Driving forces in the origins of life.

What were the physico-chemical forces that drove the origins of life? We discuss four major prebiotic 'discoveries': persistent sampling of chemical reaction space; sequence-encodable foldable catalysts; assembly of functional pathways; and encapsulation and heritability. We describe how a 'proteins-first' world gives plausible mechanisms. We note the importance of hydrophobic and polar compositions of matter in these advances.

Theory for protein folding cooperativity: helix bundles.

We present a theory for protein folding stability and cooperativity for helix bundle proteins. We treat the individual helices with a Schellman-Zimm-Bragg-like approach, using nucleation and propagation quantities, and we treat the hydrophobic and van der Waals contacts between the helices as a binding equilibrium. Predictions are in good agreement with experiments on both thermal and urea-induced transitions of (1) molecules that can undergo single helix-to-coil transitions for various chain lengths and (2) three-helix-bundle proteins A and alpha3C. The present model addresses a problem raised by Kaya and Chan that proteins fold more cooperatively than previous models predict. The present model correctly predicts the experimentally observed two-state cooperativities, DeltaH(van't Hoff)/DeltaH(cal) approximately 1, for helix-bundle proteins. The predicted folding cooperativity is greater than that of helix formation alone, or collapse alone, because of the nonlinear coupling between the tertiary interactions and the helical interactions.

Explicit-water molecular dynamics study of a short-chain 3,3 ionene in solutions with sodium halides.

Ionenes are alkyl polymer chains in which hydrophobic groups are separated by ionic charges. They are useful for studying the properties of water as a solvent because they demonstrate a sufficiently complex combination of hydrophobicity, charge interactions, and specific-ion effects that some properties cannot be predicted by implicit-solvation theories. On the other hand, they are simple enough that their molecular structures can be varied and controlled in systematic experiments. In particular, implicit-solvent models predict that all such solutes will have negative enthalpies of dilution, whereas experiments show that enthalpies of dilution are positive for the chaotropic counterions. Here, we study ionenes that are short chains (six monomer units) in solutions of different counterions, with sodium as the coion by molecular dynamics simulations in explicit water. We explore the pair distributions of various atoms within the system at three different temperatures: T=278, 298, and 318 K. We find (i) that the molecular dynamics simulations are consistent with the experimental trends for the osmotic coefficients and enthalpies of dilution, (ii) that the fluorine-nitrogen and fluorine-carbon correlations decrease with decreasing temperature, (iii) while the opposite behavior is found for iodine ions, and (iv) that in the counterion-Na(+) pair distributions, too, fluorine ions behave oppositely to iodine ions upon temperature increase.

Designing polymers that mimic biomolecules.

A new field is emerging. Chemists are beginning to synthesize polymers with properties that are similar to those of proteins and RNA. Recent studies have identified oligomer backbones that form stable secondary structures. It is now possible to assemble specific sequences of diverse monomer sets into chain lengths that are nearly sufficient for tertiary structure formation. Such molecules will teach us how natural biopolymers fold; they will also enable us to design synthetic heteropolymers with novel structures and desirable functions.

Confined water: a Mercedes-Benz model study.

We study water that is confined within small geometric spaces. We use the Mercedes-Benz (MB) model of water, in NVT and muVT Monte Carlo computer simulations. For MB water molecules between two planes separated by a distance d, we explore the structures, hydrogen bond networks, and thermodynamics as a function of d, temperature T, and water chemical potential mu. We find that squeezing the planes close enough together leads to a vaporization of waters out of the cavity. This vaporization transition has a corresponding peak in the heat capacity of the water. We also find that, in small pores, hydrogen bonding is not isotropic but, rather, it preferentially forms chains along the axis of the cavity. This may be relevant for fast proton transport in pores. Our simulations show oscillations in the forces between the inert plates, due to water structure, even for plate separations of 5-10 water diameters, consistent with experiments by Israelachvili et al. [Nature 1983, 306, 249]. Finally, we find that confinement affects water's heat capacity, consistent with recent experiments of Tombari et al. on Vycor nanopores [J. Chem. Phys. 2005, 122, 104712].

Statistical potentials extracted from protein structures: how accurate are they?

"Statistical potentials" are energies widely used in computer algorithms to fold, dock, or recognize protein structures. They are derived from: (1) observed pairing frequencies of the 20 amino acids in databases of known protein structures, and (2) approximations and assumptions about the physical process that these quantities measure. Using exact lattice models, we construct a rigorous test of those assumptions and approximations. We find that statistical potentials often correctly rank-order the relative strengths of interresidue interactions, but they do not reflect the true underlying energies because of systematic errors arising from the neglect of excluded volume in proteins. We find that complex residue-residue distance dependences observed in statistical potentials, even those among charged groups, can be largely explained as an indirect consequence of the burial of non-polar groups. Our results suggest that current statistical potentials may have limited value in protein folding algorithms and wherever they are used to provide energy-like quantities.

Hydrogen bonding in globular proteins.

A global census of the hydrogen bonds in 42 X-ray-elucidated proteins was taken and the following demographic trends identified: (1) Most hydrogen bonds are local, i.e. between partners that are close in sequence, the primary exception being hydrogen-bonded ion pairs. (2) Most hydrogen bonds are between backbone atoms in the protein, an average of 68%. (3) All proteins studied have extensive hydrogen-bonded secondary structure, an average of 82%. (4) Almost all backbone hydrogen bonds are within single elements of secondary structure. An approximate rule of thirds applies: slightly more than one-third (37%) form i----i--3 hydrogen bonds, almost one-third (32%) form i----i--4 hydrogen bonds, and slightly less than one-third (26%) reside in paired strands of beta-sheet. The remaining 5% are not wholly within an individual helix, turn or sheet. (5) Side-chain to backbone hydrogen bonds are clustered at helix-capping positions. (6) An extensive network of hydrogen bonds is present in helices. (7) To a close approximation, the total number of hydrogen bonds is a simple function of a protein's helix and sheet content. (8) A unique quantity, termed the reduced number of hydrogen bonds, is defined as the maximum number of hydrogen bonds possible when every donor:acceptor pair is constrained to be 1:1. This quantity scales linearly with chain length, with 0.71 reduced hydrogen bond per residue. Implications of these results for pathways of protein folding are discussed.

Does compactness induce secondary structure in proteins? A study of poly-alanine chains computed by distance geometry.

A few years ago, lattice model studies indicated that compactness could induce polymer chains to develop protein-like secondary structures. Subsequent off-lattice studies have found the amounts of induced structure to be relatively small. Here we use distance geometry to generate random conformations of compact poly-alanine chains of various chain lengths. The poly-alanine chains are subjected only to compactness and excluded volume constraints; no other energies or conformational propensities are included in the chain generation procedure. We find that compactness leads to considerable stabilization of secondary structure, but the absolute amount of secondary structure depends strongly on the criteria used to define helices and sheets. By loose criteria, much secondary structure arises from compactness, but by strict criteria, little does. The stabilization free energy of secondary structure provided by compactness, however, appears to be independent of criteria. Since real helices and sheets in proteins can be identified by strict criteria, we introduced small energy perturbations to compact poly-alanine chains using the AMBER force field. Small refinements produced good alpha-helices. For beta-sheets, however, larger refinements are necessary. Compactness appears to impart stability, but not much structural specificity, to secondary structures in proteins. Compactness acts more like diffusion as a force, a result of ensemble statistics, than like pair interactions such as hydrogen bonding.

Polymer principles and protein folding.

This paper surveys the emerging role of statistical mechanics and polymer theory in protein folding. In the polymer perspective, the folding code is more a solvation code than a code of local phipsi propensities. The polymer perspective resolves two classic puzzles: (1) the Blind Watchmaker's Paradox that biological proteins could not have originated from random sequences, and (2) Levinthal's Paradox that the folded state of a protein cannot be found by random search. Both paradoxes are traditionally framed in terms of random unguided searches through vast spaces, and vastness is equated with impossibility. But both processes are partly guided. The searches are more akin to balls rolling down funnels than balls rolling aimlessly on flat surfaces. In both cases, the vastness of the search is largely irrelevant to the search time and success. These ideas are captured by energy and fitness landscapes. Energy landscapes give a language for bridging between microscopics and macroscopics, for relating folding kinetics to equilibrium fluctuations, and for developing new and faster computational search strategies.

Families and the structural relatedness among globular proteins.

Protein structures come in families. Are families "closely knit" or "loosely knit" entities? We describe a measure of relatedness among polymer conformations. Based on weighted distance maps, this measure differs from existing measures mainly in two respects: (1) it is computationally fast, and (2) it can compare any two proteins, regardless of their relative chain lengths or degree of similarity. It does not require finding relative alignments. The measure is used here to determine the dissimilarities between all 12,403 possible pairs of 158 diverse protein structures from the Brookhaven Protein Data Bank (PDB). Combined with minimal spanning trees and hierarchical clustering methods, this measure is used to define structural families. It is also useful for rapidly searching a dataset of protein structures for specific substructural motifs. By using an analogy to distributions of Euclidean distances, we find that protein families are not tightly knit entities.

Local and nonlocal interactions in globular proteins and mechanisms of alcohol denaturation.

How important are helical propensities in determining the conformations of globular proteins? Using the two-dimensional lattice model and two monomer types, H (hydrophobic) and P (polar), we explore both nonlocal interactions, through an HH contact energy, epsilon, as developed in earlier work, and local interactions, through a helix energy, sigma. By computer enumeration, the partition functions for short chains are obtained without approximation for the full range of both types of energy. When nonlocal interactions dominate, some sequences undergo coil-globule collapse to a unique native structure. When local interactions dominate, all sequences undergo helix-coil transitions. For two different conformational properties, the closest correspondence between the lattice model and proteins in the Protein Data Bank is obtained if the model local interactions are made small compared to the HH contact interaction, suggesting that helical propensities may be only weak determinants of globular protein structures in water. For some HP sequences, varying sigma/epsilon leads to additional sharp transitions (sometimes several) and to "conformational switching" between unique conformations. This behavior resembles the transitions of globular proteins in water to helical states in alcohols. In particular, comparison with experiments shows that whereas urea as a denaturant is best modeled as weakening both local and nonlocal interactions, trifluoro-ethanol is best modeled as mainly weakening HH interactions and slightly enhancing local helical interactions.

A statistical mechanical model for hydrogen exchange in globular proteins.

We develop a statistical mechanical theory for the mechanism of hydrogen exchange in globular proteins. Using the HP lattice model, we explore how the solvent accessibilities of chain monomers vary as proteins fluctuate from their stable native conformations. The model explains why hydrogen exchange appears to involve two mechanisms under different conditions of protein stability: (1) a "global unfolding" mechanism by which all protons exchange at a similar rate, approaching that of the denatured protein, and (2) a "stable-state" mechanism by which protons exchange at rates that can differ by many orders of magnitude. There has been some controversy about the stable-state mechanism: does exchange take place inside the protein by solvent penetration, or outside the protein by the local unfolding of a subregion? The present model indicates that the stable-state mechanism of exchange occurs through an ensemble of conformations, some of which may bear very little resemblance to the native structure. Although most fluctuations are small-amplitude motions involving solvent penetration or local unfolding, other fluctuations (the conformational distant relatives) can involve much larger transient excursions to completely different chain folds.

Constraint-based assembly of tertiary protein structures from secondary structure elements.

A challenge in computational protein folding is to assemble secondary structure elements-helices and strands-into well-packed tertiary structures. Particularly difficult is the formation of beta-sheets from strands, because they involve large conformational searches at the same time as precise packing and hydrogen bonding. Here we describe a method, called Geocore-2, that (1) grows chains one monomer or secondary structure at a time, then (2) disconnects the loops and performs a fast rigid-body docking step to achieve canonical packings, then (3) in the case of intrasheet strand packing, adjusts the side-chain rotamers; and finally (4) reattaches loops. Computational efficiency is enhanced by using a branch-and-bound search in which pruning rules aim to achieve a hydrophobic core and satisfactory hydrogen bonding patterns. We show that the pruning rules reduce computational time by 10(3)- to 10(5)-fold, and that this strategy is computationally practical at least for molecules up to about 100 amino acids long.

The effect of multiple binding modes on empirical modeling of ligand docking to proteins.

A popular approach to the computational modeling of ligand/receptor interactions is to use an empirical free energy like model with adjustable parameters. Parameters are learned from one set of complexes, then used to predict another set. To improve these empirical methods requires an independent way to study their inherent errors. We introduce a toy model of ligand/receptor binding as a workbench for testing such errors. We study the errors incurred from the two state binding assumption--the assumption that a ligand is either bound in one orientation, or unbound. We find that the two state assumption can cause large errors in free energy predictions, but it does not affect rank order predictions significantly. We show that fitting parameters using data from high affinity ligands can reduce two state errors; so can using more physical models that do not use the two state assumption. We also find that when using two state models to predict free energies, errors are more severe on high affinity ligands than low affinity ligands. And we show that two state errors can be diagnosed by systematically adding new binding modes when predicting free energies: if predictions worsen as the modes are added, then the two state assumption in the fitting step may be at fault.

Side-chain entropy and packing in proteins.

What role does side-chain packing play in protein stability and structure? To address this question, we compare a lattice model with side chains (SCM) to a linear lattice model without side chains (LCM). Self-avoiding configurations are enumerated in 2 and 3 dimensions exhaustively for short chains and by Monte Carlo sampling for chains up to 50 main-chain monomers long. This comparison shows that (1) side-chain degrees of freedom increase the entropy of open conformations, but side-chain steric exclusion decreases the entropy of compact conformations, thus producing a substantial entropy that opposes folding; (2) there is side-chain "freezing" or ordering, i.e., a sharp decrease in entropy, near maximum compactness; and (3) the different types of contacts among side chains (s) and main-chain elements (m) have different frequencies, and the frequencies have different dependencies on compactness. mm contacts contribute significantly only at high densities, suggesting that main-chain hydrogen bonding in proteins may be promoted by compactness. The distributions of mm, ms, and ss contacts in compact SCM configurations are similar to the distributions in protein structures in the Brookhaven Protein Data Bank. We propose that packing in proteins is more like the packing of nuts and bolts in a jar than like the pairwise matching of jigsaw puzzle pieces.

Ligand binding to proteins: the binding landscape model.

Models of ligand binding are often based on four assumptions: (1) steric fit: that binding is determined mainly by shape complementarity; (2) native binding: that ligands mainly bind to native states; (3) locality: that ligands perturb protein structures mainly at the binding site; and (4) continuity: that small changes in ligand or protein structure lead to small changes in binding affinity. Using a generalization of the 2D HP lattice model, we study ligand binding and explore these assumptions. We first validate the model by showing that it reproduces typical binding behaviors. We observe ligand-induced denaturation, ANS and heme-like binding, and "lock-and-key" and "induced-fit" specific binding behaviors characterized by Michaelis-Menten or more cooperative types of binding isotherms. We then explore cases where the model predicts violations of the standard assumptions. For example, very different binding modes can result from two ligands of identical shape. Ligands can sometimes bind highly denatured states more tightly than native states and yet have Michaelis-Menten isotherms. Even low-population binding to denatured states can cause changes in global stability, hydrogen-exchange rates, and thermal B-factors, contrary to expectations, but in agreement with experiments. We conclude that ligand binding, similar to protein folding, may be better described in terms of energy landscapes than in terms of simpler mass-action models.

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.

Folding proteins with a simple energy function and extensive conformational searching.

We describe a computer algorithm for predicting the three-dimensional structures of proteins using only their amino acid sequences. The method differs from others in two ways: (1) it uses very few energy parameters, representing hydrophobic and polar interactions, and (2) it uses a new "constraint-based exhaustive" searching method, which appears to be among the fastest and most complete search methods yet available for realistic protein models. It finds a relatively small number of low-energy conformations, among which are native-like conformations, for crambin (1CRN), avian pancreatic polypeptide (1PPT), melittin (2MLT), and apamin. Thus, the lowest-energy states of very simple energy functions may predict the native structures of globular proteins.

Predicting the structures of 18 peptides using Geocore.

We describe an extensive test of Geocore, an ab initio peptide folding algorithm. We studied 18 short molecules for which there are structures in the Protein Data Bank; chains are up to 31 monomers long. Except for the very shortest peptides, an extremely simple energy function is sufficient to discriminate the true native state from more than 10(8) lowest energy conformations that are searched explicitly for each peptide. A high incidence of native-like structures is found within the best few hundred conformations generated by Geocore for each amino acid sequence. Predictions improve when the number of discrete phi/psi choices is increased.

Modeling the effects of mutations on the denatured states of proteins.

We develop a model for the reversible denaturation of proteins and for the effects of single-site mutations on the denatured states. The model is based on short chains of sequences of H (hydrophobic) and P (other) monomers configured as self-avoiding walks on the two-dimensional square lattice. The N (native) state is defined as the unique conformation of lowest contact energy, whereas the D (denatured) state is defined as the collection of all other conformations. With this model we are able to determine the exact partition function, and thus the exact native-denatured equilibrium for various solvent conditions, using the computer to exhaustively enumerate every possible configuration. Previous studies confirm that this model shows many aspects of protein-like behavior. The present study attempts to model how the denatured state (1) depends on the amino acid sequence, and (2) is changed by single-site mutations. The model accounts for two puzzling experimental results: (1) the replacement of a polar residue by a hydrophobic amino acid on the surface of a protein can destabilize a native protein, and (2) the "denaturant slope," m = partial delta G/partial c (where c is the concentration of denaturant--urea, guanidine hydrochloride), can sometimes change by as much as 30% due to a single mutation. The principal conclusion of the present study is that, under strong folding conditions, the denatured conformations that are in equilibrium with the native state are not open random configurations. Instead, they are an ensemble of highly compact conformations with a distribution that depends on the residue sequence and that can be substantially altered by single mutations. Most importantly, we conclude that mutations can exert their dominant effects on protein stability by changing the entropy of folding.