Toward absolute density of states calculations for proteins

The density of states (DOS), which gives the number of conformations with a particular energy E, is a prerequisite in computing most thermodynamic quantities and in elucidating important biological processes such as the mechanism of protein folding. However, current methods for computing DOS of large systems such as proteins generally yield only the ratios of microstate counts for different energies, which could yield absolute conformation counts if the total number of conformations in phase space is known, thus motivating this work. Here, the total number of energy minima of 50-mer polyalanine, whose size corresponds to naturally occurring small proteins, was estimated under an all-atom potential energy function based on the cumulative distribution function (CDF) of conformational differences to be approximately 10(38). This estimate can place any DOS function, such as the Gaussian DOS distribution in the random energy model, on an absolute scale. Comparing the absolute conformational counts from a Gaussian DOS function with those from the CDF derived from quenched molecular dynamics ensembles shows that the former are far greater than the latter, indicating far fewer low-energy minima actually exist. In addition to showing how CDF and relative DOS calculations can yield absolute DOS for a discrete system, we also show how they can yield absolute DOS for continuous variable systems to a specified atomic variance. In the context of protein folding, knowing this phase-space "volume" of conformations in a DOS function, as well as characteristic transition times, constrains the set of possible folding mechanisms.     

The beta-peptide hairpin in solution: conformational study of a beta-hexapeptide in methanol by NMR spectroscopy and MD simulation.

The structural and thermodynamic properties of a 6-residue beta-peptide that was designed to form a hairpin conformation have been studied by NMR spectroscopy and MD simulation in methanol solution. The predicted hairpin would be characterized by a 10-membered hydrogen-bonded turn involving residues 3 and 4, and two extended antiparallel strands. The interproton distances and backbone torsional dihedral angles derived from the NMR experiments at room temperature are in general terms compatible with the hairpin conformation. Two trajectories of system configurations from 100-ns molecular-dynamics simulations of the peptide in solution at 298 and 340 K have been analyzed. In both simulations reversible folding to the hairpin conformation is observed. Interestingly, there is a significant conformational overlap between the unfolded state of the peptide at each of the temperatures. As already observed in previous studies of peptide folding, the unfolded state is composed of a (relatively) small number of predominant conformers and in this case lacks any type of secondary-structure element. The trajectories provide an excellent ground for the interpretation of the NMR-derived data in terms of ensemble averages and distributions as opposed to single-conformation interpretations. From this perspective, a relative population of the hairpin conformation of 20% to 30% would suffice to explain the NMR-derived data. Surprisingly, however, the ensemble of structures from the simulation at 340 K reproduces more accurately the NMR-derived data than the ensemble from the simulation at 298 K, a question that needs further investigation.     

Interpreting NMR data for beta-peptides using molecular dynamics simulations

NMR is one of the most used techniques to resolve structure of proteins and peptides in solution. However, inconsistencies may occur due to the fact that a polypeptide may adopt more than one conformation. Since the NOE distance bounds and (3)J-values used in such structure determination represent a nonlinear average over the total ensemble of conformers, imposition of NOE or (3)J-value restraints to obtain one unique conformation is not an appropriate procedure in such cases. Here, we show that unrestrained MD simulation of a solute in solution using a high-quality force field yields a conformational ensemble that is largely compatible with the experimental NMR data on the solute. Four 100 ns MD simulations of two forms of a nine-residue beta-peptide in methanol at two temperatures produced conformational ensembles that were used to interpret the NMR data on this molecule and resolve inconsistencies between the experimental NOEs. The protected and unprotected forms of the beta-peptide adopt predominantly a 12/10-helix in agreement with the qualitative interpretation of the NMR data. However, a particular NOE was not compatible with this helix indicating the presence of other conformations. The simulations showed that 3(14)()-helical structures were present in the ensemble of the unprotected form and that their presence correlates with the fulfillment of the particular NOE. Additionally, all inter-hydrogen distances were calculated to compare NOEs predicted by the simulations to the ones observed experimentally. The MD conformational ensembles allowed for a detailed and consistent interpretation of the experimental data and showed the small but specific conformational differences between the protected and unprotected forms of the peptide.     

Structure and dynamics of the homologous series of alanine peptides: a joint molecular dynamics/NMR study

The phi,psi backbone angle distribution of small homopolymeric model peptides is investigated by a joint molecular dynamics (MD) simulation and heteronuclear NMR study. Combining the accuracy of the measured scalar coupling constants and the atomistic detail of the all-atom MD simulations with explicit solvent, the thermal populations of the peptide conformational states are determined with an uncertainty of <5 %. Trialanine samples mainly ( approximately 90%) a poly-l-proline II helix-like structure, some ( approximately 10%) beta extended structure, but no alphaR helical conformations. No significant change in the distribution of conformers is observed with increasing chain length (Ala(3) to Ala(7)). Trivaline samples all three major conformations significantly. Triglycine samples the four corner regions of the Ramachandran space and exists in a slow conformational equilibrium between the cis and trans conformation of peptide bonds. The backbone angle distribution was also studied for the segment Ala3 surrounded by either three or eight amino acids on both N- and C-termini from a sequence derived from the protein hen egg white lysozyme. While the conformational distribution of the central three alanine residues in the 9mer is similar to that for the small peptides Ala(3)-Ala(7), major differences are found for the 19mer, which significantly (30-40%) samples alphaR helical stuctures.     

Hidden protein folding pathways in free-energy landscapes uncovered by network analysis

A network analysis is used to uncover hidden folding pathways in free-energy landscapes usually defined in terms of such arbitrary order parameters as root-mean-square deviation from the native structure, radius of gyration, etc. The analysis has been applied to molecular dynamics (MD) trajectories of the B-domain of staphylococcal protein A, generated with the coarse-grained united-residue (UNRES) force field in a broad range of temperatures (270K ≤ T ≤ 325K). Thousands of folding pathways have been identified at each temperature. Out of these many folding pathways, several most probable ones were selected for investigation of the conformational transitions during protein folding. Unlike other conformational space network (CSN) methods, a node in the CSN variant implemented in this work is defined according to the nativelikeness class of the structure, which defines the similarity of segments of the compared structures in terms of secondary-structure, contact-pattern, and local geometry, as well as the overall geometric similarity of the conformation under consideration to that of the reference (experimental) structure. Our previous findings, regarding the folding model and conformations found at the folding-transition temperature for protein A (Maisuradze et al., J. Am. Chem. Soc. 132, 9444, 2010), were confirmed by the conformational space network analysis. In the methodology and in the analysis of the results, the shortest path identified by using the shortest-path algorithm corresponds to the most probable folding pathway in the conformational space network.     

Conformational Dynamics of apo‐GlnBP Revealed by Experimental and Computational Analysis

The glutamine binding protein (GlnBP) binds l ‐glutamine and cooperates with its cognate transporters during glutamine uptake. Crystal structure analysis has revealed an open and a closed conformation for apo‐ and holo‐GlnBP, respectively. However, the detailed conformational dynamics have remained unclear. Herein, we combined NMR spectroscopy, MD simulations, and single‐molecule FRET techniques to decipher the conformational dynamics of apo‐GlnBP. The NMR residual dipolar couplings of apo‐GlnBP were in good agreement with a MD‐derived structure ensemble consisting of four metastable states. The open and closed conformations are the two major states. This four‐state model was further validated by smFRET experiments and suggests the conformational selection mechanism in ligand recognition of GlnBP.     

Enhanced sampling method in molecular simulations using genetic algorithm for biomolecular systems

We propose a molecular simulation method using genetic algorithm (GA) for biomolecular systems to obtain ensemble averages efficiently. In this method, we incorporate the genetic crossover, which is one of the operations of GA, to any simulation method such as conventional molecular dynamics (MD), Monte Carlo, and other simulation methods. The genetic crossover proposes candidate conformations by exchanging parts of conformations of a target molecule between a pair of conformations during the simulation. If the candidate conformations are accepted, the simulation resumes from the accepted ones. While conventional simulations are based on local update of conformations, the genetic crossover introduces global update of conformations. As an example of the present approach, we incorporated genetic crossover to MD simulations. We tested the validity of the method by calculating ensemble averages and the sampling efficiency by using two kinds of peptides, ALA3 and (AAQAA)₃. The results show that for ALA3 system, the distribution probabilities of backbone dihedral angles are in good agreement with those of the conventional MD and replica-exchange MD simulations. In the case of (AAQAA)₃ system, our method showed lower structural correlation of α-helix structures than the other two methods and more flexibility in the backbone ψ angles than the conventional MD simulation. These results suggest that our method gives more efficient conformational sampling than conventional simulation methods based on local update of conformations. © 2018 Wiley Periodicals, Inc.     

Atomistic detailed free‐energy landscape of intrinsically disordered protein studied by multi‐scale divide‐and‐conquer molecular dynamics simulation

Calcineurin (CaN) is a eukaryotic serine/threonine protein phosphatase activated by both Ca²⁺ and calmodulin (CaM), including intrinsically disordered region (IDR). The region undergoes folding into an α‐helix form in the presence Ca²⁺‐loaded CaM. To sample the ordered structure of the IDR by conventional all atom model (AAM) molecular dynamics (MD) simulation, the IDR and Ca²⁺‐loaded CaM must be simultaneously treated. However, it is time‐consuming task because the coupled folding and binding should include repeated binding and dissociation. Then, in this study, we propose novel multi‐scale divide‐and‐conquer MD (MSDC‐MD), which combines AAM‐MD and coarse‐grained model MD (CGM‐MD). To speed up the conformation sampling, MSDC‐MD simulation first treats the IDR by CGM to sample conformations from wide conformation space; then, multiple AAM‐MD in a limited area is initiated using the resultant CGM conformation, which is reconstructed by homology modeling method. To investigate performance, we sampled the ordered conformation of the IDR using MSDC‐MD; the root‐mean‐square distance (RMSD) with respect to the experimental structure was 2.23 Å.     

The relationship between peptide plane rotation (PPR) and similar conformations

Currently, several energy functions and conformational search methods have been developed that are based on the observed distribution of phi and psi angles in protein structures. The definition of phi and psi angles is directly related to the orientation of the peptide plane (CA CO NH CA). Starting from one conformation and rotating a single peptide plane, the angles psi for one residue and phi for the consecutive residue that are linked by the peptide plane, display a continuous range of values within one global conformation. When peptide plane rotation is analyzed in several different conformations generated from a restricted conformation database, a large number of these conformations are related. Based on these observations, a new simplified all-atom representation for protein folding simulations is presented where only one torsion angle variable is required for each residue. The underlying theme of this article is that conformational search methods using phi and psi torsion space, search through many redundant conformations. These conformations are related by anticorrelated torsion changes of peptide plane rotations. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 947–955, 1999     

Validation of the GROMOS 54A7 Force Field Regarding Mixed α/β‐Peptide Molecules

A molecular‐dynamics (MD) simulation study of two heptapeptides containing α‐ and β‐amino acid residues is presented. According to NMR experiments, the two peptides differ in dominant fold when solvated in MeOH: peptide 3 adopts predominantly β‐hairpin‐like conformations, while peptide 8 adopts a 14/15‐helical fold. The MD simulations largely reproduce the experimental data. Application of NOE atom? atom distance restraining improves the agreement with experimental data, but reduces the conformational sampling. Peptide 3 shows a variety of conformations, while still agreeing with the NOE and ³J‐coupling data, whereas the conformational ensemble of peptide 8 is dominated by one helical conformation. The results confirm the suitability of the GROMOS 54A7 force field for simulation or structure refinement of mixed α/β‐peptides in MeOH.     

The native ensemble and folding of a protein molten-globule: functional consequence of downhill folding

The continually emerging functional significance of intrinsic disorder and conformational flexibility in proteins has challenged the long-standing dogma of a well-defined structure contributing to a specific function. Molten-globular states, a class of proteins with significant secondary-structure but a fluid hydrophobic core, is one such example. They have however been difficult to characterize due to the complexity of experimental data and lack of computational avenues. Here, we dissect the folding mechanism of the α-helical molten-globular protein NCBD from three fundamentally different approaches: statistical-mechanical variable barrier model, C(α)-based Gō-model and explicit water all-atom molecular dynamics (MD) simulations. We find that NCBD displays the characteristics of a one-state globally downhill folder but is significantly destabilized. Using simulation techniques, we generate a highly constrained but a heterogeneous native ensemble of the molten-globule for the first time that is consistent with experimental data including small angle X-ray scattering (SAXS), circular dichroism (CD), and nuclear magnetic resonance (NMR). The resulting native ensemble populates conformations reported in other bound-forms providing direct evidence to the mechanism of conformational selection for binding multiple partners in this domain. Importantly, our simulations reveal a connection between downhill folding and large conformational flexibility in this domain that has been evolutionarily selected and functionally exploited resulting in large binding promiscuity. Finally, the multimodel approach we employ here serves as a powerful methodology to study mechanisms and suggests that the thermodynamic features of molten-globules fall within the array of folding mechanisms available to small single-domain proteins.     

Darobactin B Stabilises a Lateral-Closed Conformation of the BAM Complex in E. coli Cells

The β-barrel assembly machinery (BAM complex) is essential for outer membrane protein (OMP) folding in Gram-negative bacteria, and represents a promising antimicrobial target. Several conformational states of BAM have been reported, but all have been obtained under conditions which lack the unique features and complexity of the outer membrane (OM). Here, we use Pulsed Electron-Electron Double Resonance (PELDOR, or DEER) spectroscopy distance measurements to interrogate the conformational ensemble of the BAM complex in E. coli cells. We show that BAM adopts a broad ensemble of conformations in the OM, while in the presence of the antibiotic darobactin B (DAR-B), BAM′s conformational equilibrium shifts to a restricted ensemble consistent with the lateral closed state. Our in-cell PELDOR findings are supported by new cryoEM structures of BAM in the presence and absence of DAR-B. This work demonstrates the utility of PELDOR to map conformational changes in BAM within its native cellular environment.     

Helical Folding Competing with Unfolded Aggregation in Phenylene Ethynylene Foldamers

The folding and aggregation behavior of a pair of oligo(phenylene ethynylene) (OPE) foldamers are investigated by means of UV/Vis absorption and circular dichroism spectroscopy. With identical OPE backbones, two foldamers, 1 with alkyl side groups and 2 with triethylene glycol side chains, manifest similar helical conformations in solutions in n‐hexane and methanol, respectively. However, disparate and competing folding and aggregation processes are observed in alternative solvents. In cyclohexane, oligomer 1 initially adopts the helical conformation, but the self‐aggregation of unfolded chains, as a minor component, gradually drives the folding–unfolding transition eventually to the unfolded aggregate state completely. In contrast, in aqueous solution (CH₃OH/H₂O) both folded and unfolded oligomer 2 appear to undergo self‐association; aggregates of the folded chains are thermodynamically more stable. In solutions with a high H₂O content, self‐aggregation among unfolded oligomers is kinetically favored; these oligomers very slowly transform into aggregates of helical structures with greater thermodynamic stability. The folded–unfolded conformational switch thus takes place with the free (nonaggregated) molecules, and the very slow folding transition is due to the low concentration of molecularly dispersed oligomers.     

Measuring pK(a) values in protein folding transition state ensembles by NMR spectroscopy

Protein folding kinetic data have been obtained for the marginally stable N-terminal SH3 domain of the Drosophila protein drk as a function of pH in order to investigate the electrostatic properties of Asp8 in the folding transition state ensemble. The slow exchange between folded and unfolded forms of the protein gives rise to separate NMR resonances for both folded and unfolded states at equilibrium. As a result, kinetic data can be derived from magnetization transfer between these two states without the need for denaturants. Using the fact that ionization of Asp8 dominates the electrostatic behavior of the protein between pH 2 and 3, along with pKa values for titrating groups in both folded and unfolded states that have been determined in a previous study, values of 2.9 +/- 0.1 and 3.3 +/- 0.2 are obtained for the pKa of Asp8 in the transition state for the wild-type protein and for a His7Ala mutant, respectively. The data are consistent with the partial formation in the transition state ensemble of an Asp8 side chain carboxylate-a Lys21 backbone amide interaction that represents a highly conserved contact in folded SH3 domains.     

Influence of an Unnatural Amino Acid Side Chain on the Conformational Dynamics of Peptides

In this work, a non‐natural amino acid, H‐propargylglycine‐OH (Pra), is chosen to examine the side‐chain effect on the backbone conformation of small peptides. The conformations of two synthesized Pra‐containing tripeptides, Ac‐Pra‐Pra‐NH₂ (PPTP) and Ac‐Pra‐Ala‐NH₂ (PATP), are examined by infrared (IR) spectroscopy in combination with molecular dynamics (MD) simulations and quantum chemical computations. By analyzing the joint distributions of backbone torsional angles, several significant conformations can be identified for the two tripeptides solvated in D₂O. At room temperature, 44 % of PPTP exists in the α‐α conformation and 33 % of PATP exists in the α‐polyproline‐II conformation. Larger structural inhomogeneity is seen in both cases by MD simulations at elevated temperatures. Thus even a small side chain, such as the propargyl group can significantly alter the peptide backbone conformations. The results suggest that there is no overwhelming conformational propensity of the Pra residue in short peptides. IR spectra simulated in the amide‐I region using two different methods, reasonably reproduce the experimental IR spectra and their temperature dependence.     

Kinetics of cooperative protein folding involving two separate conformational families

The master equation that describes the kinetics of protein folding is solved numerically for a portion of Staphylococcal Protein A by a Laplace transformation. The calculations are carried out with 50 local-minimum conformations belonging to two conformational families. The master equation allows for transitions among all the 50 conformations in the evolution toward the final folded equilibrium distribution of conformations. It is concluded that the native protein folds in a fast cooperative process. The global energy minimum of a native protein can be reached after a sufficiently long folding time regardless of the initial state and the existence of a large number of local energy minima. Conformations representing non-native states of the protein can transform to the native state even if they do not belong to the native conformational family. Given a starting conformation, the protein molecule can fold to its final conformation through different paths. Finally, when the folding reaches the equilibrium distribution, the protein molecule adopts a set of conformations in which the global minimum has the largest average probability.     

Involvement of Long‐Lived Intermediate States in the Complex Folding Pathway of the Human Telomeric G‐Quadruplex

The energy landscapes of human telomeric G‐quadruplexes are complex, and their folding pathways have remained largely unexplored. By using real‐time NMR spectroscopy, we investigated the K⁺‐induced folding of the human telomeric DNA sequence 5′‐TTGGG(TTAGGG)₃A‐3′. Three long‐lived states were detected during folding: a major conformation (hybrid‐1), a previously structurally uncharacterized minor conformation (hybrid‐2), and a partially unfolded state. The minor hybrid‐2 conformation is formed faster than the more stable hybrid‐1 conformation. Equilibration of the two states is slow and proceeds via a partially unfolded intermediate state, which can be described as an ensemble of hairpin‐like structures.     

RNA unrestrained molecular dynamics ensemble improves agreement with experimental NMR data compared to single static structure: a test case

Nuclear magnetic resonance (NMR) provides structural and dynamic information reflecting an average, often non-linear, of multiple solution-state conformations. Therefore, a single optimized structure derived from NMR refinement may be misleading if the NMR data actually result from averaging of distinct conformers. It is hypothesized that a conformational ensemble generated by a valid molecular dynamics (MD) simulation should be able to improve agreement with the NMR data set compared with the single optimized starting structure. Using a model system consisting of two sequence-related self-complementary ribonucleotide octamers for which NMR data was available, 0.3 ns particle mesh Ewald MD simulations were performed in the AMBER force field in the presence of explicit water and counterions. Agreement of the averaged properties of the molecular dynamics ensembles with NMR data such as homonuclear proton nuclear Overhauser effect (NOE)-based distance constraints, homonuclear proton and heteronuclear ¹H–³¹P coupling constant (J) data, and qualitative NMR information on hydrogen bond occupancy, was systematically assessed. Despite the short length of the simulation, the ensemble generated from it agreed with the NMR experimental constraints more completely than the single optimized NMR structure. This suggests that short unrestrained MD simulations may be of utility in interpreting NMR results. As expected, a 0.5 ns simulation utilizing a distance dependent dielectric did not improve agreement with the NMR data, consistent with its inferior exploration of conformational space as assessed by 2-D RMSD plots. Thus, ability to rapidly improve agreement with NMR constraints may be a sensitive diagnostic of the MD methods themselves.     

Molecular dynamics simulation reveals structural and thermodynamic features of kinase activation by cancer mutations within the epidermal growth factor receptor

The epidermal growth factor receptor (EGFR) is a major target for drugs in treating lung carcinoma as it promotes cell growth and tumor progression. Structural studies have demonstrated that EGFR exists in an equilibrium between catalytically active and inactive forms, and dramatic conformational transitions occur during its activation. It is known that EGFR mutations promote such conformational changes that affect its activation and drug efficacy. The most common point mutation in lung cancer patients is a leucine to arginine substitution at amino acid 834 (L834R). In a recent article, we have studied changes in drug binding affinities due to cancer mutations of EGFR using ensemble molecular dynamics (MD) simulations. Here, we address an enhanced activation mechanism thought to be associated with this mutation. Using extended timescale MD simulations, the structural and energetic properties are studied for both active and inactive conformations of EGFR. The thermodynamic stabilities of these two conformations are characterized by free energy landscapes estimated from molecular mechanics/Poisson-Boltzmann solvent area calculations. Our study reveals that the L834R mutation introduces conformational changes in both states, adjusting the relative stabilities of active and inactive conformations and hence the activation of the EGFR kinase.