Toward correct protein folding potentials.

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

Constructing smooth potential functions for protein folding.

A protein folding potential function ideally has several properties: it favors the native conformations for a number of protein sequences over a variety of nonnative folds; it can guide the search over conformations for the native state; it reflects changes in stability of the native fold due to changes in sequence; and it is relatively insensitive to small changes in conformation. While these are not mutually incompatible goals, attaining one property does not ensure that the others are satisfied. Examples are given of simple potentials having one property but lacking others. A new functional form of a folding potential is described where interactions between all nonhydrogen atoms are used to estimate interresidue interactions and implicit solvation. Its parameters can be adjusted to satisfy the above properties at least for barnase and a few other proteins.

Evaluation of ligand overlap by atomic parameters.

The overlap of ligand atoms has been analyzed for 32 common enzyme systems. The ligand alignment was determined by superposition of the experimentally determined protein structures. Comparison of the overlapping atoms in terms of atomic contribution to partition coefficient and molar refractivity shows that in most cases ligand atoms overlap with atoms of a similar nature, as should be expected. A new statistic, the mean maximal atomic deviation (MeMAD), is determined and validated for each system studied by comparison to randomized data. In almost all cases, the MeMAD is well separated from the randomized outcome. This result indicates the validity of the atom-based physicochemical parameters in 3-D QSAR methods.

A Gaussian statistical mechanical model for the equilibrium thermodynamics of barnase folding.

Given an all non-hydrogen-atom potential function that implicitly includes solvation effects, it is possible to adjust its parameters to favor the correct native structure for several proteins over decoys produced by ungapped threading. It is also possible to further train it to reproduce the experimental free energy of unfolding in aqueous solution at 298 K for wild-type barnase and 66 mutants. For this, the native state is represented by the crystal structure at a single energy level with a calculated low degeneracy; the denatured state is represented by the extended conformation and a high calculated degeneracy. The same two-state model can be extended to account for the stability of all 67 sequences toward urea denaturation at 298 K by building in a solvation term that depends on urea concentration. With the addition of one more parameter set to give the correct heat capacity of unfolded barnase in solution, it is possible to approximate the experimental thermodynamics of barnase thermal denaturation: melting temperature, width of thermal transition, deltaG, deltaH, deltaS, and deltaCp. This requires a novel sort of statistical mechanical model where the two states each have a Gaussian density of microscopic state distribution as a function of energy.

Disulfide recognition in an optimized threading potential.

An energy potential is constructed and trained to succeed in fold recognition for the general population of proteins as well as an important class which has previously been problematic: small, disulfide-bearing proteins. The potential is modeled on solvation, with the energy a function of side chain burial and the number of disulfide bonds. An accurate disulfide recognition algorithm identifies cysteine pairs which have the appropriate orientation to form a disulfide bridge. The potential has 22 energy parameters which are optimized so the Protein Data Bank (PDB) structure for each sequence in a training set is the lowest in energy out of thousands of alternative structures. One parameter per amino acid type reflects burial preference and a single parameter is used in an overpacking term. Additionally, one optimized parameter provides a favorable contribution for each disulfide identified in a given protein structure. With little training, the potential is >80% accurate in ungapped threading tests using a variety of proteins. The same level of accuracy is observed in a threading test of small proteins which have disulfide bonds. Importantly, the energy potential is also successful with proteins having uncrosslinked cysteines.

Potential energy function for continuous state models of globular proteins.

One of the approaches to protein structure prediction is to obtain energy functions which can recognize the native conformation of a given sequence among a zoo of conformations. The discriminations can be done by assigning the lowest energy to the native conformation, with the guarantee that the native is in the zoo. Well-adjusted functions, then, can be used in the search for other (near-) natives. Here the aim is the discrimination at relatively high resolution (RMSD difference between the native and the closest nonnative is around 1 A) by pairwise energy potentials. The potential is trained using the experimentally determined native conformation of only one protein, instead of the usual large survey over many proteins. The novel feature is that the native structure is compared to a vastly wider and more challenging array of nonnative structures found not only by the usual threading procedure, but by wide-ranging local minimization of the potential. Because of this extremely demanding search, the native is very close to the apparent global minimum of the potential function. The global minimum property holds up for one other protein having 60% sequence identity, but its performance on completely dissimilar proteins is of course much weaker.

The measurement of molecular diversity by receptor site interaction simulation.

The assembly of large compound libraries for the purpose of screening against various receptor targets to identify chemical leads for drug discovery programs has created a need for methods to measure the molecular diversity of such libraries. The method described here, for which we propose the acronym RESIS (for Receptor Site Interaction Simulation), relates directly to this use. A database is built of three-dimensional representations of the compounds in the library and a set of three-point three-dimensional theoretical receptor sites is generated based on putative hydrophobic and polar interactions. A series of flexible, three-dimensional searches is then performed over the database, using each of the theoretical sites as the basis for one such search. The resulting pattern of hits across the grid of theoretical receptor sites provides a measure of the molecular diversity of the compound library. This can be conveniently displayed as a density map which provides a readily comprehensible visual impression of the library diversity characteristics. A library of 7500 drug compounds derived from the CIPSLINEPC databases was characterized with respect to molecular diversity using the RESIS method. Some specific uses for the information obtained from application of the method are discussed. A comparison was made of the results from the RESIS method with those from a recently published two-dimensional approach for assessing molecular diversity using sets of compounds from the Maybridge database (MAY).

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.

Validation of EGSITE2, a mixed integer program for deducing objective site models for experimental binding data.

EGSITE2 represents a substantial advance in a long series of methods for calculating receptor site models given only specific binding data. Compared to our most recently reported technique, EGSITE [Schnitker et al. J. Comput.-Aided Mol. Des. 1997, 11, 93-110] the user no longer has to simplify the structures of the molecules in the training set by clustering the atoms into a few superatoms. The only remaining source of subjectivity is the user's choice of compounds for the training set, which can be surprisingly few in number. Then EGSITE2 automatically produces typically several different models that explain the observed binding without outliers. The models are remarkably simple but have substantial predictive power for any sort of test compound, with an estimation of the uncertainty of the prediction. Validation of the method is reported for four standard test cases: triazines and pyrimidines binding to dihydrofolate reductase, steroids binding to corticosteroid-binding globulin and to testosterone-binding globulin, and peptides binding to angiotensin-converting enzyme.

Objective models for steroid binding sites of human globulins.

We report the application of a recently developed alignment-free 3D QSAR method [Crippen, G.M., J. Comput. Chem., 16 (1995) 486] to a benchmark-type problem. The test system involves the binding of 31 steroid compounds to two kinds of human carrier protein. The method used not only allows for arbitrary binding modes, but also avoids the problems of traditional least-squares techniques with regard to the implicit neglect of informative outlying data points. It is seen that models of considerable predictive power can be obtained even with a very vague binding site description. Underlining a systematic, but usually ignored, problem of the QSAR approach, there is not one unique type of model but, rather, an entire manifold of distinctly different models that are all compatible with the experimental information. For a given model, there is also a considerable variation in the found binding modes, illustrating the problems that are inherent in the need for 'correct' molecular alignment in conventional 3D QSAR methods.

Protein folding and fold recognition for square lattice models.

Protein folding and inverse protein folding problems are examined for the extremely simplified model of short self-avoiding square lattice walks involving only two or three residue types. Simple interresidue contact free energy functions are given and are used to determine which sequences fold uniquely to which conformations. Contrary to general theories of protein folding, this model system shows little correlation between free energy and conformational distance from the native, nor is there any marked energy gap between the native and the best non-native structures. Furthermore, even the given free energy function sometimes fails to identify which sequences fold to a particular target structure. If current ideas about protein folding and structure/sequence compatibility fail in this model system, it is unclear why they should be valid for real proteins.

Failures of inverse folding and threading with gapped alignment.

To calculate the tertiary structure of a protein from its amino acid sequence, the thermodynamic approach requires a potential function of sequence and conformation that has its global minimum at the native conformation for many different proteins. Here we study the behavior of such functions for the simplest model system that still has some of the features of the protein folding problem, namely two-dimensional square lattice chain configurations involving two residue types. First we show that even the given contact potential, which by definition is used to identify the folding sequences and their unique native conformations, cannot always correctly select which sequences will fold to a given structure. Second, we demonstrate that the given contact potential is not always able to favor the native alignment of a native sequence on its own native conformation over other gapped alignments of different folding sequences onto that same conformation. Because of these shortcomings, even in this simple model system in which all conformations and all native sequences are known and determined directly by the given potential, we must reexamine our expectations for empirical potentials used for inverse folding and gapped alignment on more realistic representations of proteins.

Easily searched protein folding potentials.

In order to calculate the tertiary structure of a protein from its amino acid sequence, the thermodynamic approach requires a potential function of sequence and conformation that has its global minimum at the native conformation for many different proteins. Here we study the behavior of such functions for the simplest model system that still has the essential features of the protein folding problem, namely two-dimensional square lattice chain configurations involving two residue types. First we demonstrate a method for accurately recovering the given contact potential from only a knowledge of which sequences fold to which structures and what the non-native structures are. Second, we show how to derive from the same information more general potential functions having much better positive correlations between potential function value and conformational deviation from the native. These functions consequently permit faster and more reliable searches for the native conformation, given the native sequence. Furthermore, the method for finding such potentials is easily applied to more realistic protein models.

(Probably) all possible protein folds at low resolution.

For decades, a large number of investigators have been sifting the database of experimentally determined three-dimensional protein structures to discover recurring patterns of all types. Now that there are over a thousand such structures available, the natural question is whether we have seen all substantially different protein folds, and if not, how many have yet to be discovered? Answering the question can be broken down into three steps: (1) choose the range and domain for a similarity function, then (2) choose a particular similarity function, and (3) construct a corresponding protein model space that can be searched for dissimilar structures. In our analysis of the problem, we first chose to examine different conformations of the same protein, taking into account only C alpha atomic coordinates. In particular, we do not compare proteins of different chain lengths on the basis of some kind of gapped alignment. Secondly, we use a measure of conformational similarity based on rigid body superposition that emphasizes overall geometric resemblance, rather than agreement in secondary structure, for example. Third, we employed the discrete cosine transform to construct exhaustive sets of globular self-avoiding C alpha traces that were all different from each other by a given level. These sets of artificial structures were not too large to explicitly enumerate as long as the level of dissimilarity was high, and the chain flexibility was low. For chains flexible enough to match all experimental structures of 170 residue or less that are not beta-barrels, we find 128 artificial structures, of which 28 resemble nothing in the Protein Data Bank.

How many protein folding motifs are there?

As the three-dimensional structures of more and more proteins are determined by experiment, discovering substantially novel folding motifs becomes ever rarer. The natural question is how many motifs are there and how many have already been found? In order to answer this in at least one plausible and well-defined sense, we have chosen a quantitative measure of conformational similarity, rho (based on optimal rigid body superposition), and a means of generating all possible three-dimensional chain conformations using the discrete cosine transform. How many different folding motifs there are then depends on the specified cutoff in rho and on the flexibility allowed for the model polypeptide chain. For single chain proteins having no more than about 170 residues and which are not beta-barrels, there are only about 128 motifs that differ by rho > 1.0 (an extremely vague level of similarity), of which so far only 100 have been seen experimentally. The remaining 28 can be viewed as very low-resolution models of either undiscovered novel folds or violations of unknown principles of protein folding.

Size-independent comparison of protein three-dimensional structures.

Protein structures are routinely compared by their root-mean-square deviation (RMSD) in atomic coordinates after optimal rigid body superposition. What is not so clear is the significance of different RMSD values, particularly above the customary arbitrary cutoff for obvious similarity of 2-3 A. Our earlier work argued for an intrinsic cutoff for protein similarity that varied with the number of residues in the polypeptide chains being compared. Here we introduce a new measure, rho, of structural similarity based on RMSD that is independent of the sizes of the molecules involved, or of any other special properties of molecules. When rho is less than 0.4-0.5, protein structures are visually recognized to be obviously similar, but the mathematically pleasing intrinsic cutoff of rho < 1.0 corresponds to overall similarity in folding motif at a level not usually recognized until smoothing of the polypeptide chain path makes it striking. When the structures are scaled to unit radius of gyration and equal principle moments of inertia, the comparisons are even more universal, since they are no longer obscured by differences in overall size and ellipticity. With increasing chain length, the distribution of rho for pairs of random structures is skewed to higher values, but the value for the best 1% of the comparisons rises only slowly with the number of residues. This level is close to an intrinsic cutoff between similar and dissimilar comparisons, namely the maximal scaled rho possible for the two structures to be more similar to each other than one is to the other's mirror image. The intrinsic cutoff is independent of the number of residues or points being compared. For proteins having fewer than 100 residues, the 1% rho falls below the intrinsic cutoff, so that for very small proteins, geometrically significant similarity can often occur by chance. We believe these ideas will be helpful in judging success in NMR structure determination and protein folding modeling.

Learning about protein folding via potential functions.

Over the last few years we have developed an empirical potential function that solves the protein structure recognition problem: given the sequence for an n-residue globular protein and a collection of plausible protein conformations, including the native conformation for that sequence, identify the correct, native conformation. Having determined this potential on the basis of only some 6500 native/nonnative pairs of structures for 58 proteins, we find it recognizes the native conformation for essentially all compact, soluble, globular proteins having known native conformations in comparisons with 10(4) to 10(6) reasonable alternative conformations apiece. In this sense, the potential encodes nearly all the essential features of globular protein conformational preference. In addition it "knows" about many additional factors in protein folding, such as the stabilization of multimeric proteins, quaternary structure, the role of disulfide bridges and ligands, proproteins vs. processed proteins, and minimal strand lengths in globular proteins. Comparisons are made with other sorts of protein folding problems, and applications in protein conformational determination and prediction are discussed.

Significance of root-mean-square deviation in comparing three-dimensional structures of globular proteins.

In the study of globular protein conformations, one customarily measures the similarity in three-dimensional structure by the root-mean-square deviation (RMSD) of the C alpha atomic coordinates after optimal rigid body superposition. Even when the two protein structures each consist of a single chain having the same number of residues so that the matching of C alpha atoms is obvious, it is not clear how to interpret the RMSD. A very large value means they are dissimilar, and zero means they are identical in conformation, but at what intermediate values are they particularly similar or clearly dissimilar? While many workers in the field have chosen arbitrary cutoffs, and others have judged values of RMSD according to the observed distribution of RMSD for random structures, we propose a self-referential, non-statistical standard. We take two conformers to be intrinsically similar if their RMSD is smaller than that when one of them is mirror inverted. Because the structures considered here are not arbitrary configurations of point atoms, but are compact, globular, polypeptide chains, our definition is closely related to similarity in radius of gyration and overall chain folding patterns. Being strongly similar in our sense implies that the radii of gyration must be nearly identical, the root-mean-square deviation in interatomic distances is linearly related to RMSD, and the two chains must have the same general fold. Only when the RMSD exceeds this level can parts of the polypeptide chain undergo nontrivial rearrangements while remaining globular. This enables us to judge when a prediction of a protein's conformation is "correct except for minor perturbations", or when the ensemble of protein structures deduced from NMR experiments are "basically in mutual agreement".

Analysis of cocaine receptor site ligand binding by three-dimensional Voronoi site modeling approach.

The Voronoi approach has been used to obtain a three-dimensional model for the binding of the cocaine analogues at the cocaine receptor site. The method has been used to determine the geometric details and the physicochemical properties of the binding regions in the receptor site. With only eight compounds in the training set, the Voronoi site model, consisting of four regions, not only fully explains the binding affinity of the input compounds but is also successful in correctly predicting another eight compounds of the test set. The phenyl substituent at the 3-position of the tropane ring of cocaine was found to be the most significant functionality relevant for activity, while moderate contribution results from the hydrophobic interactions of the tropane ring with the binding regions. Some of the problems associated with the approach are discussed, and we report a new procedure for evaluating the validity of the model obtained from our approach.

Voronoi modeling: the binding of triazines and pyrimidines to L. casei dihydrofolate reductase.

Vorom is a computer-aided method of drug design which can model a biological receptor given only binding data of known ligands. Using the binding energies of known competitive, reversible ligands of a biological macromolecule, vorom can make predictions about the binding energies and conformations of other small molecules binding to that receptor as well as provide information about the geometry and physicochemical characteristics of the binding site. One such model of L. casei dihydrofolate reductase was made. The model was able to predict the binding energies of 31 pyrimidine and triazine inhibitors out of a total set of 47, using only eight of the molecules (four pyrimidines and four triazines) as input. The binding energy of methotrexate, which is neither a pyrimidine nor a triazine, was correctly predicted. The binding mode of methotrexate predicted by vorom is entirely consistent with known X-ray data. The predicted binding modes of the pyrimidine inhibitors and the geometry of the site model are also consistent with published NMR and crystallographic data.