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.

Modeling stochastic dynamics in biochemical systems with feedback using maximum caliber

Complex feedback systems are ubiquitous in biology. Modeling such systems with mass action laws or master equations requires information rarely measured directly. Thus rates and reaction topologies are often treated as adjustable parameters. Here we present a general stochastic modeling method for small chemical and biochemical systems with emphasis on feedback systems. The method, Maximum Caliber (MaxCal), is more parsimonious than others in constructing dynamical models requiring fewer model assumptions and parameters to capture the effects of feedback. MaxCal is the dynamical analogue of Maximum Entropy. It uses average rate quantities and correlations obtained from short experimental trajectories to construct dynamical models. We illustrate the method on the bistable genetic toggle switch. To test our method, we generate synthetic data from an underlying stochastic model. MaxCal reliably infers the statistics of the stochastic bistability and other full dynamical distributions of the simulated data, without having to invoke complex reaction schemes. The method should be broadly applicable to other systems.

Dynamical fluctuations in biochemical reactions and cycles

We develop theory for the dynamics and fluctuations in some cyclic and linear biochemical reactions. We use the approach of maximum caliber, which computes the ensemble of paths taken by the system, given a few experimental observables. This approach may be useful for interpreting single-molecule or few-particle experiments on molecular motors, enzyme reactions, ion-channels, and phosphorylation-driven biological clocks. We consider cycles where all biochemical states are observable. Our method shows how: (1) the noise in cycles increases with cycle size and decreases with the driving force that spins the cycle and (2) provides a recipe for estimating small-number features, such as probability of backward spin in small cycles, from experimental data. The back-spin probability diminishes exponentially with the deviation from equilibrium. We believe this method may also be useful for other few-particle nonequilibrium biochemical reaction systems.

Molecular organization in micelles and vesicles

The configurations of the hydrocarbon chains in micelles are severely constrained by the space-filling requirements of the chain segments, by the continuity of the chains, and by the micellar geometry. A statistical theory that takes full account of these constraints is developed by using a lattice model. The chain ends are deduced to be nonuniformly distributed, with maximal incidence approximately midway between the center of the micelle and the outer surface. Whereas the chain disorder near the outside of the hydrophobic core may approach that of a liquid, crowding of the chains near the core center imposes a degree of order approaching that in a crystal. These results are at variance with the prevailing view that the micellar interior resembles a "liquid hydrocarbon droplet." Also discussed are the effects of curvature on the chain configurations in monolayers and bilayers. It is found, for example, that the "disorder gradients" in inner and outer half-bilayers of small vesicles should be substantially different. Implications of these results are discussed.

Interphases of chain molecules: Monolayers and lipid bilayer membranes

Using the lattice model for a liquid, we treat the packing of short-chain molecules in interphases such as bilayer membranes. The constant density in the interphase imposes intermolecular constraints on the configurations of the flexible chains. The statistical theory here presented predicts a diffuse distribution of chain ends near the bilayer midplane; no adjustable parameters are required. Inasmuch as some of the chains terminate relatively near the polar interface, the number of chains reaching deeper planar layers is diminished. Consequently, configurational freedom increases with depth. This is the source of the well-known disorder gradient.

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.

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.

Theory for the solvation of nonpolar solutes in water

We recently developed an angle-dependent Wertheim integral equation theory (IET) of the Mercedes-Benz (MB) model of pure water [Silverstein et al., J. Am. Chem. Soc. 120, 3166 (1998)]. Our approach treats explicitly the coupled orientational constraints within water molecules. The analytical theory offers the advantage of being less computationally expensive than Monte Carlo simulations by two orders of magnitude. Here we apply the angle-dependent IET to studying the hydrophobic effect, the transfer of a nonpolar solute into MB water. We find that the theory reproduces the Monte Carlo results qualitatively for cold water and quantitatively for hot water.

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].

Stabilization of proteins in confined spaces

We present theory showing that confining a protein to a small inert space (a "cage") should stabilize the protein against reversible unfolding. Examples of such spaces might include the pores within chromatography columns, the Anfinsen cage in chaperonins, the interiors of ribosomes, or regions of steric occlusion inside cells. Confinement eliminates some expanded configurations of the unfolded chain, shifting the equilibrium from the unfolded state toward the native state. The partition coefficient for a protein in a confined space is predicted to decrease significantly when the solvent is changed from native to denaturing conditions. Small cages are predicted to increase the stability of the native state by as much as 15 kcal/mol. Confinement may also increase the rates of protein or RNA folding.

Transition states and the meaning of Phi-values in protein folding kinetics

What is the mechanism of two-state protein folding? The rate-limiting step is typically explored through a Phi-value, which is the mutation-induced change in the transition state free energy divided by the change in the equilibrium free energy of folding. Phi-values ranging from 0 to 1 have been interpreted as meaning the transition state is denatured-like (0), native-like (1) or in-between. But there is no classical interpretation for the experimental Phi-values that are negative or >1. Using a rigorous method to identity transition states via an exact lattice model, we find that nonclassical Phi-values can arise from parallel microscopic flow processes, such as those in funnel-shaped energy landscapes. Phi < 0 results when a mutation destabilizes a slow flow channel, causing a backflow into a faster flow channel. Phi > 1 implies the reverse: a backflow from a fast channel into a slow one. Using a 'landscape mapping' method, we find that Phi correlates with the acceleration/deceleration of folding induced by mutations, rather than with the degree of nativeness of the transition state.

Are proteins well-packed?

The average packing density inside proteins is as high as in crystalline solids. Does this mean proteins are well-packed? We go beyond average densities, and look at the full distribution functions of free volumes inside proteins. Using a new and rigorous Delaunay triangulation method for parsing space into empty and filled regions, we introduce formal definitions of interior and surface packing densities. Although proteins look like organic crystals by the criterion of average density, they look more like liquids and glasses by the criterion of their free volume distributions. The distributions are broad, and the scalings of volume-to-surface, volume-to-cluster-radius, and numbers of void versus volume show that the interiors of proteins are more like randomly packed spheres near their percolation threshold than like jigsaw puzzles. We find that larger proteins are packed more loosely than smaller proteins. And we find that the enthalpies of folding (per amino acid) are independent of the packing density of a protein, indicating that van der Waals interactions are not a dominant component of the folding forces.

Chiral N-substituted glycines can form stable helical conformations

BACKGROUND

Short sequence-specific heteropolymers of N-substituted glycines (peptoids) have emerged as promising tools for drug discovery. Recent work on medium-length peptoids containing chiral centers in their sidechains has demonstrated the existence of stable chiral conformations in solution. In this report, we explore the conformational properties of these N alpha chiral peptoids by molecular mechanics calculations and we propose a model for the solution conformation of an octamer of (S)-N-(1-phenylethyl)glycine.

RESULTS

Molecular mechanics calculations indicate that the presence of N-substituents in which the N alpha carbons are chiral centers has a dramatic impact on the available backbone conformations. These results are supported by semi-empirical quantum mechanical calculations and coincide qualitatively with simple steric considerations. They suggest that an octamer of (S)-N-(1-phenylethyl)glycine should form a right-handed helix with cis amide bonds, similar to the polyproline type I helix. This model is consistent with circular dichorism studies of these molecules.

CONCLUSIONS

Peptoid oligomers containing chiral centers in their sidechains present a new structural paradigm that has promising implications for the design of stably folded molecules. We expect that their novel structure may provide a scaffold to create heteropolymers with useful functionality.

A method for parameter optimization in computational biology

Models in computational biology, such as those used in binding, docking, and folding, are often empirical and have adjustable parameters. Because few of these models are yet fully predictive, the problem may be nonoptimal choices of parameters. We describe an algorithm called ENPOP (energy function parameter optimization) that improves-and sometimes optimizes-the parameters for any given model and for any given search strategy that identifies the stable state of that model. ENPOP iteratively adjusts the parameters simultaneously to move the model global minimum energy conformation for each of m different molecules as close as possible to the true native conformations, based on some appropriate measure of structural error. A proof of principle is given for two very different test problems. The first involves three different two-dimensional model protein molecules having 12 to 37 monomers and four parameters in common. The parameters converge to the values used to design the model native structures. The second problem involves nine bumpy landscapes, each having between 4 and 12 degrees of freedom. For the three adjustable parameters, the globally optimal values are known in advance. ENPOP converges quickly to the correct parameter set.

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.

RNA folding energy landscapes

Using a statistical mechanical treatment, we study RNA folding energy landscapes. We first validate the theory by showing that, for the RNA molecules we tested having only secondary structures, this treatment (i) predicts about the same native structures as the Zuker method, and (ii) qualitatively predicts the melting curve peaks and shoulders seen in experiments. We then predict thermodynamic folding intermediates. For one hairpin sequence, unfolding is a simple unzipping process. But for another sequence, unfolding is more complex. It involves multiple stable intermediates and a rezipping into a completely non-native conformation before unfolding. The principle that emerges, for which there is growing experimental support, is that although protein folding tends to involve highly cooperative two-state thermodynamic transitions, without detectable intermediates, the folding of RNA secondary structures may involve rugged landscapes, often with more complex intermediate states.

Predicting conformational switches in proteins

We describe a new computational technique to predict conformationally switching elements in proteins from their amino acid sequences. The method, called ASP (Ambivalent Structure Predictor), analyzes results from a secondary structure prediction algorithm to identify regions of conformational ambivalence. ASP identifies ambivalent regions in 16 test protein sequences for which function involves substantial backbone rearrangements. In the test set, all sites previously described as conformational switches are correctly predicted to be structurally ambivalent regions. No such regions are predicted in three negative control protein sequences. ASP may be useful as a guide for experimental studies on protein function and motion in the absence of detailed three-dimensional structural data.

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.

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.

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.

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.

Native protein fluctuations: the conformational-motion temperature and the inverse correlation of protein flexibility with protein stability

We study the fluctuations of native proteins by exact enumeration using the HP lattice model. The model fluctuations increase with temperature. We observe a low-temperature point, below which large fluctuations are frozen out. This prediction is consistent with the observation by Tilton et al. [R. F. Tilton, Jr., J. C. Dewan, and G. A. Petsko, Biochemistry 31, 2469 (1992)], that the thermal motions of ribonuclease A increase sharply above about 200 K. We also explore protein "flexibility" as defined by Debye-Waller-like factors and solvent accessibilities of core residues to hydrogen exchange. We find that proteins having greater stability tend to have fewer large fluctuations, and hence lower flexibilities. If flexibility is necessary for enzyme catalysis, this could explain why proteins from thermophilic organisms, which are exceptionally stable, may be catalytically inactive at normal temperatures.

NMR determination of the major solution conformation of a peptoid pentamer with chiral side chains

Polymers of N-substituted glycines ("peptoids") containing chiral centers at the alpha position of their side chains can form stable structures in solution. We studied a prototypical peptoid, consisting of five para-substituted (S)-N-(1-phenylethyl)glycine residues, by NMR spectroscopy. Multiple configurational isomers were observed, but because of extensive signal overlap, only the major isomer containing all cis-amide bonds was examined in detail. The NMR data for this molecule, in conjunction with previous CD spectroscopic results, indicate that the major species in methanol is a right-handed helix with cis-amide bonds. The periodicity of the helix is three residues per turn, with a pitch of approximately 6 A. This conformation is similar to that anticipated by computational studies of a chiral peptoid octamer. The helical repeat orients the amide bond chromophores in a manner consistent with the intensity of the CD signal exhibited by this molecule. Many other chiral polypeptoids have similar CD spectra, suggesting that a whole family of peptoids containing chiral side chains is capable of adopting this secondary structure motif. Taken together, our experimental and theoretical studies of the structural properties of chiral peptoids lay the groundwork for the rational design of more complex polypeptoid molecules, with a variety of applications, ranging from nanostructures to nonviral gene delivery systems.

Sequence-specific polypeptoids: a diverse family of heteropolymers with stable secondary structure

We have synthesized and characterized a family of structured oligo-N-substituted-glycines (peptoids) up to 36 residues in length by using an efficient solid-phase protocol to incorporate chemically diverse side chains in a sequence-specific fashion. We investigated polypeptoids containing side chains with a chiral center adjacent to the main chain nitrogen. Some of these sequences have stable secondary structure, despite the achirality of the polymer backbone and its lack of hydrogen bond donors. In both aqueous and organic solvents, peptoid oligomers as short as five residues give rise to CD spectra that strongly resemble those of peptide alpha-helices. Differential scanning calorimetry and CD measurements show that polypeptoid secondary structure is highly stable and that unfolding is reversible and cooperative. Thermodynamic parameters obtained for unfolding are similar to those obtained for the alpha-helix to coil transitions of peptides. This class of biomimetic polymers may enable the design of self-assembling macromolecules with novel structures and functions.

Protein folding in the landscape perspective: chevron plots and non-Arrhenius kinetics

We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have "chevron plot" dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponential behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process.

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.

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.

Solvation: how to obtain microscopic energies from partitioning and solvation experiments

Oil-water partitioning, solubilities, and vapor pressure experiments on small-molecule compounds are often used as models to obtain energies for biomolecular modeling. For example, measured partition coefficients, K, are often inserted into the formula -RT in K to obtain quantities thought to represent microscopic contact interaction free energies. We review evidence here that this procedure does not always give microscopically meaningful free energies. Some partitioning processes, particularly involving polymeric solvents such as octanol or hexadecane, are governed not only by translational entropies and contact interactions, but also by free energies resulting from changes in the conformations of the polymer chains upon solute insertion. The Flory-Huggins theory is more suitable for these situations than is the classical approach. We discuss the physical bases for both approaches.

From Levinthal to pathways to funnels

While the classical view of protein folding kinetics relies on phenomenological models, and regards folding intermediates in a structural way, the new view emphasizes the ensemble nature of protein conformations. Although folding has sometimes been regarded as a linear sequence of events, the new view sees folding as parallel microscopic multi-pathway diffusion-like processes. While the classical view invoked pathways to solve the problem of searching for the needle in the haystack, the pathway idea was then seen as conflicting with Anfinsen's experiments showing that folding is pathway-independent (Levinthal's paradox). In contrast, the new view sees no inherent paradox because it eliminates the pathway idea: folding can funnel to a single stable state by multiple routes in conformational space. The general energy landscape picture provides a conceptual framework for understanding both two-state and multi-state folding kinetics. Better tests of these ideas will come when new experiments become available for measuring not just averages of structural observables, but also correlations among their fluctuations. At that point we hope to learn much more about the real shapes of protein folding landscapes.

An iterative method for extracting energy-like quantities from protein structures

We present a method (ENERGI) for extracting energy-like quantities from a data base of protein structures. In this paper, we use the method to generate pairwise additive amino acid "energy" scores. These scores are obtained by iteration until they correctly discriminate a set of known protein folds from decoy conformations. The method succeeds in lattice model tests and in the gapless threading problem as defined by Maiorov and Crippen [Maiorov, V. N. & Crippen, G. M. (1992) J. Mol. Biol. 227, 876-888]. A more challenging test of threading a larger set of test proteins derived from the representative set of Hobohm and Sander [Hobohm, U. & Sander, C. (1994) Protein Sci. 3, 522-524] is used as a "workbench" for exploring how the ENERGI scores depend on their parameter sets.

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.

Binding of ionic ligands to polyelectrolytes

Ionic ligands can bind to polyelectrolytes such as DNA or charged polysaccharides. We develop a Poisson-Boltzmann treatment to compute binding constants as a function of ligand charge and salt concentration in the limit of low ligand concentration. For flexible chain ligands, such as oligopeptides, we treat their conformations using lattice statistics. The theory predicts the salt dependence and binding free energies, of Mg(2+) ions to polynucleotides, of hexamine cobalt(III) to calf thymus DNA, of polyamines to T7 DNA, of oligolysines to poly(U) and poly(a), and of tripeptides to heparin, a charged polysaccharide. One parameter is required to obtain absolute binding constants, the distance of closest separation of the ligand to the polyion. Some, but not all, of the binding entropies and enthalpies are also predicted accurately by the model.

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.

Comparing folding codes for proteins and polymers

Proteins fold to unique compact native structures. Perhaps other polymers could be designed to fold in similar ways. The chemical nature of the monomer "alphabet" determines the "energy matrix" of monomer interactions-which defines the folding code, the relationship between sequence and structure. We study two properties of energy matrices using two-dimensional lattice models: uniqueness, the number of sequences that fold to only one structure, and encodability, the number of folds that are unique lowest-energy structures of certain monomer sequences. For the simplest model folding code, involving binary sequences of H (hydrophobic) and P (polar) monomers, only a small fraction of sequences fold uniquely, and not all structures can be encoded. Adding strong repulsive interactions results in a folding code with more sequences folding uniquely and more designable folds. Some theories suggest that the quality of a folding code depends only on the number of letters in the monomer alphabet, but we find that the energy matrix itself can be at least as important as the size of the alphabet. Certain multi-letter codes, including some with 20 letters, may be less physical or protein-like than codes with smaller numbers of letters because they neglect correlations among inter-residue interactions, treat only maximally compact conformations, or add arbitrary energies to the energy matrix.

A simple model of chaperonin-mediated protein folding

Chaperonins are oligomeric proteins that help other proteins fold. They act, according to the "Anfinsen cage" or "box of infinite dilution" model, to provide private space, protected from aggregation, where a protein can fold. Recent evidence indicates, however, that proteins are often ejected from the GroEL chaperonin in nonnative conformations, and repeated cycles of binding and ejection are needed for successful folding. Some experimental evidence suggests that GroEL chaperonins can act as folding "catalysts" in an ATP-dependent manner even when no aggregation takes place. This implies that chaperonins must somehow recognize the kinetically trapped intermediate states of a protein. A central puzzle is how a chaperonin can catalyze the folding reaction of a broad spectrum of different proteins. We propose a physical mechanism by which chaperonins can flatten the energy barriers to folding in a nonspecific way. Using a lattice model, we illustrate how a chaperonin could provide a sticky surface that helps pull apart an incorrectly folded protein so it can try again to fold. Depending on the relative sizes of the protein and the chaperonin cavity, folding can proceed both inside and outside the chaperonin. Consistent with experiments, we find that the folding rate and amount of native protein can be considerably enhanced, or sometimes reduced, depending on the amino acid sequence, the chaperonin size, and the binding and ejection rates from the chaperonin.

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.

Designing amino acid sequences to fold with good hydrophobic cores

We present two methods for designing amino acid sequences of proteins that will fold to have good hydrophobic cores. Given the coordinates of the desired target protein or polymer structure, the methods generate sequences of hydrophobic (H) and polar (P) monomers that are intended to fold to these structures. One method designs hydrophobic inside, polar outside; the other minimizes an energy function in a sequence evolution process. The sequences generated by these methods agree at the level of 60-80% of the sequence positions in 20 proteins in the Protein Data Bank. A major challenge in protein design is to create sequences that can fold uniquely, i.e. to a single conformation rather than to many. While an earlier lattice-based sequence evolution method was shown not to design unique folders, our method generates unique folders in lattice model tests. These methods may also be useful in designing other types of foldable polymer not based on amino acids.

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.

A simple protein folding algorithm using a binary code and secondary structure constraints

We describe an algorithm to predict tertiary structures of small proteins. In contrast to most current folding algorithms, it uses very few energy parameters. Given the secondary structural elements in the sequence--alpha-helices and beta-strands--the algorithm searches the remaining conformational space of a simplified real-space representation of chains to find a minimum energy of an exceedingly simple potential function. The potential is based only on a single type of favorable interaction between hydrophobic residues, an unfavorable excluded volume term of spatial overlaps and, for sheet proteins, an interstrand hydrogen bond interaction. Where appropriate, the known disulfide bonds are constrained by a square-law potential. Conformations are searched by a genetic algorithm. The model predicts reasonably well the known tertiary folds of seven out of the 10 small proteins we consider. We draw two conclusions. First, for the proteins we tested, this exceedingly simple potential function is no worse than others having hundreds of energy parameters in finding the right general tertiary structures. Second, despite its simplicity, the potential function is not the weak link in this algorithm. Differences between our predicted structures and the correct targets can be ascribed to shortcomings in our search strategy. This potential function may be useful for testing other conformational search strategies.

Models of cooperativity in protein folding

What is the basis for the two-state cooperativity of protein folding? Since the 1950s, three main models have been put forward. 1. In 'helix-coil' theory, cooperativity is due to local interactions among near neighbours in the sequence. Helix-coil cooperativity is probably not the principal basis for the folding of globular proteins because it is not two-state, the forces are weak, it does not account for sheet proteins, and there is no evidence that helix formation precedes the formation of a hydrophobic core in the following pathways. 2. In the 'sidechain packing' model, cooperativity is attributed to the jigsaw-puzzle-like complementary fits of sidechains. This too is probably not the basis of folding cooperativity because exact models and experiments on homopolymers with sidechains give no evidence that sidechain freezing is two-state, sidechain complementarities in proteins are only weak trends, and the molten globule model predicted by this model is far more native-like than experiments indicate. 3. In the 'hydrophobic core collapse' model, cooperativity is due to the assembly of non-polar residues into a good core. Exact model studies show that this model gives two-state behaviour for some sequences of hydrophobic and polar monomers. It is based on strong forces. There is considerable experimental evidence for the kinetics this model predicts: the development of hydrophobic clusters and cores is concurrent with secondary structure formation. It predicts compact denatured states with sizes and degrees of disorder that are in reasonable agreement with experiments.

Principles of protein folding--a perspective from simple exact models

General principles of protein structure, stability, and folding kinetics have recently been explored in computer simulations of simple exact lattice models. These models represent protein chains at a rudimentary level, but they involve few parameters, approximations, or implicit biases, and they allow complete explorations of conformational and sequence spaces. Such simulations have resulted in testable predictions that are sometimes unanticipated: The folding code is mainly binary and delocalized throughout the amino acid sequence. The secondary and tertiary structures of a protein are specified mainly by the sequence of polar and nonpolar monomers. More specific interactions may refine the structure, rather than dominate the folding code. Simple exact models can account for the properties that characterize protein folding: two-state cooperativity, secondary and tertiary structures, and multistage folding kinetics--fast hydrophobic collapse followed by slower annealing. These studies suggest the possibility of creating "foldable" chain molecules other than proteins. The encoding of a unique compact chain conformation may not require amino acids; it may require only the ability to synthesize specific monomer sequences in which at least one monomer type is solvent-averse.