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.     

Analysis of peptides from known proteins: Clusterization in sequence space

A combinatorial sequence space (CSS) model was introduced to represent sequences as a set of overlapping k-tuples of some fixed length which correspond to points in the CSS. The aim was to analyze clusterization of protein sequences in the CSS and to test various hypotheses about the possible evolutionary basis of this clusterization. The authors developed an easy-to-use technique which can reveal and analyze such a clusterization in a multidimensional CSS. Application of the technique led to an unexpectedly high clusterization of points in the CSS corresponding to k-tuples from known proteins. The clusterization could not be inferred from nonuniform amino acid frequencies or be explained by the influence of homologous data. None of the tested possible evolutionary and structural factors could explain the clusterization observed either. It looked as if certain protein sequence variations occurred and were fixed in the early course of evolution. Subsequent evolution (predominantly neutral) allowed only a limited number of changes and permitted new variants which led to preservation of certain k-tuples during the course of evolution. This was consistent with the theory of exon shuffling and protein block structure evolution. Possible applications of sequence space features found were also discussed.Correspondence to: H.A. Lim     

Missense meanderings in sequence space: a biophysical view of protein evolution

Proteins are finicky molecules; they are barely stable and are prone to aggregate, but they must function in a crowded environment that is full of degradative enzymes bent on their destruction. It is no surprise that many common diseases are due to missense mutations that affect protein stability and aggregation. Here we review the literature on biophysics as it relates to molecular evolution, focusing on how protein stability and aggregation affect organismal fitness. We then advance a biophysical model of protein evolution that helps us to understand phenomena that range from the dynamics of molecular adaptation to the clock-like rate of protein evolution.     

Lectin-like proteins in model organisms: implications for evolution of carbohydrate-binding activity.

Classes of intracellular lectins that recognize core-type structures and mediate intracellular glycoprotein trafficking are present in vertebrates, model invertebrates such as Caenorhabditis elegans and Drosophila melanogaster, plants, and yeasts. Lectins that recognize more complex structures at the cell surface, such as C-type lectins and galectins, are also found in invertebrate organisms as well as vertebrates, but the functions of these proteins have evolved differently in different animal lineages.     

Understanding relationship between sequence and functional evolution in yeast proteins

The underlying relationship between functional variables and sequence evolutionary rates is often assessed by partial correlation analysis. However, this strategy is impeded by the difficulty of conducting meaningful statistical analysis using noisy biological data. A recent study suggested that the partial correlation analysis is misleading when data is noisy and that the principal component regression analysis is a better tool to analyze biological data. In this paper, we evaluate how these two statistical tools (partial correlation and principal component regression) perform when data are noisy. Contrary to the earlier conclusion, we found that these two tools perform comparably in most cases. Furthermore, when there is more than one ‘true’ independent variable, partial correlation analysis delivers a better representation of the data. Employing both tools may provide a more complete and complementary representation of the real data. In this light, and with new analyses, we suggest that protein length and gene dispensability play significant, independent roles in yeast protein evolution. Electronic supplementary material Supplementary material is available in the online version of this article at and is accessible for authorized users.     

A model of DNA sequence evolution

Statistical studies of gene populations on the purine/pyrimidine alphabet have shown that the mean occurrence probability of thei-motif YRY(N) _i YRY (R=purine, Y=pyrimidine, N=R or Y) is not uniform by varyingi in the range [1,99], but presents a maximum ati=6 in the following populations: protein coding genes of eukaryotes, prokaryotes, chloroplasts and mitrochondria, and also viral introns, ribosomal RNA genes and transfer RNA genes (Arquès and Michel, 1987b,J. theor. Biol. 128, 457–461). From the “universality” of this observation, we suggested that the oligonucleotide YRY(N)₆ is a primitive one and that it has a central function in DNA sequence evolution (Arquès and Michel, 1987b,J. theor. Biol. 128, 457–461). Following this idea, we introduce a concept of a model of DNA sequence evolution which will be validated according to a shema presented in three parts. In the first part, using the last version of the gene database, the YRY(N)₆YRY preferential occurrence (maximum ati=6) is confirmed for the populations mentioned above and is extended to some newly analysed populations: chloroplast introns, chloroplast 5′ regions, mitochondrial 5′ regions and small nuclear RNA genes. On the other hand, the YRY(N)₆YRY preferential occurrence and periodicities are used in order to classify 18 gene populations. In the second part, we will demonstrate that several statistical features characterizing different gene populations (in particular the YRY(N)₆YRY preferential occurrence and the periodicities) can be retrieved from a simple Markov model based on the mixing of the two oligonucleotides YRY(N)₆ and YRY(N)₃ and based on the percentages of RYR and YRY in the unspecified trinucleotides (N)₃ of YRY(N)₆ and YRY(N)₃. Several properties are identified and prove in particular that the oligonucleotide mixing is an independent process and that several different features are functions of a unique parameter. In the third part, the return of the model to the reality shows a strong correlation between reality and simulation concerning the presence of large alternating purine/pyrimidine stretches and of periodicities. It also contributes to a greater understanding of biological reality, e.g. the presence or the absence of large alternating purine/pyrimidine stretches can be explained as being a simple consequence of the mixing of two particular oligonucleotides. Finally, we believe that such an approach is the first step toward a unified model of DNA sequence evolution allowing the molecular understanding of both the origin of life and the actual biological reality.     

iUbiq-Lys: prediction of lysine ubiquitination sites in proteins by extracting sequence evolution information via a gray system model

As one of the most important posttranslational modifications (PTMs), ubiquitination plays an important role in regulating varieties of biological processes, such as signal transduction, cell division, apoptosis, and immune response. Ubiquitination is also named “lysine ubiquitination” because it occurs when an ubiquitin is covalently attached to lysine (K) residues of targeting proteins. Given an uncharacterized protein sequence that contains many lysine residues, which one of them is the ubiquitination site, and which one is of non-ubiquitination site? With the avalanche of protein sequences generated in the postgenomic age, it is highly desired for both basic research and drug development to develop an automated method for rapidly and accurately annotating the ubiquitination sites in proteins. In view of this, a new predictor called “iUbiq-Lys” was developed based on the evolutionary information, gray system model, as well as the general form of pseudo-amino acid composition. It was demonstrated via the rigorous cross-validations that the new predictor remarkably outperformed all its counterparts. As a web-server, iUbiq-Lys is accessible to the public at http://www.jci-bioinfo.cn/iUbiq-Lys. For the convenience of most experimental scientists, we have further provided a protocol of step-by-step guide, by which users can easily get their desired results without the need to follow the complicated mathematics that were presented in this paper just for the integrity of its development process.     

A structured population model with diffusion in structure space

Journal of Mathematical Biology - A structured population model is described and analyzed, in which individual dynamics is stochastic. The model consists of a PDE of advection-diffusion type in the...     

Towards a quantitative model of immunogenicity: counting pathways in sequence space

One of the fundamental aims of structural biology is the identification of high-affinity ligands for arbitrary receptors. The maturation of the antibody repertoire elegantly and robustly solves this problem through an evolutionary mechanism comprising repeated cycles of mutation and preferential replication. To understand better the limitations and biases of this process, we developed an interpretation of antibody maturation within the framework of sequence space and fitness landscapes. Several well-described phenomena can be directly derived from this framework, and new predictions can be made. Ultimately, this reconceptualization of the clonal selection process suggests a quantitative, testable model of immunogenicity.     

The evolution of slow dispersal rates: a reaction diffusion model

 We consider n phenotypes of a species in a continuous but heterogeneous environment. It is assumed that the phenotypes differ only in their diffusion rates. With haploid genetics and a small rate of mutation, it is shown that the only nontrivial equilibrium is a population dominated by the slowest diffusing phenotype. We also prove that if there are only two possible phenotypes, then this equilibrium is a global attractor and conjecture that this is true in general. Numerical simulations supporting this conjecture and suggesting that this is a robust phenomenon are also discussed. Received: 29 January 1997 / Revised version: 23 September 1997     

A diffusion model for the evolution of metalloporphyrin

We give a mathematical model of the evolution of enzymes, the molecular structure of which is like metalloporphyrins or chlorophylls. We show, for this model, that even a small amount of these enzymes at the first stage is sufficient to increase and dominate the majority in a cell (like phenomena of gene fixation). For this purpose we use Kimura's equation, which has been explored for the study of evolution of genetics and has been known as a neutral theory of molecular evolution. Our model is a non-linear, non-equilibrium and non-closed (open to the external world) model.     

Non-divergence theory of evolution: sequence comparison of some proteins from snakes and bacteria

A "non-divergence theory" is proposed for the mechanism of evolution. The theory is based on the observation that comparison of the amino acid sequences of related proteins in various organisms gives inconsistent results from one type of protein to another, and on the occurrence of significant gene transfer among living organisms. Special attention is focused on the sequence comparisons of short- and long-chain neurotoxins and phospholipases A2 from the venoms of proteroglyphous snakes and those of microbial ferredoxins, rubredoxins, and flavodoxins.     

What limits the primary sequence space of natural proteins?

Abstract

Number of naturally occurring primary sequences of proteins is an infinitesimally small subset of the possible number of primary sequences that can be synthesized using 20 amino acids. Prevailing views ascribe this to slow and incremental mutational/selection evolutionary mechanisms. However, considering the large number of avenues available in form of diversity of emerging/evolving and/or disappearing living systems for exploring the primary sequence space over the evolutionary time scale of ～3.5 billion years, this remains a conjecture. Therefore, to investigate primary sequence space limitations, we carried out a systematic study for finding primary sequences absent in nature. We report the discovery of the smallest peptide sequence “Cysteine-Glutamine-Tryptophan-Tryptophan” that is not found in over half-a-million curated protein sequences in the Uniprot (Swiss-Prot) database. Additionally, we report a library of 83605 pentapeptides that are not found in any of the known protein sequences. Compositional analyses of these absent primary sequences yield a remarkably strong power relationship between the percentage occurrence of individual amino acids in all known protein sequences and their respective frequency of occurrence in the absent peptides, regardless of their specific position in the sequences. If random evolutionary mechanisms were responsible for limitations to the primary sequence space, then one would not expect any relationship between compositions of available and absent primary sequences. Thus, we conclusively show that stoichiometric constraints on amino acids limit the primary sequence space of proteins in nature. We discuss the possibly profound implications of our findings in both evolutionary and synthetic biology.

Communicated by Ramaswamy H. Sarma     

Invertibility of the TKF model of sequence evolution

We consider character sequences evolving on a phylogenetic tree under the TKF91 model. We show that as the sequence lengths tend to infinity the topology of the phylogenetic tree and the edge lengths are determined by any one of (a) the alignment of sequences (b) the collection of sequence lengths. We also show that the probability of any homology structure on a collection of sequences related by a TKF91 process on a tree is independent of the root location.     

Inching toward reality: An improved likelihood model of sequence evolution

Summary Our previous evolutionary model is generalized to permit approximate treatment of multiple-base insertions and deletions as well as regional heterogeneity of substitution rates. Parameter estimation and alignment procedures that incorporate these generalizations are developed. Simulations are used to assess the accuracy of the parameter estimation procedure and an example of an inferred alignment is included. Offprint requests to: J.L. Thorne     

Disorder and sequence repeats in hub proteins and their implications for network evolution

Protein interaction networks display approximate scale-free topology, in which hub proteins that interact with a large number of other proteins determine the overall organization of the network. In this study, we aim to determine whether hubs are distinguishable from other networked proteins by specific sequence features. Proteins of different connectednesses were compared in the interaction networks of Saccharomyces cerevisiae, Drosophila melanogaster, Caenorhabditis elegans, and Homo sapienswith respect to the distribution of predicted structural disorder, sequence repeats, low complexity regions, and chain length. Highly connected proteins ("hub proteins") contained significantly more of, and greater proportion of, these sequence features and tended to be longer overall as compared to less connected proteins. These sequence features provide two different functional means for realizing multiple interactions: (1) extended interaction surface and (2) flexibility and adaptability, providing a mechanism for the same region to bind distinct partners. Our view contradicts the prevailing view that scaling in protein interactomes arose from gene duplication and preferential attachment of equivalent proteins. We propose an alternative evolutionary network specialization process, in which certain components of the protein interactome improved their fitness for binding by becoming longer or accruing regions of disorder and/or internal repeats and have therefore become specialized in network organization.     

Bootstrap tests for overdispersion in a zero-inflated Poisson regression model

Ridout, Hinde, and Demétrio (2001, Biometrics 57, 219-223) derived a score test for testing a zero-inflated Poisson (ZIP) regression model against zero-inflated negative binomial (ZINB) alternatives. They mentioned that the score test using the normal approximation might underestimate the nominal significance level possibly for small sample cases. To remedy this problem, a parametric bootstrap method is proposed. It is shown that the bootstrap method keeps the significance level close to the nominal one and has greater power uniformly than the existing normal approximation for testing the hypothesis.     

Challenges of the genetic code for exploring sequence space in directed protein evolution

Directed protein evolution is the most versatile method for studying protein structure-function relationships, and for tailoring a protein's properties to the needs of industrial applications. In this review, we performed a statistical analysis on the genetic code to study the extent and consequence of the organization of the genetic code on amino acid substitution patterns generated in directed evolution experiments. In detail, we analyzed amino acid substitution patterns caused by (a) a single nucleotide (nt) exchange at each position of all 64 codons, and (b) two subsequent nt exchanges (first and second nt, first and third nt, second and third nt). Additionally, transitions and transversions mutations were compared at the level of amino acid substitution patterns. The latter analysis showed that single nucleotide substitution in a codon generates only 39.5% of the natural diversity on the protein level with 5.2-7 amino acid substitutions per codon. Transversions generate more complex amino acid substitution patterns (increased number and chemically more diverse amino acid substitutions) than transitions. Simultaneous nt exchanges at both first and second nt of a codon generates very diverse amino acid substitution patterns, achieving 83.2% of the natural diversity. The statistical analysis described in this review sets the objectives for novel random mutagenesis methods that address the consequences of the organization of the genetic code. Random mutagenesis methods that favor transversions or introduce consecutive nt exchanges can contribute in this regard.