Modification and optimization of the united-residue (UNRES) potential energy function for canonical simulations. I. Temperature dependence of the effective energy function and tests of the optimization method with single training proteins

We report the modification and parametrization of the united-residue (UNRES) force field for energy-based protein structure prediction and protein folding simulations. We tested the approach on three training proteins separately: 1E0L (beta), 1GAB (alpha), and 1E0G (alpha + beta). Heretofore, the UNRES force field had been designed and parametrized to locate native-like structures of proteins as global minima of their effective potential energy surfaces, which largely neglected the conformational entropy because decoys composed of only lowest-energy conformations were used to optimize the force field. Recently, we developed a mesoscopic dynamics procedure for UNRES and applied it with success to simulate protein folding pathways. However, the force field turned out to be largely biased toward -helical structures in canonical simulations because the conformational entropy had been neglected in the parametrization. We applied the hierarchical optimization method, developed in our earlier work, to optimize the force field; in this method, the conformational space of a training protein is divided into levels, each corresponding to a certain degree of native-likeness. The levels are ordered according to increasing native-likeness; level 0 corresponds to structures with no native-like elements, and the highest level corresponds to the fully native-like structures. The aim of optimization is to achieve the order of the free energies of levels, decreasing as their native-likeness increases. The procedure is iterative, and decoys of the training protein(s) generated with the energy function parameters of the preceding iteration are used to optimize the force field in a current iteration. We applied the multiplexing replica-exchange molecular dynamics (MREMD) method, recently implemented in UNRES, to generate decoys; with this modification, conformational entropy is taken into account. Moreover, we optimized the free-energy gaps between levels at temperatures corresponding to a predominance of folded or unfolded structures, as well as to structures at the putative folding-transition temperature, changing the sign of the gaps at the transition temperature. This enabled us to obtain force fields characterized by a single peak in the heat capacity at the transition temperature. Furthermore, we introduced temperature dependence to the UNRES force field; this is consistent with the fact that it is a free-energy and not a potential energy function. beta     

Enhanced free-energy calculation using multiscale simulation

We propose a multiscale simulation method combining the efficiency of a coarse-grained model (CGM) and the accuracy of an all-atom model (AAM) for free-energy landscape calculation of protein systems. A protein's conformation space is quickly searched first using CGM. Then the obtained information is incorporated into AAM simulations. The free-energy landscape is subsequently obtained from AAM simulations. This method was tested on chignolin folding. The results demonstrated that the computational time was reduced by as much as 90%.     

Thermodynamics of Go-type models for protein folding

Go-type potentials, based on the inter-residue contacts present in the native structure of a protein, are frequently used to predict dynamic and structural features of the folding pathways through computer simulations. However, the mathematical form used to define the model interactions includes several arbitrary choices, whose consequences are not usually analyzed. In this work, we use a simple off-lattice protein model and a parallel tempering Monte Carlo simulation technique to carry out such analysis, centered in the thermodynamic characteristics of the folding transition. We show how the definition of a native contact has a deep impact on the presence of simple or complex transitions, with or without thermodynamic intermediates. In addition, we have checked that the width of the attractive wells has a profound effect on the free-energy barrier between the folded and unfolded states, mainly through its influence on the entropy of the denatured state.     

Folding kinetics of a lattice protein via a forward flux sampling approach

We implement a forward flux sampling approach [R. J. Allen et al., J. Chem. Phys. 124, 194111 (2006)] for calculating transition rate constants and for sampling paths of protein folding events. The algorithm generates trajectories for the transition between the unfolded and folded states as chains of partially connected paths, which can be used to obtain the transition-state ensemble and the properties that characterize these intermediates. We apply this approach to Monte Carlo simulations of a model lattice protein in open space and in confined spaces of varying dimensions. We study the effect of confinement on both protein thermodynamic stability and folding kinetics; the former by mapping free-energy landscapes and the latter by the determination of rate constants and mechanistic details of the folding pathway. Our results show that, for the range of temperatures where the native state is stable, confinement of a protein destabilizes the unfolded state by reducing its entropy, resulting in increased thermodynamic stability of the folded state. Relative to the folding in open space, we find that the kinetics can be accelerated at temperatures above the temperature at which the unconfined protein folds fastest and that the rate constant increases with the number of constrained dimensions. By examining the statistical properties of the transition-state ensemble, we detect signs of a classical nucleation folding mechanism for a core of native contacts formed at an early stage of the process. This nucleus acts as folding foci and is composed of those residues that have higher probability to form native contacts in the transition-state intermediates, which can vary depending on the confinement conditions of the system.     

Estimation of protein folding rate from Monte Carlo simulations and entropy capacity

The problem of protein self-organization is one of the most important problems of molecular biology nowadays. Despite the recent success in the understanding of general principles of protein folding, details of this process are yet to be elucidated. Moreover, the prediction of protein folding rates has its own practical value due to the fact that aggregation directly depends on the rate of protein folding. The time of folding has been calculated for 67 proteins with known experimental data at the point of thermodynamic equilibrium between unfolded and native states using a Monte Carlo model where each residue is considered to be either folded as in the native state or completely disordered. The times of folding for 67 proteins which reach the native state within the limit of 10(8) Monte Carlo steps are in a good correlation with the experimentally measured folding rate at the mid-transition point (the correlation coefficient is -0.82). Theoretical consideration of a capillarity model for the process of protein folding demonstrates that the difference in the folding rate for proteins sharing more spherical and less spherical folds is the result of differences in the conformational entropy due to a larger surface of the boundary between folded and unfolded phases in the transition state for proteins with more spherical fold. The capillarity model allows us to predict the folding rate at the same level of correlation as by Monte Carlo simulations. The calculated model entropy capacity (conformational entropy per residue divided by the average contact energy per residue) for 67 proteins correlates by about 78% with the experimentally measured folding rate at the mid-transition point.     

Basin hopping simulations for all-atom protein folding

We investigate different protocols of the basin hopping technique for de novo protein folding. Using the protein free-energy force field PFF01 we report the reproducible all-atom folding of the 20-amino-acid tryptophan-cage protein [Protein Data Bank (PDB) code: 112y] and of the recently discovered 26-amino-acid potassium channel blocker (PDB code: 1wqc), which exhibits an unusual fold. We find that simulations with increasing cycle length and random starting temperatures perform best in comparison with other parametrizations. The basin hopping technique emerges as a simple but very efficient and robust workhorse for all-atom protein folding.     

Exploration of two-dimensional hydrophobic-polar lattice model by combining local search with elastic net algorithm

Determining the functional conformation of a protein from its amino acid sequence remains a central problem in computational biology. In this paper, we establish the mathematical optimal model of protein folding problem (PFP) on two-dimensional space based on the minimal energy principle. A novel hybrid of elastic net algorithm and local search method (ENLS) is applied successfully to simulations of protein folding on two-dimensional hydrophobic-polar (HP) lattice model. Eight HP benchmark instances with up to 64 amino acids are tested to verify the effectiveness of proposed approach and model. In several cases, the ENLS method finds new lower energy states. The numerical results show that it is drastically superior to other methods in finding the ground state of a protein.     

Folding transition-state and denatured-state ensembles of FSD-1 from folding and unfolding simulations

Characterization of the folding transition-state ensemble and the denatured-state ensemble is an important step toward a full elucidation of protein folding mechanisms. We report herein an investigation of the free-energy landscape of FSD-1 protein by a total of four sets of folding and unfolding molecular dynamics simulations with explicit solvent. The transition-state ensemble was initially identified from unfolding simulations at 500 K and was verified by simulations at 300 K starting from the ensemble structures. The denatured-state ensemble and the early-stage folding were studied by a combination of unfolding simulations at 500 K and folding simulations at 300 K starting from the extended conformation. A common feature of the transition-state ensemble was the substantial formation of the native secondary structures, including both the alpha-helix and beta-sheet, with partial exposure of the hydrophobic core in the solvent. Both the native and non-native secondary structures were observed in the denatured-state ensemble and early-stage folding, consistent with the smooth experimental melting curve. Interestingly, the contact orders of the transition-state ensemble structures were similar to that of the native structure and were notably lower than those of the compact structures found in early-stage folding, implying that chain and topological entropy might play significant roles in protein folding. Implications for FSD-1 folding mechanisms and the rate-limiting step are discussed. Analyses further revealed interesting non-native interactions in the denatured-state ensemble and early-stage folding and the possibility that destabilization of these interactions could help to enhance the stability and folding rate of the protein.     

Theoretical analysis on changes in thermodynamic quantities upon protein folding: essential role of hydration

The free energy change associated with the coil-to-native structural transition of protein G in aqueous solution is calculated by using the molecular theory of solvation, also known as the three-dimensional reference interaction site model theory, to uncover the molecular mechanism of protein folding. The free energy is decomposed into the protein intramolecular energy, the hydration energy, and the hydration entropy. The folding is accompanied with a large gain in the protein intramolecular energy. However, it is almost canceled by the correspondingly large loss in the hydration energy due to the dehydration, resulting in the total energy gain about an order of magnitude smaller than might occur in vacuum. The hydration entropy gain is found to be a substantial driving force in protein folding. It is comparable with or even larger than the total energy gain. The total energy gain coupled with the hydration entropy gain is capable of suppressing the conformational entropy loss in the folding. Based on careful analysis of the theoretical results, the authors present a challenging physical picture of protein folding where the overall folding process is driven by the water entropy effect.     

Discrete state model and accurate estimation of loop entropy of RNA secondary structures

Conformational entropy makes important contribution to the stability and folding of RNA molecule, but it is challenging to either measure or compute conformational entropy associated with long loops. We develop optimized discrete k-state models of RNA backbone based on known RNA structures for computing entropy of loops, which are modeled as self-avoiding walks. To estimate entropy of hairpin, bulge, internal loop, and multibranch loop of long length (up to 50), we develop an efficient sampling method based on the sequential Monte Carlo principle. Our method considers excluded volume effect. It is general and can be applied to calculating entropy of loops with longer length and arbitrary complexity. For loops of short length, our results are in good agreement with a recent theoretical model and experimental measurement. For long loops, our estimated entropy of hairpin loops is in excellent agreement with the Jacobson-Stockmayer extrapolation model. However, for bulge loops and more complex secondary structures such as internal and multibranch loops, we find that the Jacobson-Stockmayer extrapolation model has large errors. Based on estimated entropy, we have developed empirical formulae for accurate calculation of entropy of long loops in different secondary structures. Our study on the effect of asymmetric size of loops suggest that loop entropy of internal loops is largely determined by the total loop length, and is only marginally affected by the asymmetric size of the two loops. Our finding suggests that the significant asymmetric effects of loop length in internal loops measured by experiments are likely to be partially enthalpic. Our method can be applied to develop improved energy parameters important for studying RNA stability and folding, and for predicting RNA secondary and tertiary structures. The discrete model and the program used to calculate loop entropy can be downloaded at http://gila.bioengr.uic.edu/resources/RNA.html.     

Peptide chain dynamics in light and heavy water: zooming in on internal friction

Frictional effects due to the chain itself, rather than the solvent, may have a significant effect on protein dynamics. Experimentally, such "internal friction" has been investigated by studying folding or binding kinetics at varying solvent viscosity; however, the molecular origin of these effects is hard to pinpoint. We consider the kinetics of disordered glycine-serine and α-helix forming alanine peptides and a coarse-grained protein folding model in explicit-solvent molecular dynamics simulations. By varying the solvent mass over more than two orders of magnitude, we alter only the solvent viscosity and not the folding free energy. Folding dynamics at the near-vanishing solvent viscosities accessible by this approach suggests that solvent and internal friction effects are intrinsically entangled. This finding is rationalized by calculation of the polymer end-to-end distance dynamics from a Rouse model that includes internal friction. An analysis of the friction profile along different reaction coordinates, extracted from the simulation data, demonstrates that internal as well as solvent friction varies substantially along the folding pathways and furthermore suggests a connection between friction and the formation of hydrogen bonds upon folding.     

Changes in thermodynamic quantities upon contact of two solutes in solvent under isochoric and isobaric conditions

The changes in excess thermodynamic quantities upon the contact of two solutes immersed in a solvent are analyzed using the radial-symmetric and three-dimensional versions of the integral equation theory. A simple model mimicking a solute in water is employed. The solute-solute interaction energy is not included in the calculations. Under the isochoric condition, the solute contact always leads to a positive entropy change irrespective of the solute solvophobicity or solvophilicity. The energy change is negative for solvophobic solutes while it is positive for solvophilic ones. Under the isobaric condition, the contact of solvophobic solutes results in system-volume compression but that of solvophilic ones gives rise to expansion. Effects of the compression and expansion on the changes in enthalpy and entropy are enlarged with rising temperature. When the solute solvophobicity is sufficiently high, the entropy change (multiplied by the absolute temperature) can become negative due to the compression, except at low temperatures with the result of an even larger, negative enthalpy change. The expansion in the case of solvophilic solutes leads to a large, positive entropy change accompanied by an even larger, positive enthalpy change. The changes in enthalpy and entropy are strongly dependent on the temperature. However, the changes in enthalpy and entropy are largely cancelled out and the temperature dependency of the free-energy change is much weaker. The authors also discuss possible relevance to the enthalpy-entropy compensation experimentally known for a variety of physicochemical processes in aqueous solution such as protein folding.     

Dynamic control of protein folding pathway with a polymer of tunable hydrophobicity

While the knowledge of protein folding in a dilute solution is now well-advanced, little is known of the influence of surrounding conditions on the folding kinetics, in particular when the protein is in a dynamically responsive environment. Here we report a new procedure to control the pathways of protein folding by using a thermally responsive polymer that varies its hydrophobicity concomitant with the protein structural changes. The advantages of folding in a dynamic environment have been demonstrated first by Langevin dynamics simulations on the basis of coarse-grained models for both the protein and polymer and then by experiments for lysozyme refolding in the presence of poly(N-isopropylacrylamide-co-N-tert-butylacrylamide), a thermal responsive polymer that varies its hydrophobicity in response to temperature. The simulation suggests that decreasing the polymer hydrophobicity during the folding process may result in an optimized free-energy landscape that enhances both the folding yield and kinetics. The experiments affirm that an optimal folding condition can be identified when structural transitions of the protein collaborate with the polymer hydrophobicity tuned by variation of temperature.     

QM/MM-PBSA method to estimate free energies for reactions in proteins

We have developed a method to estimate free energies of reactions in proteins, called QM/MM-PBSA. It estimates the internal energy of the reactive site by quantum mechanical (QM) calculations, whereas bonded, electrostatic, and van der Waals interactions with the surrounding protein are calculated at the molecular mechanics (MM) level. The electrostatic part of the solvation energy of the reactant and the product is estimated by solving the Poisson-Boltzmann (PB) equation, and the nonpolar part of the solvation energy is estimated from the change in solvent-accessible surface area (SA). Finally, the change in entropy is estimated from the vibrational frequencies. We test this method for five proton-transfer reactions in the active sites of [Ni,Fe] hydrogenase and copper nitrite reductase. We show that QM/MM-PBSA reproduces the results of a strict QM/MM free-energy perturbation method with a mean absolute deviation (MAD) of 8-10 kJ/mol if snapshots from molecular dynamics simulations are used and 4-14 kJ/mol if a single QM/MM structure is used. This is appreciably better than the original QM/MM results or if the QM energies are supplemented with a point-charge model, a self-consistent reaction field, or a PB model of the protein and the solvent, which give MADs of 22-36 kJ/mol for the same test set.     

Thermodynamics of catalytic formation of dimethyl ether from methanol in acidic zeolites

We present a theoretical study of the formation of the first intermediate, dimethyl ether, in the methanol to gasoline conversion within the framework of an ab initio molecular dynamics approach. The study is performed under conditions that closely resemble the reaction conditions in the zeolite catalyst including the full topology of the framework. The use of the method of thermodynamic integration allows us to extract the free-energy profile along the reaction coordinate. We find that the entropic contribution qualitatively alters the free-energy profile relative to the total energy profile. Different transition states are found from the internal and free energy profiles. The entropy contribution varies significantly along the reaction coordinate and is responsible for stabilizing the products and for lowering the energy barrier. The hugely inhomogeneous variation of the entropy can be understood in terms of elementary processes that take place during the chemical reaction. Our simulations provide new insights into the complex nature of this chemical reaction.     

The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics

Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.     

Thermodynamic functions of conformational changes: conformational network of glycine diamide folding, entropy lowering, and informational accumulation

A refined grid of a conformational potential energy surface (PES) and a conformational entropy surface for glycine diamide was generated by ab initio molecular computations. The possible network of reaction paths was recognized in terms of the linear combinations of internal coordinates corresponding to conrotatory and disrotatory modes of motions. Such a Woodward-Hoffmann-like path selection principle was detected for the folding of this peptide from extended to some virtually cyclic structure. It seemed reasonable to assume that this principle (or its generalized form) might be applicable to protein folding. A reaction path network was projected on the potential energy, and a continuous entropy surface was constructed under the condition of reduced dimensionality. The low entropy of the folded conformation indicated an information accumulation between 326% and 1414% with respect to the fully extended or unfolded structure. It is found that the location of existing and 'latent' critical points on the surface is revealed by the extrema and inflection points of the entropy curve.     

Solvent viscosity dependence of the protein folding dynamics

Solvent viscosity has been frequently adopted as an adjustable parameter in various computational studies (e.g., protein folding simulations) with implicit solvent models. A common approach is to use low viscosities to expedite simulations. While using viscosities lower than that of aqueous is unphysical, such treatment is based on observations that the viscosity affects the kinetics (rates) in a well-defined manner as described by Kramers' theory. Here, we investigate the effect of viscosity on the detailed dynamics (mechanism) of protein folding. On the basis of a simple mathematical model, we first show that viscosity may indeed affect the dynamics in a complex way. By applying the model to the folding of a small protein, we demonstrate that the detailed dynamics is affected rather pronouncedly especially at unphysically low viscosities, cautioning against using such viscosities. In this regard, our model may also serve as a diagnostic tool for validating low-viscosity simulations. It is also suggested that the viscosity dependence can be further exploited to gain information about the protein folding mechanism.