Side-chain conformational entropy in protein folding.

An important, but often neglected, contribution to the thermodynamics of protein folding is the loss of entropy that results from restricting the number of accessible side-chain conformers in the native structure. Conformational entropy changes can be found by comparing the number of accessible rotamers in the unfolded and folded states, or by estimating fusion entropies. Comparison of several sets of results using different techniques shows that the mean conformational free energy change (T delta S) is 1 kcal.mol-1 per side chain or 0.5 kcal.mol-1 per bond. Changes in vibrational entropy appear to be negligible compared to the entropy change resulting from the loss of accessible rotamers. Side-chain entropies can help rationalize alpha-helix propensities, predict protein/inhibitor complex structures, and account for the distribution of side chains on the protein surface or interior.     

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.     

Computational protein design with side-chain conformational entropy

Recent advances in modeling protein structures at the atomic level have made it possible to tackle "de novo" computational protein design. Most procedures are based on combinatorial optimization using a scoring function that estimates the folding free energy of a protein sequence on a given main-chain structure. However, the computation of the conformational entropy in the folded state is generally an intractable problem, and its contribution to the free energy is not properly evaluated. In this article, we propose a new automated protein design methodology that incorporates such conformational entropy based on statistical mechanics principles. We define the free energy of a protein sequence by the corresponding partition function over rotamer states. The free energy is written in variational form in a pairwise approximation and minimized using the Belief Propagation algorithm. In this way, a free energy is associated to each amino acid sequence: we use this insight to rescore the results obtained with a standard minimization method, with the energy as the cost function. Then, we set up a design method that directly uses the free energy as a cost function in combination with a stochastic search in the sequence space. We validate the methods on the design of three superficial sites of a small SH3 domain, and then apply them to the complete redesign of 27 proteins. Our results indicate that accounting for entropic contribution in the score function affects the outcome in a highly nontrivial way, and might improve current computational design techniques based on protein stability.     

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.     

Side-chain entropy effects on protein secondary structure formation

Loss of conformational entropy is one of the primary factors opposing protein folding. Both the backbone and side-chain of each residue in a protein will have their freedom of motion restricted in the final folded structure. The type of secondary structure of which a residue is part will have a significant impact on how much side-chain entropy is lost. Side-chain conformational entropies have previously been determined for folded proteins, simple models of unfolded proteins, alpha-helices, and a dipeptide model for beta-strands, but not for polyproline II (PII) helices. In this work, we present side-chain conformational estimates for the three regular secondary structure types: alpha-helices, beta-strands, and PII helices. Entropies are estimated from Monte Carlo computer simulations. Beta-strands are modeled as two structures, parallel and antiparallel beta-strands. Our data indicate that restraining a residue to the PII helix or antiparallel beta-strand conformations results in side-chain entropies equal to or higher than those obtained by restraining residues to the parallel beta-strand conformation. Side-chains in the alpha-helix conformation have the lowest side-chain entropies. The observation that extended structures retain the most side-chain entropy suggests that such structures would be entropically favored in unfolded proteins under folding conditions. Our data indicate that the PII helix conformation would be somewhat favored over beta-strand conformations, with antiparallel beta-strand favored over parallel. Notably, our data imply that, under some circumstances, residues may gain side-chain entropy upon folding. Implications of our findings for protein folding and unfolded states are discussed.     

Influence of glycosidic linkage on solution conformational entropy of oligosaccharides: Malto- vs. isomalto- and cello- vs. laminarioligosaccharides

Carbohydrate flexibility can influence a variety of recognition, processing, and end-use properties, at both the polymeric and oligomeric levels. The influence of glycosidic linkage, in particular, on carbohydrate flexibility is manifested in properties such as bacterial selectivity, solution viscosity, and the ability to regulate the spread of disease. Here, we apply size-exclusion chromatography, an entropically controlled technique, to determine the solution conformational entropy (ΔS) of various oligosaccharide series. The aim of the present study is to highlight how, for a given anomeric configuration, glycosidic linkage affects ΔS, and to do so quantitatively as a function of degree of polymerization (DP). To this end, we compare ΔS values for DP 1-7 for malto- and isomaltooligosaccharides, and for DP 1-5 for cello- and laminarioligosaccharides. To do so, we realize previously unattainable separations of disaccharides via a strict size-exclusion mechanism. Also given here are the requirements for extending our method to other oligomers, as well as to biopolymers     

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.     

A study of deleterious gene structure in plants using Markov chain Monte Carlo

The characteristics of deleterious genes have been of great interest in both theory and practice in genetics. Because of the complex genetic mechanism of these deleterious genes, most current studies try to estimate the overall magnitude of mortality effects on a population, which is characterized classically by the number of lethal equivalents. This number is a combination of several parameters, each of which has a distinct biological effect on genetic mortality. In conservation and breeding programs, it is important to be able to distinguish among different combinations of these parameters that lead to the same number of lethal equivalents, such as a large number of mildly deleterious genes or a few lethal genes, The ability to distinguish such parameter combinations requires more than one generation of mating. We propose a model for survival data from a two-generation mating experiment on the plant species Brassica rapa, and we enable inference with Markov chain Monte Carlo. This computational strategy is effective because a vast amount of missing genotype information must be accounted for. In addition to the lethal equivalents, the two-generation data provide separate information on the average intensity of mortality and the average number of deleterious genes carried by an individual. In our Markov chain Monte Carlo algorithm, we use a vector proposal distribution to overcome inefficiency of a single-site Gibbs sampler. Information about environmental effects is obtained from an outcrossing experiment conducted in parallel with the two-generation mating experiments.     

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.     

樟子松人工林树冠结构的分形分析

基于樟子松人工林7块固定标准地中的31株解析木的树冠体积和叶量,以幂函数关系(F=Av^(D/3))建立了预估树冠表面积的分形维数。同时根据生物量实测数据,建立预估叶量的生物模型Lw=0.180397D^3045903H^-1.67348。基于枝解析、树干解析数据,动态地预估了一年、二年、三年前的树冠体积,并结合树冠体积、叶量的这种幂函数关系可以动态地预估一年、二年、三年前树冠表面积的分形维数,从而反映出树冠结构的动态变化规律。为了了解不同分级样木的分维数变化情况,利用2003年调查的4块生物量标准地数据,根据单株树木各个枝条占据的空间体积与该枝条的带叶枝干重的关系,计算了各标准地不同分级样木树冠的分维数。为探讨单株样木树冠的分维数的计算提供了一种可行方法。树冠的分维数作为表征树冠的动态生长变化是一有用和可靠的指标。