Proteins with selected sequences fold into unique native conformation

Sequence-dependent and -independent information in a combined random energy model for protein folding and coding

Random energy models (REMs) provide a simple description of the energy landscapes that guide protein folding and evolution. The requirement of a large energy gap between the native structure and unfolded conformations, considered necessary for cooperative, protein-like, folding behavior, indicates that proteins differ markedly from random heteropolymers. It has been suggested, therefore, that natural selection might have acted to choose nonrandom amino acid sequences satisfying this particular condition, implying that a large fraction of possible, unselected random sequences, would not fold to any structure. From an informational perspective, however, this scenario could indicate that protein structures, regarded as messages to be transmitted through a communication channel, would not be efficiently encoded in amino acid sequences, regarded as the communication channel for this transmission, since a large fraction of possible channel states would not be used. Here, we use a combined REM for conformations and sequences, with previously estimated parameters for natural proteins, to explore an alternative possibility in which the appropriate shape of the landscape results mainly from the deviation from randomness of possible native structures instead of sequences. We observe that this situation emerges naturally if the distribution of conformational energies happens to arise from two independent contributions corresponding to sequence-dependent and -independent terms. This construction is consistent with the hypothesis of a protein burial folding code, with native structures being determined by a modest amount of sequence-dependent atomic burial information with sequence-independent constraints imposed by unspecific hydrogen bond formation. More generally, an appropriate combination of sequence-dependent and -independent information accommodates the possibility of an efficient structural encoding with the main physical requirement for folding, providing possible insight not only on the folding process but also on several aspects sequence evolution such as neutral networks, conformational coverage, and de novo gene emergence.

Key aspects of the past 30 years of protein design

Proteins are the workhorse of life. They are the building infrastructure of living systems; they are the most efficient molecular machines known, and their enzymatic activity is still unmatched in versatility by any artificial system. Perhaps proteins' most remarkable feature is their modularity. The large amount of information required to specify each protein's function is analogically encoded with an alphabet of just ∼20 letters. The protein folding problem is how to encode all such information in a sequence of 20 letters. In this review, we go through the last 30 years of research to summarize the state of the art and highlight some applications related to fundamental problems of protein evolution.

Rules of Physical Mathematics Govern Intrinsically Disordered Proteins

In stark contrast to foldable proteins with a unique folded state, intrinsically disordered proteins and regions (IDPs) persist in perpetually disordered ensembles. Yet an IDP ensemble has conformational features-even when averaged-that are specific to its sequence. In fact, subtle changes in an IDP sequence can modulate its conformational features and its function. Recent advances in theoretical physics reveal a set of elegant mathematical expressions that describe the intricate relationships among IDP sequences, their ensemble conformations, and the regulation of their biological functions. These equations also describe the molecular properties of IDP sequences that predict similarities and dissimilarities in their functions and facilitate classification of sequences by function, an unmet challenge to traditional bioinformatics. These physical sequence-patterning metrics offer a promising new avenue for advancing synthetic biology at a time when multiple novel functional modes mediated by IDPs are emerging.

Native state of natural proteins optimizes local entropy

The differing ability of polypeptide conformations to act as the native state of proteins has long been rationalized in terms of differing kinetic accessibility or thermodynamic stability. Building on the successful applications of physical concepts and sampling algorithms recently introduced in the study of disordered systems, in particular artificial neural networks, we quantitatively explore how well a quantity known as the local entropy describes the native state of model proteins. In lattice models and all-atom representations of proteins, we are able to efficiently sample high local entropy states and to provide a proof of concept of enhanced stability and folding rate. Our methods are based on simple and general statistical-mechanics arguments, and thus we expect that they are of very general use.

Force Dependence of Proteins' Transition State Position and the Bell-Evans Model

Single-molecule force spectroscopy has opened a new field of research in molecular biophysics and biochemistry. Pulling experiments on individual proteins permit us to monitor conformational transitions with high temporal resolution and measure their free energy landscape. The force–extension curves of single proteins often present large hysteresis, with unfolding forces that are higher than refolding ones. Therefore, the high energy of the transition state (TS) in these molecules precludes kinetic rates measurements in equilibrium hopping experiments. In irreversible pulling experiments, force-dependent kinetic rates measurements show a systematic discrepancy between the sum of the folding and unfolding TS distances derived by the kinetic Bell–Evans model and the full molecular extension predicted by elastic models. Here, we show that this discrepancy originates from the force-induced movement of TS. Specifically, we investigate the highly kinetically stable protein barnase, using pulling experiments and the Bell–Evans model to characterize the position of its kinetic barrier. Experimental results show that while the TS stays at a roughly constant distance relative to the native state, it shifts with force relative to the unfolded state. Interestingly, a conversion of the protein extension into amino acid units shows that the TS position follows the Leffler–Hammond postulate: the higher the force, the lower the number of unzipped amino acids relative to the native state. The results are compared with the quasi-reversible unfolding–folding of a short DNA hairpin.

A Specific Set of Heterogeneous Native Interactions Yields Efficient Knotting in Protein Folding

Native interactions are crucial for folding, and non-native interactions appear to be critical for efficiently knotting proteins. Therefore, it is important to understand both their roles in the folding of knotted proteins. It has been proposed that non-native interactions drive the correct order of contact formation, which is essential to avoid backtracking and efficiently self-tie. In this study, we ask if non-native interactions are strictly necessary to tangle a protein or if the correct order of contact formation can be assured by a specific set of native, but otherwise heterogeneous (i.e., having distinct energies), interactions. In order to address this problem, we conducted extensive Monte Carlo simulations of lattice models of protein-like sequences designed to fold into a preselected knotted conformation embedding a trefoil knot. We were able to identify a specific set of heterogeneous native interactions that drives efficient knotting and is able to fold the protein when combined with the remaining native interactions modeled as homogeneous. This specific set of heterogeneous native interactions is strictly enough to efficiently self-tie. A distinctive feature of these native interactions is that they do not backtrack because their energies ensure the correct order of contact formation. Furthermore, they stabilize a knotted intermediate state, which is en route to the native structure. Our results thus show that-at least in the context of the adopted model-non-native interactions are not necessary to knot a protein. However, when they are taken into account in protein energetics, it is possible to find specific, nonlocal non-native interactions that operate as a scaffold that assists the knotting step.

The hydrophobic effect characterises the thermodynamic signature of amyloid fibril growth

Many proteins have the potential to aggregate into amyloid fibrils, protein polymers associated with a wide range of human disorders such as Alzheimer’s and Parkinson’s disease. The thermodynamic stability of amyloid fibrils, in contrast to that of folded proteins, is not well understood: the balance between entropic and enthalpic terms, including the chain entropy and the hydrophobic effect, are poorly characterised. Using a combination of theory, in vitro experiments, simulations of a coarse-grained protein model and meta-data analysis, we delineate the enthalpic and entropic contributions that dominate amyloid fibril elongation. Our prediction of a characteristic temperature-dependent enthalpic signature is confirmed by the performed calorimetric experiments and a meta-analysis over published data. From these results we are able to define the necessary conditions to observe cold denaturation of amyloid fibrils. Overall, we show that amyloid fibril elongation is associated with a negative heat capacity, the magnitude of which correlates closely with the hydrophobic surface area that is buried upon fibril formation, highlighting the importance of hydrophobicity for fibril stability.

Most proteins fold in the cell into stable, compact structures. Nevertheless, many proteins also have the ability to stick together, forming long fibrillar structures that are associated with a wide range of human disorders including Alzheimer’s and Parkinson’s disease. The exact nature of the amyloid-causing stickiness is not well understood, nevertheless amyloid fibrils show some very specific thermodynamic characteristics. Some fibrils even destabilise at low temperatures. In this work we translate hydrophobic theory previously used to model protein folding to fibril formation. We combine this theory with experimental measurements, simulations and meta-data analysis of different types of fibrils. This allowed us to unravel the nature of the stickiness in amyloid fibrils by observing the effect of temperature changes, specifically at low temperatures, on hydrophobicity.

Identification of Protein Functional Regions

Protein sequence stores the information relative to both functionality and stability, thus making it difficult to disentangle the two contributions. However, the identification of critical residues for function and stability has important implications for the mapping of the proteome interactions, as well as for many pharmaceutical applications, e. g. the identification of ligand binding regions for targeted pharmaceutical protein design. In this work, we propose a computational method to identify critical residues for protein functionality and stability and to further categorise them in strictly functional, structural and intermediate. We evaluate single site conservation and use Direct Coupling Analysis (DCA) to identify co-evolved residues both in natural and artificial evolution processes. We reproduce artificial evolution using protein design and base our approach on the hypothesis that artificial evolution in the absence of any functional constraint would exclusively lead to site conservation and co-evolution events of the structural type. Conversely, natural evolution intrinsically embeds both functional and structural information. By comparing the lists of conserved and co-evolved residues, outcomes of the analysis on natural and artificial evolution, we identify the functional residues without the need of any a priori knowledge of the biological role of the analysed protein.

A method for partitioning the information contained in a protein sequence between its structure and function

Proteins employ the information stored in the genetic code and translated into their sequences to carry out well-defined functions in the cellular environment. The possibility to encode for such functions is controlled by the balance between the amount of information supplied by the sequence and that left after that the protein has folded into its structure. We study the amount of information necessary to specify the protein structure, providing an estimate that keeps into account the thermodynamic properties of protein folding. We thus show that the information remaining in the protein sequence after encoding for its structure (the 'information gap') is very close to what needed to encode for its function and interactions. Then, by predicting the information gap directly from the protein sequence, we show that it may be possible to use these insights from information theory to discriminate between ordered and disordered proteins, to identify unknown functions, and to optimize artificially-designed protein sequences.

Stability of the Influenza Virus Hemagglutinin Protein Correlates with Evolutionary Dynamics

One of the constraints on fast-evolving viruses, such as influenza virus, is protein stability, or how strongly the folded protein holds together. Despite the importance of this protein property, there has been limited investigation of the impact of the stability of the influenza virus hemagglutinin protein—the primary antibody target of the immune system—on its evolution. Using a combination of computational estimates of stability and experiments, our analysis found that viruses with more-stable hemagglutinin proteins were associated with long-term persistence in the population. There are two potential reasons for the observed persistence. One is that more-stable proteins tolerate destabilizing mutations that less-stable proteins could not, thus increasing opportunities for immune escape. The second is that greater stability increases the fitness of the virus through increased production of infectious particles. Further research on the relative importance of these mechanisms could help inform the annual influenza vaccine composition decision process.

Protein thermodynamics are an integral determinant of viral fitness and one of the major drivers of protein evolution. Mutations in the influenza A virus (IAV) hemagglutinin (HA) protein can eliminate neutralizing antibody binding to mediate escape from preexisting antiviral immunity. Prior research on the IAV nucleoprotein suggests that protein stability may constrain seasonal IAV evolution; however, the role of stability in shaping the evolutionary dynamics of the HA protein has not been explored. We used the full coding sequence of 9,797 H1N1pdm09 HA sequences and 16,716 human seasonal H3N2 HA sequences to computationally estimate relative changes in the thermal stability of the HA protein between 2009 and 2016. Phylogenetic methods were used to characterize how stability differences impacted the evolutionary dynamics of the virus. We found that pandemic H1N1 IAV strains split into two lineages that had different relative HA protein stabilities and that later variants were descended from the higher-stability lineage. Analysis of the mutations associated with the selective sweep of the higher-stability lineage found that they were characterized by the early appearance of highly stabilizing mutations, the earliest of which was not located in a known antigenic site. Experimental evidence further suggested that H1N1 HA stability may be correlated with in vitro virus production and infection. A similar analysis of H3N2 strains found that surviving lineages were also largely descended from viruses predicted to encode more-stable HA proteins. Our results suggest that HA protein stability likely plays a significant role in the persistence of different IAV lineages.

IMPORTANCE One of the constraints on fast-evolving viruses, such as influenza virus, is protein stability, or how strongly the folded protein holds together. Despite the importance of this protein property, there has been limited investigation of the impact of the stability of the influenza virus hemagglutinin protein—the primary antibody target of the immune system—on its evolution. Using a combination of computational estimates of stability and experiments, our analysis found that viruses with more-stable hemagglutinin proteins were associated with long-term persistence in the population. There are two potential reasons for the observed persistence. One is that more-stable proteins tolerate destabilizing mutations that less-stable proteins could not, thus increasing opportunities for immune escape. The second is that greater stability increases the fitness of the virus through increased production of infectious particles. Further research on the relative importance of these mechanisms could help inform the annual influenza vaccine composition decision process.

Optimising minimal building blocks for addressable self-assembly

Addressable structures are characterised by the set of unique components from which they are built and by the specific location that each component occupies. For an addressable structure to self-assemble, its constituent building blocks must be encoded with sufficient information to define their positions with respect to each other and to enable them to navigate to those positions. DNA, with its vast scope for encoding specific interactions, has been successfully used to synthesise addressable systems of several hundred components. In this work we examine the complementary question of the minimal requirements for building blocks to undergo addressable self-assembly driven by a controlled temperature quench. Our testbed is an idealised model of cubic particles patterned with attractive interactions. We introduce a scheme for optimising the interactions using a variant of basin-hopping and a negative design principle. The designed building blocks are tested dynamically in simple target structures to establish how their complexity affects the limits of reliable self-assembly.

The role of directional interactions in the designability of generalized heteropolymers

Heteropolymers are important examples of self-assembling systems. However, in the design of artificial heteropolymers the control over the single chain self-assembling properties does not reach that of the natural bio-polymers, and in particular proteins. Here, we introduce a sufficiency criterion to identify polymers that can be designed to adopt a predetermined structure and show that it is fulfilled by polymers made of monomers interacting through directional (anisotropic) interactions. The criterion is based on the appearance of a particular peak in the radial distribution function, that we show being a universal feature of all designable heteropolymers, as it is present also in natural proteins. Our criterion can be used to engineer new self-assembling modular polymers that will open new avenues for applications in materials science.

Structure-Based Prediction of Protein-Folding Transition Paths

We propose a general theory to describe the distribution of protein-folding transition paths. We show that transition paths follow a predictable sequence of high-free-energy transient states that are separated by free-energy barriers. Each transient state corresponds to the assembly of one or more discrete, cooperative units, which are determined directly from the native structure. We show that the transition state on a folding pathway is reached when a small number of critical contacts are formed between a specific set of substructures, after which folding proceeds downhill in free energy. This approach suggests a natural resolution for distinguishing parallel folding pathways and provides a simple means to predict the rate-limiting step in a folding reaction. Our theory identifies a common folding mechanism for proteins with diverse native structures and establishes general principles for the self-assembly of polymers with specific interactions.

Computational protein design: a review

Proteins are one of the most versatile modular assembling systems in nature. Experimentally, more than 110 000 protein structures have been identified and more are deposited every day in the Protein Data Bank. Such an enormous structural variety is to a first approximation controlled by the sequence of amino acids along the peptide chain of each protein. Understanding how the structural and functional properties of the target can be encoded in this sequence is the main objective of protein design. Unfortunately, rational protein design remains one of the major challenges across the disciplines of biology, physics and chemistry. The implications of solving this problem are enormous and branch into materials science, drug design, evolution and even cryptography. For instance, in the field of drug design an effective computational method to design protein-based ligands for biological targets such as viruses, bacteria or tumour cells, could give a significant boost to the development of new therapies with reduced side effects. In materials science, self-assembly is a highly desired property and soon artificial proteins could represent a new class of designable self-assembling materials. The scope of this review is to describe the state of the art in computational protein design methods and give the reader an outline of what developments could be expected in the near future.

Limiting the valence: advancements and new perspectives on patchy colloids, soft functionalized nanoparticles and biomolecules

Artistic representation of limited valance units consisting of a soft core (in blue) and a small number of flexible bonding patches (in orange).

Control globular structure formation of a copolymer chain through inverse design

A copolymer chain in dilute solution can exhibit various globular structures with characteristic morphologies, which makes it a potentially useful candidate for artificial materials design. However, the chain has a huge conformation space and may not naturally form the globular structure we desire. An ideal way to control globular structure formation should be inverse design, i.e., starting from the target structure and finding out what kind of polymers can effectively generate it. To accomplish this, we propose an inverse design procedure, which is combined with Wang-Landau Monte Carlo to fully and precisely explore the huge conformation space of the chain. Starting from a desired target structure, all the geometrically possible sequences are exactly enumerated. Interestingly, reasonable interaction strengths are obtained and found to be not specified for only one sequence. Instead, they can be combined with many other sequences and also achieve a relatively high yield for target structure, although these sequences may be rather different. These results confirm the possibility of controlling globular structure formation of a copolymer chain through inverse design and pave the way for targeted materials design.

A theoretical method to compute sequence dependent configurational properties in charged polymers and proteins

A general formalism to compute configurational properties of proteins and other heteropolymers with an arbitrary sequence of charges and non-uniform excluded volume interaction is presented. A variational approach is utilized to predict average distance between any two monomers in the chain. The presented analytical model, for the first time, explicitly incorporates the role of sequence charge distribution to determine relative sizes between two sequences that vary not only in total charge composition but also in charge decoration (even when charge composition is fixed). Furthermore, the formalism is general enough to allow variation in excluded volume interactions between two monomers. Model predictions are benchmarked against the all-atom Monte Carlo studies of Das and Pappu [Proc. Natl. Acad. Sci. U. S. A. 110, 13392 (2013)] for 30 different synthetic sequences of polyampholytes. These sequences possess an equal number of glutamic acid (E) and lysine (K) residues but differ in the patterning within the sequence. Without any fit parameter, the model captures the strong sequence dependence of the simulated values of the radius of gyration with a correlation coefficient of R(2) = 0.9. The model is then applied to real proteins to compare the unfolded state dimensions of 540 orthologous pairs of thermophilic and mesophilic proteins. The excluded volume parameters are assumed similar under denatured conditions, and only electrostatic effects encoded in the sequence are accounted for. With these assumptions, thermophilic proteins are found-with high statistical significance-to have more compact disordered ensemble compared to their mesophilic counterparts. The method presented here, due to its analytical nature, is capable of making such high throughput analysis of multiple proteins and will have broad applications in proteomic studies as well as in other heteropolymeric systems.

Consistent Treatment of Hydrophobicity in Protein Lattice Models Accounts for Cold Denaturation

The hydrophobic effect stabilizes the native structure of proteins by minimizing the unfavorable interactions between hydrophobic residues and water through the formation of a hydrophobic core. Here, we include the entropic and enthalpic contributions of the hydrophobic effect explicitly in an implicit solvent model. This allows us to capture two important effects: a length-scale dependence and a temperature dependence for the solvation of a hydrophobic particle. This consistent treatment of the hydrophobic effect explains cold denaturation and heat capacity measurements of solvated proteins.

Information and redundancy in the burial folding code of globular proteins within a wide range of shapes and sizes

Recent ab initio folding simulations for a limited number of small proteins have corroborated a previous suggestion that atomic burial information obtainable from sequence could be sufficient for tertiary structure determination when combined to sequence-independent geometrical constraints. Here, we use simulations parameterized by native burials to investigate the required amount of information in a diverse set of globular proteins comprising different structural classes and a wide size range. Burial information is provided by a potential term pushing each atom towards one among a small number L of equiprobable concentric layers. An upper bound for the required information is provided by the minimal number of layers L(min) still compatible with correct folding behavior. We obtain L(min) between 3 and 5 for seven small to medium proteins with 50 ≤ Nr ≤ 110 residues while for a larger protein with Nr = 141 we find that L ≥ 6 is required to maintain native stability. We additionally estimate the usable redundancy for a given L ≥ L(min) from the burial entropy associated to the largest folding-compatible fraction of "superfluous" atoms, for which the burial term can be turned off or target layers can be chosen randomly. The estimated redundancy for small proteins with L = 4 is close to 0.8. Our results are consistent with the above-average quality of burial predictions used in previous simulations and indicate that the fraction of approachable proteins could increase significantly with even a mild, plausible, improvement on sequence-dependent burial prediction or on sequence-independent constraints that augment the detectable redundancy during simulations.

Anomalous diffusion in neutral evolution of model proteins

Protein evolution is frequently explored using minimalist polymer models, however, little attention has been given to the problem of structural drift, or diffusion. Here, we study neutral evolution of small protein motifs using an off-lattice heteropolymer model in which individual monomers interact as low-resolution amino acids. In contrast to most earlier models, both the length and folded structure of the polymers are permitted to change. To describe structural change, we compute the mean-square distance (MSD) between monomers in homologous folds separated by n neutral mutations. We find that structural change is episodic, and, averaged over lineages (for example, those extending from a single sequence), exhibits a power-law dependence on n. We show that this exponent depends on the alignment method used, and we analyze the distribution of waiting times between neutral mutations. The latter are more disperse than for models required to maintain a specific fold, but exhibit a similar power-law tail.

Interplay of Coil-Globule Transition and Surface Adsorption of a Lattice HP Protein Model

An end-grafted hydrophobic-polar (HP) model protein chain with alternating H and P monomers is studied to examine interactions between the critical adsorption transition due to surface attraction and the collapse transition due to pairwise attractive H–H interactions. We find that the critical adsorption phenomenon can always be observed; however, the critical adsorption temperature T_CAP is influenced by the attractive H–H interactions in some cases. When the collapse temperature T_c is lower than T_CAP, the critical adsorption of the HP chain is similar to that of a homopolymer without intrachain attractions and T_CAP remains unchanged, whereas the collapse transition is suppressed by the adsorption. In contrast, for cases where T_c is close to or higher than T_CAP, T_CAP of the HP chain is increased, indicating that a collapsed chain is more easily adsorbed on the surface. The strength of the H–H attraction also influences the statistical size and shape of the polymer, with strong H–H attractions resulting in adsorbed and collapsed chains adopting two-dimensional, circular conformations.

Transferable coarse-grained potential for de novo protein folding and design

Protein folding and design are major biophysical problems, the solution of which would lead to important applications especially in medicine. Here we provide evidence of how a novel parametrization of the Caterpillar model may be used for both quantitative protein design and folding. With computer simulations it is shown that, for a large set of real protein structures, the model produces designed sequences with similar physical properties to the corresponding natural occurring sequences. The designed sequences require further experimental testing. For an independent set of proteins, previously used as benchmark, the correct folded structure of both the designed and the natural sequences is also demonstrated. The equilibrium folding properties are characterized by free energy calculations. The resulting free energy profiles not only are consistent among natural and designed proteins, but also show a remarkable precision when the folded structures are compared to the experimentally determined ones. Ultimately, the updated Caterpillar model is unique in the combination of its fundamental three features: its simplicity, its ability to produce natural foldable designed sequences, and its structure prediction precision. It is also remarkable that low frustration sequences can be obtained with such a simple and universal design procedure, and that the folding of natural proteins shows funnelled free energy landscapes without the need of any potentials based on the native structure.

Molecular interactions within the halophilic, thermophilic, and mesophilic prokaryotic ribosomal complexes: clues to environmental adaptation

Using the available crystal structures of 50S ribosomal subunits from three prokaryotic species: Escherichia coli (mesophilic), Thermus thermophilus (thermophilic), and Haloarcula marismortui (halophilic), we have analyzed different structural features of ribosomal RNAs (rRNAs), proteins, and of their interfaces. We have correlated these structural features with the environmental adaptation strategies of the corresponding species. While dense intra-rRNA packing is observed in thermophilic, loose intra-rRNA packing is observed in halophilic (both compared to mesophilic). Interestingly, protein-rRNA interfaces of both the extremophiles are densely packed compared to that of the mesophilic. The intersubunit bridge regions are almost devoid of cavities, probably ensuring the proper formation of each bridge (by not allowing any loosely packed region nearby). During rRNA binding, the ribosomal proteins experience some structural transitions. Here, we have analyzed the intrinsically disordered and ordered regions of the ribosomal proteins, which are subjected to such transitions. The intrinsically disordered and disorder-to-order transition sites of the thermophilic and mesophilic ribosomal proteins are simultaneously (i) highly conserved and (ii) slowly evolving compared to rest of the protein structure. Although high conservation is observed at such sites of halophilic ribosomal proteins, but slow rate of evolution is absent. Such differences between thermophilic, mesophilic, and halophilic can be explained from their environmental adaptation strategy. Interestingly, a universal biophysical principle evident by a linear relationship between the free energy of interface formation, interface area, and structural changes of r-proteins during assembly is always maintained, irrespective of the environmental conditions.

Rare-event trajectory ensemble analysis reveals metastable dynamical phases in lattice proteins

We explore the dynamical large deviations of a lattice heteropolymer model of a protein by means of path sampling of trajectories. We uncover the existence of nonequilibrium dynamical phase transitions in ensembles of trajectories between active and inactive dynamical phases, whose nature depends on the properties of the interaction potential. We consider three potentials: two heterogeneous interaction potentials and a homogeneous Gō potential. When preserving the full heterogeneity of interactions due to a given amino acid sequence, either in a fully interacting model or in a native contacts interacting model (heterogeneous Gō model), the observed dynamic transitions occur between equilibrium highly native states and highly native but kinetically trapped states. A native activity is defined that allows us to distinguish these dynamic phases. In contrast, for the homogeneous Gō model, where all native interaction energies are uniform and the amino acid sequence plays no role, the dynamical transition is a direct consequence of the static bistability between the unfolded and the native state. In the two heterogeneous interaction models the native-active and native-inactive states, despite their thermodynamic similarity, have widely varying dynamical properties, and the transition between them occurs even in lattice proteins whose sequences are designed to make them optimal folders.

The energy landscape of a protein switch

Protein switches are made of highly similar sequences that fold to dramatically different structures. A structural switching system with 31 sequence variants for α and α+β folds has been illustrated experimentally by He et al., Structure, 2012, 20, 283 and is investigated computationally in the present study. Methods to assign a sequence to one of the two folds are reported and analyzed. A fast and accurate protocol to identify the correct fold of the 31 sequences is based on enriching modeled structures using short molecular dynamics (MD) trajectories and scoring these structures with coarse-grained energy functions. We examine five coarse-grained energy functions and illustrate that the Hinds-Levitt potential works the best for this task. We show that enrichment by MD significantly enhances prediction accuracy. Finally, we find that melting temperature correlates well with the energy difference between the two folds (correlation coefficient ∼-0.7). The correlation reduces dramatically (∼0.4) if the absolute energy of the correct fold is considered. Moreover, prediction of melting temperature is sensitive to the structural templates. We emphasize in our analyses the use of native structures as templates since these folds are more readily available from structural biology experiments.

A simple lattice model that captures protein folding, aggregation and amyloid formation

The ability of many proteins to convert from their functional soluble state to amyloid fibrils can be attributed to inter-molecular beta strand formation. Such amyloid formation is associated with neurodegenerative disorders like Alzheimer's and Parkinson's. Molecular modelling can play a key role in providing insight into the factors that make proteins prone to fibril formation. However, fully atomistic models are computationally too expensive to capture the length and time scales associated with fibril formation. As the ability to form fibrils is the rule rather than the exception, much insight can be gained from the study of coarse-grained models that capture the key generic features associated with amyloid formation. Here we present a simple lattice model that can capture both protein folding and beta strand formation. Unlike standard lattice models, this model explicitly incorporates the formation of hydrogen bonds and the directionality of side chains. The simplicity of our model makes it computationally feasible to investigate the interplay between folding, amorphous aggregation and fibril formation, and maintains the capability of classic lattice models to simulate protein folding with high specificity. In our model, the folded proteins contain structures that resemble naturally occurring beta-sheets, with alternating polar and hydrophobic amino acids. Moreover, fibrils with intermolecular cross-beta strand conformations can be formed spontaneously out of multiple short hydrophobic peptide sequences. Both the formation of hydrogen bonds in folded structures and in fibrils is strongly dependent on the amino acid sequence, indicating that hydrogen-bonding interactions alone are not strong enough to initiate the formation of beta sheets. This result agrees with experimental observations that beta sheet and amyloid formation is strongly sequence dependent, with hydrophobic sequences being more prone to form such structures. Our model should open the way to a systematic study of the interplay between the factors that lead to amyloid formation.

Structural and energetic determinants of co-translational folding

We performed extensive lattice Monte Carlo simulations of ribosome-bound stalled nascent chains (RNCs) to explore the relative roles of native topology and non-native interactions in co-translational folding of small proteins. We found that the formation of a substantial part of the native structure generally occurs towards the end of protein synthesis. However, multi-domain structures, which are rich in local interactions, are able to develop gradually during chain elongation, while those with proximate chain termini require full protein synthesis to fold. A detailed assessment of the conformational ensembles populated by RNCs with different lengths reveals that the directionality of protein synthesis has a fine-tuning effect on the probability to populate low-energy conformations. In particular, if the participation of non-native interactions in folding energetics is mild, the formation of native-like conformations is majorly determined by the properties of the contact map around the tethering terminus. Likewise, a pair of RNCs differing by only 1-2 residues can populate structurally well-resolved low energy conformations with significantly different probabilities. An interesting structural feature of these low-energy conformations is that, irrespective of native structure, their non-native interactions are always long-ranged and marginally stabilizing. A comparison between the conformational spectra of RNCs and chain fragments folding freely in the bulk reveals drastic changes amongst the two set-ups depending on the native structure. Furthermore, they also show that the ribosome may enhance (up to 20%) the population of low energy conformations for chains folding to native structures dominated by local interactions. In contrast, a RNC folding to a non-local topology is forced to remain largely unstructured but can attain low energy conformations in bulk.

The role of site-directed point mutations in protein misfolding

Mutations inducing higher clashing and lower matching residue pairs lead to misfolding.

The loop hypothesis: contribution of early formed specific non-local interactions to the determination of protein folding pathways

The extremely fast and efficient folding transition (in seconds) of globular proteins led to the search for some unifying principles embedded in the physics of the folding polypeptides. Most of the proposed mechanisms highlight the role of local interactions that stabilize secondary structure elements or a folding nucleus as the starting point of the folding pathways, i.e., a "bottom-up" mechanism. Non-local interactions were assumed either to stabilize the nucleus or lead to the later steps of coalescence of the secondary structure elements. An alternative mechanism was proposed, an "up-down" mechanism in which it was assumed that folding starts with the formation of very few non-local interactions which form closed long loops at the initiation of folding. The possible biological advantage of this mechanism, the "loop hypothesis", is that the hydrophobic collapse is associated with ordered compactization which reduces the chance for degradation and misfolding. In the present review the experiments, simulations and theoretical consideration that either directly or indirectly support this mechanism are summarized. It is argued that experiments monitoring the time-dependent development of the formation of specifically targeted early-formed sub-domain structural elements, either long loops or secondary structure elements, are necessary. This can be achieved by the time-resolved FRET-based "double kinetics" method in combination with mutational studies. Yet, attempts to improve the time resolution of the folding initiation should be extended down to the sub-microsecond time regime in order to design experiments that would resolve the classes of proteins which first fold by local or non-local interactions.

A one-shot germinal center model under protein structural stability constraints

The germinal center reaction is the process by which low-affinity B cells evolve into potent, immunoglobulin-secreting plasma and memory B cells. Since the recycling hypothesis was created, experimental studies have both tracked movement of a small population of B cells from the light zone into the dark zone, supporting the recycling model, and parallel to the light zone-dark zone interface, indicating a one-way trajectory. We present a novel, sequence-based ab initio model of protein stability and protein interactions. Our model contains a dark zone region of clonal expansion and somatic hypermutation and a light zone site of antigenic selection. We show not only that a one-shot model is sufficient to achieve biologically-realistic rates of affinity growth, population dynamics, and silent:non-silent mutation ratios in the complementary determining region and framework region of antibodies, but also that a stochastic recycling program with or without realistic constraints on the structural stabilities of GC antibodies cannot produce biologically-observed affinity growth, population dynamics or silent:non-silent mutation profiles. The effect of recycling erases affinity gains made by potent antibodies cycling back from the light zone and causes B cells to pool in the dark zone under high replication rates.

Toward correct protein folding potentials

Empirical protein folding potentialfunctions should have a global minimum nearthe native conformationof globular proteins that fold stably, andthey should give the correct free energy offolding. We demonstrate that otherwise verysuccessful potentials fail to have even alocal minimumanywhere near the native conformation, anda seemingly well validated method ofestimatingthe thermodynamic stability of the nativestate is extremely sensitive to smallperturbations inatomic coordinates. These are bothindicative of fitting a great deal ofirrelevant detail. Here weshow how to devise a robust potentialfunction that succeeds very well at bothtasks, at least for alimited set of proteins, and this involvesdeveloping a novel representation of thedenatured state.Predicted free energies of unfolding for 25mutants of barnase are in close agreementwith theexperimental values, while for 17 mutantsthere are substantial discrepancies.

Selecting fast-folding proteins by their rate of convergence

We propose a general method for predicting potentially good folders from a given number of amino acid sequences. Our approach is based on the calculation of the rate of convergence of each amino acid chain towards the native structure using only the very initial parts of the dynamical trajectories. It does not require any preliminary knowledge of the native state and can be applied to different kinds of models, including atomistic descriptions. We tested the method within both the lattice and off-lattice model frameworks and obtained several so far unknown good folders. The unbiased algorithm also allows one to determine the optimal folding temperature and takes at least 3-4 orders of magnitude fewer time steps than those needed to compute folding times.

Topography of funneled landscapes determines the thermodynamics and kinetics of protein folding

The energy landscape approach has played a fundamental role in advancing our understanding of protein folding. Here, we quantify protein folding energy landscapes by exploring the underlying density of states. We identify three quantities essential for characterizing landscape topography: the stabilizing energy gap between the native and nonnative ensembles δE, the energetic roughness ΔE, and the scale of landscape measured by the entropy S. We show that the dimensionless ratio between the gap, roughness, and entropy of the system Λ=δE/(ΔE√(2S)) accurately predicts the thermodynamics, as well as the kinetics of folding. Large Λ implies that the energy gap (or landscape slope towards the native state) is dominant, leading to more funneled landscapes. We investigate the role of topological and energetic roughness for proteins of different sizes and for proteins of the same size, but with different structural topologies. The landscape topography ratio Λ is shown to be monotonically correlated with the thermodynamic stability against trapping, as characterized by the ratio of folding temperature versus trapping temperature. Furthermore, Λ also monotonically correlates with the folding kinetic rates. These results provide the quantitative bridge between the landscape topography and experimental folding measurements.

Finding low-energy conformations of lattice protein models by quantum annealing

Lattice protein folding models are a cornerstone of computational biophysics. Although these models are a coarse grained representation, they provide useful insight into the energy landscape of natural proteins. Finding low-energy threedimensional structures is an intractable problem even in the simplest model, the Hydrophobic-Polar (HP) model. Description of protein-like properties are more accurately described by generalized models, such as the one proposed by Miyazawa and Jernigan (MJ), which explicitly take into account the unique interactions among all 20 amino acids. There is theoretical and experimental evidence of the advantage of solving classical optimization problems using quantum annealing over its classical analogue (simulated annealing). In this report, we present a benchmark implementation of quantum annealing for lattice protein folding problems (six different experiments up to 81 superconducting quantum bits). This first implementation of a biophysical problem paves the way towards studying optimization problems in biophysics and statistical mechanics using quantum devices.

Hydrophobic forces and the length limit of foldable protein domains

To find the native conformation (fold), proteins sample a subspace that is typically hundreds of orders of magnitude smaller than their full conformational space. Whether such fast folding is intrinsic or the result of natural selection, and what is the longest foldable protein, are open questions. Here, we derive the average conformational degeneracy of a lattice polypeptide chain in water and quantitatively show that the constraints associated with hydrophobic forces are themselves sufficient to reduce the effective conformational space to a size compatible with the folding of proteins up to approximately 200 amino acids long within a biologically reasonable amount of time. This size range is in general agreement with the experimental protein domain length distribution obtained from approximately 1,200 proteins. Molecular dynamics simulations of the Trp-cage protein confirm this picture on the free energy landscape. Our analytical and computational results are consistent with a model in which the length and time scales of protein folding, as well as the modular nature of large proteins, are dictated primarily by inherent physical forces, whereas natural selection determines the native state.

Why do protein folding rates correlate with metrics of native topology?

For almost 15 years, the experimental correlation between protein folding rates and the contact order parameter has been under scrutiny. Here, we use a simple simulation model combined with a native-centric interaction potential to investigate the physical roots of this empirical observation. We simulate a large set of circular permutants, thus eliminating dependencies of the folding rate on other protein properties (e.g. stability). We show that the rate-contact order correlation is a consequence of the fact that, in high contact order structures, the contact order of the transition state ensemble closely mirrors the contact order of the native state. This happens because, in these structures, the native topology is represented in the transition state through the formation of a network of tertiary interactions that are distinctively long-ranged.

Tertiary DNA structure in the single-stranded hTERT promoter fragment unfolds and refolds by parallel pathways via cooperative or sequential events

The discovery of G-quadruplexes and other DNA secondary elements has increased the structural diversity of DNA well beyond the ubiquitous double helix. However, it remains to be determined whether tertiary interactions can take place in a DNA complex that contains more than one secondary structure. Using a new data analysis strategy that exploits the hysteresis region between the mechanical unfolding and refolding traces obtained by a laser-tweezers instrument, we now provide the first convincing kinetic and thermodynamic evidence that a higher order interaction takes place between a hairpin and a G-quadruplex in a single-stranded DNA fragment that is found in the promoter region of human telomerase. During the hierarchical unfolding or refolding of the DNA complex, a 15-nucleotide hairpin serves as a common species among three intermediates. Moreover, either a mutant that prevents this hairpin formation or the addition of a DNA fragment complementary to the hairpin destroys the cooperative kinetic events by removing the tertiary interaction mediated by the hairpin. The coexistence of the sequential and the cooperative refolding events provides direct evidence for a unifying kinetic partition mechanism previously observed only in large proteins and complex RNA structures. Not only does this result rationalize the current controversial observations for the long-range interaction in complex single-stranded DNA structures, but also this unexpected complexity in a promoter element provides additional justification for the biological function of these structures in cells.

Super folds, networks, and barriers

Exhaustive enumeration of sequences and folds is conducted for a simple lattice model of conformations, sequences, and energies. Examination of all foldable sequences and their nearest connected neighbors (sequences that differ by no more than a point mutation) illustrates the following: (i) There exist unusually large number of sequences that fold into a few structures (super-folds). The same observation was made experimentally and computationally using stochastic sampling and exhaustive enumeration of related models. (ii) There exist only a few large networks of connected sequences that are not restricted to one fold. These networks cover a significant fraction of fold spaces (super-networks). (iii) There exist barriers in sequence space that prevent foldable sequences of the same structure to "connect" through a series of single point mutations (super-barrier), even in the presence of the sequence connection between folds. While there is ample experimental evidence for the existence of super-folds, evidence for a super-network is just starting to emerge. The prediction of a sequence barrier is an intriguing characteristic of sequence space, suggesting that the overall sequence space may be disconnected. The implications and limitations of these observations for evolution of protein structures are discussed.

Equilibrium properties of realistic random heteropolymers and their relevance for globular and naturally unfolded proteins

Random heteropolymers do not display the typical equilibrium properties of globular proteins, but are the starting point to understand the physics of proteins and, in particular, to describe their non-native states. So far, they have been studied with mean-field models in the thermodynamic limit, or with computer simulations of very small chains on lattice. After describing a self-adjusting parallel-tempering technique to sample efficiently the low-energy states of frustrated systems without the need of tuning the system-dependent parameters of the algorithm, we apply it to random heteropolymers moving in continuous space. We show that if the mean interaction between monomers is negative, the usual description through the random-energy model is nearly correct, provided that it is extended to account for noncompact conformations. If the mean interaction is positive, such a simple description breaks out and the system behaves in a way more similar to Ising spin glasses. The former case is a model for the denatured state of globular proteins, the latter of naturally unfolded proteins, whose equilibrium properties thus result as qualitatively different.