A fast conformational search strategy for finding low energy structures of model proteins

We describe a new computer algorithm for finding low-energy conformations of proteins. It is a chain-growth method that uses a heuristic bias function to help assemble a hydrophobic core. We call it the Core-directed chain Growth method (CG). We test the CG method on several well-known literature examples of HP lattice model proteins [in which proteins are modeled as sequences of hydrophobic (H) and polar (P) monomers], ranging from 20-64 monomers in two dimensions, and up to 88-mers in three dimensions. Previous nonexhaustive methods--Monte Carlo, a Genetic Algorithm, Hydrophobic Zippers, and Contact Interactions--have been tried on these same model sequences. CG is substantially better at finding the global optima, and avoiding local optima, and it does so in comparable or shorter times. CG finds the global minimum energy of the longest HP lattice model chain for which the global optimum is known, a 3D 88-mer that has only been reachable before by the CHCC complete search method. CG has the potential advantage that it should have nonexponential scaling with chain length. We believe this is a promising method for conformational searching in protein folding algorithms.     

An evolutionary approach to folding small alpha-helical proteins that uses sequence information and an empirical guiding fitness function

Three short protein sequences have been guided by computer to folds resembling their crystal structures. Initially, peptide fragment conformations ranging in size from 9 to 25 residues were selected from a database of known protein structures. A fragment was selected if it was compatible with a segment of the sequence to be folded, as judged by three-dimensional profile scores. By linking the selected fragment conformations together, hundreds of trial structures were generated of the same length and sequence as the protein to be folded. These starting trial structures were then improved by an evolutionary algorithm. Selection pressure for improving the structures was provided by an energy function that was designed to guide the conformational search procedure toward the correct structure. We find that by evolution of only 400 structures for fewer than 1400 generations, the overall fold of some small helical proteins can be computed from the sequence, with deviations from observed structures of 2.5-4.0 A for C alpha atoms.     

Antiretroviral therapies

We describe an algorithm which enables us to search the conformational space of the side chains of a protein to identify the global minimum energy combination of side chain conformations as well as all other conformations within a specified energy cutoff of the global energy minimum. The program is used to explore the side chain conformational energy surface of a number of proteins, to investigate how this surface varies with the energy model used to describe the interactions within the system and the rotamer library. Enumeration of the rotamer combinations enables us to directly evaluate the partition function, and thus calculate the side chain contribution to the conformational entropy of the folded protein. An investigation of these conformations and the relationships between them shows that most of the conformations near to the global energy minimum arise from changes in side chain conformations that are essentially independent; very few result from a concerted change in conformation of two or more residues. Some of the limitations of the approach are discussed.     

Local rules for protein folding on a triangular lattice and generalized hydrophobicity in the HP model

We consider the problem of determining the three-dimensional folding of a protein given its one-dimensional amino acid sequence. We use the HP model for protein folding proposed by Dill (1985), which models protein as a chain of amino acid residues that are either hydrophobic or polar, and hydrophobic interactions are the dominant initial driving force for the protein folding. Hart and Istrail (1996a) gave approximation algorithms for folding proteins on the cubic lattice under the HP model. In this paper, we examine the choice of a lattice by considering its algorithmic and geometric implications and argue that the triangular lattice is a more reasonable choice. We present a set of folding rules for a triangular lattice and analyze the approximation ratio they achieve. In addition, we introduce a generalization of the HP model to account for residues having different levels of hydrophobicity. After describing the biological foundation for this generalization, we show that in the new model we are able to achieve similar constant factor approximation guarantees on the triangular lattice as were achieved in the standard HP model. While the structures derived from our folding rules are probably still far from biological reality, we hope that having a set of folding rules with different properties will yield more interesting folds when combined.     

Perturbations of the denatured state ensemble: modeling their effects on protein stability and folding kinetics

By considering the denatured state of a protein as an ensemble of conformations with varying numbers of sequence-specific interactions, the effects on stability, folding kinetics, and aggregation of perturbing these interactions can be predicted from changes in the molecular partition function. From general considerations, the following conclusions are drawn: (1) A perturbation that enhances a native interaction in denatured state conformations always increases the stability of the native state. (2) A perturbation that promotes a non-native interaction in the denatured state always decreases the stability of the native state. (3) A change in the denatured state ensemble can alter the kinetics of aggregation and folding. (4) The loss (or increase) in stability accompanying two mutations, each of which lowers (or raises) the free energy of the denatured state, will be less than the sum of the effects of the single mutations, except in cases where both mutations affect the same set of partially folded conformations. By modeling the denatured state as the ensemble of all non-native conformations of hydrophobic-polar (HP) chains configured on a square lattice, it can be shown that the stabilization obtained from enhancement of native interactions derives in large measure from the avoidance of non-native interactions in the D state. In addition, the kinetic effects of fixing single native contacts in the denatured state or imposing linear gradients in the HH contact probabilities are found, for some sequences, to significantly enhance the efficiency of folding by a simple hydrophobic zippering algorithm. Again, the dominant mechanism appears to be avoidance of non-native interactions. These results suggest stabilization of native interactions and imposition of gradients in the stability of local structure are two plausible mechanisms involving the denatured state that could play a role in the evolution of protein folding and stability.     

Assembly of protein structure from sparse experimental data: an efficient Monte Carlo model

A new, efficient method for the assembly of protein tertiary structure from known, loosely encoded secondary structure restraints and sparse information about exact side chain contacts is proposed and evaluated. The method is based on a new, very simple method for the reduced modeling of protein structure and dynamics, where the protein is described as a lattice chain connecting side chain centers of mass rather than Calphas. The model has implicit built-in multibody correlations that simulate short- and long-range packing preferences, hydrogen bonding cooperativity and a mean force potential describing hydrophobic interactions. Due to the simplicity of the protein representation and definition of the model force field, the Monte Carlo algorithm is at least an order of magnitude faster than previously published Monte Carlo algorithms for structure assembly. In contrast to existing algorithms, the new method requires a smaller number of tertiary restraints for successful fold assembly; on average, one for every seven residues as compared to one for every four residues. For example, for smaller proteins such as the B domain of protein G, the resulting structures have a coordinate root mean square deviation (cRMSD), which is about 3 A from the experimental structure; for myoglobin, structures whose backbone cRMSD is 4.3 A are produced, and for a 247-residue TIM barrel, the cRMSD of the resulting folds is about 6 A. As would be expected, increasing the number of tertiary restraints improves the accuracy of the assembled structures. The reliability and robustness of the new method should enable its routine application in model building protocols based on various (very sparse) experimentally derived structural restraints.     

Role of hydrophobicity and solvent-mediated charge-charge interactions in stabilizing alpha-helices

A theoretical study to identify the conformational preferences of lysine-based oligopeptides has been carried out. The solvation free energy and free energy of ionization of the oligopeptides have been calculated by using a fast multigrid boundary element method that considers the coupling between the conformation of the molecule and the ionization equilibria explicitly, at a given pH value. It has been found experimentally that isolated alanine and lysine residues have somewhat small intrinsic helix-forming tendencies; however, results from these simulations indicate that conformations containing right-handed alpha-helical turns are energetically favorable at low values of pH for lysine-based oligopeptides. Also, unusual patterns of interactions among lysine side chains with large hydrophobic contacts and close proximity (5-6 A) between charged NH3+ groups are observed. Similar arrangements of charged groups have been seen for lysine and arginine residues in experimentally determined structures of proteins available from the Protein Data Bank. The lowest-free-energy conformation of the sequence Ac-(LYS)6-NMe from these simulations showed large pKalpha shifts for some of the NH3+ groups of the lysine residues. Such large effects are not observed in the lowest-energy conformations of oligopeptide sequences with two, three, or four lysine residues. Calculations on the sequence Ac-LYS-(ALA)4-LYS-NMe also reveal low-energy alpha-helical conformations with interactions of one of the LYS side chains with the helix backbone in an arrangement quite similar to the one described recently by (Proc. Natl. Acad. Sci. U.S.A. 93:4025-4029). The results of this study provide a sound basis with which to discuss the nature of the interactions, such as hydrophobicity, charge-charge interaction, and solvent polarization effects, that stabilize right-handed alpha-helical conformations.     

How evolution makes proteins fold quickly

Sequences of fast-folding model proteins (48 residues long on a cubic lattice) were generated by an evolution-like selection toward fast folding. We find that fast-folding proteins exhibit a specific folding mechanism in which all transition state conformations share a smaller subset of common contacts (folding nucleus). Acceleration of folding was accompanied by dramatic strengthening of interactions in the folding nucleus whereas average energy of nonnucleus interactions remained largely unchanged. Furthermore, the residues involved in the nucleus are the most conserved ones within families of evolved sequences. Our results imply that for each protein structure there is a small number of conserved positions that are key determinants of fast folding into that structure. This conjecture was tested on two protein superfamilies: the first having the classical monophosphate binding fold (CMBF; 98 families) and the second having type-III repeat fold (47 families). For each superfamily, we discovered a few positions that exhibit very strong and statistically significant "conservatism of conservatism"-amino acids in those positions are conserved within every family whereas the actual types of amino acids varied from family to family. Those amino acids are in spatial contact with each other. The experimental data of Serrano and coworkers [Lopez-Hernandez, E. & Serrano, L. (1996) Fold. Des. (London) 1, 43-55]. for one of the proteins of the CMBF superfamily (CheY) show that residues identified this way indeed belong to the folding nucleus. Further analysis revealed deep connections between nucleation in CMBF proteins and their function.     

Molecular modeling of interleukin-8 receptor beta and analysis of the receptor-ligand interaction

A structural model of interleukin-8 receptor type beta (IL-8R-beta) was constructed based on the structure of bacteriorhodopsin. High temperature molecular dynamics simulations were performed to search the possible conformations of loop regions in IL-8R-beta which recognize the ligand. The crystal structure of interleukin 8 (IL-8) was used as a geometric constraint of the extracellular loop regions of IL-8R-beta in the conformational search. 500 complex structures were extracted from the dynamics trajectory and five plausible models were selected based on the binding energy and known experimental data. To study further the interaction between IL-8R-beta and its ligands, the complex of IL-8R-beta and platelet factor 4 (PF4) C-terminal peptide was also modeled by molecular dynamics simulations. From these models, the N-terminus, extracellular domain 3 and extracellular domain 4 of IL-8R-beta were found to be important for ligand binding. Key residues of these regions involved in ligand binding were characterized. These models provide insight into the structural basis of biological activity of IL-8 and PF4 and may guide the design of potential therapeutic agents targeting IL-8 receptors. Furthermore, the approach developed from this study may have implications for the understanding of other chemokine receptor-ligand interactions that have been recently suggested to be involved in HIV infection.     

Protein structure and energy landscape dependence on sequence using a continuous energy function

We have recently described a new conformational search strategy for protein folding algorithms called the CGU (convex global underestimator) method. Here we use a simplified protein chain representation and a differentiable form of the Sun/Thomas/Dill energy function to test the CGU method. Standard search methods, such as Monte Carlo and molecular dynamics are slowed by kinetic traps. That is, the computer time depends more strongly on the shape of the energy landscape (dictated by the amino acid sequence) than on the number of degrees of freedom (dictated by the chain length). The CGU method is not subject to this limitation, since it explores the underside of the energy landscape, not the top. We find that the CGU computer time is largely independent of the monomer sequence for different chain folds and scales as O(n4) with chain length. By using different starting points, we show that the method appears to find global minima. Since we can currently find stable states of 36-residue chains in 2.4 hours, the method may be practical for small proteins.     

Serum sensitivity and cell surface hydrophobicity of Klebsiella pneumoniae treated with gentamicin, tobramycin and amikacin

The individual chains in the triple helix of collagen occur in a conformation related to polyproline II because of the presence of large number of imino peptide bonds. However, these residues are not evenly distributed in the collagen molecule which also contains many non-imino residues. These non-imino regions of collagen may be expected to show preference for other than triple helical conformations. The appearance of several Raman bands in solution phase at 65 degrees C raises the possibility of non-uniform triple helical structure in collagen. Raman spectroscopic studies on collagen in the solid state and in solution at a temperature greater than its denaturation temperature, reported here suggest that denatured collagen may exhibit an ensemble of conformational states with yet unknown implications to the biochemical interactions of this important protein component of connective tissues.     

Sequence database searches via de novo peptide sequencing by tandem mass spectrometry

A method is described for searching protein sequence databases using tandem mass spectra of tryptic peptides. The approach uses a de novo sequencing algorithm to derive a short list of possible sequence candidates which serve as query sequences in a subsequent homology-based database search routine. The sequencing algorithm employs a graph theory approach similar to previously described sequencing programs. In addition, amino acid composition, peptide sequence tags and incomplete or ambiguous Edman sequence data can be used to aid in the sequence determinations. Although sequencing of peptides from tandem mass spectra is possible, one of the frequently encountered difficulties is that several alternative sequences can be deduced from one spectrum. Most of the alternative sequences, however, are sufficiently similar for a homology-based sequence database search to be possible. Unfortunately, the available protein sequence database search algorithms (e.g. Blast or FASTA) require a single unambiguous sequence as input. Here we describe how the publicly available FASTA computer program was modified in order to search protein databases more effectively in spite of the ambiguities intrinsic in de novo peptide sequencing algorithms.     

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.     

Use of quantitative structure-property relationships to predict the folding ability of model proteins

We investigate the folding of a 125-bead heteropolymer model for proteins subject to Monte Carlo dynamics on a simple cubic lattice. Detailed study of a few sequences revealed a folding mechanism consisting of a rapid collapse followed by a slow search for a stable core that served as the transition state for folding to a near-native intermediate. Rearrangement from the intermediate to the native state slowed folding further because it required breaking native-like local structure between surface monomers so that those residues could condense onto the core. We demonstrate here the generality of this mechanism by a statistical analysis of a 200 sequence database using a method that employs a genetic algorithm to pick the sequence attributes that are most important for folding and an artificial neural network to derive the corresponding functional dependence of folding ability on the chosen sequence attributes [quantitative structure-property relationships (QSPRs)]. QSPRs that use three sequence attributes yielded substantially more accurate predictions than those that use only one. The results suggest that efficient search for the core is dependent on both the native state's overall stability and its amount of kinetically accessible, cooperative structure, whereas rearrangement from the intermediate is facilitated by destabilization of contacts between surface monomers. Implications for folding and design are discussed.     

Flexible algorithm for direct multiple alignment of protein structures and sequences

The recently described equivalence between the alignment of two proteins and a conformation of a lattice chain on a two-dimensional square lattice is extended to multiple alignments. The search for the optimal multiple alignment between several proteins, which is equivalent to finding the energy minimum in the conformational space of a multi-dimensional lattice chain, is studied by the Monte Carlo approach. This method, while not deterministic, and for two-dimensional problems slower than dynamic programming, can accept arbitrary scoring functions, including non-local ones, and its speed decreases slowly with increasing number of dimensions. For the local scoring functions, the MC algorithm can also reproduce known exact solutions for the direct multiple alignments. As illustrated by examples, both for structure- and sequence-based alignments, direct multi-dimensional alignments are able to capture weak similarities between divergent families much better than ones built from pairwise alignments by a hierarchical approach.     

Epithelioid angiomyolipoma of the kidney: a report of five cases with a prominent and diagnostically confusing epithelioid smooth muscle component

We suggest and test potentials for the modeling of protein structure on coarse lattices. The coarser the lattice, the more complete and faster is the exploration of the conformational space of a molecule. However, there are inevitable energy errors in lattice modeling caused by distortions in distances between interacting residues; the coarser the lattice, the larger are the energy errors. It is generally believed that an improvement in the accuracy of lattice modelling can be achieved only by reducing the lattice spacing. We reduce the errors on coarse lattices with lattice-adapted potentials. Two methods are used: in the first approach, 'lattice-derived' potentials are obtained directly from a database of lattice models of protein structure; in the second approach, we derive 'lattice-adjusted' potentials using our previously developed method of statistical adjustment of the 'off-lattice' energy functions for lattices. The derivation of off-lattice Calpha atom-based distance-dependent pairwise potentials has been reported previously. The accuracy of 'lattice-derived', 'lattice-adjusted' and 'off-lattice' potentials is estimated in threading tests. It is shown that 'lattice-derived' and 'lattice-adjusted' potentials give virtually the same accuracy and ensure reasonable protein fold recognition on the coarsest considered lattice (spacing 3.8 A), however, the 'off-lattice' potentials, which efficiently recognize off-lattice folds, do not work on this lattice, mainly because of the errors in short-range interactions between neighboring residues.     

Exploring the folding funnel of a polypeptide chain by biophysical studies on protein fragments

We are examining possible roles of native and non-native interactions in early events in protein folding by a systematic analysis of the structures of fragments of proteins whose folding pathways are well characterised. Seven fragments of the 110-residue protein barnase, corresponding to the progressive elongation from its N terminus, have been characterised by a battery of biophysical and spectroscopic methods. Barnase is a multi-modular protein that folds via an intermediate in which the C-terminal region of its major alpha-helix (alpha-helix1, residues Thr6-His18) is substantially formed as is also its anti-parallel beta-sheet, centred around a beta-hairpin (residues Ser92-Leu95). Fragments up to, and including, residues 1-95 (fragment B95), appeared to be mainly disordered, although a small amount of helical secondary structure in each was inferred from far-UV CD experiments, and fluorescence studies indicated some native-like tertiary interactions in B95. The largest fragment (residues 1-105, B105) is compactly folded. The secondary structure in alpha-helix1 in the seven fragments was found by NMR to increase with increasing chain length faster than the build-up of tertiary interactions, indicating that alpha-helix1 is being stabilised by non-native interactions. This behaviour contrasts with that in fragments of the 64-residue chymotrypsin inhibitor 2 (CI2), in which tertiary and secondary structures build up in parallel with increasing length. CI2 consists of a single module of structure that folds without a detectable intermediate. The largest fragment of barnase, B105, has interactions that resemble its folding intermediate, whereas one of the largest fragments of CI2 (residues 1-60) resembles the folding transition state. The folding pathways of both proteins are consistent with a scheme in which there are low levels of native-like secondary structure in the denatured state that become stabilised by long-range interactions as folding proceeds. Neither protein forms a stable fold when lacking the last ten residues at the C terminus. Since at least 20 amino acid residues are bound to the ribosome during protein biosynthesis, these small proteins do not fold until they have left the ribosome, and so the studies of the folding of such proteins in vitro may be relevant to their folding in vivo, especially as the molecular chaperone GroEL binds only weakly to denatured CI2 and does not discernibly alter the folding mechanism of barnase.     

Constructing lattice models of protein chains with side groups

An algorithm to construct lattice models of polymers with side chains is presented. A search for the global minimum of the error function for a given lattice-to-chain orientation is done by dynamic programming, making the search both fast and complete. Application of the algorithm is illustrated by constructing lattice models for 12 proteins of different sizes and structural types.     

Molecular mechanisms for cooperative folding of proteins

The folding of single-domain globular proteins exhibits the character of first-order or two-state thermodynamics. The origin of such high cooperativity in relatively small polymer systems is still not well understood. Recently, the statistical mechanics of protein folding has been studied extensively with simple protein models such as short cubic-lattice chains with contact-based interactions. While many valuable insights about protein folding were gained with such models, some concerns have also arisen, viz. that they lack the character of protein backbones whose interactions would limit the folding patterns of proteins. Here, a comparative study of the conventional cubic-lattice chain model and a fine-grained more realistic lattice protein model with both backbone and side-chain interactions is carried out. It is found that, even though both types of models exhibit a cooperative two-state folding transition to the native structure with optimized force fields, the character and origin of cooperativity of the two models are different. In the simple contact-based model, the free-energy barrier occurs at the low end of the energy scale, and the cooperativity arises from a concerted formation of native contacts among many residues in a compact state. In the other more complicated model, the free-energy barrier occurs in the intermediate energy region, and the folding cooperativity arises from collective orientational arrangements of locally structured units in semi-open conformational states. On the basis of these results, two limiting molecular mechanisms for protein folding emerge, which can be used for analyzing the folding process of real proteins.