Short-range conformational energies,secondary structure propensities,and recognition of correct sequence-structure matches

A statistical analysis of known structures is made for an assessment of the utility of short-range energy considerations. For each type of amino acid, the potentials governing (1) the torsions and bond angle changes of virtual C^α-C^α bonds and (2) the coupling between torsion and bond angle changes are derived. These contribute approximately −2 RT per residue to the stability of native proteins, approximately half of which is due to coupling effects. The torsional potentials for the α-helical states of different residues are verified to be strongly correlated with the free-energy change measurements made upon single-site mutations at solvent-exposed regions. Likewise, a satisfactory correlation is shown between the β-sheet potentials of different amino acids and the scales from free-energy measurements, despite the role of tertiary context in stabilizing β-sheets. Furthermore, there is excellent agreement between our residue-specific potentials for α-helical state and other thermodynamic based scales. Threading experiments performed by using an inverse folding protocol show that 50 of 62 test structures correctly recognize their native sequence on the basis of short-range potentials. The performance is improved to 55, upon simultaneous consideration of short-range potentials and the nonbonded interaction potentials between sequentially distant residues. Interactions between near residues along the primary structure, i.e., the local or short-range interactions, are known to be insufficient, alone, for understanding the tertiary structural preferences of proteins alone. Yet, knowledge of short-range conformational potentials permits rationalizing the secondary structure propensities and aids in the discrimination between correct and incorrect tertiary folds. Proteins 29:292–308, 1997. © 1997 Wiley-Liss, Inc.     

On the characterization of energy networks of proteins

The construction of a realistic theoretical model of proteins is determinant for improving the computational simulations of their structural and functional aspects. Modeling proteins as a network of non-covalent connections between the atoms of amino acid residues has shown valuable insights into these macromolecules. The energy-related properties of protein structures are known to be very important in molecular dynamics. However, these same properties have been neglected when the protein structures are modeled as networks of atoms and amino acid residues. A new approach for the construction of protein models based on a network of atoms is presented. This method, based on interatomic interaction, takes into account the energy and geometric aspects of the protein structures that were not employed before, such as atomic occlusion inside the protein, the use of solvation, protein modeling and analysis, and the use of energy potentials to estimate the energies of interatomic non-covalent contacts. As a result, we achieved a more realistic network model of proteins. This model has the virtue of being more robust in face of different unknown variables that usually are arbitrarily estimated. We were able to determine the most connected residues of all the proteins studied, so that we are now in a better condition to study their structural role.     

Use of a potential of mean force to analyze free energy contributions in protein folding.

A method for calculation of the free energy of residues as a function of residue burial is proposed. The method is based on the potential of mean force, with a reaction coordinate expressed by residue burial. Residue burials are calculated from high-resolution protein structures. The largest individual contributions to the free energy of a residue are found to be due to the hydrophobic interactions of the nonpolar atoms, interactions of the main chain polar atoms, and interactions of the charged groups of residues Arg and Lys. The contribution to the free energy of folding due to the uncharged side chain polar atoms is small. The contribution to the free energy of folding due to the main chain polar atoms is favorable for partially buried residues and less favorable or unfavorable for fully buried residues. Comparison of the accessible surface areas of proteins and model spheres shows that proteins deviate considerably from a spherical shape and that the deviations increase with the size of a protein. The implications of these results for protein folding are also discussed.     

Conformational analysis of endothelin-1: Effects of solvation free energy

In order to investigate conformational preferences of the 21-residue peptide hormone endothelin-1 (ET-1), an extensive conformational search was carried out in vacuo using a combination of high temperature molecular dynamics / annealing and a Monte Carlo / minimization search in torsion angle space. Fully minimized conformations from the search were grouped into families using a clustering technique based on rms fitting over the Cartesian coordinates of the atoms of the peptide backbone of the ring region. A wide range of local energy minima were identified even though two disulfide bridges (Cys¹-Cys¹⁵ and Cys³-Cys¹¹) constrain the structure of the peptide. Low energy conformers of ET-1 as a nonionized species in vacuo arestabilized by intramolecular interaction of the ring region (residues 1-15) with the tail (residues 16–21). Strained conformations for individual residues are observed. Conformational similarity to protein loops is established by matching to protein crystal structures In order to assess the influence of aqueous environment on conformational preference, the electrostatic contribution to the solvation energy was calculated for ET-1 as a fully ionized species (Asp⁸, Lys⁹, Glu¹⁰, Asp¹⁸, N- and C-terminus) using a continuum electrostatics model (DelPhi) for each of the conformed generated in vacuo, and the total solvation free energy was estimated by adding a hydrophobic contribution proportional to solvent accessible surface area. Solvation dramatically alters the relative energetics of ET-1 conformers from that calculated in vacuo. Conformers of ET-1 favored by the electrostatic salvation energy in water include conformers with helical secondary structure in the region of residues 9–15. Perhaps of most importance, it was demonstrated that the contribution tosolvation by an individual charge depends not only on its solvent accessibility but on the proximity of other charges, i.e., it is a cooperative effect. This was shown by the calculation of electrostatic solvation energy as afunction of conformation with individual charges systematically turned “on” and “off”. The cooperative effect of multiple charges on solvation demonstrated in this manner calls into question models that relate solvation energysimply to solvent accessibility by atom or residue alone. © 1995 John Wiley & Sons, Inc.     

Rebuilding flavodoxin from C alpha coordinates: a test study

The tertiary structure of flavodoxin has been model built from only the X-ray crystallographic alpha-carbon coordinates. Main-chain atoms were generated from a dictionary of backbone structures. Side-chain conformations were initially set according to observed statistical distributions, clashes were resolved with reference to other knowledge-based parameters, and finally, energy minimization was applied. The RMSD of the model was 1.7 A across all atoms to the native structure. Regular secondary structural elements were modeled more accurately than other regions. About 40% of the chi 1 torsional angles were modeled correctly. Packing of side chains in the core was energetically stable but diverged significantly from the native structure in some regions. The modeling of protein structures is increasing in popularity but relatively few checks have been applied to determine the accuracy of the approach. In this work a variety of parameters have been examined. It was found that close contacts, and hydrogen-bonding patterns could identify poorly packed residues. These tests, however, did not indicate which residues had a conformation different from the native structure or how to move such residues to bring them into agreement. To assist in the modeling of interacting side chains a database of known interactions has been prepared.     

An empirical energy potential with a reference state for protein fold and sequence recognition.

We consider modifications of an empirical energy potential for fold and sequence recognition to represent approximately the stabilities of proteins in various environments. A potential used here includes a secondary structure potential representing short-range interactions for secondary structures of proteins, and a tertiary structure potential consisting of a long-range, pairwise contact potential and a repulsive packing potential. This potential is devised to evaluate together the total conformational energy of a protein at the coarse grained residue level. It was previously estimated from the observed frequencies of secondary structures, from contact frequencies between residues, and from the distributions of the number of residues in contact in known protein structures by regarding those distributions as the equilibrium distributions with the Boltzmann factor of these interaction energies. The stability of native structures is assumed as a primary requirement for proteins to fold into their native structures. A collapse energy is subtracted from the contact energies to remove the protein size dependence and to represent protein stabilities for monomeric and multimeric states. The free energy of the whole ensemble of protein conformations that is subtracted from the conformational energy to represent protein stability is approximated as the average energy expected for a typical native structure with the same amino acid composition. This term may be constant in fold recognition but essentially varies in sequence recognition. A simple test of threading sequences into structures without gaps is employed to demonstrate the importance of the present modifications that permit the same potential to be utilized for both fold and sequence recognition. Proteins 1999;36:357-369. Published 1999 Wiley-Liss, Inc.     

Polar and nonpolar atomic environments in the protein core: Implications for folding and binding

Hydrophobic interactions are believed to play an important role in protein folding and stability. Semi-empirical attempts to estimate these interactions are usually based on a model of solvation, whose contribution to the stability of proteins is assumed to be proportional to the surface area buried upon folding. Here we propose an extension of this idea by defining an environment free energy that characterizes the environment of each atom of the protein, including solvent, polar or nonpolar atoms of the same protein or of another molecule that interacts with the protein. In our model, the difference of this environment free energy between the folded state and the unfolded (extended) state of a protein is shown to be proportional to the area buried by nonpolar atoms upon folding. General properties of this environment free energy are derived from statistical studies on a database of 82 well-refined protein structures. This free energy is shown to be able to discriminate misfolded from correct structural models, to provide an estimate of the stabilization due to oligomerization, and to predict the stability of mutants in which hydrophobic residues have been substituted by site-directed mutagenesis, provided that no large structural modifications occur. © 1994 Wiley-Liss, Inc.     

Comparison of atomic solvation parametric sets: applicability and limitations in protein folding and binding.

Atomic solvation parameters (ASP) are widely used to estimate the solvation contribution to the thermodynamic stability of proteins as well as the free energy of association for protein-ligand complexes. They are also included in several molecular mechanics computer programs. In this work, a total of eight atomic solvation parametric sets has been employed to calculate the solvation contribution to the free energy of folding delta Gs for 17 proteins. A linear correlation between delta Gs and the number of residues in each protein was found for each ASP set. The calculations also revealed a great variety in the absolute value and in the sign of delta Gs values such that certain ASP sets predicted the unfolded state to be more stable than the folded, whereas others yield precisely the opposite. Further, the solvation contribution to the free energy of association of helix pairs and to the disassociation of loops (connection between secondary structural elements in proteins) from the protein tertiary structures were computed for each of the eight ASP sets and discrepancies were evident among them.     

Core formation in apomyoglobin: probing the upper reaches of the folding energy landscape

An acid-destabilized form of apomyoglobin, the so-called E state, consists of a set of heterogeneous structures that are all characterized by a stable hydrophobic core composed of 30-40 residues at the intersection of the A, G, and H helices of the protein, with little other secondary structure and no other tertiary structure. Relaxation kinetics studies were carried out to characterize the dynamics of core melting and formation in this protein. The unfolding and/or refolding response is induced by a laser-induced temperature jump between the folded and unfolded forms of E, and structural changes are monitored using the infrared amide I' absorbance at 1648-1651 cm(-1) that reports on the formation of solvent-protected, native-like helix in the core and by fluorescence emission changes from apomyoglobin's Trp14, a measure of burial of the indole group of this residue. The fluorescence kinetics data are monoexponential with a relaxation time of 14 micros. However, infrared kinetics data are best fit to a biexponential function with relaxation times of 14 and 59 micros. These relaxation times are very fast, close to the limits placed on folding reactions by diffusion. The 14 micros relaxation time is weakly temperature dependent and thus represents a pathway that is energetically downhill. The appearance of this relaxation time in both the fluorescence and infrared measurements indicates that this folding event proceeds by a concomitant formation of compact secondary and tertiary structures. The 59 micros relaxation time is much more strongly temperature dependent and has no fluorescence counterpart, indicating an activated process with a large energy barrier wherein nonspecific hydrophobic interactions between helix A and the G and H helices cause some helix burial but Trp14 remains solvent exposed. These results are best fit by a multiple-pathway kinetic model when U collapses to form the various folded core structures of E. Thus, the results suggest very robust dynamics for core formation involving multiple folding pathways and provide significant insight into the primary processes of protein folding.     

Folding type-specific secondary structure propensities of amino acids,derived from α-helical, β-sheet, α/β, and α+β proteins of known structures

Folding type-specific secondary structure propensities of 20 naturally occurring amino acids have been derived from α-helical, β-sheet, α/β, and α+β proteins of known structures. These data show that each residue type of amino acids has intrinsic propensities in different regions of secondary structures for different folding types of proteins. Each of the folding types shows markedly different rank ordering, indicating folding type-specific effects on the secondary structure propensities of amino acids. Rigorous statistical tests have been made to validate the folding type-specific effects. It should be noted that α and β proteins have relatively small α-helices and β-strands forming propensities respectively compared with those of α+β and α/β proteins. This may suggest that, with more complex architectures than α and β proteins, α+β and α/β proteins require larger propensities to distinguish from interacting α-helices and β-strands. Our finding of folding type-specific secondary structure propensities suggests that sequence space accessible to each folding type may have differing features. Differing sequence space features might be constrained by topological requirement for each of the folding types. Almost all strong β-sheet forming residues are hydrophobic in character regardless of folding types, thus suggesting the hydrophobicities of side chains as a key determinant of β-sheet structures. In contrast, conformational entropy of side chains is a major determinant of the helical propensities of amino acids, although other interactions such as hydrophobicities and charged interactions cannot be neglected. These results will be helpful to protein design, class-based secondary structure prediction, and protein folding. © 1998 John Wiley & Sons, Inc. Biopoly 45: 35–49, 1998     

Structures of apomyoglobin's various acid-destabilized forms

The structures and the cold and hot melting thermodynamics of the acid- and salt-destabilized states of horse heart apomyoglobin (apoMb), including the E (extended) and various I forms, are studied using probes of tertiary structure (tryptophan fluorescence and FTIR spectroscopy) and secondary structure (far-UV CD and FTIR spectroscopy). These forms likely resemble early structures in the folding of the largely helical protein. Both the I and E forms retain the AGH core whereby the two ends of the protein are tied together with sufficient numbers of tertiary contacts, involving a number of hydrophobic residues, to show cooperative melting. The melting thermodynamics of E and I are distinctly different. E contains no other tertiary structure and probably little other secondary structure apart from the core. The more destabilized E form appears to contain "random" buried runs of polypeptide backbone which convert to alpha-helix in the I form(s). Most interestingly, E consists not of a single structure but is composed of a heterogeneous mixture of conformations, all showing corelike cooperative melting characteristics, and consisting presumably of varying contacts between the A portion of apomyoglobin and the G-H hairpin. These results bear on the energy landscape and structural features of the early part of apomyoglobin's folding pathway.     

Contact prediction for beta and alpha‐beta proteins using integer linear optimization and its impact on the first principles 3D structure prediction method ASTRO‐FOLD

An integer linear optimization model is presented to predict residue contacts in β, α + β, and α/β proteins. The total energy of a protein is expressed as sum of a C^α? C^α distance dependent contact energy contribution and a hydrophobic contribution. The model selects contact that assign lowest energy to the protein structure as satisfying a set of constraints that are included to enforce certain physically observed topological information. A new method based on hydrophobicity is proposed to find the β‐sheet alignments. These β‐sheet alignments are used as constraints for contacts between residues of β‐sheets. This model was tested on three independent protein test sets and CASP8 test proteins consisting of β, α + β, α/β proteins and it was found to perform very well. The average accuracy of the predictions (separated by at least six residues) was ～61%. The average true positive and false positive distances were also calculated for each of the test sets and they are 7.58 Å and 15.88 Å, respectively. Residue contact prediction can be directly used to facilitate the protein tertiary structure prediction. This proposed residue contact prediction model is incorporated into the first principles protein tertiary structure prediction approach, ASTRO‐FOLD. The effectiveness of the contact prediction model was further demonstrated by the improvement in the quality of the protein structure ensemble generated using the predicted residue contacts for a test set of 10 proteins. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

An analysis of non-bonded energy of proteins

Non-bonded energy of 16 proteins was calculated using the atomic co-ordinates obtained by X-ray crystallography. The curve of total energy against the number of atoms in proteins is approximately linear with a slight concaved shape. According to a linear equation to fit the curve, the extrapolated length of a polypeptide chain of a globular shape is expected to be 18 residues, which corresponds conceivably to an approximate size of nucleus for a folding of the polypeptide chain. Contributions from short-range and medium-range energies are always much greater than those from long-range energy for all the proteins and there seems to exist a change of each contribution in a range from 1200 to 1700 atoms. The energies with a lag less than four residues are a major part of the total energy and the contribution of energy from main-chain atoms is considerably higher than that from side-chain atoms. Side-chain atoms of a residue have a tendency to interact more strongly with main-chain atoms of N-terminal, than with those of C-terminal side of the residue, indicating asymmetry of the interaction in a protein. Amino acid residues in proteins may be divided into three groups by the order of strength of average energy. The first group exhibiting strong interaction consists mainly of hydrophobic amino acids and the third group consists of hydrophilic ones corresponding to the location in a protein molecule. Cys, val, leu and met are important for medium-range and long-range energies; gly and ala for medium-range energy; ile, trp, phe, tyr and arg for long-range energy. One simple application of the average energy of amino acid residues is illustrated to estimate local energy of a segment of nine residues given by a protein sequence. There is a good correlation between the curve computed by the average energy and the experimental curve for myoglobin.     

Calculation of the total electrostatic energy of a macromolecular system: solvation energies, binding energies, and conformational analysis

In this report we describe an accurate numerical method for calculating the total electrostatic energy of molecules of arbitrary shape and charge distribution, accounting for both Coulombic and solvent polarization terms. In addition to the solvation energies of individual molecules, the method can be used to calculate the electrostatic energy associated with conformational changes in proteins as well as changes in solvation energy that accompany the binding of charged substrates. The validity of the method is examined by calculating the hydration energies of acetate, methyl ammonium, ammonium, and methanol. The method is then used to study the relationship between the depth of a charge within a protein and its interaction with the solvent. Calculations of the relative electrostatic energies of crystal and misfolded conformations of Themiste dyscritum hemerythrin and the VL domain of an antibody are also presented. The results indicate that electrostatic charge-solvent interactions strongly favor the crystal structures. More generally, it is found that charge-solvent interactions, which are frequently neglected in protein structure analysis, can make large contributions to the total energy of a macromolecular system.     

Refolding molecular dynamics simulations of small- and middle-sized proteins in an explicit solvent

In order to elucidate the protein folding problem, we performed molecular dynamics simulations for small- and middle-sized two unfolding and six refolding proteins in an explicit solvent. Histidine-containing phosphocarrier protein and small designed protein were chosen for the simulations. We found that the protein folding process of these proteins was divided into three phases: an α -helix formation phase, a packing phase and a β -sheet formation phase. In the α -helix formation phase, an α -helix was developed from a β -turn structure through a 3₁₀-helix state. In the packing phase, proteins became compact, and tertiary structures (α / α or pre- β / β packing) were formed. Formation of a hydrophobic nucleus occurred concomitant with the α -helix formation and packing phase. Finally, in the β -sheet formation phase, a β -sheet was developed owing to the sequential formation of hydrogen bonds between two neighbouring strands, just like a "closing zipper".     

Thermal unfolding simulations of a multimeric protein--transition state and unfolding pathways

The folding of an oligomeric protein poses an extra challenge to the folding problem because the protein not only has to fold correctly; it has to avoid nonproductive aggregation. We have carried out over 100 molecular dynamics simulations using an implicit solvation model at different temperatures to study the unfolding of one of the smallest known tetramers, p53 tetramerization domain (p53tet). We found that unfolding started with disruption of the native tetrameric hydrophobic core. The transition state for the tetramer to dimer transition was characterized as a diverse ensemble of different structures using Phi value analysis in quantitative agreement with experimental data. Despite the diversity, the ensemble was still native-like with common features such as partially exposed tetramer hydrophobic core and shifts in the dimer-dimer arrangements. After passing the transition state, the secondary and tertiary structures continued to unfold until the primary dimers broke free. The free dimer had little secondary structure left and the final free monomers were random-coil like. Both the transition states and the unfolding pathways from these trajectories were very diverse, in agreement with the new view of protein folding. The multiple simulations showed that the folding of p53tet is a mixture of the framework and nucleation-condensation mechanisms and the folding is coupled to the complex formation. We have also calculated the entropy and effective energy for the different states along the unfolding pathway and found that the tetramerization is stabilized by hydrophobic interactions.     

Probing protein fold space with a simplified model

We probe the stability and near-native energy landscape of protein fold space using powerful conformational sampling methods together with simple reduced models and statistical potentials. Fold space is represented by a set of 280 protein domains spanning all topological classes and having a wide range of lengths (33-300 residues) amino acid composition and number of secondary structural elements. The degrees of freedom are taken as the loop torsion angles. This choice preserves the native secondary structure but allows the tertiary structure to change. The proteins are represented by three-point per residue, three-dimensional models with statistical potentials derived from a knowledge-based study of known protein structures. When this space is sampled by a combination of parallel tempering and equi-energy Monte Carlo, we find that the three-point model captures the known stability of protein native structures with stable energy basins that are near-native (all α: 4.77 Å, all β: 2.93 Å, α/β: 3.09 Å, α+β: 4.89 Å on average and within 6 Å for 71.41%, 92.85%, 94.29% and 64.28% for all-α, all-β, α/β and α+β, classes, respectively). Denatured structures also occur and these have interesting structural properties that shed light on the different landscape characteristics of α and β folds. We find that α/β proteins with alternating α and β segments (such as the β-barrel) are more stable than proteins in other fold classes.     

Sequence-structure specificity of a knowledge based energy function at the secondary structure level

MOTIVATION: This paper investigates the sequence-structure specificity of a representative knowledge based energy function by applying it to threading at the level of secondary structures of proteins. Assessing the strengths and weaknesses of an energy function at this fundamental level provides more detailed and insightful information than at the tertiary structure level and the results obtained can be useful in tertiary level threading. RESULTS: We threaded each of the 293 non-redundant proteins onto the secondary structures contained in its respective native protein (host template). We also used 68 pairs of proteins with similar folds and low sequence identity. For each pair, we threaded the sequence of one protein onto the secondary structures of the other protein. The discerning power of the total energy function and its one-body, pairwise, and mutation components is studied. We then applied our energy function to a recent study which demonstrated how a designed 11-amino acid sequence can replace distinct segments (one segment is an alpha-helix, the other is a beta-sheet) of a protein without changing its fold. We conducted random mutations of the designed sequence to determine the patterns for favorable mutations. We also studied the sequence-structure specificity at the boundaries of a secondary structure. Finally, we demonstrated how to speed up tertiary level threading by filtering out alignments found to be energetically unfavorable during the secondary structure threading. AVAILABILITY: The program is available on request from the authors. CONTACT: xud@ornl.gov     

Engineering a compact non-native state of intestinal fatty acid-binding protein

The last three C-terminal residues (129-131) of intestinal fatty acid-binding protein (IFABP) participate in four main-chain hydrogen bonds and two electrostatic interactions to sequentially distant backbone and side-chain atoms. To assess if these interactions are involved in the final adjustment of the tertiary structure during folding, we engineered an IFABP variant truncated at residue 128. An additional mutation, Trp-6-->Phe, was introduced to simplify the conformational analysis by optical methods. Although the changes were limited to a small region of the protein surface, they resulted in an IFABP with altered secondary and tertiary structure. Truncated IFABP retains some cooperativity, is monomeric, highly compact, and has the molecular dimensions and shape of the native protein. Our results indicated that residues 129-131 are part of a crucial conformational determinant in which several long-range interactions, essential for the acquisition of the native state, are established. This work suggests that carefully controlled truncation can populate equilibrium non-native states under physiological conditions. These non-native states hold a great promise as experimental models for protein folding.