that can distinguish a native protein from decoy/modelled structures. This is a rigorous extension of an earlier study performed at I_min???0% (Chatterjee, Bhattacharyya et al., 2012). Network properties such as the size of the largest cluster (SLClu), largest k-2 communities (ComSk2) and clustering coefficients (CCoe) are analysed for 5422 native structures and 29543 decoy/modelled structures. Steeper transition profile of the native structures as a function of I_min is consistently observed (see Figure) . The network properties generated at different I_min and main-chain hydrogen bonds (MHB) are integrated into support vector machine to build a classifier, giving an accuracy of 94.11%. The uniqueness of the protein structures through side-chain interactions are captured by the network parameters, while MHB represents the backbone packing. Quality predictions for the recently concluded CASP 10 predicted models are also performed using the model with the selected ones showing RMSD values?<?2.5?Å with respect to the native structures. Amongst the network properties, ComSk2 is maximally able to capture the transition properties of the structures. Importance of ComSk2 has earlier been implicated to capture the percolating behaviour of a protein structure (Deb & Vishveshwara, 2009). Overall, a robust classifier is obtained, and patterns specific to native structures have been analysed and discussed. The study highlights the importance of side-chain interactions at different I_mins, along with backbone level interactions.     

Connectivity of Neutral Networks,Overdispersion, and Structural Conservation in Protein Evolution

Abstract Protein structures are much more conserved than sequences during evolution. Based on this observation, we investigate the consequences of structural conservation on protein evolution. We study seven of the most studied protein folds, determining that an extended neutral network in sequence space is associated with each of them. Within our model, neutral evolution leads to a non-Poissonian substitution process, due to the broad distribution of connectivities in neutral networks. The observation that the substitution process has non-Poissonian statistics has been used to argue against the original Kimura neutral theory, while our model shows that this is a generic property of neutral evolution with structural conservation. Our model also predicts that the substitution rate can strongly fluctuate from one branch to another of the evolutionary tree. The average sequence similarity within a neutral network is close to the threshold of randomness, as observed for families of sequences sharing the same fold. Nevertheless, some positions are more difficult to mutate than others. We compare such structurally conserved positions to positions conserved in protein evolution, suggesting that our model can be a valuable tool to distinguish structural from functional conservation in databases of protein families. These results indicate that a synergy between database analysis and structurally based computational studies can increase our understanding of protein evolution.