Ab initio construction of protein tertiary structures using a hierarchical approach

We present a hierarchical method to predict protein tertiary structure models from sequence. We start with complete enumeration of conformations using a simple tetrahedral lattice model. We then build conformations with increasing detail, and at each step select a subset of conformations using empirical energy functions with increasing complexity. After enumeration on lattice, we select a subset of low energy conformations using a statistical residue-residue contact energy function, and generate all-atom models using predicted secondary structure. A combined knowledge-based atomic level energy function is then used to select subsets of the all-atom models. The final predictions are generated using a consensus distance geometry procedure. We test the feasibility of the procedure on a set of 12 small proteins covering a wide range of protein topologies. A rigorous double-blind test of our method was made under the auspices of the CASP3 experiment, where we did ab initio structure predictions for 12 proteins using this approach. The performance of our methodology at CASP3 is reasonably good and completely consistent with our initial tests.     

Calculation of protein backbone geometry from alpha-carbon coordinates based on peptide-group dipole alignment.

An algorithm is proposed for the conversion of a virtual-bond polypeptide chain (connected C alpha atoms) to an all-atom backbone, based on determining the most extensive hydrogen-bond network between the peptide groups of the backbone, while maintaining all of the backbone atoms in energetically feasible conformations. Hydrogen bonding is represented by aligning the peptide-group dipoles. These peptide groups are not contiguous in the amino acid sequence. The first dipoles to be aligned are those that are both sufficiently close in space to be arranged in approximately linear arrays termed dipole paths. The criteria used in the construction of dipole paths are: to assure good alignment of the greatest possible number of dipoles that are close in space; to optimize the electrostatic interactions between the dipoles that belong to different paths close in space; and to avoid locally unfavorable amino acid residue conformations. The equations for dipole alignment are solved separately for each path, and then the remaining single dipoles are aligned optimally with the electrostatic field from the dipoles that belong to the dipole-path network. A least-squares minimizer is used to keep the geometry of the alpha-carbon trace of the resulting backbone close to that of the input virtual-bond chain. This procedure is sufficient to convert the virtual-bond chain to a real chain; in applications to real systems, however, the final structure is obtained by minimizing the total ECEPP/2 (empirical conformational energy program for peptides) energy of the system, starting from the geometry resulting from the solution of the alignment equations. When applied to model alpha-helical and beta-sheet structures, the algorithm, followed by the ECEPP/2 energy minimization, resulted in an energy and backbone geometry characteristic of these alpha-helical and beta-sheet structures. Application to the alpha-carbon trace of the backbone of the crystallographic 5PTI structure of bovine pancreatic trypsin inhibitor, followed by ECEPP/2 energy minimization with C alpha-distance constraints, led to a structure with almost as low energy and root mean square deviation as the ECEPP/2 geometry analog of 5PTI, the best agreement between the crystal and reconstructed backbone being observed for the residues involved in the dipole-path network.     

Protein docking and complementarity

Predicting the structures of protein-protein complexes is a difficult problem owing to the topographical and thermodynamic complexity of these structures. Past efforts in this area have focussed on fitting the interacting proteins together using rigid body searches, usually with the conformations of the proteins as they occur in crystal structure complexes. Here we present work which uses a rigid body docking method to generate the structures of three known protein complexes, using both the bound and unbound conformations of the interacting molecules. In all cases we can regenerate the geometry of the crystal complexes to high accuracy. We also are able to find geometries that do not resemble the crystal structure but nevertheless are surprisingly reasonable both mechanistically and by some simple physical criteria. In contrast to previous work in this area, we find that simple methods for evaluating the complementarity at the protein-protein interface cannot distinguish between the configurations that resemble the crystal structure complex and those that do not. Methods that could not distinguish between such similar and dissimilar configurations include surface area burial, solvation free energy, packing and mechanism-based filtering. Evaluations of the total interaction energy and the electrostatic interaction energy of the complexes were somewhat better. Of the techniques that we tried, energy minimization distinguished most clearly between the "true" and "false" positives, though even here the energy differences were surprisingly small. We found the lowest total interaction energy from amongst all of the putative complexes generated by docking was always within 5 A root-mean-square of the crystallographic structure. There were, however, several putative complexes that were very dissimilar to the crystallographic structure but had energies that were close to that of the low energy structure. The magnitude of the error in energy calculations has not been established in macromolecular systems, and thus the reliability of the small differences in energy remains to be determined. The ability of this docking method to regenerate the crystallographic configurations of the interacting proteins using their unbound conformations suggests that it will be a useful tool in predicting the structures of unsolved complexes.     

HOPE: a homotopy optimization method for protein structure prediction.

We use a homotopy optimization method, HOPE, to minimize the potential energy associated with a protein model. The method uses the minimum energy conformation of one protein as a template to predict the lowest energy structure of a query sequence. This objective is achieved by following a path of conformations determined by a homotopy between the potential energy functions for the two proteins. Ensembles of solutions are produced by perturbing conformations along the path, increasing the likelihood of predicting correct structures. Successful results are presented for pairs of homologous proteins, where HOPE is compared to a variant of Newton's method and to simulated annealing.     

The Lipid Bilayer Modulates the Structure and Function of an ATP-binding Cassette Exporter

ATP-binding cassette exporters use the energy of ATP hydrolysis to transport substrates across membranes by switching between inward- and outward-facing conformations. Essentially all structural studies of these proteins have been performed with the proteins in detergent micelles, locked in specific conformations and/or at low temperature. Here, we used luminescence resonance energy transfer spectroscopy to study the prototypical ATP-binding cassette exporter MsbA reconstituted in nanodiscs at 37 °C while it performs ATP hydrolysis. We found major differences when comparing MsbA in these native-like conditions with double electron-electron resonance data and the crystal structure of MsbA in the open inward-facing conformation. The most striking differences include a significantly smaller separation between the nucleotide-binding domains and a larger fraction of molecules with associated nucleotide-binding domains in the nucleotide-free apo state. These studies stress the importance of studying membrane proteins in an environment that approaches physiological conditions.     

Native atomic burials, supplemented by physically motivated hydrogen bond constraints, contain sufficient information to determine the tertiary structure of small globular proteins

We investigate the possibility that atomic burials, as measured by their distances from the structural geometrical center, contain sufficient information to determine the tertiary structure of globular proteins. We report Monte Carlo simulated annealing results of all-atom hard-sphere models in continuous space for four small proteins: the all-beta WW-domain 1E0L, the alpha/beta protein-G 1IGD, the all-alpha engrailed homeo-domain 1ENH, and the alpha + beta engineered monomeric form of the Cro protein 1ORC. We used as energy function the sum over all atoms, labeled by i, of |R(i) - R(i) (*)|, where R(i) is the atomic distance from the center of coordinates, or central distance, and R(i) (*) is the "ideal" central distance obtained from the native structure. Hydrogen bonds were taken into consideration by the assignment of two ideal distances for backbone atoms forming hydrogen bonds in the native structure depending on the formation of a geometrically defined bond, independently of bond partner. Lowest energy final conformations turned out to be very similar to the native structure for the four proteins under investigation and a strong correlation was observed between energy and distance root mean square deviation (DRMS) from the native in the case of all-beta 1E0L and alpha/beta 1IGD. For all alpha 1ENH and alpha + beta 1ORC the overall correlation between energy and DRMS among final conformations was not as high because some trajectories resulted in high DRMS but low energy final conformations in which alpha-helices adopted a non-native mutual orientation. Comparison between central distances and actual accessible surface areas corroborated the implicit assumption of correlation between these two quantities. The Z-score obtained with this native-centric potential in the discrimination of native 1ORC from a set of random compact structures confirmed that it contains a much smaller amount of native information when compared to a traditional contact Go potential but indicated that simple sequence-dependent burial potentials still need some improvement in order to attain a similar discriminability. Taken together, our results suggest that central distances, in conjunction to physically motivated hydrogen bond constraints, contain sufficient information to determine the native conformation of these small proteins and that a solution to the folding problem for globular proteins could arise from sufficiently accurate burial predictions from sequence followed by minimization of a burial-dependent energy function.     

Statistical mechanics of protein folding by exhaustive enumeration

It is hard to construct theories for the folding of globular proteins because they are large and complicated molecules having enormous numbers of nonnative conformations and having native states that are complicated to describe. Statistical mechanical theories of protein folding are constructed around major simplifying assumptions about the energy as a function of conformation and/or simplifications of the representation of the polypeptide chain, such as one point per residue on a cubic lattice. It is not clear how the results of these theories are affected by their various simplifications. Here we take a very different simplification approach where the chain is accurately represented and the energy of each conformation is calculated by a not unreasonable empirical function. However, the set of amino acid sequences and allowed conformations is so restricted that it becomes computationally feasible to examine them all. Hence we are able to calculate melting curves for thermal denaturation as well as the detailed kinetic pathway of refolding. Such calculations are based on a novel representation of the conformations as points in an abstract 12-dimensional Euclidean conformation space. Fast folding sequences have relatively high melting temperatures, native structures with relatively low energies, small kinetic barriers between local minima, and relatively many conformations in the global energy minimum's watershed. In contrast to other folding theories, these models show no necessary relationship between fast folding and an overall funnel shape to the energy surface, or a large energy gap between the native and the lowest nonnative structure, or the depth of the native energy minimum compared to the roughness of the energy landscape. Proteins 32:425–437, 1998. © 1998 Wiley-Liss, Inc.     

A new computational model for protein folding based on atomic solvation.

A new model for calculating the solvation energy of proteins is developed and tested for its ability to identify the native conformation as the global energy minimum among a group of thousands of computationally generated compact non-native conformations for a series of globular proteins. In the model (called the WZS model), solvation preferences for a set of 17 chemically derived molecular fragments of the 20 amino acids are learned by a training algorithm based on maximizing the solvation energy difference between native and non-native conformations for a training set of proteins. The performance of the WZS model confirms the success of this learning approach; the WZS model misrecognizes (as more stable than native) only 7 of 8,200 non-native structures. Possible applications of this model to the prediction of protein structure from sequence are discussed.     

Reversible scaling of dihedral angle barriers during molecular dynamics to improve structure prediction of cyclic peptides.

Peptide cyclization or chemical cross-linking has frequently been used to restrict the conformational freedom of a peptide, for example, to enhance its capacity for selective binding to a target receptor molecule. Structure prediction of cyclic peptides is important to evaluate possible conformations prior to synthesis. Because of the conformational constraints imposed by cyclization low energy conformations of cyclic peptides can be separated by large energy barriers. In order to improve the conformational search properties of molecular dynamics (MD) simulations a potential scaling method has been designed. The approach consists of several consecutive MD simulations with a specific lowering of dihedral energy barriers and reduced nonbonded interactions between atoms separated by three atoms followed by gradually scaling the potential until the original barriers are reached. Application to four cyclic penta- and hexa-peptide test cases and a protein loop of known structure indicates that the potential scaling method is more efficient and faster in locating low energy conformations than standard MD simulations. Combined with a generalized Born implicit solvation model the low energy cyclic peptide conformations and the loop structure are in good agreement with experiment. Applications in the presence of explicit water molecules during the simulations showed also improved convergence to structures close to experiment compared with regular MD.     

Investigating energy‐based pool structure selection in the structure ensemble modeling with experimental distance constraints: The example from a multidomain protein Pub1

The structural variations of multidomain proteins with flexible parts mediate many biological processes, and a structure ensemble can be determined by selecting a weighted combination of representative structures from a simulated structure pool, producing the best fit to experimental constraints such as interatomic distance. In this study, a hybrid structure‐based and physics‐based atomistic force field with an efficient sampling strategy is adopted to simulate a model di‐domain protein against experimental paramagnetic relaxation enhancement (PRE) data that correspond to distance constraints. The molecular dynamics simulations produce a wide range of conformations depicted on a protein energy landscape. Subsequently, a conformational ensemble recovered with low‐energy structures and the minimum‐size restraint is identified in good agreement with experimental PRE rates, and the result is also supported by chemical shift perturbations and small‐angle X‐ray scattering data. It is illustrated that the regularizations of energy and ensemble‐size prevent an arbitrary interpretation of protein conformations. Moreover, energy is found to serve as a critical control to refine the structure pool and prevent data overfitting, because the absence of energy regularization exposes ensemble construction to the noise from high‐energy structures and causes a more ambiguous representation of protein conformations. Finally, we perform structure‐ensemble optimizations with a topology‐based structure pool, to enhance the understanding on the ensemble results from different sources of pool candidates.     

Discrimination of the native from misfolded protein models with an energy function including implicit solvation.

An essential requirement for theoretical protein structure prediction is an energy function that can discriminate the native from non-native protein conformations. To date most of the energy functions used for this purpose have been extracted from a statistical analysis of the protein structure database, without explicit reference to the physical interactions responsible for protein stability. The use of the statistical functions has been supported by the widespread belief that they are superior for such discrimination to physics-based energy functions. An effective energy function which combined the CHARMM vacuum potential with a Gaussian model for the solvation free energy is tested for its ability to discriminate the native structure of a protein from misfolded conformations; the results are compared with those obtained with the vacuum CHARMM potential. The test is performed on several sets of misfolded structures prepared by others, including sets of about 650 good decoys for six proteins, as well as on misfolded structures of chymotrypsin inhibitor 2. The vacuum CHARMM potential is successful in most cases when energy minimized conformations are considered, but fails when applied to structures relaxed by molecular dynamics. With the effective energy function the native state is always more stable than grossly misfolded conformations both in energy minimized and molecular dynamics-relaxed structures. The present results suggest that molecular mechanics (physics-based) energy functions, complemented by a simple model for the solvation free energy, should be tested for use in the inverse folding problem, and supports their use in studies of the effective energy surface of proteins in solution. Moreover, the study suggests that the belief in the superiority of statistical functions for these purposes may be ill founded.     

Mechanisms of protein folding

The strong correlation between protein folding rates and the contact order suggests that folding rates are largely determined by the topology of the native structure. However, for a given topology, there may be several possible low free energy paths to the native state and the path that is chosen (the lowest free energy path) may depend on differences in interaction energies and local free energies of ordering in different parts of the structure. For larger proteins whose folding is assisted by chaperones, such as the Escherichia coli chaperonin GroEL, advances have been made in understanding both the aspects of an unfolded protein that GroEL recognizes and the mode of binding to the chaperonin. The possibility that GroEL can remove non-native proteins from kinetic traps by unfolding them either during polypeptide binding to the chaperonin or during the subsequent ATP-dependent formation of folding-active complexes with the co-chaperonin GroES has also been explored.     

Energy embedding of trypsin inhibitor

Energy embedding has been shown recently to be a useful extension of the distance geometry approach to conformational calculations in the case of very small molecules and simple energy functions. This paper tests the ability of energy embedding to locate low energy conformations satisfying both weak and strong geometric constraints when the molecule is the small protein, bovine pancreatic trypsin inhibitor, and the energy function is the complicated Oobatake-Crippen residue–residue potential. Using the potential function alone, the algorithm reaches a structure with energy lower than that of the native conformation, but with little resemblance to it. Aided by numerous geometric constraints, such as preformed secondary structure segments, the algorithm again finds a local minimum with energy better than that of the native, and with only 3.3 Å rms deviation from it. This is significantly closer to the native value than can be obtained using standard distance geometry and the geometric constraints alone. Thus, energy embedding using the Oobatake-Crippen potential function is a significant help in finding native conformations of proteins. However, additional trials on a hairpin bend fragment of trypsin inhibitor demonstrate the potential's shortcomings in encouraging proper secondary structure.     

Using robotics to fold proteins and dock ligands

The problems of protein folding and ligand docking have been explored largely using molecular dynamics or Monte Carlo methods. These methods are very compute intensive because they often explore a much wider range of energies, conformations and time than necessary. In addition, Monte Carlo methods often get trapped in local minima. We initially showed that robotic motion planning permitted one to determine the energy of binding and dissociation of ligands from protein binding sites (Singh et al., 1999). The robotic motion planning method maps complicated three-dimensional conformational states into a much simpler, but higher dimensional space in which conformational rearrangements can be represented as linear paths. The dimensionality of the conformation space is of the same order as the number of degrees of conformational freedom in three-dimensional space. We were able to determine the relative energy of association and dissociation of a ligand to a protein by calculating the energetics of interaction for a few thousand conformational states in the vicinity of the protein and choosing the best path from the roadmap. More recently, we have applied roadmap planning to the problem of protein folding (Apaydin et al., 2002a). We represented multiple conformations of a protein as nodes in a compact graph with the edges representing the probability of moving between neighboring states. Instead of using Monte Carlo simulation to simulate thousands of possible paths through various conformational states, we were able to use Markov methods to calculate the steady state occupancy of each conformation, needing to calculate the energy of each conformation only once. We referred to this Markov method of representing multiple conformations and transitions as stochastic roadmap simulation or SRS. We demonstrated that the distribution of conformational states calculated with exhaustive Monte Carlo simulations asymptotically approached the Markov steady state if the same Boltzman energy distribution was used in both methods. SRS permits one to calculate contributions from all possible paths simultaneously with far fewer energy calculations than Monte Carlo or molecular dynamics methods. The SRS method also permits one to represent multiple unfolded starting states and multiple, near-native, folded states and all possible paths between them simultaneously. The SRS method is also independent of the function used to calculate the energy of the various conformational states. In a paper to be presented at this conference (Apaydin et al., 2002b) we have also applied SRS to ligand docking in which we calculate the dynamics of ligand-protein association and dissociation in the region of various binding sites on a number of proteins. SRS permits us to determine the relative times of association to and dissociation from various catalytic and non-catalytic binding sites on protein surfaces. Instead of just following the best path in a roadmap, we can calculate the contribution of all the possible binding or dissociation paths and their relative probabilities and energies simultaneously.     

Role of unfolded state heterogeneity and en-route ruggedness in protein folding kinetics

In order to improve our understanding of the physical bases of protein folding, there is a compelling need for better connections between experimental and computational approaches. This work addresses the role of unfolded state conformational heterogeneity and en-route intermediates, as an aid for planning and interpreting protein folding experiments. The expected kinetics were modeled for different types of energy landscapes, including multiple parallel folding routes, preferential paths dominated by one primary folding route, and distributed paths with a wide spectrum of microscopic folding rate constants. In the presence of one or more preferential routes, conformational exchange among unfolded state populations slows down the observed rates for native protein formation. We find this to be a general phenomenon, taking place even when unfolded conformations interconvert much faster than the "escape" rate constants to folding. Dramatic kinetic deceleration is expected in the presence of an increasing number of folding-incompetent unfolded conformations. This argues for the existence of parallel folding paths involving several folding-competent unfolded conformations, during the early stages of protein folding. Deviations from single-exponential behavior are observed for unfolded conformations exchanging at comparable rates or more slowly than folding events. Analysis of the effect of en-route (on-path) intermediate formation and landscape ruggedness on folding kinetics leads to the following unexpected conclusions: (1) intermediates, which often retard native state formation, may in some cases accelerate folding, and (2) rugged landscapes, usually associated with stretched exponentials, display single-exponential behavior in the presence of late high-friction paths.     

Evaluation of the conformational free energies of loops in proteins

In this paper we discuss the problem of including solvation free energies in evaluating the relative stabilities of loops in proteins. A conformational search based on a gas-phase potential function is used to generate a large number of trial conformations. As has been found previously, the energy minimization step in this process tends to pack charged and polar side chains against the protein surface, resulting in conformations which are unstable in the aqueous phase. Various solvation models can easily identify such structures. In order to provide a more severe test of solvation models, gas phase conformations were generated in which side chains were kept extended so as to maximize their interaction with the solvent. The free energies of these conformations were compared to that calculated for the crystal structure in three loops of the protein E. coli RNase H, with lengths of 7, 8, and 9 residues. Free energies were evaluated with a finite difference Poisson-Boltzmann (FDPB) calculation for electrostatics and a surface area-based term for nonpolar contributions. These were added to a gas-phase potential function. A free energy function based on atomic solvation parameters was also tested. Both functions were quite successful in selecting, based on a free energy criterion, conformations quite close to the crystal structure for two of the three loops. For one loop, which is involved in crystal contacts, conformations that are quite different from the crystal structure were also selected. A method to avoid precision problems associated with using the FDPB method to evaluate conformational free energies in proteins is described. © 1994 John Wiley & Sons, Inc.     

The Structural Basis of ATP as an Allosteric Modulator

Adenosine-5’-triphosphate (ATP) is generally regarded as a substrate for energy currency and protein modification. Recent findings uncovered the allosteric function of ATP in cellular signal transduction but little is understood about this critical behavior of ATP. Through extensive analysis of ATP in solution and proteins, we found that the free ATP can exist in the compact and extended conformations in solution, and the two different conformational characteristics may be responsible for ATP to exert distinct biological functions: ATP molecules adopt both compact and extended conformations in the allosteric binding sites but conserve extended conformations in the substrate binding sites. Nudged elastic band simulations unveiled the distinct dynamic processes of ATP binding to the corresponding allosteric and substrate binding sites of uridine monophosphate kinase, and suggested that in solution ATP preferentially binds to the substrate binding sites of proteins. When the ATP molecules occupy the allosteric binding sites, the allosteric trigger from ATP to fuel allosteric communication between allosteric and functional sites is stemmed mainly from the triphosphate part of ATP, with a small number from the adenine part of ATP. Taken together, our results provide overall understanding of ATP allosteric functions responsible for regulation in biological systems.     

Noninteracting local-structure model of folding and unfolding transition in globular proteins. I. Formulation

A statistical-mechanical model (a noninteracting local structure model) of folding and unfolding transition in globular proteins is described and a formulation is given to calculate the partition function. The process of transition is discussed in this model within the framework of equilibrium statistical mechanics. In order to clarify the range of applicability of such an approach, the characteristics of the folding and unfolding transition in globular proteins are analyzed from the statistical-physical point of view. A theoretical advantage is pointed out in studying folding and unfolding processes taking place as conformational fluctuations in individual protein molecules under macroscopic equilibrium at the melting temperature. In this case, paths of folding and unfolding are shown to be identical in the statistical sense. A key to the noninteracting local structure model lies in the concept of local structures and the assumption of the absence of interactions between local structures. A local structure is defined as a continuous section of the chain which takes the same or similar local conformation as in the native conformation. The assumption of the absence of inter-actions between local structures endows the model with the remarkable character that its partition function can be calculated exactly; thereby the equilibrium population of various conformations along the folding and unfolding paths can be discussed only by a knowledge of the folded native conformation.