A novel exhaustive search algorithm for predicting the conformation of polypeptide segments in proteins

Highly Accurate and Efficient Deep Learning Paradigm for Full-Atom Protein Loop Modeling with KarmaLoop

Protein loop modeling is a challenging yet highly nontrivial task in protein structure prediction. Despite recent progress, existing methods including knowledge-based, ab initio, hybrid, and deep learning (DL) methods fall substantially short of either atomic accuracy or computational efficiency. To overcome these limitations, we present KarmaLoop, a novel paradigm that distinguishes itself as the first DL method centered on full-atom (encompassing both backbone and side-chain heavy atoms) protein loop modeling. Our results demonstrate that KarmaLoop considerably outperforms conventional and DL-based methods of loop modeling in terms of both accuracy and efficiency, with the average RMSDs of 1.77 and 1.95 Å for the CASP13+14 and CASP15 benchmark datasets, respectively, and manifests at least 2 orders of magnitude speedup in general compared with other methods. Consequently, our comprehensive evaluations indicate that KarmaLoop provides a state-of-the-art DL solution for protein loop modeling, with the potential to hasten the advancement of protein engineering, antibody-antigen recognition, and drug design.

Inverse Kinematics in Biology: The Protein Loop Closure Problem

Assembling fragments from known protein structures is a widely used approach to construct structural models for new proteins. We describe an application of this idea to an important inverse kinematics problem in structural biology: the loop closure problem. We have developed an algorithm for generating the conformations of candidate loops that fit in a gap of given length in a protein structure framework. Our method proceeds by concatenating small fragments of protein chosen from small libraries of representative fragments. Our approach has the advantages of ab initio methods since we are able to enumerate all candidate loops in the discrete approximation of the conformational space accessible to the loop, as well as the advantages of database search approach since the use of fragments of known protein structures guarantees that the backbone conformations are physically reasonable. We test our approach on a set of 427 loops, varying in length from four residues to 14 residues. The quality of the candidate loops is evaluated in terms of global coordinate root mean square (cRMS). The top predictions vary between 0.3 and 4.2 Å for four-residue loops and between 1.5 and 3.1 Å for 14-residue loops, respectively.

LoopIng: a template-based tool for predicting the structure of protein loops

Motivation: Predicting the structure of protein loops is very challenging, mainly because they are not necessarily subject to strong evolutionary pressure. This implies that, unlike the rest of the protein, standard homology modeling techniques are not very effective in modeling their structure. However, loops are often involved in protein function, hence inferring their structure is important for predicting protein structure as well as function.

Results: We describe a method, LoopIng, based on the Random Forest automated learning technique, which, given a target loop, selects a structural template for it from a database of loop candidates. Compared to the most recently available methods, LoopIng is able to achieve similar accuracy for short loops (4–10 residues) and significant enhancements for long loops (11–20 residues). The quality of the predictions is robust to errors that unavoidably affect the stem regions when these are modeled. The method returns a confidence score for the predicted template loops and has the advantage of being very fast (on average: 1 min/loop).

Availability and implementation:www.biocomputing.it/looping

Contact:anna.tramontano@uniroma1.it

Supplementary information:Supplementary data are available at Bioinformatics online.

ROTAS: a rotamer-dependent, atomic statistical potential for assessment and prediction of protein structures

Background

Multibody potentials accounting for cooperative effects of molecular interactions have shown better accuracy than typical pairwise potentials. The main challenge in the development of such potentials is to find relevant structural features that characterize the tightly folded proteins. Also, the side-chains of residues adopt several specific, staggered conformations, known as rotamers within protein structures. Different molecular conformations result in different dipole moments and induce charge reorientations. However, until now modeling of the rotameric state of residues had not been incorporated into the development of multibody potentials for modeling non-bonded interactions in protein structures.

Results

In this study, we develop a new multibody statistical potential which can account for the influence of rotameric states on the specificity of atomic interactions. In this potential, named “rotamer-dependent atomic statistical potential” (ROTAS), the interaction between two atoms is specified by not only the distance and relative orientation but also by two state parameters concerning the rotameric state of the residues to which the interacting atoms belong. It was clearly found that the rotameric state is correlated to the specificity of atomic interactions. Such rotamer-dependencies are not limited to specific type or certain range of interactions. The performance of ROTAS was tested using 13 sets of decoys and was compared to those of existing atomic-level statistical potentials which incorporate orientation-dependent energy terms. The results show that ROTAS performs better than other competing potentials not only in native structure recognition, but also in best model selection and correlation coefficients between energy and model quality.

Conclusions

A new multibody statistical potential, ROTAS accounting for the influence of rotameric states on the specificity of atomic interactions was developed and tested on decoy sets. The results show that ROTAS has improved ability to recognize native structure from decoy models compared to other potentials. The effectiveness of ROTAS may provide insightful information for the development of many applications which require accurate side-chain modeling such as protein design, mutation analysis, and docking simulation.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-307) contains supplementary material, which is available to authorized users.

Conformational sampling in template-free protein loop structure modeling: an overview

Accurately modeling protein loops is an important step to predict three-dimensional structures as well as to understand functions of many proteins. Because of their high flexibility, modeling the three-dimensional structures of loops is difficult and is usually treated as a "mini protein folding problem" under geometric constraints. In the past decade, there has been remarkable progress in template-free loop structure modeling due to advances of computational methods as well as stably increasing number of known structures available in PDB. This mini review provides an overview on the recent computational approaches for loop structure modeling. In particular, we focus on the approaches of sampling loop conformation space, which is a critical step to obtain high resolution models in template-free methods. We review the potential energy functions for loop modeling, loop buildup mechanisms to satisfy geometric constraints, and loop conformation sampling algorithms. The recent loop modeling results are also summarized.

Exploring the energy landscapes of flexible molecular loops using higher-dimensional continuation

The conformational space of a flexible molecular loop includes the set of conformations fulfilling the geometric loop-closure constraints and its energy landscape can be seen as a scalar field defined on this implicit set. Higher-dimensional continuation tools, recently developed in dynamical systems and also applied to robotics, provide efficient algorithms to trace out implicitly defined sets. This article describes these tools and applies them to obtain full descriptions of the energy landscapes of short molecular loops that, otherwise, can only be partially explored, mainly via sampling. Moreover, to deal with larger loops, this article exploits the higher-dimensional continuation tools to find local minima and minimum energy transition paths between them, without deviating from the loop-closure constraints. The proposed techniques are applied to previously studied molecules revealing the intricate structure of their energy landscapes.

SMS 2.0: an updated database to study the structural plasticity of short peptide fragments in non-redundant proteins

The function of a protein molecule is greatly influenced by its three-dimensional (3D) structure and therefore structure prediction will help identify its biological function. We have updated Sequence, Motif and Structure (SMS), the database of structurally rigid peptide fragments, by combining amino acid sequences and the corresponding 3D atomic coordinates of non-redundant (25%) and redundant (90%) protein chains available in the Protein Data Bank (PDB). SMS 2.0 provides information pertaining to the peptide fragments of length 5-14 residues. The entire dataset is divided into three categories, namely, same sequence motifs having similar, intermediate or dissimilar 3D structures. Further, options are provided to facilitate structural superposition using the program structural alignment of multiple proteins (STAMP) and the popular JAVA plug-in (Jmol) is deployed for visualization. In addition, functionalities are provided to search for the occurrences of the sequence motifs in other structural and sequence databases like PDB, Genome Database (GDB), Protein Information Resource (PIR) and Swiss-Prot. The updated database along with the search engine is available over the World Wide Web through the following URL http://cluster.physics.iisc.ernet.in/sms/.

Near-native protein loop sampling using nonparametric density estimation accommodating sparcity

Unlike the core structural elements of a protein like regular secondary structure, template based modeling (TBM) has difficulty with loop regions due to their variability in sequence and structure as well as the sparse sampling from a limited number of homologous templates. We present a novel, knowledge-based method for loop sampling that leverages homologous torsion angle information to estimate a continuous joint backbone dihedral angle density at each loop position. The φ,ψ distributions are estimated via a Dirichlet process mixture of hidden Markov models (DPM-HMM). Models are quickly generated based on samples from these distributions and were enriched using an end-to-end distance filter. The performance of the DPM-HMM method was evaluated against a diverse test set in a leave-one-out approach. Candidates as low as 0.45 Å RMSD and with a worst case of 3.66 Å were produced. For the canonical loops like the immunoglobulin complementarity-determining regions (mean RMSD <2.0 Å), the DPM-HMM method performs as well or better than the best templates, demonstrating that our automated method recaptures these canonical loops without inclusion of any IgG specific terms or manual intervention. In cases with poor or few good templates (mean RMSD >7.0 Å), this sampling method produces a population of loop structures to around 3.66 Å for loops up to 17 residues. In a direct test of sampling to the Loopy algorithm, our method demonstrates the ability to sample nearer native structures for both the canonical CDRH1 and non-canonical CDRH3 loops. Lastly, in the realistic test conditions of the CASP9 experiment, successful application of DPM-HMM for 90 loops from 45 TBM targets shows the general applicability of our sampling method in loop modeling problem. These results demonstrate that our DPM-HMM produces an advantage by consistently sampling near native loop structure. The software used in this analysis is available for download at http://www.stat.tamu.edu/~dahl/software/cortorgles/.

A protein's structure consists of elements of regular secondary structure connected by less regular stretches of loop segments. The irregularity of the loop structure makes loop modeling quite challenging. More accurate sampling of these loop conformations has a direct impact on protein modeling, design, function classification, as well as protein interactions. A method has been developed that extends a more comprehensive knowledge-based approach to producing models of the loop regions of protein structure. Most physical models cannot adequately sample the large conformational space, while the more discrete knowledge based libraries are conformationally limited. To address both of these problems, we introduce a novel statistical method that produces a continuous yet weighted estimation of loop conformational space from a discrete library of structures by using a Dirichlet process mixture of hidden Markov models (DPM-HMM). Applied to loop structure sampling, the results of a number of tests demonstrate that our approach quickly generates large numbers of candidates with near native loop conformations. Most significantly, in the cases where the template sampling is sparse and/or far from native conformations, the DPM-HMM method samples close to the native space and produces a population of accurate loop structures.

Development of a new physics-based internal coordinate mechanics force field and its application to protein loop modeling

We report the development of internal coordinate mechanics force field (ICMFF), new force field parameterized using a combination of experimental data for crystals of small molecules and quantum mechanics calculations. The main features of ICMFF include: (a) parameterization for the dielectric constant relevant to the condensed state (ε = 2) instead of vacuum, (b) an improved description of hydrogen-bond interactions using duplicate sets of van der Waals parameters for heavy atom-hydrogen interactions, and (c) improved backbone covalent geometry and energetics achieved using novel backbone torsional potentials and inclusion of the bond angles at the C(α) atoms into the internal variable set. The performance of ICMFF was evaluated through loop modeling simulations for 4-13 residue loops. ICMFF was combined with a solvent-accessible surface area solvation model optimized using a large set of loop decoys. Conformational sampling was carried out using the biased probability Monte Carlo method. Average/median backbone root-mean-square deviations of the lowest energy conformations from the native structures were 0.25/0.21 Å for four residues loops, 0.84/0.46 Å for eight residue loops, and 1.16/0.73 Å for 12 residue loops. To our knowledge, these results are significantly better than or comparable with those reported to date for any loop modeling method that does not take crystal packing into account. Moreover, the accuracy of our method is on par with the best previously reported results obtained considering the crystal environment. We attribute this success to the high accuracy of the new ICM force field achieved by meticulous parameterization, to the optimized solvent model, and the efficiency of the search method.

Loop simulations

Loop modeling is crucial for high-quality homology model construction outside conserved secondary structure elements. Dozens of loop modeling protocols involving a range of database and ab initio search algorithms and a variety of scoring functions have been proposed. Knowledge-based loop modeling methods are very fast and some can successfully and reliably predict loops up to about eight residues long. Several recent ab initio loop simulation methods can be used to construct accurate models of loops up to 12-13 residues long, albeit at a substantial computational cost. Major current challenges are the simulations of loops longer than 12-13 residues, the modeling of multiple interacting flexible loops, and the sensitivity of the loop predictions to the accuracy of the loop environment.

MOLS sampling and its applications in structural biophysics

This review describes the MOLS method and its applications. This computational method has been developed in our laboratory primarily to explore the conformational space of small peptides and identify features of interest, particularly the minima, i.e., the low energy conformations. A systematic "brute-force" search through the vast conformational space for such features faces the insurmountable problem of combinatorial explosion, whilst other techniques, e.g., Monte Carlo searches, are somewhat limited in their region of exploration and may be considered inexhaustive. The MOLS method, on the other hand, uses a sampling technique commonly employed in experimental design theory to identify a small sample of the conformational space that nevertheless retains information about the entire space. The information is extracted using a technique that is a variant of the self-consistent mean field technique, which has been used to identify, for example, the optimal set of side-chain conformations in a protein. Applications of the MOLS method to understand peptide structure, predict the structures of loops in proteins, predict three-dimensional structures of small proteins, and arrive at the best conformation, orientation, and positions of a small molecule ligand in a protein receptor site have all yielded satisfactory results.

New computational method for prediction of interacting protein loop regions

Flexible loop regions of proteins play a crucial role in many biological functions such as protein-ligand recognition, enzymatic catalysis, and protein-protein association. To date, most computational methods that predict the conformational states of loops only focus on individual loop regions. However, loop regions are often spatially in close proximity to one another and their mutual interactions stabilize their conformations. We have developed a new method, titled CorLps, capable of simultaneously predicting such interacting loop regions. First, an ensemble of individual loop conformations is generated for each loop region. The members of the individual ensembles are combined and are accepted or rejected based on a steric clash filter. After a subsequent side-chain optimization step, the resulting conformations of the interacting loops are ranked by the statistical scoring function DFIRE that originated from protein structure prediction. Our results show that predicting interacting loops with CorLps is superior to sequential prediction of the two interacting loop regions, and our method is comparable in accuracy to single loop predictions. Furthermore, improved predictive accuracy of the top-ranked solution is achieved for 12-residue length loop regions by diversifying the initial pool of individual loop conformations using a quality threshold clustering algorithm.

FREAD revisited: Accurate loop structure prediction using a database search algorithm

Loops are the most variable regions of protein structure and are, in general, the least accurately predicted. Their prediction has been approached in two ways, ab initio and database search. In recent years, it has been thought that ab initio methods are more powerful. In light of the continued rapid expansion in the number of known protein structures, we have re-evaluated FREAD, a database search method and demonstrate that the power of database search methods may have been underestimated. We found that sequence similarity as quantified by environment specific substitution scores can be used to significantly improve prediction. In fact, FREAD performs appreciably better for an identifiable subset of loops (two thirds of shorter loops and half of the longer loops tested) than the ab initio methods of MODELLER, PLOP, and RAPPER. Within this subset, FREAD's predictive ability is length independent, in general, producing results within 2A RMSD, compared to an average of over 10A for loop length 20 for any of the other tested methods. We also benchmarked the prediction protocols on a set of 212 loops from the model structures in CASP 7 and 8. An extended version of FREAD is able to make predictions for 127 of these, it gives the best prediction of the methods tested in 61 of these cases. In examining FREAD's ability to predict in the model environment, we found that whole structure quality did not affect the quality of loop predictions.

Local descriptors of protein structure: a systematic analysis of the sequence-structure relationship in proteins using short- and long-range interactions

Local protein structure representations that incorporate long-range contacts between residues are often considered in protein structure comparison but have found relatively little use in structure prediction where assembly from single backbone fragments dominates. Here, we introduce the concept of local descriptors of protein structure to characterize local neighborhoods of amino acids including short- and long-range interactions. We build a library of recurring local descriptors and show that this library is general enough to allow assembly of unseen protein structures. The library could on average re-assemble 83% of 119 unseen structures, and showed little or no performance decrease between homologous targets and targets with folds not represented among domains used to build it. We then systematically evaluate the descriptor library to establish the level of the sequence signal in sets of protein fragments of similar geometrical conformation. In particular, we test whether that signal is strong enough to facilitate correct assignment and alignment of these local geometries to new sequences. We use the signal to assign descriptors to a test set of 479 sequences with less than 40% sequence identity to any domain used to build the library, and show that on average more than 50% of the backbone fragments constituting descriptors can be correctly aligned. We also use the assigned descriptors to infer SCOP folds, and show that correct predictions can be made in many of the 151 cases where PSI-BLAST was unable to detect significant sequence similarity to proteins in the library. Although the combinatorial problem of simultaneously aligning several fragments to sequence is a major bottleneck compared with single fragment methods, the advantage of the current approach is that correct alignments imply correct long range distance constraints. The lack of these constraints is most likely the major reason why structure prediction methods fail to consistently produce adequate models when good templates are unavailable or undetectable. Thus, we believe that the current study offers new and valuable insight into the prediction of sequence-structure relationships in proteins.

Prediction of protein loop structures using a local move Monte Carlo approach and a grid-based force field

We have developed an improved local move Monte Carlo (LMMC) loop sampling approach for loop predictions. The method generates loop conformations based on simple moves of the torsion angles of side chains and local moves of backbone of loops. To reduce the computational costs for energy evaluations, we developed a grid-based force field to represent the protein environment and solvation effect. Simulated annealing has been used to enhance the efficiency of the LMMC loop sampling and identify low-energy loop conformations. The prediction quality is evaluated on a set of protein loops with known crystal structure that has been previously used by others to test different loop prediction methods. The results show that this approach can reproduce the experimental results with the root mean square deviation within 1.8 A for all the test cases. The LMMC loop prediction approach developed here could be useful for improvement in the quality the loop regions in homology models, flexible protein-ligand and protein-protein docking studies.

Efficient algorithms to explore conformation spaces of flexible protein loops

Several applications in biology - e.g., incorporation of protein flexibility in ligand docking algorithms, interpretation of fuzzy X-ray crystallographic data, and homology modeling - require computing the internal parameters of a flexible fragment (usually, a loop) of a protein in order to connect its termini to the rest of the protein without causing any steric clash. One must often sample many such conformations in order to explore and adequately represent the conformational range of the studied loop. While sampling must be fast, it is made difficult by the fact that two conflicting constraints - kinematic closure and clash avoidance - must be satisfied concurrently. This paper describes two efficient and complementary sampling algorithms to explore the space of closed clash-free conformations of a flexible protein loop. The "seed sampling" algorithm samples broadly from this space, while the "deformation sampling" algorithm uses seed conformations as starting points to explore the conformation space around them at a finer grain. Computational results are presented for various loops ranging from 5 to 25 residues. More specific results also show that the combination of the sampling algorithms with a functional site prediction software (FEATURE) makes it possible to compute and recognize calcium-binding loop conformations. The sampling algorithms are implemented in a toolkit (LoopTK), which is available at https://simtk.org/home/looptk.

Prediction of protein loop conformations using multiscale modeling methods with physical energy scoring functions

This article examines ab initio methods for the prediction of protein loops by a computational strategy of multiscale conformational sampling and physical energy scoring functions. Our approach consists of initial sampling of loop conformations from lattice-based low-resolution models followed by refinement using all-atom simulations. To allow enhanced conformational sampling, the replica exchange method was implemented. Physical energy functions based on CHARMM19 and CHARMM22 parameterizations with generalized Born (GB) solvent models were applied in scoring loop conformations extracted from the lattice simulations and, in the case of all-atom simulations, the ensemble of conformations were generated and scored with these models. Predictions are reported for 25 loop segments, each eight residues long and taken from a diverse set of 22 protein structures. We find that the simulations generally sampled conformations with low global root-mean-square-deviation (RMSD) for loop backbone coordinates from the known structures, whereas clustering conformations in RMSD space and scoring detected less favorable loop structures. Specifically, the lattice simulations sampled basins that exhibited an average global RMSD of 2.21 +/- 1.42 A, whereas clustering and scoring the loop conformations determined an RMSD of 3.72 +/- 1.91 A. Using CHARMM19/GB to refine the lattice conformations improved the sampling RMSD to 1.57 +/- 0.98 A and detection to 2.58 +/- 1.48 A. We found that further improvement could be gained from extending the upper temperature in the all-atom refinement from 400 to 800 K, where the results typically yield a reduction of approximately 1 A or greater in the RMSD of the detected loop. Overall, CHARMM19 with a simple pairwise GB solvent model is more efficient at sampling low-RMSD loop basins than CHARMM22 with a higher-resolution modified analytical GB model; however, the latter simulation method provides a more accurate description of the all-atom energy surface, yet demands a much greater computational cost.

Exploring conformational space using a mean field technique with MOLS sampling

The computational identification of all the low energy structures of a peptide given only its sequence is not an easy task even for small peptides,due to the multiple-minima problem and combinatorial explosion. We have developed an algorithm, called the MOLS technique,that addresses this problem, and have applied it to a number of different aspects of the study of peptide and protein structure. Conformational studies of oligopeptides, including loop sequences in proteins have been carried out using this technique. In general the calculations identified all the folds determined by previous studies,and in addition picked up other energetically favorable structures. The method was also used to map the energy surface of the peptides. In another application, we have combined the MOLS technique, using it to generate a library of low energy structures of an oligopeptide, with a genetic algorithm to predict protein structures. The method has also been applied to explore the conformational space of loops in protein structures.Further, it has been applied to the problem of docking a ligand in its receptor site, with encouraging results.

Genetic algorithms for protein conformation sampling and optimization in a discrete backbone dihedral angle space

We have investigated protein conformation sampling and optimization based on the genetic algorithm and discrete main chain dihedral state model. An efficient approach combining the genetic algorithm with local minimization and with a niche technique based on the sharing function is proposed. Using two different types of potential energy functions, a Go-type potential function and a knowledge-based pairwise potential energy function, and a test set containing small proteins of varying sizes and secondary structure compositions, we demonstrated the importance of local minimization and population diversity in protein conformation optimization with genetic algorithms. Some general properties of the sampled conformations such as their native-likeness and the influences of including side-chains are discussed.

Exploring the conformational space of protein loops using a mean field technique with MOLS sampling

We have recently developed a computational technique that uses mutually orthogonal Latin square sampling to explore the conformational space of oligopeptides in an exhaustive manner. In this article, we report its use to analyze the conformational spaces of 120 protein loop sequences in proteins, culled from the PDB, having the length ranging from 5 to 10 residues. The force field used did not have any information regarding the sequences or structures that flanked the loop. The results of the analyses show that the native structure of the loop, as found in the PDB falls at one of the low energy points in the conformational landscape of the sequences. Thus, a large portion of the structural determinants of the loop may be considered intrinsic to the sequence, regardless of either adjacent sequences or structures, or the interactions that the atoms of the loop make with other residues in the protein or in neighboring proteins.

Modeling protein conformational ensembles: From missing loops to equilibrium fluctuations

Characterizing protein flexibility is an important goal for understanding the physical-chemical principles governing biological function. This paper presents a Fragment Ensemble Method to capture the mobility of a protein fragment such as a missing loop and its extension into a Protein Ensemble Method to characterize the mobility of an entire protein at equilibrium. The underlying approach in both methods is to combine a geometric exploration of conformational space with a statistical mechanics formulation to generate an ensemble of physical conformations on which thermodynamic quantities can be measured as ensemble averages. The Fragment Ensemble Method is validated by applying it to characterize loop mobility in both instances of strongly stable and disordered loop fragments. In each instance, fluctuations measured over generated ensembles are consistent with data from experiment and simulation. The Protein Ensemble Method captures the mobility of an entire protein by generating and combining ensembles of conformations for consecutive overlapping fragments defined over the protein sequence. This method is validated by applying it to characterize flexibility in ubiquitin and protein G. Thermodynamic quantities measured over the ensembles generated for both proteins are fully consistent with available experimental data. On these proteins, the method recovers nontrivial data such as order parameters, residual dipolar couplings, and scalar couplings. Results presented in this work suggest that the proposed methods can provide insight into the interplay between protein flexibility and function.

A multibody, whole-residue potential for protein structures, with testing by Monte Carlo simulated annealing

A new multibody, whole-residue potential for protein tertiary structure is described. The potential is based on the local environment surrounding each main-chain alpha carbon (CA), defined as the set of all residues whose CA coordinates lie within a spherical volume of set radius in 3-dimensional (3D) space surrounding that position. It is shown that the relative positions of the CAs in these local environments belong to a set of preferred templates. The templates are derived by cluster analysis of the presently available database of over 3000 protein chains (750,000 residues) having not more than 30% sequence similarity. For each template is derived also a set of residue propensities for each topological position in the template. Using lookup tables of these derived templates, it is then possible to calculate an energy for any conformation of a given protein sequence. The application of the potential to ab initio protein tertiary structure prediction is evaluated by performing Monte Carlo simulated annealing on test protein sequences.

Optimization of the GB/SA solvation model for predicting the structure of surface loops in proteins

Implicit solvation models are commonly optimized with respect to experimental data or Poisson-Boltzmann (PB) results obtained for small molecules, where the force field is sometimes not considered. In previous studies, we have developed an optimization procedure for cyclic peptides and surface loops in proteins based on the entire system studied and the specific force field used. Thus, the loop has been modeled by the simplified solvation function E(tot) = E(FF) (epsilon = 2r) + Sigma(i) sigma(i)A(i), where E(FF) (epsilon = nr) is the AMBER force field energy with a distance-dependent dielectric function, epsilon = nr, A(i) is the solvent accessible surface area of atom i, and sigma(i) is its atomic solvation parameter. During the optimization process, the loop is free to move while the protein template is held fixed in its X-ray structure. To improve on the results of this model, in the present work we apply our optimization procedure to the physically more rigorous solvation model, the generalized Born with surface area (GB/SA) (together with the all-atom AMBER force field) as suggested by Still and co-workers (J. Phys. Chem. A 1997, 101, 3005). The six parameters of the GB/SA model, namely, P(1)-P(5) and the surface area parameter, sigma (programmed in the TINKER package) are reoptimized for a "training" group of nine loops, and a best-fit set is defined from the individual sets of optimized parameters. The best-fit set and Still's original set of parameters (where Lys, Arg, His, Glu, and Asp are charged or neutralized) were applied to the training group as well as to a "test" group of seven loops, and the energy gaps and the corresponding RMSD values were calculated. These GB/SA results based on the three sets of parameters have been found to be comparable; surprisingly, however, they are somewhat inferior (e.g, of larger energy gaps) to those obtained previously from the simplified model described above. We discuss recent results for loops obtained by other solvation models and potential directions for future studies.

Recapitulation and design of protein binding peptide structures and sequences

An important objective of computational protein design is the generation of high affinity peptide inhibitors of protein-peptide interactions, both as a precursor to the development of therapeutics aimed at disrupting disease causing complexes, and as a tool to aid investigators in understanding the role of specific complexes in the cell. We have developed a computational approach to increase the affinity of a protein-peptide complex by designing N or C-terminal extensions which interact with the protein outside the canonical peptide binding pocket. In a first in silico test, we show that by simultaneously optimizing the sequence and structure of three to nine residue peptide extensions starting from short (1-6 residue) peptide stubs in the binding pocket of a peptide binding protein, the approach can recover both the conformations and the sequences of known binding peptides. Comparison with phage display and other experimental data suggests that the peptide extension approach recapitulates naturally occurring peptide binding specificity better than fixed backbone design, and that it should be useful for predicting peptide binding specificities from crystal structures. We then experimentally test the approach by designing extensions for p53 and dystroglycan-based peptides predicted to bind with increased affinity to the Mdm2 oncoprotein and to dystrophin, respectively. The measured increases in affinity are modest, revealing some limitations of the method. Based on these in silico and experimental results, we discuss future applications of the approach to the prediction and design of protein-peptide interactions.

A hierarchical approach to all-atom protein loop prediction

The application of all-atom force fields (and explicit or implicit solvent models) to protein homology-modeling tasks such as side-chain and loop prediction remains challenging both because of the expense of the individual energy calculations and because of the difficulty of sampling the rugged all-atom energy surface. Here we address this challenge for the problem of loop prediction through the development of numerous new algorithms, with an emphasis on multiscale and hierarchical techniques. As a first step in evaluating the performance of our loop prediction algorithm, we have applied it to the problem of reconstructing loops in native structures; we also explicitly include crystal packing to provide a fair comparison with crystal structures. In brief, large numbers of loops are generated by using a dihedral angle-based buildup procedure followed by iterative cycles of clustering, side-chain optimization, and complete energy minimization of selected loop structures. We evaluate this method by using the largest test set yet used for validation of a loop prediction method, with a total of 833 loops ranging from 4 to 12 residues in length. Average/median backbone root-mean-square deviations (RMSDs) to the native structures (superimposing the body of the protein, not the loop itself) are 0.42/0.24 A for 5 residue loops, 1.00/0.44 A for 8 residue loops, and 2.47/1.83 A for 11 residue loops. Median RMSDs are substantially lower than the averages because of a small number of outliers; the causes of these failures are examined in some detail, and many can be attributed to errors in assignment of protonation states of titratable residues, omission of ligands from the simulation, and, in a few cases, probable errors in the experimentally determined structures. When these obvious problems in the data sets are filtered out, average RMSDs to the native structures improve to 0.43 A for 5 residue loops, 0.84 A for 8 residue loops, and 1.63 A for 11 residue loops. In the vast majority of cases, the method locates energy minima that are lower than or equal to that of the minimized native loop, thus indicating that sampling rarely limits prediction accuracy. The overall results are, to our knowledge, the best reported to date, and we attribute this success to the combination of an accurate all-atom energy function, efficient methods for loop buildup and side-chain optimization, and, especially for the longer loops, the hierarchical refinement protocol.

Modeling structurally variable regions in homologous proteins with rosetta

A major limitation of current comparative modeling methods is the accuracy with which regions that are structurally divergent from homologues of known structure can be modeled. Because structural differences between homologous proteins are responsible for variations in protein function and specificity, the ability to model these differences has important functional consequences. Although existing methods can provide reasonably accurate models of short loop regions, modeling longer structurally divergent regions is an unsolved problem. Here we describe a method based on the de novo structure prediction algorithm, Rosetta, for predicting conformations of structurally divergent regions in comparative models. Initial conformations for short segments are selected from the protein structure database, whereas longer segments are built up by using three- and nine-residue fragments drawn from the database and combined by using the Rosetta algorithm. A gap closure term in the potential in combination with modified Newton's method for gradient descent minimization is used to ensure continuity of the peptide backbone. Conformations of variable regions are refined in the context of a fixed template structure using Monte Carlo minimization together with rapid repacking of side-chains to iteratively optimize backbone torsion angles and side-chain rotamers. For short loops, mean accuracies of 0.69, 1.45, and 3.62 A are obtained for 4, 8, and 12 residue loops, respectively. In addition, the method can provide reasonable models of conformations of longer protein segments: predicted conformations of 3A root-mean-square deviation or better were obtained for 5 of 10 examples of segments ranging from 13 to 34 residues. In combination with a sequence alignment algorithm, this method generates complete, ungapped models of protein structures, including regions both similar to and divergent from a homologous structure. This combined method was used to make predictions for 28 protein domains in the Critical Assessment of Protein Structure 4 (CASP 4) and 59 domains in CASP 5, where the method ranked highly among comparative modeling and fold recognition methods. Model accuracy in these blind predictions is dominated by alignment quality, but in the context of accurate alignments, long protein segments can be accurately modeled. Notably, the method correctly predicted the local structure of a 39-residue insertion into a TIM barrel in CASP 5 target T0186.

High-resolution prediction of protein helix positions and orientations

We have developed a new method for predicting helix positions in globular proteins that is intended primarily for comparative modeling and other applications where high precision is required. Unlike helix packing algorithms designed for ab initio folding, we assume that knowledge is available about the qualitative placement of all helices. However, even among homologous proteins, the corresponding helices can demonstrate substantial differences in positions and orientations, and for this reason, improperly positioned helices can contribute significantly to the overall backbone root-mean-square deviation (RMSD) of comparative models. A helix packing algorithm for use in comparative modeling must obtain high precision to be useful, and for this reason we utilize an all-atom protein force field (OPLS) and a Generalized Born continuum solvent model. To reduce the computational expense associated with using a detailed, physics-based energy function, we have developed new hierarchical and multiscale algorithms for sampling the helices and flanking loops. We validate the method using a test suite of 33 cases, which are drawn from a diverse set of high-resolution crystal structures. The helix positions are reproduced with an average backbone RMSD of 0.6 A, while the average backbone RMSD of the complete loop-helix-loop region (i.e., the helix with the surrounding loops, which are also repredicted) is 1.3 A.

Efficient methods for filtering and ranking fragments for the prediction of structurally variable regions in proteins

The prediction of protein 3D structures close to insertions and deletions or, more generally, loop prediction, is still one of the major challenges in homology modeling projects. In this article, we developed ranking criteria and selection filters to improve knowledge-based loop predictions. These criteria were developed and optimized for a test data set containing 678 insertions and deletions. The examples are, in principle, predictable from the used loop database with an RMSD < 1 A and represent realistic modeling situations. Four noncorrelated criteria for the selection of fragments are evaluated. A fast prefilter compares the distance between the anchor groups in the template protein with the stems of the fragments. The RMSD of the anchor groups is used for fitting and ranking of the selected loop candidates. After fitting, repulsive close contacts of loop candidates with the template protein are used for filtering, and fragments with backbone torsion angles, which are unfavorable according to a knowledge-based potential, are eliminated. By the combined application of these filter criteria to the test set, it was possible to increase the percentage of predictions with a global RMSD < 1 A to over 50% among the first five ranks, with average global RMSD values for the first rank candidate that are between 1.3 and 2.2 A for different loop lengths. Compared to other examples described in the literature, our large numbers of test cases are not self-predictions, where loops are placed in a protein after a peptide loop has been cut out, but are attempts to predict structural changes that occur in evolution when a protein is affected by insertions and deletions.

Sampling protein conformations and pathways

Protein flexibility and rigidity can be analyzed using constraint theory, which views proteins as 3D networks of constraints involving covalent bonds and also including hydrophobic interactions and hydrogen bonds. This article describes an algorithm, ROCK (Rigidity Optimized Conformational Kinetics), which generates new conformations for these complex networks with many interlocked rings while maintaining the constraints. These new conformations are tracked for the flexible regions of a protein, while leaving the rigid regions undisturbed. An application to HIV protease demonstrates how large the flap motion can be. The algorithm is also used to generate conformational pathways between two distinct protein conformations. As an example, directed trajectories between the closed and the occluded conformations of the protein dihydrofolate reductase are determined.

Ab initio construction of polypeptide fragments: Accuracy of loop decoy discrimination by an all-atom statistical potential and the AMBER force field with the Generalized Born solvation model

The accuracy of model selection from decoy ensembles of protein loop conformations was explored by comparing the performance of the Samudrala-Moult all-atom statistical potential (RAPDF) and the AMBER molecular mechanics force field, including the Generalized Born/surface area solvation model. Large ensembles of consistent loop conformations, represented at atomic detail with idealized geometry, were generated for a large test set of protein loops of 2 to 12 residues long by a novel ab initio method called RAPPER that relies on fine-grained residue-specific phi/psi propensity tables for conformational sampling. Ranking the conformers on the basis of RAPDF scores resulted in selected conformers that had an average global, non-superimposed RMSD for all heavy mainchain atoms ranging from 1.2 A for 4-mers to 2.9 A for 8-mers to 6.2 A for 12-mers. After filtering on the basis of anchor geometry and RAPDF scores, ranking by energy minimization of the AMBER/GBSA potential energy function selected conformers that had global RMSD values of 0.5 A for 4-mers, 2.3 A for 8-mers, and 5.0 A for 12-mers. Minimized fragments had, on average, consistently lower RMSD values (by 0.1 A) than their initial conformations. The importance of the Generalized Born solvation energy term is reflected by the observation that the average RMSD accuracy for all loop lengths was worse when this term is omitted. There are, however, still many cases where the AMBER gas-phase minimization selected conformers of lower RMSD than the AMBER/GBSA minimization. The AMBER/GBSA energy function had better correlation with RMSD to native than the RAPDF. When the ensembles were supplemented with conformations extracted from experimental structures, a dramatic improvement in selection accuracy was observed at longer lengths (average RMSD of 1.3 A for 8-mers) when scoring with the AMBER/GBSA force field. This work provides the basis for a promising hybrid approach of ab initio and knowledge-based methods for loop modeling.

Ab initio construction of polypeptide fragments: efficient generation of accurate, representative ensembles

We describe a novel method to generate ensembles of conformations of the main-chain atoms [N, C(alpha), C, O, Cbeta] for a sequence of amino acids within the context of a fixed protein framework. Each conformation satisfies fundamental stereo-chemical restraints such as idealized geometry, favorable phi/psi angles, and excluded volume. The ensembles include conformations both near and far from the native structure. Algorithms for effective conformational sampling and constant time overlap detection permit the generation of thousands of distinct conformations in minutes. Unlike previous approaches, our method samples dihedral angles from fine-grained phi/psi state sets, which we demonstrate is superior to exhaustive enumeration from coarse phi/psi sets. Applied to a large set of loop structures, our method samples consistently near-native conformations, averaging 0.4, 1.1, and 2.2 A main-chain root-mean-square deviations for four, eight, and twelve residue long loops, respectively. The ensembles make ideal decoy sets to assess the discriminatory power of a selection method. Using these decoy sets, we conclude that quality of anchor geometry cannot reliably identify near-native conformations, though the selection results are comparable to previous loop prediction methods. In a subsequent study (de Bakker et al.: Proteins 2003;51:21-40), we demonstrate that the AMBER forcefield with the Generalized Born solvation model identifies near-native conformations significantly better than previous methods.

Positioning of anchor groups in protein loop prediction: the importance of solvent accessibility and secondary structure elements

The prediction of loop regions in the process of protein structure prediction by homology is still an unsolved problem. In an earlier publication, we could show that the correct placement of the amino acids serving as an anchor group to be connected by a loop fragment with a predicted geometry is a highly important step and an essential requirement within the process (Lessel and Schomburg, Proteins 1999; 37:56-64). In this article, we present an analysis of the quality of possible loop predictions with respect to gap length, fragment length, amino acid type, secondary structure, and solvent accessibility. For 550 insertions and 544 deletions, we test all possible positions for anchor groups with an inserted loop of a length between 3 and 12 amino acids. We could show that approximately 80% of the indel regions could be predicted within 1.5 A RMSD from a knowledge-based loop data base if criteria for the correct localization of anchor groups could be found and the loops can be sorted correctly. From our analysis, several conclusions regarding the optimal placement of anchor groups become obvious: (1) The correct placement of anchor groups is even more important for longer gap lengths, (2) medium length fragments (length 5-8) perform better than short or long ones, (3) the placement of anchor groups at hydrophobic amino acids gives a higher chance to include the best possible loop, (4) anchor groups within secondary structure elements, in particular beta-sheets are suitable, (5) amino acids with lower solvent accessibility are better anchor group. A preliminary test using a combination of the anchor group positioning criteria deduced from our analysis shows very promising results.

A divide and conquer approach to fast loop modeling

We describe a fast ab initio method for modeling local segments in protein structures. The algorithm is based on a divide and conquer approach and uses a database of precalculated look-up tables, which represent a large set of possible conformations for loop segments of variable length. The target loop is recursively decomposed until the resulting conformations are small enough to be compiled analytically. The algorithm, which is not restricted to any specific loop length, generates a ranked set of loop conformations in 20-180 s on a desktop PC. The prediction quality is evaluated in terms of global RMSD. Depending on loop length the top prediction varies between 1.06 A RMSD for three-residue loops and 3.72 A RMSD for eight-residue loops. Due to its speed the method may also be useful to generate alternative starting conformations for complex simulations.

Structures of the two 3D domain-swapped RNase A trimers

When concentrated in mildly acidic solutions, bovine pancreatic ribonuclease (RNase A) forms long-lived oligomers including two types of dimer, two types of trimer, and higher oligomers. In previous crystallographic work, we found that the major dimeric component forms by a swapping of the C-terminal beta-strands between the monomers, and that the minor dimeric component forms by swapping the N-terminal alpha-helices of the monomers. On the basis of these structures, we proposed that a linear RNase A trimer can form from a central molecule that simultaneously swaps its N-terminal helix with a second RNase A molecule and its C-terminal strand with a third molecule. Studies by dissociation are consistent with this model for the major trimeric component: the major trimer dissociates into both the major and the minor dimers, as well as monomers. In contrast, the minor trimer component dissociates into the monomer and the major dimer. This suggests that the minor trimer is cyclic, formed from three monomers that swap their C-terminal beta-strands into identical molecules. These conclusions are supported by cross-linking of lysyl residues, showing that the major trimer swaps its N-terminal helix, and the minor trimer does not. We verified by X-ray crystallography the proposed cyclic structure for the minor trimer, with swapping of the C-terminal beta-strands. This study thus expands the variety of domain-swapped oligomers by revealing the first example of a protein that can form both a linear and a cyclic domain-swapped oligomer. These structures permit interpretation of the enzymatic activities of the RNase A oligomers on double-stranded RNA.

Sequence analyses and comparative modeling of fly and worm fibroblast growth factor receptors indicate that the determinants for FGF and heparin binding are retained in evolution

The presence of a large number of fibroblast growth factors (FGFs) and multiple splice forms of their receptors (FGFRs) in higher vertebrates makes the three-dimensional (3D) analysis of FGF interactions with their receptors a formidable task. The situation differs in Caenorhabditis elegans (worm) and Drosophila melanogaster (fruit fly), where only one or two FGF and FGFR sequences have been identified. Structural studies of the FGF-FGFR complexes in such primitive organisms should reveal the basic features of the ligand-receptor interactions as they first emerged through evolution. We have analysed the sequences of worm and fly FGFs and FGFRs and used the recently determined crystal structure of the human FGF1-FGFR2-heparin ternary complex [Pellegrini, L., Burke, D.F., von Delft, F., Mulloy, B. and Blundell, T.L. (2000) Nature 407, 1029-34] to construct 3D models of the homologous complexes. In spite of a low sequence similarity with their human counterparts, key structural features required for ligand-receptor and protein-heparin binding in humans are conserved in the fly and worm FGF-FGFR-heparin complexes. Analyses of the models show that tertiary interactions that are not conserved in sequence are maintained through novel interactions or complementary mutations in the fly and worm sequences. The overall charge distributions observed in the human FGF-FGFR-heparin complex are retained in the fly and worm models. The arginine residue at position 253 in the linker region between the Ig-like domains D2 and D3 in the wild type fly and worm sequences is particularly striking, as the Pro253Arg mutation in humans is responsible for Apert syndrome. This change may enhance the affinity of receptors for their FGF molecules as observed in Apert mutants.

Improved protein loop prediction from sequence alone

The SLoop database of supersecondary fragments, first described by Donate et al. (Protein Sci., 1996, 5, 2600-2616), contains protein loops, classified according to structural similarity. The database has recently been updated and currently contains over 10 000 loops up to 20 residues in length, which cluster into over 560 well populated classes. The database can be found at http://www-cryst.bioc.cam.ac.uk/~sloop. In this paper, we identify conserved structural features such as main chain conformation and hydrogen bonding. Using the original approach of Rufino and co-workers (1997), the correct structural class is predicted with the highest SLoop score for 35% of loops. This rises to 65% by considering the three highest scoring class predictions and to 75% in the top five scoring class predictions. Inclusion of residues from the neighbouring secondary structures and use of substitution tables derived using a reduced definition of secondary structure increase these prediction accuracies to 58, 78 and 85%, respectively. This suggests that capping residues can stabilize the loop conformation as well as that of the secondary structure. Further increases are achieved if only well-populated classes are considered in the prediction. These results correspond to an average loop root mean square deviation of between 0.4 and 2.6 A for loops up to five residues in length.

Modeling of loops in protein structures

Comparative protein structure prediction is limited mostly by the errors in alignment and loop modeling. We describe here a new automated modeling technique that significantly improves the accuracy of loop predictions in protein structures. The positions of all nonhydrogen atoms of the loop are optimized in a fixed environment with respect to a pseudo energy function. The energy is a sum of many spatial restraints that include the bond length, bond angle, and improper dihedral angle terms from the CHARMM-22 force field, statistical preferences for the main-chain and side-chain dihedral angles, and statistical preferences for nonbonded atomic contacts that depend on the two atom types, their distance through space, and separation in sequence. The energy function is optimized with the method of conjugate gradients combined with molecular dynamics and simulated annealing. Typically, the predicted loop conformation corresponds to the lowest energy conformation among 500 independent optimizations. Predictions were made for 40 loops of known structure at each length from 1 to 14 residues. The accuracy of loop predictions is evaluated as a function of thoroughness of conformational sampling, loop length, and structural properties of native loops. When accuracy is measured by local superposition of the model on the native loop, 100, 90, and 30% of 4-, 8-, and 12-residue loop predictions, respectively, had <2 A RMSD error for the mainchain N, C(alpha), C, and O atoms; the average accuracies were 0.59 +/- 0.05, 1.16 +/- 0.10, and 2.61 +/- 0.16 A, respectively. To simulate real comparative modeling problems, the method was also evaluated by predicting loops of known structure in only approximately correct environments with errors typical of comparative modeling without misalignment. When the RMSD distortion of the main-chain stem atoms is 2.5 A, the average loop prediction error increased by 180, 25, and 3% for 4-, 8-, and 12-residue loops, respectively. The accuracy of the lowest energy prediction for a given loop can be estimated from the structural variability among a number of low energy predictions. The relative value of the present method is gauged by (1) comparing it with one of the most successful previously described methods, and (2) describing its accuracy in recent blind predictions of protein structure. Finally, it is shown that the average accuracy of prediction is limited primarily by the accuracy of the energy function rather than by the extent of conformational sampling.