Structural basis of hierarchical multiple substates of a protein. III: Side chain and main chain local conformations

An analysis is carried out of differences in the minimum energy conformations obtained in the previous paper by energy minimization starting from conformations sampled by a Monte Carlo simulation of conformational fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor. Main conformational differences in each pair of energy minima are found usually localized in several side chains and in a few local main chain segments. Such side chains and local main chain segments are found to take a few distinct local conformations in the minimum energy conformations. Energy minimum conformations can thus be described in terms of combinations of these multiple local conformations.     

The backbone conformational entropy of protein folding: experimental measures from atomic force microscopy

The energy dissipated during the atomic force microscopy-based mechanical unfolding and extension of proteins is typically an order of magnitude greater than their folding free energy. The vast majority of the "excess" energy dissipated is thought to arise due to backbone conformational entropy losses as the solvated, random-coil unfolded state is stretched into an extended, low-entropy conformation. We have investigated this hypothesis in light of recent measurements of the energy dissipated during the mechanical unfolding of "polyproteins" comprised of multiple, homogeneous domains. Given the assumption that backbone conformational entropy losses account for the vast majority of the energy dissipated (an assumption supported by numerous lines of experimental evidence), we estimate that approximately 19(+/-2)J/(mol K residue) of entropy is lost during the extension of three mechanically stable beta-sheet polyproteins. If, as suggested by measured peak-to-peak extension distances, pulling proceeds to near completion, this estimate corresponds to the absolute backbone conformational entropy of the unfolded state. As such, it is exceedingly close to previous theoretical and semi-empirical estimates that place this value at approximately 20J/(mol K residue). The estimated backbone conformational entropy lost during the extension of two helical polyproteins, which, in contrast to the mechanically stable beta-sheet polyproteins, rupture at very low applied forces, is three- to sixfold less. Either previous estimates of the backbone conformational entropy are significantly in error, or the reduced mechanical strength of the helical proteins leads to the rupture of a subsequent domain before full extension (and thus complete entropy loss) is achieved.     

Conformational stability of biological polyelectrolytes: Evaluation of enthalpy and entropy changes of conformational transitions

A new method is proposed for the determination of the enthalpy and entropy changes of nonionic origin upon conformational transition of linear biopolyelectrolytes in solution. For all transition midpoints, defined by given temperature and ionic strength, the total free energy change of the system is zero, which means that the nonionic contribution to the free energy change is equal in value and opposite in sign to the polyelectrolytic one. The counterion condensation theory of linear polyelectrolytes provides for the appropriate analytical expression to be used in such calculations. Linear plots of the proper functions of the calculated free energy changes vs the proper functions of temperature allows for the determination of the enthalpic and entropic terms of the nonionic free energy change of transition. The method has been applied to the extensive available data of the ion-induced conformational change of κ-carrageenan, a linear sulfated galactan extracted from seaweeds. The method has proved very successful, with the results showing a remarkable convergency of the enthalpy values for different monovalent counterions. On the other hand, the above approach has made it possible to explain the known effect of counterion specificity on the transition by a small difference in the nonionic entropic contributions. © 1998 John Wiley & Sons, Inc. Biopoly 45: 203–216, 1998     

Investigations into sequence and conformational dependence of backbone entropy,inter-basin dynamics and the Flory isolated-pair hypothesis for peptides

The populations and transitions between Ramachandran basins are studied for combinations of the standard 20 amino acids in monomers, dimers and trimers using an implicit solvent Langevin dynamics algorithm and employing seven commonly used force-fields. Both the basin populations and inter-conversion rates are influenced by the nearest neighbor's conformation and identity, contrary to the Flory isolated-pair hypothesis. This conclusion is robust to the choice of force-field, even though the use of different force-fields produces large variations in the populations and inter-conversion rates between the dominant helical, extended beta, and polyproline II basins. The computed variation of conformational and dynamical properties with different force-fields exceeds the difference between explicit and implicit solvent calculations using the same force-field. For all force-fields, the inter-basin transitions exhibit a directional dependence, with most transitions going through extended beta conformation, even when it is the least populated basin. The implications of these results are discussed in the context of estimates for the backbone entropy of single residues, and for the ability of all-atom simulations to reproduce experimental protein folding data.     

Monte Carlo simulations of a protein molecule with and without hydration energy calculated by the hydration-shell model

Monte Carlo simulations of a small protein, carmbin, were carried out with and without hydration energy. The methodology presented here is characterized, as compared with the other similar simulations of proteins in solution, by two points: (1) protein conformations are treated in fixed geometry so that dihedral angles are independent variables rather than cartesian coordinates of atoms; and (2) instead of treating water molecules explicitly in the calculation, hydration energy is incorporated in the conformational energy function in the form of g _i A _i, whereA _i is the accessible surface area of an atomic groupi in a given conformation, andg _i is the free energy of hydration per unit surface area of the atomic group (i.e., hydration-shell model). Reality of this model was tested by carrying out Monte Carlo simulations for the two kinds of starting conformations, native and unfolded ones, and in the two kinds of systems,in vacuo and solution. In the simulations starting from the native conformation, the differences between the mean propertiesin vacuo and solution simulations are not very large, but their fluctuations around the mean conformation during the simulation are relatively smaller in solution thanin vacuo. On the other hand, in the simulations starting from the unfolded conformation, the molecule fluctuates much more largely in solution thanin vacuo, and the effects of taking into account the hydration energy are pronounced very much. The results suggest that the method presented in this paper is useful for the simulations of proteins in solution.     

Optimal ratio between the average conformational entropy and the average energy of interaction between residues for fast protein folding

Based on available experimental data and using a theoretical model of protein folding, we demonstrate that there is an optimal ratio between the average conformational entropy and the average contact energy per residue for fast protein folding. A statistical analysis of the conformational entropy and the number of contacts per residue for 5829 protein domains from four main classes (α, β, α/β, α+β) shows that each class has its own characteristic average number of contacts per residue and average conformational entropy per residue. These class-specific characteristics determine the protein folding rates: α-proteins are the fastest to fold, β-proteins are the second fastest, α+β-proteins are the third, and α/β-proteins are the last to fold.     

Atomistic simulations of the effects of polyglutamine chain length and solvent quality on conformational equilibria and spontaneous homodimerization

Aggregation of expanded polyglutamine tracts is associated with nine different neurodegenerative diseases, including Huntington's disease. Experiments and computer simulations have demonstrated that monomeric forms of polyglutamine molecules sample heterogeneous sets of collapsed structures in water. The current work focuses on a mechanistic characterization of polyglutamine homodimerization as a function of chain length and temperature. These studies were carried out using molecular simulations based on a recently developed continuum solvation model that was designed for studying conformational and binding equilibria of intrinsically disordered molecules such as polyglutamine systems. The main results are as follows: Polyglutamine molecules form disordered, collapsed globules in aqueous solution. These molecules spontaneously associate at conditions approaching those of typical in vitro experiments for chains of length N ≥ 15. The spontaneity of these homotypic associations increases with increasing chain length and decreases with increasing temperature. Similar and generic driving forces govern both collapse and spontaneous homodimerization of polyglutamine in aqueous milieus. Collapse and dimerization maximize self-interactions and reduce the interface between polyglutamine molecules and the surrounding solvent. Other than these generic considerations, there do not appear to be any specific structural requirements for either chain collapse or chain dimerization; that is, both collapse and dimerization are nonspecific in that disordered globules form disordered dimers. In fact, it is shown that the driving force for intermolecular associations is governed by spontaneous conformational fluctuations within monomeric polyglutamine. These results suggest that polyglutamine aggregation is unlikely to follow a homogeneous nucleation mechanism with the monomer as the critical nucleus. Instead, the results support the formation of disordered, non-β-sheet-like soluble molten oligomers as early intermediates—a proposal that is congruent with recent experimental data.     

Monoclonal antibody higher order structure analysis by high throughput protein conformational array

The elucidation of antibody higher order structure (HOS) is critical in therapeutic antibody development. Since HOS determines the protein bioactivity and chemo-physical properties, this knowledge can help to ensure that the safety and efficacy attributes are not compromised. Protein conformational array (PCA) is a novel method for determining the HOS of monoclonal antibodies. Previously, we successfully utilized an enzyme-linked immunosorbent assay (ELISA)-based PCA along with other bioanalytical tools to elucidate the structures of antibody aggregates. In this study, applying a new multiplex-based PCA with 48-fold higher throughput than the ELISA-based one we revealed structural differences between different antibody molecules and antibody structure changes affected by various processing conditions. The PCA analysis of antibody molecules clearly demonstrated significant differences between IgG1 and IgG4 subclasses in epitope exposure and folding status. Furthermore, we applied small angle X-ray scattering to decipher mechanistic insights of PCA technology and validate structural information obtained using PCA. These findings enhance our fundamental understanding of mAbs' HOS in general. The PCA analysis of antibody samples from various processing conditions also revealed that antibody aggregation caused significantly higher exposure of antibody epitopes, which potentially led to a “foreign” molecule that could cause immunogenicity. The PCA data correlated well with protein stability results from traditional methods such as size-exclusion chromatography and protein thermal shift assay. Our study demonstrated that high throughput PCA is a suitable method for HOS analysis in the discovery and development of therapeutic antibodies.     

Bayesian phylogenetic model selection using reversible jump Markov chain Monte Carlo

A common problem in molecular phylogenetics is choosing a model of DNA substitution that does a good job of explaining the DNA sequence alignment without introducing superfluous parameters. A number of methods have been used to choose among a small set of candidate substitution models, such as the likelihood ratio test, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and Bayes factors. Current implementations of any of these criteria suffer from the limitation that only a small set of models are examined, or that the test does not allow easy comparison of non-nested models. In this article, we expand the pool of candidate substitution models to include all possible time-reversible models. This set includes seven models that have already been described. We show how Bayes factors can be calculated for these models using reversible jump Markov chain Monte Carlo, and apply the method to 16 DNA sequence alignments. For each data set, we compare the model with the best Bayes factor to the best models chosen using AIC and BIC. We find that the best model under any of these criteria is not necessarily the most complicated one; models with an intermediate number of substitution types typically do best. Moreover, almost all of the models that are chosen as best do not constrain a transition rate to be the same as a transversion rate, suggesting that it is the transition/transversion rate bias that plays the largest role in determining which models are selected. Importantly, the reversible jump Markov chain Monte Carlo algorithm described here allows estimation of phylogeny (and other phylogenetic model parameters) to be performed while accounting for uncertainty in the model of DNA substitution.     

Fractal analysis and its applications in human palaentology

Fractal analysis was applied in human palaentology by Gibert and Palmqvist to estimate the value of the fractal dimension obtained from the cranial sutures preserved in the fragment of occipital bone (VM-0) found at the Venta Micena site. This paper also estimated the values of fractal dimension for different specimens in order to establish a taxonomy. Although that initial study demonstrated that the technique could be useful in human palaentology, the large variability of sutures observed in the VM-0 sample requires a mechanism that makes it possible to automatically obtain an objective plot of the suture to be analyzed.     

Exploring the conformational space of protein side chains using dead-end elimination and the A* algorithm

We describe an algorithm which enables us to search the conformational space of the side chains of a protein to identify the global minimum energy combination of side chain conformations as well as all other conformations within a specified energy cutoff of the global energy minimum. The program is used to explore the side chain conformational energy surface of a number of proteins, to investigate how this surface varies with the energy model used to describe the interactions within the system and the rotamer library. Enumeration of the rotamer combinations enables us to directly evaluate the partition function, and thus calculate the side chain contribution to the conformational entropy of the folded protein. An investigation of these conformations and the relationships between them shows that most of the conformations near to the global energy minimum arise from changes in side chain conformations that are essentially independent; very few result from a concerted change in conformation of two or more residues. Some of the limitations of the approach are discussed. Proteins 33:227–239, 1998. © 1998 Wiley-Liss, Inc.     

Folding small proteins via annealing stochastic approximation Monte Carlo

Recently, the stochastic approximation Monte Carlo algorithm has been proposed by Liang et al. (2007) as a general-purpose stochastic optimization and simulation algorithm. An annealing version of this algorithm was developed for real small protein folding problems. The numerical results indicate that it outperforms simulated annealing and conventional Monte Carlo algorithms as a stochastic optimization algorithm. We also propose one method for the use of secondary structures in protein folding. The predicted protein structures are rather close to the true structures.     

Structural basis of hierarchical multiple substates of a protein. I: Introduction

A computer experiment of protein dynamics is carried out, which consists of two steps: (1) A Monte Carlo simulation of thermal fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor; and (2) a simulation of the quick freezing of fluctuating conformations into energy minima by minimization of the energy of a number of conformations sampled in the Monte Carlo simulation. From the analysis of results of the computer experiment is obtained the following picture of protein dynamics: multiple energy minima exist in the native state, and they are distributed in clusters in the conformational space. The dynamics has a hierarchical structure which has at least two levels. In the first level, dynamics is restricted within one of the clusters of minima. In the second, transitions occur among the clusters. Local parts of a protein molecule, side chains and local main chain segments, can take multiple locally stable conformations in the native state. Many minima result from combinations of these multiple local conformations. The hierarchical structure in the dynamics comes from interactions among the local parts. Protein molecules have two types of flexibility, each associated with elastic and plastic deformations, respectively.     

Polypeptide folding with off-lattice Monte Carlo dynamics: the method

The MC dynamics of an off-lattice all-atom protein backbone model with rigid amide planes are studied. The only degrees of freedom are the dihedral angle pairs of the C-atoms. Conformational changes are generated by Monte Carlo (MC) moves. The MC moves considered are single rotations (simple moves, SM's) giving rise to global conformational changes or, alternatively, cooperative rotations in a window of amide planes (window moves, WM's) generating local conformational changes in the window. Outside the window the protein conformation is kept invariant by constraints. These constraints produce a bias in the distribution of dihedral angles. The WM's are corrected for this bias by suitable Jacobians. The energy function used is derived from the CHARMM force field. In a first application to polyalanine it is demonstrated that WM's sample the conformational space more efficiently than SM's.Abbreviations CPU Central Processing Unit - MC Monte Carlo - MCD Monte Carlo Dynamics - MD Molecular Dynamics - RMS Root-Mean-Square - RMSD Root-Mean-Square-Deviation - SM Simple Move - WM Window Move     

Structural basis of hierarchical multiple substates of a protein. V: Nonlocal deformations

Distances between centers of gravity of individual residues are compared among the minimum energy conformations derived from the record of the Monte Carlo simulation of conformational fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor. It is found that local deformations originating from the multiplicity of local conformations cause deformations of the whole structure of the molecule in various ways, which can be classified into two types. Type 1: When a local deformation occurs in a region consisting of a few residues near the surface of the molecule, the whole shape of the molecule responds by deforming elastically. The magnitude of this deformation is in the range of thermal fluctuations calculated by the harmonic approximation around a single minimum. Type 2: We have observed one case belonging to the second type in which local deformations occur cooperatively in an extended region. This region goes across the whole molecule and divide the remaining parts into two. Atom packing changes in and around the extended region of local deformations. For this reason deformation in this region is plastic. Relative location and orientation between the divided two parts change very much. Deformation of the whole shape in this case, associated with the plastic deformation in an extended region, demonstrates that protein molecules have a flexibility beyond the harmonic limit.     

Structural basis of hierarchical multiple substates of a protein. IV: Rearrangements in atom packing and local deformations

Differences in atom packing are studied in the minimum energy conformations derived from the record of the Monte Carlo simulation of conformational fluctuation in the native state of a globular protein, bovine pancreatic trypsin inhibitor. It is found that local deformations observed among the minima which are found in the previous paper are accompanied by rearrangement of atom packing. Spatial locations of the local deformations in the three-dimensional folded structure are also studied. It is found that the local deformations are distributed in space in several clusters in the folded structure. The size and location of the clusters characterize the respective fluctuations of the first and the second levels observed in the simulation. In the fluctuations of the first level local deformations, each of which usually involves a few side chains and one main chain local segment, are thermally exited independently of each other near the surface of the molecule. The observed fluctuation of the second level involves a cooperative deformation involving many side chains and local main chain segments all in one cluster, which goes though the core of the molecule. The collective local deformations observed both in the first and second levels are plastic in the sense that they are accompanied with rearrangement of atom packing.     

Effect of short- and long-range interactions on protein folding

The relative importance of short- and long-range interactions is examined using a Monte Carlo simulation of protein folding on bovine pancreatic trypsin inhibitor. The model of the protein and the interaction energies were parametrized using X-ray structures of 30 native proteins. A nearest neighbor Ising model is used to determine the conformational state at each stage of the Monte Carlo procedure. Long-range interactions are simulated by contact free energies which become effective as two residues, separated by four or more residues along the chain, approach each other, and by disulfide-bond energies. Short-range interactions for residues separated by one, two, or three residues along the chain are also modeled by contact free energies and by -helical hydrogen bonds. A hard-sphere model is used to represent repulsive interactions. The ratios of short- to long-range interactions studied are 1:1, 2:1, 1:2, 0:1, and 1:0; e.g., for the 2:1 ratio, short-range interactions are weighted twice as much as long-range interactions, and for the 1:0 ratio, long-range interactions are omitted. For each ratio of short- to long-range interactions, a native conformation is found by a Monte Carlo procedure, a segment of 11 residues (residue numbers 1–11) is then rotated away from the rest of the molecule [breaking the 5–55 native disulfide bond, and moving this segment so that the distance between the sulfur atoms of the 5 and 55 cystine side chains (averaged for all native conformations) increases from 3.9 to 7.3 Å], and the Monte Carlo simulation is carried out (allowing the conformation of the whole molecule to change) until equilibrium is attained. For each ratio, the refolded conformation is compared to the native one using triangular distance maps and differential geometry distance criteria. With ratios of short- to long-range interaction energies of 1:1 and 0:1, the native disulfide bond could be re-formed; with ratios of 2:1 and 1:2 it did not; and with the 1:0 ratio, even a stable native conformation was not achieved. Therefore, long-range interactions (in addition to short-range ones) are required to bring remote parts of the protein together and to stabilize its native conformation.NIH Postdoctoral Fellow, 1977–1978.     

Dynamic fluorescence of extrinsic fluorophores as a tool for studying protein conformational substates

Summary The fluorescence lifetime distribution of 2-p-toluidinyl-6-naphthalene sulfonic acids (TNS) bound to the heme site of apomyoglobin has been examined. The results were compared to those observed for the free fluorophore in isotropic nonviscous solvent. Two different excitation wavelengths were used, i.e. 290 and 350 nm. The results showed that the distribution of TNS bound to apomyoglobin is wider than that of the free fluorophore, thus indicating the existence of a large number of conformational substates originating from the interaction between TNS and the protein matrix. The comparison of the distribution obtained at two different excitation wavelengths allowed the emission arising from conformational substates, in which the excited state of fluorophore moiety has a higher probability to be populated by Forster energy transfer mechanism, to be distinguished.