Side-chain conformational entropy in protein unfolded states

The largest force disfavoring the folding of a protein is the loss of conformational entropy. A large contribution to this entropy loss is due to the side-chains, which are restricted, although not immobilized, in the folded protein. In order to accurately estimate the loss of side-chain conformational entropy that occurs upon folding it is necessary to have accurate estimates of the amount of entropy possessed by side-chains in the ensemble of unfolded states. A new scale of side-chain conformational entropies is presented here. This scale was derived from Monte Carlo computer simulations of small peptide models. It is demonstrated that the entropies are independent of host peptide length. This new scale has the advantage over previous scales of being more precise with low standard errors. Better estimates are obtained for long (e.g., Arg and Lys) and rare (e.g., Trp and Met) side-chains. Excellent agreement with previous side-chain entropy scales is achieved, indicating that further advancements in accuracy are likely to be small at best. Strikingly, longer side-chains are found to possess a smaller fraction of the theoretical maximum entropy available than short side-chains. This indicates that rotations about torsions after chi(2) are significantly affected by side-chain interactions with the polypeptide backbone. This finding invalidates previous assumptions about side-chain-backbone interactions. Proteins 2000;40:443-450.     

Physiological time-series analysis using approximate entropy and sample entropy

Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series.     

Accounting for conformational entropy in predicting binding free energies of protein-protein interactions

Protein-protein interactions are governed by the change in free energy upon binding, ΔG = ΔH - TΔS. These interactions are often marginally stable, so one must examine the balance between the change in enthalpy, ΔH, and the change in entropy, ΔS, when investigating known complexes, characterizing the effects of mutations, or designing optimized variants. To perform a large-scale study into the contribution of conformational entropy to binding free energy, we developed a technique called GOBLIN (Graphical mOdel for BiomoLecular INteractions) that performs physics-based free energy calculations for protein-protein complexes under both side-chain and backbone flexibility. Goblin uses a probabilistic graphical model that exploits conditional independencies in the Boltzmann distribution and employs variational inference techniques that approximate the free energy of binding in only a few minutes. We examined the role of conformational entropy on a benchmark set of more than 700 mutants in eight large, well-studied complexes. Our findings suggest that conformational entropy is important in protein-protein interactions--the root mean square error (RMSE) between calculated and experimentally measured ΔΔGs decreases by 12% when explicit entropic contributions were incorporated. GOBLIN models all atoms of the protein complex and detects changes to the binding entropy along the interface as well as positions distal to the binding interface. Our results also suggest that a variational approach to entropy calculations may be quantitatively more accurate than the knowledge-based approaches used by the well-known programs FOLDX and Rosetta--GOBLIN's RMSEs are 10 and 36% lower than these programs, respectively.     

Shortening a loop can increase protein native state entropy

Protein loops are essential structural elements that influence not only function but also protein stability and folding rates. It was recently reported that shortening a loop in the AcP protein may increase its native state conformational entropy. This effect on the entropy of the folded state can be much larger than the lower entropic penalty of ordering a shorter loop upon folding, and can therefore result in a more pronounced stabilization than predicted by polymer model for loop closure entropy. In this study, which aims at generalizing the effect of loop length shortening on native state dynamics, we use all‐atom molecular dynamics simulations to study how gradual shortening a very long or solvent‐exposed loop region in four different proteins can affect their stability. For two proteins, AcP and Ubc7, we show an increase in native state entropy in addition to the known effect of the loop length on the unfolded state entropy. However, for two permutants of SH3 domain, shortening a loop results only with the expected change in the entropy of the unfolded state, which nicely reproduces the observed experimental stabilization. Here, we show that an increase in the native state entropy following loop shortening is not unique to the AcP protein, yet nor is it a general rule that applies to all proteins following the truncation of any loop. This modification of the loop length on the folded state and on the unfolded state may result with a greater effect on protein stability. Proteins 2015; 83:2137–2146. © 2015 Wiley Periodicals, Inc.     

Effects of exercise and chills on entropy production in human body

Entropy flows and changes of entropy content for naked subjects in the respiration calorimeter in exercise and chills are calculated from the energetic data given by Hardy et al. (1938, J. Nutr. 16, 477) and Du Bois (1939, Bull. N.Y. Acad. Med. 15, 143). By use of these values, entropy productions in the human body in exercise and chills are estimated. The entropy production in mild exercise is 1.5-2.4 times as great as that in basal conditions. The entropy production in violent exercise is six to eight times as great as that before exercise. The entropy production in chills in cold environments is about twice as large as that in basal conditions. The entropy production in a malarial chill is about four times of that in normal subjects. These increases in entropy production will be due to the increase in heat production within the body. It seems that there is a parallel between energy and entropy viewpoints for human physiology.     

The concept of structural entropy in tissue-based diagnosis

The concept of entropy is described and its characteristics discussed as applied in tissue-based diagnosis. The concept of entropy includes at least 2 points of view--thermodynamic and informatics perspectives. Entropy can be defined by various methods: a measure of nonreversible energy or of system heterogeneity or as information content of a message. It is a statistical measure and system feature composed of macrosystems and microsystems. The structural entropy of macrosystems relies on definition of individual events and built-in microsystems. It depends on interaction of events and probability distribution (e.g., Gibbs-Boltzmann). The more generalized q-entropy involves account interaction of neighboring events. The thermodynamic concept of structural entropy can be expanded according to the theorem of Prigogine, introducing entropy flow. In biology, cells usually serve for events in the thermodynamic entropy approach. Entropy has been successfully used to describe tissue sections, nuclei and nuclear substructures such as DNA content, chromosomes and AgNORs. The concept of entropy reveals a close relationship of structural entropy and prognosis-associated diagnosis of malignancies. It is useful in prognosis-associated, tissue-based diagnosis in breast, prostate, bladder and lung cancer and is a promising expansion of image analysis in diagnostic agnosis in breast, prostate, bladder and lung cancer and is a promising expansion of image analysis in diagnostic pathology.     

Nuclear magnetic resonance titration curves of histidine ring protons. The effect of temperature on ribonuclease A.

The ionization constants of 3 of the histidine residues of ribonuclease A have beenobtained at 5 temperatures from the nuclear magnetic resonance titration curves of the imidazole C2 proton resonances. Thermodynamic parameters derived from the ionization constants indicate that histidine residues 105 and 119 are fairly well exposed to solvent, while histidine residue 12 is in a somewhat more restricted environment. Measurements of the low pH inflection present in the titration curve of histidine-12 yield a large negative entropy value, indicating that the group givine rise to this inflection is also buried.     

Photosynthesis and negative entropy production

The widely held view that the maximum efficiency of a photosynthetic pigment system is given by the Carnot cycle expression (1-T/Tr) for energy transfer from a hot bath (radiation at temperature Tr) to a cold bath (pigment system at temperature T) is critically examined and demonstrated to be inaccurate when the entropy changes associated with the microscopic process of photon absorption and photochemistry at the level of single photosystems are considered. This is because entropy losses due to excited state generation and relaxation are extremely small (DeltaS < T/Tr) and are essentially associated with the absorption-fluorescence Stokes shift. Total entropy changes associated with primary photochemistry for single photosystems are shown to depend critically on the thermodynamic efficiency of the process. This principle is applied to the case of primary photochemistry of the isolated core of higher plant photosystem I and photosystem II, which are demonstrated to have maximal thermodynamic efficiencies of xi > 0.98 and xi > 0.92 respectively, and which, in principle, function with negative entropy production. It is demonstrated that for the case of xi > (1-T/Tr) entropy production is always negative and only becomes positive when xi < (1-T/Tr).     

Numerical study of the entropy loss of dimerization and the folding thermodynamics of the GCN4 leucine zipper

A lattice-based model of a protein and the Monte Carlo simulation method are used to calculate the entropy loss of dimerization of the GCN4 leucine zipper. In the representation used, a protein is a sequence of interaction centers arranged on a cubic lattice, with effective interaction potentials that are both of physical and statistical nature. The Monte Carlo simulation method is then used to sample the partition functions of both the monomer and dimer forms as a function of temperature. A method is described to estimate the entropy loss upon dimerization, a quantity that enters the free energy difference between monomer and dimer, and the corresponding dimerization reaction constant. As expected, but contrary to previous numerical studies, we find that the entropy loss of dimerization is a strong function of energy (or temperature), except in the limit of large energies in which the motion of the two dimer chains becomes largely uncorrelated. At the monomer-dimer transition temperature we find that the entropy loss of dimerization is approximately five times smaller than the value that would result from ideal gas statistics, a result that is qualitatively consistent with a recent experimental determination of the entropy loss of dimerization of a synthetic peptide that also forms a two-stranded alpha-helical coiled coil.     

Buried surface area,conformational entropy,and protein stability

In this paper, how to calculate the loss of conformational entropy owing to protein folding from data on protein stability, the disulfide bonding pattern and the buried surface area is suggested. The average values of the initiation constant and entropy loss per residue for the helix-coil transition, calculated according to the suggested approach, are in a good agreement with available data. It is shown that the loss of conformational entropy for proteins and protein fragments, thus calculated, can be quite accurately approximated by three straight lines, each representing the loss of conformational entropy in a given range of molecular weights. The first line corresponds to the formation of α helices, the second to proteins with molecular weights below 10,000, and the third to proteins with molecular weights above this value. The standard error varied from 1.3 kcal/mol, for the first region, to between ～3 and 5 kcal/mol for the third. Use of these approximating linear functions is proposed for the calculation of protein stabilities from x-ray data. Such calculations, performed for more than 20 proteins and their fragments, have led to at least qualitative agreement between the calculated stabilities and available data for more than 90% of the cases. While quantitative predictions are at the limit of the method's accuracy, predictions of the probabilities of proteins or their fragments to be stable are expected to give a correct answer in ～95% of the cases.     

Dynamics and entropy of a calmodulin-peptide complex studied by NMR and molecular dynamics

All-atom, explicit water molecular dynamics simulations of calcium-loaded calmodulin complexed with a peptide corresponding to the smooth muscle myosin light chain kinase target were carried out at 295 and 346 K. Amide and side chain methyl angular generalized order parameters were calculated and analyzed in the context of the protein's structure and dynamics. The agreement between amide order parameters measured by NMR and those from the simulations was found to be good, especially at the higher temperature, indicating both better convergence for the latter and excellent transferrability of the CHARMM parameters to the higher temperature. Subtle dynamical features such as helix fraying were reproduced. A large range of order parameters for the nine calmodulin methionines was observed in the NMR, and reproduced quite well in the simulations. The major determinant of the methionine order parameter was found to be the proximity to side chains of aromatic residues. An upper bound estimate of the difference in backbone entropy between loop and helical regions was extracted from the order parameters using a model of motion in an effective potential. Although loop regions are more flexible than helical regions, it was found that the entropy loss per residue upon folding was only approximately 20% less for loops than for helices. Pairwise correlated motions, which could significantly lower entropy estimates obtained from order parameter analysis alone, were found to be largely absent.     

Conformational constraint in protein ligand design and the inconsistency of binding entropy

It is an accepted practice in ligand design to introduce conformational constraint with the expectation of improving affinity, justified by the theoretical possibility that an unfavorable change in binding entropy will be reduced. This rationale of minimizing the entropic penalty through imposing structural constraints upon a ligand, however, has been voiced more often than verified. Here we examine three modified cyclic peptides, along with multiple versions of their linear control analogs, and determine their thermodynamic parameters when binding the same host, the third PDZ domain (PDZ3) of the mammalian postsynaptic density-95 (PSD-95) protein. To begin a two-stage investigation, the initial evaluation involved solution binding studies with isothermal titration calorimetry (ITC), which provided the changes in Gibbs free energy (DeltaG), enthalpy (DeltaH), and entropy (TDeltaS) upon formation of the protein-ligand complex. In the second stage, a selected macrocycle along with two matched linear controls were subjected to more rigorous analysis by ITC, which included (1) change in heat of buffer ionization (DeltaH(ion)) titrations, to examine the role of proton transfer events; (2) change in heat capacity (DeltaC(p)) determinations, to indirectly probe the nature of the binding surface; and (3) osmotic stress experiments, to evaluate desolvation effects and quantitate water release. Together, these demonstrate that the entropic relationship between a macrocyclic ligand and a linear counterpart can be a complex one that is difficult to rationalize. Further, the addition of constraint can, counterintuitively, lead to a less favorable change in binding entropy. This underscores the need to use matched linear control ligands to assure that comparisons are made in a meaningful manner.     

Side-chain entropy and packing in proteins.

What role does side-chain packing play in protein stability and structure? To address this question, we compare a lattice model with side chains (SCM) to a linear lattice model without side chains (LCM). Self-avoiding configurations are enumerated in 2 and 3 dimensions exhaustively for short chains and by Monte Carlo sampling for chains up to 50 main-chain monomers long. This comparison shows that (1) side-chain degrees of freedom increase the entropy of open conformations, but side-chain steric exclusion decreases the entropy of compact conformations, thus producing a substantial entropy that opposes folding; (2) there is a side-chain “freezing” or ordering, i.e., a sharp decrease in entropy, near maximum compactness; and (3) the different types of contacts among side chains (s) and main-chain elements (m) have different frequencies, and the frequencies have different dependencies on compactness. mm contacts contribute significantly only at high densities, suggesting that main-chain hydrogen bonding in proteins may be promoted by compactness. The distributions of mm, ms, and ss contacts in compact SCM configurations are similar to the distributions in protein structures in the Brookhaven Protein Data Bank. We propose that packing in proteins is more like the packing of nuts and bolts in a jar than like the pairwise matching of jigsaw puzzle pieces.     

Induced fit and the entropy of structural adaptation in the complexation of CAP and lambda-repressor with cognate DNA sequences

Molecular dynamics (MD) simulations of 5 ns on protein-DNA complexes of catabolite-activator protein (CAP), lambda-repressor, and their corresponding uncomplexed protein and DNA, are reported. These cases represent two extremes of DNA bending, with CAP DNA bent severely and the lambda-operator nearly straight when complexed with protein. The calculations were performed using the AMBER suite of programs and the parm94 force field, validated for these studies by good agreement with experimental nuclear magnetic resonance data on DNA. An explicit computational model of structural adaptation and computation of the quasiharmonic entropy of association were obtained from the MD. The results indicate that, with respect to canonical B-form DNA, the extreme bending of the DNA in the complex with CAP is approximately 60% protein-induced and 40% intrinsic to the sequence-dependent structure of the free oligomer. The DNA in the complex is an energetically strained form, and the MD results are consistent with a conformational-capture mechanism. The calculated quasiharmonic entropy change accounts for the entropy difference between the two cases. The calculated entropy was decomposed into contributions from protein adaptation, DNA adaptation, and protein-DNA structural correlations. The origin of the entropy difference between CAP and lambda-repressor complexation arises more from the additional protein adaptation in the case of lambda, than to DNA bending and entropy contribution from DNA bending. The entropy arising from protein DNA cross-correlations, a contribution not previously discussed, is surprisingly large.     

Side-chain conformational entropy at protein-protein interfaces

Protein-protein interactions are the key to many biological processes. How proteins selectively and correctly associate with their required protein partner(s) is still unclear. Previous studies of this "protein-docking problem" have found that shape complementarity is a major determinant of interaction, but the detailed balance of energy contributions to association remains unclear. This study estimates side-chain conformational entropy (per unit solvent accessible area) for various protein surface regions, using a self-consistent mean field calculation of rotamer probabilities. Interfacial surface regions were less flexible than the rest of the protein surface for calculations with monomers extracted from homodimer datasets in 21 of 25 cases, and in 8 of 9 for the large protomer from heterodimer datasets. In surface patch analysis, based on side-chain conformational entropy, 68% of true interfaces were ranked top for the homodimer set and 66% for the large protomer/heterodimer set. The results indicate that addition of a side-chain entropic term could significantly improve empirical calculations of protein-protein association.     

Sample entropy analysis of neonatal heart rate variability

Abnormal heart rate characteristics of reduced variability and transient decelerations are present early in the course of neonatal sepsis. To investigate the dynamics, we calculated sample entropy, a similar but less biased measure than the popular approximate entropy. Both calculate the probability that epochs of window length m that are similar within a tolerance r remain similar at the next point. We studied 89 consecutive admissions to a tertiary care neonatal intensive care unit, among whom there were 21 episodes of sepsis, and we performed numerical simulations. We addressed the fundamental issues of optimal selection of m and r and the impact of missing data. The major findings are that entropy falls before clinical signs of neonatal sepsis and that missing points are well tolerated. The major mechanism, surprisingly, is unrelated to the regularity of the data: entropy estimates inevitably fall in any record with spikes. We propose more informed selection of parameters and reexamination of studies where approximate entropy was interpreted solely as a regularity measure.     

Gene-centric genomewide association study via entropy

Genes are the functional units in most organisms. Compared to genetic variants located outside genes, genic variants are more likely to affect disease risk. The development of the human HapMap project provides an unprecedented opportunity for genetic association studies at the genomewide level for elucidating disease etiology. Currently, most association studies at the single-nucleotide polymorphism (SNP) or the haplotype level rely on the linkage information between SNP markers and disease variants, with which association findings are difficult to replicate. Moreover, variants in genes might not be sufficiently covered by currently available methods. In this article, we present a gene-centric approach via entropy statistics for a genomewide association study to identify disease genes. The new entropy-based approach considers genic variants within one gene simultaneously and is developed on the basis of a joint genotype distribution among genetic variants for an association test. A grouping algorithm based on a penalized entropy measure is proposed to reduce the dimension of the test statistic. Type I error rates and power of the entropy test are evaluated through extensive simulation studies. The results indicate that the entropy test has stable power under different disease models with a reasonable sample size. Compared to single SNP-based analysis, the gene-centric approach has greater power, especially when there is more than one disease variant in a gene. As the genomewide genic SNPs become available, our entropy-based gene-centric approach would provide a robust and computationally efficient way for gene-based genomewide association study.     

Enzyme catalysis via control of activation entropy: site-directed mutagenesis of 6,7-dimethyl-8-ribityllumazine synthase

6,7-Dimethyl-8-ribityllumazine synthase (lumazine synthase) catalyses the penultimate step in the biosynthesis of riboflavin. In Bacillus subtilis, 60 lumazine synthase subunits form an icosahedral capsid enclosing a homotrimeric riboflavin synthase unit. The ribH gene specifying the lumazine synthase subunit can be expressed in high yield. All amino acid residues exposed at the surface of the active site cavity were modified by PCR assisted mutagenesis. Polar amino acid residues in direct contact with the enzyme substrates, 5-amino-6-ribitylamino-2,4(1H,3H)-pyrimidinedione and 3,4-dihydroxy-2-butanone 4-phosphate, could be replaced with relative impunity with regard to the catalytic properties. Only the replacement of Arg127, which forms a salt bridge with the phosphate group of 3,4-dihydroxy-2-butanone 4-phosphate, reduced the catalytic rate by more than one order of magnitude. Replacement of His88, which is believed to assist in proton transfer reactions, reduced the catalytic activity by about one order of magnitude. Surprisingly, the activation enthalpy deltaH of the lumazine synthase reaction exceeds that of the uncatalysed reaction. On the other hand, the free energy of activation deltaG of the uncatalysed reaction is characterised by a large entropic term (TdeltaS) of -37.8 kJmol(-1), whereas the entropy of activation (TdeltaS) of the enzyme-catalysed reaction is -6.7 kJmol(-1). This suggests that the rate enhancement by the enzyme is predominantly achieved by establishing a favourable topological relation of the two substrates, whereas acid/base catalysis may play a secondary role.