The nature of the free energy barriers to two-state folding

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.     

Protein design simulations suggest that side-chain conformational entropy is not a strong determinant of amino acid environmental preferences

Loss of side-chain conformational entropy is an important force opposing protein folding and the relative preferences of the amino acids for being buried or solvent exposed may be partially determined by which amino acids lose more side-chain entropy when placed in the core of a protein. To investigate these preferences, we have incorporated explicit modeling of side-chain entropy into the protein design algorithm, RosettaDesign. In the standard version of the program, the energy of a particular sequence for a fixed backbone depends only on the lowest energy side-chain conformations that can be identified for that sequence. In the new model, the free energy of a single amino acid sequence is calculated by evaluating the average energy and entropy of an ensemble of structures generated by Monte Carlo sampling of amino acid side-chain conformations. To evaluate the impact of including explicit side-chain entropy, sequences were designed for 110 native protein backbones with and without the entropy model. In general, the differences between the two sets of sequences are modest, with the largest changes being observed for the longer amino acids: methionine and arginine. Overall, the identity between the designed sequences and the native sequences does not increase with the addition of entropy, unlike what is observed when other key terms are added to the model (hydrogen bonding, Lennard-Jones energies, and solvation energies). These results suggest that side-chain conformational entropy has a relatively small role in determining the preferred amino acid at each residue position in a protein.     

Does conformational free energy distinguish loop conformations in proteins?

Limitations in protein homology modeling often arise from the inability to adequately model loops. In this paper we focus on the selection of loop conformations. We present a complete computational treatment that allows the screening of loop conformations to identify those that best fit a molecular model. The stability of a loop in a protein is evaluated via computations of conformational free energies in solution, i.e., the free energy difference between the reference structure and the modeled one. A thermodynamic cycle is used for calculation of the conformational free energy, in which the total free energy of the reference state (i.e., gas phase) is the CHARMm potential energy. The electrostatic contribution of the solvation free energy is obtained from solving the finite-difference Poisson-Boltzmann equation. The nonpolar contribution is based on a surface area-based expression. We applied this computational scheme to a simple but well-characterized system, the antibody hypervariable loop (complementarity-determining region, CDR). Instead of creating loop conformations, we generated a database of loops extracted from high-resolution crystal structures of proteins, which display geometrical similarities with antibody CDRs. We inserted loops from our database into a framework of an antibody; then we calculated the conformational free energies of each loop. Results show that we successfully identified loops with a "reference-like" CDR geometry, with the lowest conformational free energy in gas phase only. Surprisingly, the solvation energy term plays a confusing role, sometimes discriminating "reference-like" CDR geometry and many times allowing "non-reference-like" conformations to have the lowest conformational free energies (for short loops). Most "reference-like" loop conformations are separated from others by a gap in the gas phase conformational free energy scale. Naturally, loops from antibody molecules are found to be the best models for long CDRs (> or = 6 residues), mainly because of a better packing of backbone atoms into the framework of the antibody model.     

The role of slow and fast protein motions in allosteric interactions

Allostery is fundamentally thermodynamic in nature. Long-range communication in proteins may be mediated not only by changes in the mean conformation with enthalpic contribution but also by changes in dynamic fluctuations with entropic contribution. The important role of protein motions in mediating allosteric interactions has been established by NMR spectroscopy. By using CAP as a model system, we have shown how changes in protein structure and internal dynamics can allosterically regulate protein function and activity. The results indicate that changes in conformational entropy can give rise to binding enhancement, binding inhibition, or have no effect in the expected affinity, depending on the magnitude and sign of enthalpy–entropy compensation. Moreover, allosteric interactions can be regulated by the modulation a low-populated conformation states that serve as on-pathway intermediates for ligand binding. Taken together, the interplay between fast internal motions, which are intimately related to conformational entropy, and slow internal motions, which are related to poorly populated conformational states, can regulate protein activity in a way that cannot be predicted on the basis of the protein’s ground-state structure.     

Accounting for conformational entropy in predicting binding free energies of protein-protein interactions

Protein-protein interactions are governed by the change in free energy upon binding, ΔG = ΔH - TΔS. These interactions are often marginally stable, so one must examine the balance between the change in enthalpy, ΔH, and the change in entropy, ΔS, when investigating known complexes, characterizing the effects of mutations, or designing optimized variants. To perform a large-scale study into the contribution of conformational entropy to binding free energy, we developed a technique called GOBLIN (Graphical mOdel for BiomoLecular INteractions) that performs physics-based free energy calculations for protein-protein complexes under both side-chain and backbone flexibility. Goblin uses a probabilistic graphical model that exploits conditional independencies in the Boltzmann distribution and employs variational inference techniques that approximate the free energy of binding in only a few minutes. We examined the role of conformational entropy on a benchmark set of more than 700 mutants in eight large, well-studied complexes. Our findings suggest that conformational entropy is important in protein-protein interactions--the root mean square error (RMSE) between calculated and experimentally measured ΔΔGs decreases by 12% when explicit entropic contributions were incorporated. GOBLIN models all atoms of the protein complex and detects changes to the binding entropy along the interface as well as positions distal to the binding interface. Our results also suggest that a variational approach to entropy calculations may be quantitatively more accurate than the knowledge-based approaches used by the well-known programs FOLDX and Rosetta--GOBLIN's RMSEs are 10 and 36% lower than these programs, respectively.     

The entropic nature of protein thermal stabilization

We performed thermodynamic analysis of temperature-induced unfolding of mesophilic and thermophilic proteins. It was shown that the variability in protein thermostability associated with pH-dependent unfolding or linked to the substitution of amino acid residues on the protein surface is evidence of the governing role of the entropy factor. Numerical values of conformational components in enthalpy, entropy and free energy which characterize protein unfolding in the “gas phase” were obtained. Based on the calculated absolute values of entropy and free energy, a model of protein unfolding is proposed in which the driving force is the conformational entropy of native protein, as an energy of the heat motion (T·S^NC) increasing with temperature and acting as an factor devaluating the energy of intramolecular weak bonds in the transition state.     

Analysis of side chain mobility among protein G B1 domain mutants with widely varying stabilities

"Host-guest" studies of the B1 domain from Streptococcal protein G have been used previously to establish a thermodynamic scale for the beta-sheet-forming propensities of the 20 common amino acids. To investigate the contribution of side chain conformational entropy to the relative stabilities of B1 domain mutants, we have determined the dynamics of side chain methyl groups in 10 of the 20 mutants used in a previous study. Deuterium relaxation rates were measured using two-dimensional NMR techniques for 13CH2D groups. Analysis of the relaxation data using the Lipari-Szabo model-free formalism showed that mutations introduced at the guest position caused small but statistically significant changes in the methyl group dynamics. In addition, there was a low level of covariation of the Lipari-Szabo order parameters among the 10 mutants. The variations in conformational free energy estimated from the order parameters were comparable in magnitude to the variations in global stability of the 10 mutants but did not correlate with the global stability of the domain or with the structural properties of the guest amino acids. The data support the view that conformational entropy in the folded state is one of many factors that can influence the folding thermodynamics of proteins.     

Designing proteins from the inside out

Globular proteins are characterized by the specific and tight packing of hydrophobic side-chains in the so-called "hydrophobic core." Formation of the core is key in folding, stabilization, and conformational specificity. The critical role of hydrophobic cores in maintaining the highly ordered structures present in natural proteins justifies the tremendous efforts devoted to their redesign. Both experimental and computational combinatorial-based approaches have been reported in the last years as powerful protein design tools. These manage to explore large regions of the sequence/conformational space, allowing the search for alternative protein core arrangements displaying native-like properties. The overall results obtained from core design projects have contributed significantly to our present knowledge of protein folding and function. In addition, core design has worked as a benchmark for the development of ambitious protein design projects that nowadays are allowing the de novo design of novel protein structures and functions.     

A fast empirical approach to binding free energy calculations based on protein interface information

Three useful variables from the interfaces of 20 protein-protein complexes were investigated. These variables are the side-chain accessible number (N(b)), the number of hydrophilic pairs (N(pair)) and buried a polar solvent accessible surface areas (DeltaDeltaASA(apol)). An empirical model based on the three variables was developed to describe the free energy of protein associations. As the results show, the side-chain accessible numbers characterize the loss of side-chain conformational entropy of protein interactions and the effective empirical function presented here has great capability for estimating the binding free energy. It was found that the variables of interface information capture most of the significant features of protein-protein association. Also, we applied the model based on the variables as a rescoring function to docking simulations and found that it has the potential to distinguish the 'true' binding mode. It is clear that the simple and empirical scale developed here is an attractive target function for calculating binding free energy for various biological processes to rational protein design.     

Side-chain conformational entropy in protein folding.

An important, but often neglected, contribution to the thermodynamics of protein folding is the loss of entropy that results from restricting the number of accessible side-chain conformers in the native structure. Conformational entropy changes can be found by comparing the number of accessible rotamers in the unfolded and folded states, or by estimating fusion entropies. Comparison of several sets of results using different techniques shows that the mean conformational free energy change (T delta S) is 1 kcal.mol-1 per side chain or 0.5 kcal.mol-1 per bond. Changes in vibrational entropy appear to be negligible compared to the entropy change resulting from the loss of accessible rotamers. Side-chain entropies can help rationalize alpha-helix propensities, predict protein/inhibitor complex structures, and account for the distribution of side chains on the protein surface or interior.     

Consistency in structural energetics of protein folding and peptide recognition.

We report a new free energy decomposition that includes structure-derived atomic contact energies for the desolvation component, and show that it applies equally well to the analysis of single-domain protein folding and to the binding of flexible peptides to proteins. Specifically, we selected the 17 single-domain proteins for which the three-dimensional structures and thermodynamic unfolding free energies are available. By calculating all terms except the backbone conformational entropy change and comparing the result to the experimentally measured free energy, we estimated that the mean entropy gain by the backbone chain upon unfolding (delta Sbb) is 5.3 cal/K per mole of residue, and that the average backbone entropy for glycine is 6.7 cal/K. Both numbers are in close agreement with recent estimates made by entirely different methods, suggesting a promising degree of consistency between data obtained from disparate sources. In addition, a quantitative analysis of the folding free energy indicates that the unfavorable backbone entropy for each of the proteins is balanced predominantly by favorable backbone interactions. Finally, because the binding of flexible peptides to receptors is physically similar to folding, the free energy function should, in principle, be equally applicable to flexible docking. By combining atomic contact energies, electrostatics, and sequence-dependent backbone entropy, we calculated a priori the free energy changes associated with the binding of four different peptides to HLA-A2, 1 MHC molecule and found agreement with experiment to within 10% without parameter adjustment.     

Triathlon for energy functions: Who is the winner for design of protein–protein interactions?

Computational prediction of stabilizing mutations into monomeric proteins has become an almost ordinary task. Yet, computational stabilization of protein–protein complexes remains a challenge. Design of protein–protein interactions (PPIs) is impeded by the absence of an energy function that could reliably reproduce all favorable interactions between the binding partners. In this work, we present three energy functions: one function that was trained on monomeric proteins, while the other two were optimized by different techniques to predict side-chain conformations in a dataset of PPIs. The performances of these energy functions are evaluated in three different tasks related to design of PPIs: predicting side-chain conformations in PPIs, recovering native binding-interface sequences, and predicting changes in free energy of binding due to mutations. Our findings show that both functions optimized on side-chain repacking in PPIs are more suitable for PPI design compared to the function trained on monomeric proteins. Yet, no function performs best at all three tasks. Comparison of the three energy functions and their performances revealed that (1) burial of polar atoms should not be penalized significantly in PPI design as in single-protein design and (2) contribution of electrostatic interactions should be increased several-fold when switching from single-protein to PPI design. In addition, the use of a softer van der Waals potential is beneficial in cases when backbone flexibility is important. All things considered, we define an energy function that captures most of the nuances of the binding energetics and hence, should be used in future for design of PPIs.     

Validating a Coarse-Grained Potential Energy Function through Protein Loop Modelling

Coarse-grained (CG) methods for sampling protein conformational space have the potential to increase computational efficiency by reducing the degrees of freedom. The gain in computational efficiency of CG methods often comes at the expense of non-protein like local conformational features. This could cause problems when transitioning to full atom models in a hierarchical framework. Here, a CG potential energy function was validated by applying it to the problem of loop prediction. A novel method to sample the conformational space of backbone atoms was benchmarked using a standard test set consisting of 351 distinct loops. This method used a sequence-independent CG potential energy function representing the protein using -carbon positions only and sampling conformations with a Monte Carlo simulated annealing based protocol. Backbone atoms were added using a method previously described and then gradient minimised in the Rosetta force field. Despite the CG potential energy function being sequence-independent, the method performed similarly to methods that explicitly use either fragments of known protein backbones with similar sequences or residue-specific /-maps to restrict the search space. The method was also able to predict with sub-Angstrom accuracy two out of seven loops from recently solved crystal structures of proteins with low sequence and structure similarity to previously deposited structures in the PDB. The ability to sample realistic loop conformations directly from a potential energy function enables the incorporation of additional geometric restraints and the use of more advanced sampling methods in a way that is not possible to do easily with fragment replacement methods and also enable multi-scale simulations for protein design and protein structure prediction. These restraints could be derived from experimental data or could be design restraints in the case of computational protein design. C++ source code is available for download from http://www.sbg.bio.ic.ac.uk/phyre2/PD2/.     

Rotamer strain energy in protein helices - quantification of a major force opposing protein folding

It is widely believed that the dominant force opposing protein folding is the entropic cost of restricting internal rotations. The energetic changes from restricting side-chain torsional motion are more complex than simply a loss of conformational entropy, however. A second force opposing protein folding arises when a side-chain in the folded state is not in its lowest-energy rotamer, giving rotameric strain. chi strain energy results from a dihedral angle being shifted from the most stable conformation of a rotamer when a protein folds. We calculated the energy of a side-chain as a function of its dihedral angles in a poly(Ala) helix. Using these energy profiles, we quantify conformational entropy, rotameric strain energy and chi strain energy for all 17 amino acid residues with side-chains in alpha-helices. We can calculate these terms for any amino acid in a helix interior in a protein, as a function of its side-chain dihedral angles, and have implemented this algorithm on a web page. The mean change in rotameric strain energy on folding is 0.42 kcal mol-1 per residue and the mean chi strain energy is 0.64 kcal mol-1 per residue. Loss of conformational entropy opposes folding by a mean of 1.1 kcal mol-1 per residue, and the mean total force opposing restricting a side-chain into a helix is 2.2 kcal mol-1. Conformational entropy estimates alone therefore greatly underestimate the forces opposing protein folding. The introduction of strain when a protein folds should not be neglected when attempting to quantify the balance of forces affecting protein stability. Consideration of rotameric strain energy may help the use of rotamer libraries in protein design and rationalise the effects of mutations where side-chain conformations change.     

Coupling between conformation and proton binding in proteins

Interest centers here on whether the use of a fixed charge distribution of a protein solute, or a treatment that considers proton-binding equilibria by solving the Poisson equation, is a better approach to discriminate native from non-native conformations of proteins. In this analysis of the charge distribution of 7 proteins, we estimate the solvation free energy contribution to the total free energy by exploring the 2(zeta) possible ionization states of the whole molecule, with zeta being the number of ionizable groups in the amino acid sequence, for every conformation in the ensembles of 7 proteins. As an additional consideration of the role of electrostatic interactions in determining the charge distribution of native folds, we carried out a comparison of alternative charge assignment models for the ionizable residues in a set of 21 native-like proteins. The results of this work indicate that (1) for 6 out of 7 proteins, estimation of solvent polarization based on the Generalized Born model with a fixed charge distribution provides the optimal trade-off between accuracy, with respect to the Poisson equation, and speed when compared to the accessible surface area model; for the seventh protein, consideration of all possible ionization states of the whole molecule appears to be crucial to discriminate the native from non-native conformations; (2) significant differences in the degree of ionization and hence the charge distribution for native folds are found between the different charge models examined; (3) the stability of the native state is determined by a delicate balance of all the energy components, and (4) conformational entropy, and hence the dynamics of folding, may play a crucial role for a successful ab initio protein folding prediction.     

Energy landscape and dynamics of the beta-hairpin G peptide and its isomers: Topology and sequences

We have investigated free energy landscape [MM/PBSA + normal modes entropy] of permutations in the G peptide (41-56) from the protein G B1 domain by studying six isomers corresponding to moving the hydrophobic cluster along the beta-strands (toward the turn: T1, AGEWTYDDKTFTVTET; T2, GEDTWDYATFTVTKTE; T3, GEDDWTYATFTVTKTE; toward the end: E1, WTYDDAGETKTFTVT; E2, WEYTGDDATKTETFTV; E3, WTYEGDDATKTETFTV). The free energy terms include molecular mechanics energy, Poisson-Boltzmann electrostatic solvation energy, surface area solvation energy, and conformational entropy estimated by using normal mode analysis. From the wild type to T1, then T3, and finally T2, we see a progressively changing energy landscape, toward a less stable beta-hairpin structure. Moving the hydrophobic cluster outside toward the end region causes a greater change in the energy landscape. alpha-Helical instead of a beta-hairpin structure was the most stable form for the E2 isomer. However, no matter how much the sequence changes, for all variants studied, ideal "native" beta-hairpin topologies remain as minima (regardless of whether global or local) in the energy landscape. In general, we find that the energy landscape is dependent on the hydrophobic cluster topology and on the sequence. Our present study indicates that the key is the relative conformational energies of the different conformations. Changes in the sequence strongly modulate the relative stabilities of topologically similar regions in the energy landscape, rather than redefine the topology space. This finding is consistent with a population redistribution in the process of protein folding. The limited variation of topological space, compared with the number of possible sequence changes, may relate to the observation that the number of known protein folds are far less than the sequential allowance.     

Effective water model for Monte Carlo simulations of proteins

We present an effective theory for water. Our goal is to formulate on accurate model for the effects of solvation on protein dynamics, without incurring the huge computational cost and the slow temporal evolution typical of molecular dynamics simulations of liquids. We replace the individual water molecules in an all-atom potential with a local dielectric density field, with self interactions given by the Landau-Ginzburg free energy and external interactions by Lennard-Jones forces at the surface of the protein atoms. We explore conformational space with finite temperature Monte Carlo dynamics, using parallel Langevin and Fourier acceleration algorithms well suited to data-parallel computer architectures such as the Connection Machine. To establish the validity of our approximations, we compare our electrostatic contribution to the solvalion energy with the results of Lim, Bashford, and Karplus using a conventional static continuum dielectric cavity model, and the non electrostatic contributions with estimates of hydrophohic surface free energy. Our model can also accommodate ionic charges and temperature fluctuations, We propose future investigations extending our effective theory of solvation to include explicit orientational entropy and hydroxen-bonding terms. © 1995 John Wiley & Sons, Inc.     

Improvement in protein functional site prediction by distinguishing structural and functional constraints on protein family evolution using computational design

The prediction of functional sites in newly solved protein structures is a challenge for computational structural biology. Most methods for approaching this problem use evolutionary conservation as the primary indicator of the location of functional sites. However, sequence conservation reflects not only evolutionary selection at functional sites to maintain protein function, but also selection throughout the protein to maintain the stability of the folded state. To disentangle sequence conservation due to protein functional constraints from sequence conservation due to protein structural constraints, we use all atom computational protein design methodology to predict sequence profiles expected under solely structural constraints, and to compute the free energy difference between the naturally occurring amino acid and the lowest free energy amino acid at each position. We show that functional sites are more likely than non-functional sites to have computed sequence profiles which differ significantly from the naturally occurring sequence profiles and to have residues with sub-optimal free energies, and that incorporation of these two measures improves sequence based prediction of protein functional sites. The combined sequence and structure based functional site prediction method has been implemented in a publicly available web server.     

Incorporation of evolutionary information into Rosetta comparative modeling

Prediction of protein structures from sequences is a fundamental problem in computational biology. Algorithms that attempt to predict a structure from sequence primarily use two sources of information. The first source is physical in nature: proteins fold into their lowest energy state. Given an energy function that describes the interactions governing folding, a method for constructing models of protein structures, and the amino acid sequence of a protein of interest, the structure prediction problem becomes a search for the lowest energy structure. Evolution provides an orthogonal source of information: proteins of similar sequences have similar structure, and therefore proteins of known structure can guide modeling. The relatively successful Rosetta approach takes advantage of the first, but not the second source of information during model optimization. Following the classic work by Andrej Sali and colleagues, we develop a probabilistic approach to derive spatial restraints from proteins of known structure using advances in alignment technology and the growth in the number of structures in the Protein Data Bank. These restraints define a region of conformational space that is high-probability, given the template information, and we incorporate them into Rosetta's comparative modeling protocol. The combined approach performs considerably better on a benchmark based on previous CASP experiments. Incorporating evolutionary information into Rosetta is analogous to incorporating sparse experimental data: in both cases, the additional information eliminates large regions of conformational space and increases the probability that energy-based refinement will hone in on the deep energy minimum at the native state.