Protein folding problem: enigma, paradox, solution.

The ability of protein chains to spontaneously form their three-dimensional structures is a long-standing mystery in molecular biology. The most conceptual aspect of this mystery is how the protein chain can find its native, "working" spatial structure (which, for not too big protein chains, corresponds to the global free energy minimum) in a biologically reasonable time, without exhaustive enumeration of all possible conformations, which would take billions of years. This is the so-called "Levinthal's paradox." In this review, we discuss the key ideas and discoveries leading to the current understanding of protein folding kinetics, including folding landscapes and funnels, free energy barriers at the folding/unfolding pathways, and the solution of Levinthal's paradox. A special role here is played by the "all-or-none" phase transition occurring at protein folding and unfolding and by the point of thermodynamic (and kinetic) equilibrium between the "native" and the "unfolded" phases of the protein chain (where the theory obtains the simplest form). The modern theory provides an understanding of key features of protein folding and, in good agreement with experiments, it (i) outlines the chain length-dependent range of protein folding times, (ii) predicts the observed maximal size of "foldable" proteins and domains. Besides, it predicts the maximal size of proteins and domains that fold under solely thermodynamic (rather than kinetic) control. Complementarily, a theoretical analysis of the number of possible protein folding patterns, performed at the level of formation and assembly of secondary structures, correctly outlines the upper limit of protein folding times.

HypercubeME: two hundred million combinatorially complete datasets from a single experiment.

MOTIVATION: Epistasis, the context-dependence of the contribution of an amino acid substitution to fitness, is common in evolution. To detect epistasis, fitness must be measured for at least four genotypes: the reference genotype, two different single mutants and a double mutant with both of the single mutations. For higher-order epistasis of the order n, fitness has to be measured for all 2n genotypes of an n-dimensional hypercube in genotype space forming a "combinatorially complete dataset". So far, only a handful of such datasets have been produced by manual curation. Concurrently, random mutagenesis experiments have produced measurements of fitness and other phenotypes in a high-throughput manner, potentially containing a number of combinatorially complete datasets. RESULTS: We present an effective recursive algorithm for finding all hypercube structures in random mutagenesis experimental data. To test the algorithm, we applied it to the data from a recent HIS3 protein dataset and found all 199,847,053 unique combinatorially complete genotype combinations of dimensionality ranging from two to twelve. The algorithm may be useful for researchers looking for higher-order epistasis in their high-throughput experimental data. AVAILABILITY: https://github.com/ivankovlab/HypercubeME.git. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape.

Characterizing the fitness landscape, a representation of fitness for a large set of genotypes, is key to understanding how genetic information is interpreted to create functional organisms. Here we determined the evolutionarily-relevant segment of the fitness landscape of His3, a gene coding for an enzyme in the histidine synthesis pathway, focusing on combinations of amino acid states found at orthologous sites of extant species. Just 15% of amino acids found in yeast His3 orthologues were always neutral while the impact on fitness of the remaining 85% depended on the genetic background. Furthermore, at 67% of sites, amino acid replacements were under sign epistasis, having both strongly positive and negative effect in different genetic backgrounds. 46% of sites were under reciprocal sign epistasis. The fitness impact of amino acid replacements was influenced by only a few genetic backgrounds but involved interaction of multiple sites, shaping a rugged fitness landscape in which many of the shortest paths between highly fit genotypes are inaccessible.

Self-consistency test reveals systematic bias in programs for prediction change of stability upon mutation.

Motivation: Computational prediction of the effect of mutations on protein stability is used by researchers in many fields. The utility of the prediction methods is affected by their accuracy and bias. Bias, a systematic shift of the predicted change of stability, has been noted as an issue for several methods, but has not been investigated systematically. Presence of the bias may lead to misleading results especially when exploring the effects of combination of different mutations. Results: Here we use a protocol to measure the bias as a function of the number of introduced mutations. It is based on a self-consistency test of the reciprocity the effect of a mutation. An advantage of the used approach is that it relies solely on crystal structures without experimentally measured stability values. We applied the protocol to four popular algorithms predicting change of protein stability upon mutation, FoldX, Eris, Rosetta and I-Mutant, and found an inherent bias. For one program, FoldX, we manage to substantially reduce the bias using additional relaxation by Modeller. Authors using algorithms for predicting effects of mutations should be aware of the bias described here. Availability and implementation: All calculations were implemented by in-house PERL scripts. Supplementary information: Supplementary data are available at Bioinformatics online. Note: The article 10.1093/bioinformatics/bty348, published alongside this paper, also addresses the problem of biases in protein stability change predictions.

There and back again: Two views on the protein folding puzzle.

The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured folding times of single-domain globular proteins range from microseconds to hours: the difference (10-11 orders of magnitude) is the same as that between the life span of a mosquito and the age of the universe. This review describes physical theories of rates of overcoming the free-energy barrier separating the natively folded (N) and unfolded (U) states of protein chains in both directions: "U-to-N" and "N-to-U". In the theory of protein folding rates a special role is played by the point of thermodynamic (and kinetic) equilibrium between the native and unfolded state of the chain; here, the theory obtains the simplest form. Paradoxically, a theoretical estimate of the folding time is easier to get from consideration of protein unfolding (the "N-to-U" transition) rather than folding, because it is easier to outline a good unfolding pathway of any structure than a good folding pathway that leads to the stable fold, which is yet unknown to the folding protein chain. And since the rates of direct and reverse reactions are equal at the equilibrium point (as follows from the physical "detailed balance" principle), the estimated folding time can be derived from the estimated unfolding time. Theoretical analysis of the "N-to-U" transition outlines the range of protein folding rates in a good agreement with experiment. Theoretical analysis of folding (the "U-to-N" transition), performed at the level of formation and assembly of protein secondary structures, outlines the upper limit of protein folding times (i.e., of the time of search for the most stable fold). Both theories come to essentially the same results; this is not a surprise, because they describe overcoming one and the same free-energy barrier, although the way to the top of this barrier from the side of the unfolded state is very different from the way from the side of the native state; and both theories agree with experiment. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control and explain the observed maximal size of the "foldable" protein domains.

Local fitness landscape of the green fluorescent protein.

Fitness landscapes depict how genotypes manifest at the phenotypic level and form the basis of our understanding of many areas of biology, yet their properties remain elusive. Previous studies have analysed specific genes, often using their function as a proxy for fitness, experimentally assessing the effect on function of single mutations and their combinations in a specific sequence or in different sequences. However, systematic high-throughput studies of the local fitness landscape of an entire protein have not yet been reported. Here we visualize an extensive region of the local fitness landscape of the green fluorescent protein from Aequorea victoria (avGFP) by measuring the native function (fluorescence) of tens of thousands of derivative genotypes of avGFP. We show that the fitness landscape of avGFP is narrow, with 3/4 of the derivatives with a single mutation showing reduced fluorescence and half of the derivatives with four mutations being completely non-fluorescent. The narrowness is enhanced by epistasis, which was detected in up to 30% of genotypes with multiple mutations and mostly occurred through the cumulative effect of slightly deleterious mutations causing a threshold-like decrease in protein stability and a concomitant loss of fluorescence. A model of orthologous sequence divergence spanning hundreds of millions of years predicted the extent of epistasis in our data, indicating congruence between the fitness landscape properties at the local and global scales. The characterization of the local fitness landscape of avGFP has important implications for several fields including molecular evolution, population genetics and protein design.

Machine Learning: How Much Does It Tell about Protein Folding Rates?

The prediction of protein folding rates is a necessary step towards understanding the principles of protein folding. Due to the increasing amount of experimental data, numerous protein folding models and predictors of protein folding rates have been developed in the last decade. The problem has also attracted the attention of scientists from computational fields, which led to the publication of several machine learning-based models to predict the rate of protein folding. Some of them claim to predict the logarithm of protein folding rate with an accuracy greater than 90%. However, there are reasons to believe that such claims are exaggerated due to large fluctuations and overfitting of the estimates. When we confronted three selected published models with new data, we found a much lower predictive power than reported in the original publications. Overly optimistic predictive powers appear from violations of the basic principles of machine-learning. We highlight common misconceptions in the studies claiming excessive predictive power and propose to use learning curves as a safeguard against those mistakes. As an example, we show that the current amount of experimental data is insufficient to build a linear predictor of logarithms of folding rates based on protein amino acid composition.

QARIP: a web server for quantitative proteomic analysis of regulated intramembrane proteolysis.

Regulated intramembrane proteolysis (RIP) is a critical mechanism for intercellular communication and regulates the function of membrane proteins through sequential proteolysis. RIP typically starts with ectodomain shedding of membrane proteins by extracellular membrane-bound proteases followed by intramembrane proteolysis of the resulting membrane-tethered fragment. However, for the majority of RIP proteases the corresponding substrates and thus, their functions, remain unknown. Proteome-wide identification of RIP protease substrates is possible by mass spectrometry-based quantitative comparison of RIP substrates or their cleavage products between different biological states. However, this requires quantification of peptides from only the ectodomain or cytoplasmic domain. Current analysis software does not allow matching peptides to either domain. Here we present the QARIP (Quantitative Analysis of Regulated Intramembrane Proteolysis) web server which matches identified peptides to the protein transmembrane topology. QARIP allows determination of quantitative ratios separately for the topological domains (cytoplasmic, ectodomain) of a given protein and is thus a powerful tool for quality control, improvement of quantitative ratios and identification of novel substrates in proteomic RIP datasets. To our knowledge, the QARIP web server is the first tool directly addressing the phenomenon of RIP. The web server is available at http://webclu.bio.wzw.tum.de/qarip/. This website is free and open to all users and there is no login requirement.

Restrictions to protein folding determined by the protein size.

Experimentally measured rates of spontaneous folding of single-domain globular proteins range from microseconds to hours: the difference (11 orders of magnitude!) is akin to the difference between the life span of a mosquito and the age of the Universe. We show that physical theory with biological constraints outlines the possible range of folding rates for single-domain globular proteins of various size and stability, and that the experimentally measured folding rates fall within this narrow "golden triangle" built without any adjustable parameters, filling it almost completely. This "golden triangle" also successfully predicts the maximal allowed size of the "foldable" protein domains, as well as the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control. In conclusion, we give a phenomenological formula for dependence of the folding rate on the size, shape and stability of the protein fold.

Golden triangle for folding rates of globular proteins.

The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured rates of spontaneous folding of single-domain globular proteins range from microseconds to hours: the difference (11 orders of magnitude) is akin to the difference between the life span of a mosquito and the age of the universe. Here, we show that physical theory with biological constraints outlines a "golden triangle" limiting the possible range of folding rates for single-domain globular proteins of various size and stability, and that the experimentally measured folding rates fall within this narrow triangle built without any adjustable parameters, filling it almost completely. In addition, the golden triangle predicts the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control. It also predicts the maximal allowed size of the "foldable" protein domains, and the size of domains found in known protein structures is in a good agreement with this limit.

Accessible surfaces of beta proteins increase with increasing protein molecular mass more rapidly than those of other proteins.

Here we present a systematic analysis of accessible surface areas and hydrogen bonds of 2554 globular proteins from four structural classes (all-α, all-ß, α/ß and α+ß proteins) that is aimed to learn in which structural class the accessible surface area increases with increasing protein molecular mass more rapidly than in other classes, and what structural peculiarities are responsible for this effect. The beta structural class of proteins was found to be the leader, with the following possible explanations of this fact. First, in beta structural proteins, the fraction of residues not included in the regular secondary structure is the largest, and second, the accessible surface area of packaged elements of the beta-structure increases more rapidly with increasing molecular mass in comparison with the alpha-structure. Moreover, in the beta structure, the probability of formation of backbone hydrogen bonds is higher than that in the alpha helix for all residues of α+ß proteins (the average probability is 0.73±0.01 for the beta-structure and 0.60±0.01 for the alpha-structure without proline) and α/ß proteins, except for asparagine, aspartic acid, glycine, threonine, and serine (0.70±0.01 for the beta-structure and 0.60±0.01 for the alpha-structure without the proline residue). There is a linear relationship between the number of hydrogen bonds and the number of amino acid residues in the protein (Number of hydrogen bonds=0.678·number of residues-3.350).

Library of disordered patterns in 3D protein structures.

Intrinsically disordered regions serve as molecular recognition elements, which play an important role in the control of many cellular processes and signaling pathways. It is useful to be able to predict positions of disordered regions in protein chains. The statistical analysis of disordered residues was done considering 34,464 unique protein chains taken from the PDB database. In this database, 4.95% of residues are disordered (i.e. invisible in X-ray structures). The statistics were obtained separately for the N- and C-termini as well as for the central part of the protein chain. It has been shown that frequencies of occurrence of disordered residues of 20 types at the termini of protein chains differ from the ones in the middle part of the protein chain. Our systematic analysis of disordered regions in PDB revealed 109 disordered patterns of different lengths. Each of them has disordered occurrences in at least five protein chains with identity less than 20%. The vast majority of all occurrences of each disordered pattern are disordered. This allows one to use the library of disordered patterns for predicting the status of a residue of a given protein to be ordered or disordered. We analyzed the occurrence of the selected patterns in three eukaryotic and three bacterial proteomes.

Coupling between properties of the protein shape and the rate of protein folding.

There are several important questions on the coupling between properties of the protein shape and the rate of protein folding. We have studied a series of structural descriptors intended for describing protein shapes (the radius of gyration, the radius of cross-section, and the coefficient of compactness) and their possible connection with folding behavior, either rates of folding or the emergence of folding intermediates, and compared them with classical descriptors, protein chain length and contact order. It has been found that when a descriptor is normalized to eliminate the influence of the protein size (the radius of gyration normalized to the radius of gyration of a ball of equal volume, the coefficient of compactness defined as the ratio of the accessible surface area of a protein to that of an ideal ball of equal volume, and relative contact order) it completely looses its ability to predict folding rates. On the other hand, when a descriptor correlates well with protein size (the radius of cross-section and absolute contact order in our consideration) then it correlates well with the logarithm of folding rates and separates reasonably well two-state folders from multi-state ones. The critical control for the performance of new descriptors demonstrated that the radius of cross-section has a somewhat higher predictive power (the correlation coefficient is -0.74) than size alone (the correlation coefficient is -0.65). So, we have shown that the numerical descriptors of the overall shape-geometry of protein structures are one of the important determinants of the protein-folding rate and mechanism.

KineticDB: a database of protein folding kinetics.

We propose here KineticDB, a systematically compiled database of protein folding kinetics, which contains about 90 unique proteins. The main goal of the KineticDB is to provide users with a diverse set of protein folding rates determined experimentally. The search for determinants of protein folding is still in progress, aimed at obtaining a new understanding of the folding process. Comparison with experimental protein folding rates has been the main tool for validation of both theoretical models and empirical relationships during the last 10 years. It is, therefore, necessary to provide a researcher with as much data as possible in a simple and easy-to-use way. At present, the KineticDB contains the results of folding kinetics measurements of single-domain proteins and separate protein domains as well as short peptides without disulfide bonds. It includes data on about 90 unique proteins and many mutants that have been systematically accumulated over the last 10 years and is the largest collection of protein folding kinetic data presented as a database. The KineticDB is available at http://kineticdb.protres.ru/db/index.pl.

Prediction of loop regions in protein sequence.

We suggest an algorithm that inputs a protein sequence and outputs a decomposition of the protein chain into a regular part including secondary structures and a nonregular part corresponding to loop regions. We have analyzed loop regions in a protein dataset of 3,769 globular domains and defined the optimal parameters for this prediction: the threshold between regular and nonregular regions and the optimal window size for averaging procedures using the scale of the expected number of contacts in a globular state and entropy scale as the number of degrees of freedom for the angles phi, psi, and chi for each amino acid. Comparison with known methods demonstrates that our method gives the same results as the well-known ALB method based on physical properties of amino acids (the percentage of true predictions is 64% against 66%), and worse prediction for regular and nonregular regions than PSIPRED (Protein Structure Prediction Server) without alignment of homologous proteins (the percentage of true predictions is 73%). The potential advantage of the suggested approach is that the predicted set of loops can be used to find patterns of rigid and flexible loops as possible candidates to play a structure/function role as well as a role of antigenic determinants.

Compactness determines protein folding type.

We have demonstrated here that protein compactness, which we define as the ratio of the accessible surface area of a protein to that of the ideal sphere of the same volume, is one of the factors determining the mechanism of protein folding. Proteins with multi-state kinetics, on average, are more compact (compactness is 1.49+/-0.02 for proteins within the size range of 101-151 amino acid residues) than proteins with two-state kinetics (compactness is 1.59+/-0.03 for proteins within the same size range of 101-151 amino acid residues). We have shown that compactness for homologous proteins can explain both the difference in folding rates and the difference in folding mechanisms.

More compact protein globules exhibit slower folding rates.

We have demonstrated that, among proteins of the same size, alpha/beta proteins have on the average a greater number of contacts per residue due to their more compact (more "spherical") structure, rather than due to tighter packing. We have examined the relationship between the average number of contacts per residue and folding rates in globular proteins according to general protein structural class (all-alpha, all-beta, alpha/beta, alpha+beta). Our analysis demonstrates that alpha/beta proteins have both the greatest number of contacts and the slowest folding rates in comparison to proteins from the other structural classes. Because alpha/beta proteins are also known to be the oldest proteins, it can be suggested that proteins have evolved to pack more quickly and into looser structures.

Trend of amino acid composition of proteins of different taxa.

Archaea, bacteria and eukaryotes represent the main kingdoms of life. Is there any trend for amino acid compositions of proteins found in full genomes of species of different kingdoms? What is the percentage of totally unstructured proteins in various proteomes? We obtained amino acid frequencies for different taxa using 195 known proteomes and all annotated sequences from the Swiss-Prot data base. Investigation of the two data bases (proteomes and Swiss-Prot) shows that the amino acid compositions of proteins differ substantially for different kingdoms of life, and this difference is larger between different proteomes than between different kingdoms of life. Our data demonstrate that there is a surprisingly small selection for the amino acid composition of proteins for higher organisms (eukaryotes) and their viruses in comparison with the "random" frequency following from a uniform usage of codons of the universal genetic code. On the contrary, lower organisms (bacteria and especially archaea) demonstrate an enhanced selection of amino acids. Moreover, according to our estimates, 12%, 3% and 2% of the proteins in eukaryotic, bacterial and archaean proteomes are totally disordered, and long (> 41 residues) disordered segments are found to occur in 16% of arhaean, 20% of eubacterial and 43% of eukaryotic proteins for 19 archaean, 159 bacterial and 17 eukaryotic proteomes, respectively. A correlation between amino acid compositions of proteins of various taxa, show that the highest correlation is observed between eukaryotes and their viruses (the correlation coefficient is 0.98), and bacteria and their viruses (the correlation coefficient is 0.96), while correlation between eukaryotes and archaea is 0.85 only.