Molecular theory of the helix-coil transition in polyamino acids, I. Formulation

The Iconic α-Helix: From Pauling to the Present

The protein folding problem dates back to Pauling's insights almost a century ago, but the first venture into actual protein structure was the Pauling-Corey-Brandson α-helix in 1951, a proposed model that was confirmed almost immediately using X-ray crystallography. Many subsequent efforts to predict protein helices from the amino acid sequence met with only partial success, as discussed here. Surprisingly, in 2021, these efforts were superseded by deep-learning artificial intelligence, especially AlphaFold2, a machine learning program based on neural nets. This approach can predict most protein structures successfully at or near atomic resolution. Deservedly, deep-learning artificial intelligence was named Science magazine's 2021 "breakthrough of the year." Today, ~200 million predicted protein structures can be downloaded from the AlphaFold2 Protein Structure Database. Deep learning represents a deep conundrum because these successfully predicted macromolecular structures are based on methods that are completely devoid of a hypothesis or of any physical chemistry. Perhaps we are now poised to transcend five centuries of reductive science.

Entropy in bimolecular simulations: A comprehensive review of atomic fluctuations-based methods

Entropy of binding constitutes a major, and in many cases a detrimental, component of the binding affinity in biomolecular interactions. While the enthalpic part of the binding free energy is easier to calculate, estimating the entropy of binding is further more complicated. A precise evaluation of entropy requires a comprehensive exploration of the complete phase space of the interacting entities. As this task is extremely hard to accomplish in the context of conventional molecular simulations, calculating entropy has involved many approximations. Most of these golden standard methods focused on developing a reliable estimation of the conformational part of the entropy. Here, we review these methods with a particular emphasis on the different techniques that extract entropy from atomic fluctuations. The theoretical formalisms behind each method is explained highlighting its strengths as well as its limitations, followed by a description of a number of case studies for each method. We hope that this brief, yet comprehensive, review provides a useful tool to understand these methods and realize the practical issues that may arise in such calculations.

My 65 years in protein chemistry

This is a tour of a physical chemist through 65 years of protein chemistry from the time when emphasis was placed on the determination of the size and shape of the protein molecule as a colloidal particle, with an early breakthrough by James Sumner, followed by Linus Pauling and Fred Sanger, that a protein was a real molecule, albeit a macromolecule. It deals with the recognition of the nature and importance of hydrogen bonds and hydrophobic interactions in determining the structure, properties, and biological function of proteins until the present acquisition of an understanding of the structure, thermodynamics, and folding pathways from a linear array of amino acids to a biological entity. Along the way, with a combination of experiment and theoretical interpretation, a mechanism was elucidated for the thrombin-induced conversion of fibrinogen to a fibrin blood clot and for the oxidative-folding pathways of ribonuclease A. Before the atomic structure of a protein molecule was determined by x-ray diffraction or nuclear magnetic resonance spectroscopy, experimental studies of the fundamental interactions underlying protein structure led to several distance constraints which motivated the theoretical approach to determine protein structure, and culminated in the Empirical Conformational Energy Program for Peptides (ECEPP), an all-atom force field, with which the structures of fibrous collagen-like proteins and the 46-residue globular staphylococcal protein A were determined. To undertake the study of larger globular proteins, a physics-based coarse-grained UNited-RESidue (UNRES) force field was developed, and applied to the protein-folding problem in terms of structure, thermodynamics, dynamics, and folding pathways. Initially, single-chain and, ultimately, multiple-chain proteins were examined, and the methodology was extended to protein-protein interactions and to nucleic acids and to protein-nucleic acid interactions. The ultimate results led to an understanding of a variety of biological processes underlying natural and disease phenomena.

Estimation of conformational entropy in protein-ligand interactions: a computational perspective

Conformational entropy is an important component of the change in free energy upon binding of a ligand to its target protein. As a consequence, development of computational techniques for reliable estimation of conformational entropies is currently receiving an increased level of attention in the context of computational drug design. Here, we review the most commonly used techniques for conformational entropy estimation from classical molecular dynamics simulations. Although by-and-large still not directly used in practical drug design, these techniques provide a golden standard for developing other, computationally less-demanding methods for such applications, in addition to furthering our understanding of protein-ligand interactions in general. In particular, we focus on the quasi-harmonic approximation and discuss different approaches that can be used to go beyond it, most notably, when it comes to treating anharmonic and/or correlated motions. In addition to reviewing basic theoretical formalisms, we provide a concrete set of steps required to successfully calculate conformational entropy from molecular dynamics simulations, as well as discuss a number of practical issues that may arise in such calculations.

From helix-coil transitions to protein folding

An evolution of procedures to simulate protein structure and folding pathways is described. From an initial focus on the helix-coil transition and on hydrogen-bonding and hydrophobic interactions, our original attempts to determine protein structure and folding pathways were based on an experimental approach. Experiments on the oxidative folding of reduced bovine pancreatic ribonuclease A (RNase A) led to a mechanism by which the molecule folded to the native structure by a minimum of four different pathways. The experiments with RNase A were followed by development of a molecular mechanics approach, first, making use of global optimization procedures and then with molecular dynamics (MD), evolving from an all-atom to a united-residue model. This hierarchical MD approach facilitated probing of the folding trajectory to longer time scales than with all-atom MD, and hence led to the determination of complete folding trajectories, thus far for a protein containing as many as 75 amino acid residues. With increasing refinement of the computational procedures, the computed results are coming closer to experimental observations, providing an understanding as to how physics directs the folding process.

Role of binding entropy in the refinement of protein-ligand docking predictions: analysis based on the use of 11 scoring functions

We present results of testing the ability of eleven popular scoring functions to predict native docked positions using a recently developed method (Ruvinsky and Kozintsev, J Comput Chem 2005, 26, 1089) for estimation the entropy contributions of relative motions to protein-ligand binding affinity. The method is based on the integration of the configurational integral over clusters obtained from multiple docked positions. We use a test set of 100 PDB protein-ligand complexes and ensembles of 101 docked positions generated by (Wang et al. J Med Chem 2003, 46, 2287) for each ligand in the test set. To test the suggested method we compared the averaged root-mean square deviations (RMSD) of the top-scored ligand docked positions, accounting and not accounting for entropy contributions, relative to the experimentally determined positions. We demonstrate that the method increases docking accuracy by 10-21% when used in conjunction with the AutoDock scoring function, by 2-25% with G-Score, by 7-41% with D-Score, by 0-8% with LigScore, by 1-6% with PLP, by 0-12% with LUDI, by 2-8% with F-Score, by 7-29% with ChemScore, by 0-9% with X-Score, by 2-19% with PMF, and by 1-7% with DrugScore. We also compared the performance of the suggested method with the method based on ranking by cluster occupancy only. We analyze how the choice of a clustering-RMSD and a low bound of dense clusters impacts on docking accuracy of the scoring methods. We derive optimal intervals of the clustering-RMSD for 11 scoring functions.

New and fast statistical-thermodynamic method for computation of protein-ligand binding entropy substantially improves docking accuracy

We present a novel method to estimate the contributions of translational and rotational entropy to protein-ligand binding affinity. The method is based on estimates of the configurational integral through the sizes of clusters obtained from multiple docking positions. Cluster sizes are defined as the intervals of variation of center of ligand mass and Euler angles in the cluster. Then we suggest a method to consider the entropy of torsional motions. We validate the suggested methods on a set of 135 PDB protein-ligand complexes by comparing the averaged root-mean square deviations (RMSD) of the top-scored ligand docked positions, accounting and not accounting for entropy contributions, relative to the experimentally determined positions. We demonstrate that the method increases docking accuracy by 10-21% when used in conjunction with the AutoDock docking program, thus reducing the percent of incorrectly docked ligands by 1.4-fold to four-fold, so that in some cases the percent of ligands correctly docked to within an RMSD of 2 A is above 90%. We show that the suggested method to account for entropy of relative motions is identical to the method based on the Monte Carlo integration over intervals of variation of center of ligand mass and Euler angles in the cluster.

Helix-coil transitions re-visited

The thermally-induced helix-coil transition in polyamino acids is a good model for determining the helix-forming propensities of amino acids but not for the two-state folding/unfolding transition in globular proteins. The equilibrium and kinetic treatments of the helix-coil transition are summarized here together with a description of applications to various types of homopolymers and copolymers. Attention is then focused on the helix-coil transition in poly-L-alanine as an example of a non-polar polyamino acid. To render such a non-polar polymer water soluble, it is necessary to introduce polar amino acids such as lysines, but care must be taken as to the location of such polar residues. If they are attached as end groups, as in a triblock copolymer, they do not perturb the helix-forming tendency of the central poly-L-alanine block significantly, but if they are introduced within the sequence of alanine residues, then the hydration properties of the lysines dominate the behavior of the resulting copolymer, thereby leading to erroneous values of the parameters characterizing the helix-forming tendency of the alanines. Neutral but polar residues, such as glutamines, also exhibit hydration-dominating properties but less so than charged lysines. Some details of the calculations for an alanine/glutamine copolymer are presented here. It is concluded that random copolymers based on a neutral water-soluble host provide reliable information about the helix-forming tendencies of amino acid residues that are introduced as guests among such neutral host residues.

Evolution of physics-based methodology for exploring the conformational energy landscape of proteins

The evolution of our physics-based computational methods for determining protein conformation without the introduction of secondary-structure predictions, homology modeling, threading, or fragment coupling is described. Initial use of a hard-sphere potential captured much of the structural properties of polypeptide chains, and subsequent more refined force fields, together with efficient methods of global optimization provide indications that progress is being made toward an understanding of the interresidue interactions that underlie protein folding.

Solvophobically driven folding of nonbiological oligomers

In solution, biopolymers commonly fold into well-defined three-dimensional structures, but only recently has analogous behavior been explored in synthetic chain molecules. An aromatic hydrocarbon backbone is described that spontaneously acquires a stable helical conformation having a large cavity. The chain does not form intramolecular hydrogen bonds, and solvophobic interactions drive the folding transition, which is sensitive to chain length, solvent quality, and temperature.

The helix-coil transition in polypeptides: a microscopic approach. II

In the framework of an earlier constructed model [N.S. Ananikyan et al. (1990) Biopolymers, Vol. 30, pp. 357-367], some analytical estimates for the correlation length and degree of helicity near the transition point were obtained in the case of an arbitrary topology of hydrogen bond closing (delta). It was shown that the Zimm-Bragg cooperativity parameter sigma is determined by the set of (delta-1) amino acid residues and so is nonlocal. An analytic expression for cooperativity parameters in a heteropolypeptide chain was obtained and numerical calculations showed that in case of heteropolypeptide with random primary structure the nonlocality of cooperativity parameter influenced the temperature dependence of helicity degree.

Helix-forming tendencies of nonpolar amino acids predicted by Monte Carlo simulated annealing

Monte Carlo simulated annealing is applied to the study of the alpha-helix-forming tendencies of seven nonpolar amino acids, Ala, Leu, Met, Phe, Ile, Val, and Gly. Homooligomers of 10 amino acids are used and the helix tendency is calculated by folding alpha-helicies from completely random initial conformations. The results of the simulation imply that Met, Ala, and Leu are helix formers and that Val, Ile, and Gly are helix breakers, while Phe comes in between the two groups. The differences between helix formers and breakers turned out to be large in agreement with the recent experiments with short peptides. It is argued from the energy distributions of the obtained conformations that the helix tendency is small for the helix breakers because of steric hindrance of side chains. Homoglycine is shown to favor a random coil conformation. The beta-strand tendencies of the same homooligomers are also considered, and they are shown to agree with the frequencies of amino acids in beta-sheet from the protein data base.

Investigation of conformational equilibrium of polypeptides by internal coordinate stochastic dynamics. Met5-enkephalin

The equilibrium population of different conformational states of a polypeptide can in principle be obtained by a very long molecular dynamics simulation. The method of internal coordinate molecular dynamics earlier developed in this laboratory (A.K. Mazur and R.A. Abagyan J. Biomol. Struct. Dyn. 6,833 (1989)) allows one to use time steps much larger than usual for computing molecular trajectories. It is shown here that the sampling of the conformational space can be additionally enhanced by adding a random component to the set of forces applied to atoms. We describe the algorithms by which the random force is introduced and also a special method which excludes the fast rotation of polar hydrogens from equations of motion but keeps them movable. As a result the task stated in the title becomes realistic. Internal coordinate stochastic dynamics is applied for scanning the conformational space of the pentapeptide Met5-enkephalin which is a common test example widely used in theoretical studies. A large number of conformational transitions is observed during the 20 ns simulation starting from the global energy minimum thus allowing us to arrive at a nearly Boltzmann distribution of populations of conformational states. A few states are found which are distinguished by high apparent configurational entropy which turn out to correspond well to experimentally observed conformations of enkephalins.

Minimization of polypeptide energy. VII. Second derivatives and statistical weights of energy minima for deca-L-alanine

The matrix of second derivatives of 21 apparent local energy minima of deca-L-alanine has been computed. The results show that all these conformations are true local energy minima and, therefore, potentially stable conformations for this peptide. Calculations of the relative statistical weights of these conformations confirm earlier theoretical considerations on the importance of the librational free energy of stable conformations of peptides. As predicted earlier, the relative statistical weights of the local energy minima do not always fall in the same order as the relative energies.

A model of alpha-helical distribution in proteins

It is shown that alpha-helical content of eleven proteins is well correlated with alanine plus leucine content. These residues, taken singly or together, are to a first approximation randomly distributed in the four proteins whose tertiary structures have been determined (i.e., myoglobin, lysozyme, ribonuclease, alpha-chymotrypsin). A model based on the concept that certain randomly distributed residues specifically participate in helix nucleation is shown to be in reasonable agreement with the presently published structures.