A comparison of two-electron chemistry performed by the manganese and iron heterodimer and homodimers

Two-electron chemistry with an iron dimer, a manganese dimer, and a manganese–iron dimer as a catalyst has been modeled using B3LYP* hybrid density functional theory. The recently discovered MnFe proteins form (at least) two functionally distinct groups, performing radical generation (class Ic ribonucleotide reductase subunit II) and substrate oxidations (subunit II-like ligand-binding oxidases, R2lox), respectively. Proteins from the latter group appear to be functionally similar to the diiron carboxylate proteins that perform two-electron oxidations of substrates, such as methane monooxygenase. To qualitatively determine the potential role of a MnFe center in R2lox, methane hydroxylation with the MnFe heterodimer and with the FeFe and MnMn homodimers is studied. The redox potential of the active state of the Mn(IV)Fe(IV) heterodimer is about 7 kcal mol⁻¹ lower than that of the active state of the Fe(IV)Fe(IV) homodimer, leading to a high barrier for the rate-limiting hydrogen abstraction with the MnFe site. If the entropy loss is not included, the barriers are lower, and the MnFe heterodimer can therefore have a role in R2lox as an oxidase for larger substrates exergonically bound to the protein. A MnMn center has a high barrier both with and without entropy loss. The higher stability of Fe(IV) in the presence of Mn(IV) in the other site compared with a second Fe(IV) suggests an explanation for the presence of the MnFe site in R2lox: to provide a metal center that is capable of two-electron chemistry, and which is more stable and less sensitive to external reductants than an Fe(IV)Fe(IV) site.     

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.     

Dynamic structure of subtilisin-eglin c complex studied by normal mode analysis

Normal mode analysis of subtilisin-eglin c complex was performed to investigate the dynamics at the interface between the enzyme and the inhibitor. The internal motions of the complex calculated from the normal modes were divided into three parts: the internal motions changing the shape of each molecule, the external rigid-body motions changing their mutual dispositions, and the coupling between the internal and external motions. From the results of the analysis, the following characteristic features were found in the dynamics at the interface regions: 1) negative correlation between the internal and external motions within each molecule, and 2) positive correlation between the external motions of the two molecules. The former decreases the apparent amplitudes of motions at the interface. The latter minimizes the interference between individual motions of the two molecules. These dynamic characteristics allow the enzyme and the inhibitor to move as freely as possible. This finding suggests that the experimental evidence of the large entropy gain on binding should be attributed not only to strong hydrophobic interactions, but also to the dynamic structure of the complex, which is found to minimize an unavoidable loss of the conformational entropy on binding. Proteins 32:324–333, 1998. © 1998 Wiley-Liss, Inc.     

Deep Learning Using Havrda-Charvat Entropy for Classification of Pulmonary Optical Endomicroscopy

1) ObjectivePulmonary optical endomicroscopy (POE) is an imaging technology in real time. It allows to examine pulmonary alveoli at a microscopic level. Acquired in clinical settings, a POE image sequence can have as much as 25% of the sequence being uninformative frames (i.e. pure-noise and motion artifacts). For future data analysis, these uninformative frames must be first removed from the sequence. Therefore, the objective of our work is to develop an automatic detection method of uninformative images in endomicroscopy images.2) Material and methodsWe propose to take the detection problem as a classification one. Considering advantages of deep learning methods, a classifier based on CNN (Convolutional Neural Network) is designed with a new loss function based on Havrda-Charvat entropy which is a parametrical generalization of the Shannon entropy. We propose to use this formula to get a better hold on all sorts of data since it provides a model more stable than the Shannon entropy.3) ResultsOur method is tested on one POE dataset including 3895 distinct images and is showing better results than using Shannon entropy and behaves better with regard to the problem of overfitting. We obtain 70% of accuracy with Shannon entropy versus 77 to 79% with Havrda-Charvat.4) ConclusionWe can conclude that Havrda-Charvat entropy is better suited for restricted and or noisy datasets due to its generalized nature. It is also more suitable for classification in endomicroscopy datasets.     

Loss of translational entropy in molecular associations

Molecular associations in solution are opposed by the loss of entropy (DeltaS) that results from the restriction of motion of each component in the complex. Theoretical estimates of DeltaS are essential for rationalizing binding affinities, as well as for calculating entropic contribution to enzyme catalysis. Recently a statistical-mechanical framework has been proposed for estimating efficiently the translational entropy loss (DeltaS(trsl)), while taking explicitly into account the complex intermolecular interactions between the solute and the solvent. This framework relates the translational entropy of a solute in solution to its "free volume," defined as the volume accessible to the center of mass of the solute in the presence of the solvent and calculated by using an extension of the cell model (CM) for condensed phases. The translational entropy of pure water, estimated with the CM algorithm, shows good agreement with the experimental information. The free volume of various solutes in water, calculated within the CM by using molecular dynamics simulations with explicit solvent, displays a strong correlation with the solutes' polar and total surface areas. This correlation is used to propose a parameterization that can be used to calculate routinely the translational entropy of a solute in water. We also applied the CM formalism to calculate the free volume and translational entropy loss (DeltaS(trsl)) on binding of benzene to a cavity in a mutant T4-lysozyme. Our results agree with previously published estimates of the binding of benzene to this mutant T4-lysozyme. These and other considerations suggest that the cell model is a simple yet efficient theoretical framework to evaluate the translational entropy loss on molecular association in solution.     

Configurational entropy change of netropsin and distamycin upon DNA minor-groove binding

Binding of a small molecule to a macromolecular target reduces its conformational freedom, resulting in a negative entropy change that opposes the binding. The goal of this study is to estimate the configurational entropy change of two minor-groove-binding ligands, netropsin and distamycin, upon binding to the DNA duplex d(CGCGAAAAACGCG).d(CGCGTTTTTCGCG). Configurational entropy upper bounds based on 10-ns molecular dynamics simulations of netropsin and distamycin in solution and in complex with DNA in solution were estimated using the covariance matrix of atom-positional fluctuations. The results suggest that netropsin and distamycin lose a significant amount of configurational entropy upon binding to the DNA minor groove. The estimated changes in configurational entropy for netropsin and distamycin are -127 J K(-1) mol(-1) and -104 J K(-1) mol(-1), respectively. Estimates of the configurational entropy contributions of parts of the ligands are presented, showing that the loss of configurational entropy is comparatively more pronounced for the flexible tails than for the relatively rigid central body.     

The human brain from above: an increase in complexity from environmental stimuli to abstractions

Contrary to common belief, the brain appears to increase the complexity from the perceived object to the idea of it. Topological models predict indeed that: (a) increases in anatomical/functional dimensions and symmetries occur in the transition from the environment to the higher activities of the brain, and (b) informational entropy in the primary sensory areas is lower than in the higher associative ones. To demonstrate this novel hypothesis, we introduce a straightforward approach to measuring island information levels in fMRI neuroimages, via Rényi entropy derived from tessellated fMRI images. This approach facilitates objective detection of entropy and corresponding information levels in zones of fMRI images generally not taken into account. We found that the Rényi entropy is higher in associative cortices than in the visual primary ones. This suggests that the brain lies in dimensions higher than the environment and that it does not concentrate, but rather dilutes messages coming from external inputs.     

Entropy Hotspots for the Binding of Intrinsically Disordered Ligands to a Receptor Domain

Proline-rich motifs (PRMs) are widely used for mediating protein-protein interactions with weak binding affinities. Because they are intrinsically disordered when unbound, conformational entropy plays a significant role for the binding. However, residue-level differences of the entropic contribution in the binding of different ligands remain not well understood. We use all-atom molecular dynamics simulation and the maximal information spanning tree formalism to analyze conformational entropy associated with the binding of two PRMs, one from the Abl kinase and the other from the nonstructural protein 1 of the 1918 Spanish influenza A virus, to the N-terminal SH3 (nSH3) domain of the CrkII protein. Side chains of the stably folded nSH3 experience more entropy change upon ligand binding than the backbone, whereas PRMs involve comparable but heterogeneous entropy changes among the backbone and side chains. In nSH3, two conserved nonpolar residues forming contacts with the PRM experience the largest side-chain entropy loss. In contrast, the C-terminal charged residues of PRMs that form polar contacts with nSH3 experience the greatest side-chain entropy loss, although their “fuzzy” nature is attributable to the backbone that remains relatively flexible. Thus, residues that form high-occupancy contacts between nSH3 and PRM do not reciprocally contribute to entropy loss. Furthermore, certain surface residues of nSH3 distal to the interface with PRMs gain entropy, indicating a nonlocal effect of ligand binding. Comparing between the PRMs from cAbl and nonstructural protein 1, the latter involves a larger side-chain entropy loss and forms more contacts with nSH3. Consistent with experiments, this indicates stronger binding of the viral ligand at the expense of losing the flexibility of side chains, whereas the backbone experiences less entropy loss. The entropy “hotspots” as identified in this study will be important for tuning the binding affinity of various ligands to a receptor.     

The role of environmental calcium in freshwater survival of the marine teleost, Lagodon rhomboides.

(1) The marine teleost fish, Lagodon rhomboides, can only tolerate fresh water (5 mM Na) if Ca is also present (10 mM). Transfer to Ca-free fresh water is followed by a substantial increase in radioactive Na efflux with little or no change in the transepithelial potential. Addition of the chelating agent EDTA (2 mM) further increases Na efflux. Fish left in Ca-free fresh water for 2-5 h die with a total body Na less than 50% of that found in animals acclimated to Ca-supplemented fresh water. (2) Rates of Na uptake were measured on either sea-water-acclimated or Ca-supplemented fresh water-acclimated fish transferred to various low Na media. In both cases Na uptake has a high Km, is saturable, inhibited by external NH4, H and amiloride, and is not related to changes in the trans-epithelial potential. (3) It is suggested that L. rhomboides is dependent upon external Ca to decrease diffusional Na loss in low salinities so that a relatively inefficient Na uptake can balance diffusional and urinary Na loss.     

Association entropy in adsorption processes

The association of two species to form a bound complex, e.g., the binding of a ligand to a protein or the adsorption of a peptide on a lipid membrane, involves an entropy loss, reflecting the conversion of free translational and rotational degrees of freedom into bound motions. Previous theoretical estimates of the standard entropy change in bimolecular binding processes, DeltaS(o), have been derived from the root-mean-square fluctuations in protein crystals, suggesting DeltaS(o) approximately -50 e.u., i.e., TDeltaS degrees approximately -25 kT = -15 kcal/mol. In this work we focus on adsorption, rather than binding processes. We first present a simple statistical-thermodynamic scheme for calculating the adsorption entropy, including its resolution into translational and rotational contributions, using the known distance-orientation dependent binding (adsorption) potential. We then utilize this scheme to calculate the free energy of interaction and entropy of pentalysine adsorption onto a lipid membrane, obtaining TDeltaS(o) approximately -1.7 kT approximately -1.3 kcal/mol. Most of this entropy change is due to the conversion of one free translation into a bound motion, the rest arising from the confinement of two rotational degrees of freedom. The smaller entropy loss in adsorption compared to binding processes arises partly because a smaller number of degrees of freedom become restricted, but mainly due to the fact that the binding potential is much "softer."     

Thermodynamic consequences of grafting enhanced affinity toward the mutated antigen onto an antibody. The case of anti-lysozyme antibody, HyHEL-10

In order to address the mechanism of enhancement of the affinity of an antibody toward an antigen from a thermodynamic viewpoint, anti-hen lysozyme (HEL) antibody HyHEL-10, which also recognize the mutated antigen turkey lysozyme (TEL) with reduced affinity, was examined. Grafting high affinity toward TEL onto HyHEL-10 was performed by saturation mutagenesis into four residues (Tyr(53), Ser(54), Ser(56), and Tyr(58)) in complementarity-determining region 2 of the heavy chain (CDR-H2) followed by selection with affinity for TEL. Several clones enriched have a Phe residue at site 58. Thermodynamic analyses showed that the clones selected had experienced a greater than 3-fold affinity increase toward TEL in comparison with wild-type Fv, originating from an increase in negative enthalpy change. Substitution of HyHEL-10 HTyr(58) with Phe led to the increase in negative enthalpy change and to almost identical affinity for TEL in comparison with mutants selected, indicating that mutations at other sites decrease the entropy loss despite little contribution to the affinity for TEL. These results suggest that the affinity of an antibody toward the antigen is enhanced by the increase in enthalpy change by some limited mutation, and excess entropy loss due to the mutation is decreased by other energetically neutral mutations.     

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.     

Toward rational protein crystallization: A Web server for the design of crystallizable protein variants

Growing well-diffracting crystals constitutes a serious bottleneck in structural biology. A recently proposed crystallization methodology for "stubborn crystallizers" is to engineer surface sequence variants designed to form intermolecular contacts that could support a crystal lattice. This approach relies on the concept of surface entropy reduction (SER), i.e., the replacement of clusters of flexible, solvent-exposed residues with residues with lower conformational entropy. This strategy minimizes the loss of conformational entropy upon crystallization and renders crystallization thermodynamically favorable. The method has been successfully used to crystallize more than 15 novel proteins, all stubborn crystallizers. But the choice of suitable sites for mutagenesis is not trivial. Herein, we announce a Web server, the surface entropy reduction prediction server (SERp server), designed to identify mutations that may facilitate crystallization. Suggested mutations are predicted based on an algorithm incorporating a conformational entropy profile, a secondary structure prediction, and sequence conservation. Minor considerations include the nature of flanking residues and gaps between mutation candidates. While designed to be used with default values, the server has many user-controlled parameters allowing for considerable flexibility. Within, we discuss (1) the methodology of the server, (2) how to interpret the results, and (3) factors that must be considered when selecting mutations. We also attempt to benchmark the server by comparing the server's predictions with successful SER structures. In most cases, the structure yielding mutations were easily identified by the SERp server. The server can be accessed at http://www.doe-mbi.ucla.edu/Services/SER.     

Effects of mutations on the thermodynamics of a protein folding reaction: implications for the mechanism of formation of the intermediate and transition states

We have measured changes in heat capacity, entropy, and enthalpy for each step in the folding reaction of CD2.d1 and evaluated the effects of core mutations on these properties. All wild-type and mutant forms fold through a rapidly formed intermediate state that precedes the rate-limiting transition state. Mutations have a pronounced effect on the enthalpy of both the intermediate and folded states, but in all cases a compensatory change in entropy results in a small net free-energy change. While the enthalpy change in the folded state can be attributed to a loss of van der Waals interactions, it has already been shown that changes in the stability of the intermediate are dominated by changes in secondary structure propensity [Lorch et al. (1999) Biochemistry 38, 1377-1385]. It follows that the thermodynamic basis of beta-propensity is enthalpic in origin. The effects of mutations on the enthalpy and entropy of the transition state are smaller than on the ground states. This relative insensitivity to mutation is discussed in the light of theories concerning the nature of the rate-limiting barrier in folding reactions.     

Entropy loss in long-distance DNA looping

The entropy loss due to the formation of one or multiple loops in circular and linear DNA chains is calculated from a scaling approach in the limit of long chain segments. The analytical results allow us to obtain a fast estimate for the entropy loss for a given configuration. Numerical values obtained for some examples suggest that the entropy loss encountered in loop closure in typical genetic switches may become a relevant factor in comparison to both k(B)T and typical bond energies in biopolymers, which has to be overcome by the released bond energy between the looping contact sites.     

Cancer-related mutations in BRCA1-BRCT cause long-range structural changes in protein-protein binding sites: a molecular dynamics study

Cancer-associated mutations in the BRCT domain of BRCA1 (BRCA1-BRCT) abolish its tumor suppressor function by disrupting interactions with other proteins such as BACH1. Many cancer-related mutations do not cause sufficient destabilization to lead to global unfolding under physiological conditions, and thus abrogation of function probably is due to localized structural changes. To explore the reasons for mutation-induced loss of function, the authors performed molecular dynamics simulations on three cancer-associated mutants, A1708E, M1775R, and Y1853ter, and on the wild type and benign M1652I mutant, and compared the structures and fluctuations. Only the cancer-associated mutants exhibited significant backbone structure differences from the wild-type crystal structure in BACH1-binding regions, some of which are far from the mutation sites. Backbone differences of the A1708E mutant from the liganded wild type structure in these regions are much larger than those of the unliganded wild type X-ray or molecular dynamics structures. These BACH1-binding regions of the cancer-associated mutants also exhibited increases in their fluctuation magnitudes compared with the same regions in the wild type and M1562I mutant, as quantified by quasiharmonic analysis. Several of the regions of increased fluctuation magnitude correspond to correlated motions of residues in contact that provide a continuous path of fluctuating amino acids in contact from the A1708E and Y1853ter mutation sites to the BACH1-binding sites with altered structure and dynamics. The increased fluctuations in the disease-related mutants suggest an increase in vibrational entropy in the unliganded state that could result in a larger entropy loss in the disease-related mutants upon binding BACH1 than in the wild type. To investigate this possibility, vibrational entropies of the A1708E and wild type in the free state and bound to a BACH1-derived phosphopeptide were calculated using quasiharmonic analysis, to determine the binding entropy difference DeltaDeltaS between the A1708E mutant and the wild type. DeltaDeltaS was determined to be -4.0 cal mol(-1) K(-1), with an uncertainty of 2 cal mol(-1) K(-1); that is, the entropy loss upon binding the peptide is 4.0 cal mol(-1) K(-1) greater for the A1708E mutant, corresponding to an entropic contribution to the DeltaDeltaG of binding (-TDeltaDeltaS) 1.1 kcal mol(-1) more positive for the mutant. The observed differences in structure, flexibility, and entropy of binding likely are responsible for abolition of BACH1 binding, and illustrate that many disease- related mutations could have very long-range effects. The methods described here have potential for identifying correlated motions responsible for other long-range effects of deleterious mutations.     

Estimates of the loss of main-chain conformational entropy of different residues on protein folding.

The average contribution of conformational entropy for individual amino acid residues towards the free energy of protein folding is not well understood. We have developed empirical scales for the loss of the main-chain (torsion angles, phi and psi) conformational entropy by taking its side-chain into account. The analysis shows that the main-chain component of the total conformational entropy loss for a residue is significant and reflects intrinsic characteristics associated with individual residues. The values have direct correlation with the hydrophobicity values and this has important bearing on the folding process. Proteins 1999;36:332-339.     

Empirical assessment of the relation between ecological connectivity and land complexity based on information-theoretic metrics

Habitat fragmentation and connectivity loss pose significant threats to biodiversity at both local and landscape levels. Strategies to increase ecological connectivity and preserve strong connectivity are important for dealing with the potential threat of habitat degradation. Various metrics have been used to measure (i.e., quantify) landscape composition and configuration in landscape ecology. However, their relationship with ecological connectivity must be understood to interpret landscape patterns comprehensively. In the present study, correlations between ecological connectivity and land complexity are examined based on information-theory metrics. Two primary questions are explored: (1) to what extent are landscape mosaic measures of entropy correlated with ecological connectivity, with landscape gradient-based measures, and with each other? (2) are landscape gradient-based entropy measures correlated with ecological connectivity more than discrete entropy measures? Results show that all information theoretic metrics are statistically significant (p < 0.05) for modelling ecological connectivity. Among categorically-based indices, the relationship between ECI and joint entropy was the most significant, while a generalized additive model indicated that Boltzmann entropy could predict the ecological connectivity index, explaining ∼60% of the variance. Therefore, configurational entropy can be used for improving ecological connectivity models.     

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.     

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.