Computation of conformational entropy from protein sequences using the machine-learning method--application to the study of the relationship between structural conservation and local structural stability

A complete protein sequence can usually determine a unique conformation; however, the situation is different for shorter subsequences--some of them are able to adopt unique conformations, independent of context; while others assume diverse conformations in different contexts. The conformations of subsequences are determined by the interplay between local and nonlocal interactions. A quantitative measure of such structural conservation or variability will be useful in the understanding of the sequence-structure relationship. In this report, we developed an approach using the support vector machine method to compute the conformational variability directly from sequences, which is referred to as the sequence structural entropy. As a practical application, we studied the relationship between sequence structural entropy and the hydrogen exchange for a set of well-studied proteins. We found that the slowest exchange cores usually comprise amino acids of the lowest sequence structural entropy. Our results indicate that structural conservation is closely related to the local structural stability. This relationship may have interesting implications in the protein folding processes, and may be useful in the study of the sequence-structure relationship.     

Entropy of dynamical social networks

Human dynamical social networks encode information and are highly adaptive. To characterize the information encoded in the fast dynamics of social interactions, here we introduce the entropy of dynamical social networks. By analysing a large dataset of phone-call interactions we show evidence that the dynamical social network has an entropy that depends on the time of the day in a typical week-day. Moreover we show evidence for adaptability of human social behavior showing data on duration of phone-call interactions that significantly deviates from the statistics of duration of face-to-face interactions. This adaptability of behavior corresponds to a different information content of the dynamics of social human interactions. We quantify this information by the use of the entropy of dynamical networks on realistic models of social interactions.     

Natural hierarchy emerges from energy dispersal

Hierarchical organization of ‘systems within systems’ is an apparent characteristic of nature. For many biotic and abiotic systems it is known how the nested structural and functional order builds up, yet the general principle why matter evolves to hierarchies has remained unfamiliar to many. We clarify that increasingly larger integrated systems result from the quest to decrease free energy according to the 2nd law of thermodynamics. The argumentation is based on the recently derived equation of motion for natural processes. Hierarchically organized energy transduction machinery emerges naturally when it provides increased rates of energy dispersal. Likewise, a hierarchical system will dismantle into its constituents when they as independent systems will provide higher rates of entropy increase. Since energy flows via interactions, decreasing strengths of interactions over increasingly larges lengths scales mark natural boundaries for nested integrated systems.     

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.     

204 Polymer translocation

The translocation of a single macromolecule through a protein pore or a solid-state nanopore involves three major stages: (1) approach of the macromolecule towards the pore, (2) capture/recognition of the macromolecule at the pore entrance, and (3) threading through the pore (see the Figure) (Muthukumar, 2011). All of these stages are controlled by conformational entropy of the macromolecule, charge decoration, and the geometry of the pore, hydrodynamics, and electrostatic interactions. Chief among the contributing factors are the entropic barrier presented by the pore to the penetration of the macromolecule, pore–polymer interactions, electro-osmotic flow, and the drift-diffusion of the macromolecule in electrolyte solutions. A unifying theory of these contributing factors will be described in the context of a few illustrative experimental data on DNA translocation and protein translocation through protein pores and solid-state nanopores. Future challenges to specific biological systems will be briefly discussed.     

On the Fractal Geometry of DNA by the Binary Image Analysis

The multifractal analysis of binary images of DNA is studied in order to define a methodological approach to the classification of DNA sequences. This method is based on the computation of some multifractality parameters on a suitable binary image of DNA, which takes into account the nucleotide distribution. The binary image of DNA is obtained by a dot-plot (recurrence plot) of the indicator matrix. The fractal geometry of these images is characterized by fractal dimension (FD), lacunarity, and succolarity. These parameters are compared with some other coefficients such as complexity and Shannon information entropy. It will be shown that the complexity parameters are more or less equivalent to FD, while the parameters of multifractality have different values in the sense that sequences with higher FD might have lower lacunarity and/or succolarity. In particular, the genome of Drosophila melanogaster has been considered by focusing on the chromosome 3r, which shows the highest fractality with a corresponding higher level of complexity. We will single out some results on the nucleotide distribution in 3r with respect to complexity and fractality. In particular, we will show that sequences with higher FD also have a higher frequency distribution of guanine, while low FD is characterized by the higher presence of adenine.     

Consistency in structural energetics of protein folding and peptide recognition.

We report a new free energy decomposition that includes structure-derived atomic contact energies for the desolvation component, and show that it applies equally well to the analysis of single-domain protein folding and to the binding of flexible peptides to proteins. Specifically, we selected the 17 single-domain proteins for which the three-dimensional structures and thermodynamic unfolding free energies are available. By calculating all terms except the backbone conformational entropy change and comparing the result to the experimentally measured free energy, we estimated that the mean entropy gain by the backbone chain upon unfolding (delta Sbb) is 5.3 cal/K per mole of residue, and that the average backbone entropy for glycine is 6.7 cal/K. Both numbers are in close agreement with recent estimates made by entirely different methods, suggesting a promising degree of consistency between data obtained from disparate sources. In addition, a quantitative analysis of the folding free energy indicates that the unfavorable backbone entropy for each of the proteins is balanced predominantly by favorable backbone interactions. Finally, because the binding of flexible peptides to receptors is physically similar to folding, the free energy function should, in principle, be equally applicable to flexible docking. By combining atomic contact energies, electrostatics, and sequence-dependent backbone entropy, we calculated a priori the free energy changes associated with the binding of four different peptides to HLA-A2, 1 MHC molecule and found agreement with experiment to within 10% without parameter adjustment.     

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.     

Thermodynamic Aspects of cAMP Dependent Protein Kinase Catalytic Subunit Allostery

Kinetics of thermal inactivation of acrylodan-labeled cAMP dependent protein kinase catalytic subunit, its binary complexes with ATP and peptide inhibitor PKI[5–24], respectively, and the ternary complex involving both of these ligands were studied at different temperatures (5–50 °C). The thermodynamic parameters ΔH and ΔS for ligand binding equilibria as well as for the allosteric interaction between the binding sites of these ligands were obtained by using the Van’t Hoff analysis. The results indicated that more inter- and intra-molecular non-covalent bonds were involved in ATP binding with the protein when compared to the peptide binding. Similarly, nucleotide and peptide binding steps were accompanied with different entropy effects, while almost no entropy change accompanied PKI[5–24] binding, suggesting that the protein flexibility was not affected in this case. Differently from the binary complex formation the ternary complex formation was accompanied by a significant entropy change and with intensive formation of new non-covalent interactions (ΔH). At the same time both ligand binding steps as well as the allosteric interaction between ligand binding sites could be described by a common entropy–enthalpy compensation plot, pointing to a similar mechanism of these phenomena. It was concluded that numerous weak interactions govern the allostery of cAMP dependent protein kinase catalytic subunit.     

Normalized mutual entropy in biology: quantifying division of labor

Division of labor is one of the primary adaptations of sociality and the focus of much theoretical work on self-organization. This work has been hampered by the lack of a quantitative measure of division of labor that can be applied across systems. We divide Shannon's mutual entropy by marginal entropy to quantify division of labor, rendering it robust over changes in number of individuals or tasks. Reinterpreting individuals and tasks makes this methodology applicable to a wide range of other contexts, such as breeding systems and predator-prey interactions.     

Thermodynamic analysis of biomolecular interactions.

Direct measurement of the thermodynamics of biomolecular interactions is now relatively easy. Interpretation of these thermodynamics in simple molecular terms is not. Recent work shows how the multiplicity of weak noncovalent interactions, and the inevitable enthalpy/entropy compensation that these interactions engender, lead to difficulties in teasing out the different components.     

The Hydrophobic Effect: A New Insight from Cold Denaturation and a Two-State Water Structure

Herein we provide a new insight into the hydrophobic effect in protein folding. Our proposition explains the molecular basis of cold denaturation, and of intermediate states in heat and their absence in cold denaturation. The exposure of non-polar surface reduces the entropy and enthalpy of the system, at low and at high temperatures. At low temperatures the favorable reduction in enthalpy overcomes the unfavorable reduction in entropy, leading to cold denaturation. At high temperatures, folding/unfolding is a two-step process: in the first, the entropy gain leads to hydrophobic collapse, in the second, the reduction in enthalpy due to protein-protein interactions leads to the native state. The different entropy and enthalpy contributions to the Gibbs energy change at each step at high, and at low, temperatures can be conveniently explained by a two-state model of the water structure. The model provides a clear view of the dominant factors in protein folding and stability. Consequently, it appears to provide a microscopic view of the hydrophobic effect and is consistently linked to macroscopic thermodynamic parameters.     

The hydrophobic effect: a new insight from cold denaturation and a two-state water structure

Herein we provide a new insight into the hydrophobic effect in protein folding. Our proposition explains the molecular basis of cold denaturation, and of intermediate states in heat and their absence in cold denaturation. The exposure of non-polar surface reduces the entropy and enthalpy of the system, at low and at high temperatures. At low temperatures the favorable reduction in enthalpy overcomes the unfavorable reduction in entropy, leading to cold denaturation. At high temperatures, folding/unfolding is a two-step process: in the first, the entropy gain leads to hydrophobic collapse, in the second, the reduction in enthalpy due to protein-protein interactions leads to the native state. The different entropy and enthalpy contributions to the Gibbs energy change at each step at high, and at low, temperatures can be conveniently explained by a two-state model of the water structure. The model provides a clear view of the dominant factors in protein folding and stability. Consequently, it appears to provide a microscopic view of the hydrophobic effect and is consistently linked to macroscopic thermodynamic parameters.     

MuTE: A MATLAB Toolbox to Compare Established and Novel Estimators of the Multivariate Transfer Entropy

A challenge for physiologists and neuroscientists is to map information transfer between components of the systems that they study at different scales, in order to derive important knowledge on structure and function from the analysis of the recorded dynamics. The components of physiological networks often interact in a nonlinear way and through mechanisms which are in general not completely known. It is then safer that the method of choice for analyzing these interactions does not rely on any model or assumption on the nature of the data and their interactions. Transfer entropy has emerged as a powerful tool to quantify directed dynamical interactions. In this paper we compare different approaches to evaluate transfer entropy, some of them already proposed, some novel, and present their implementation in a freeware MATLAB toolbox. Applications to simulated and real data are presented.     

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.     

Analysing protein energy data by a stochastic model for cooperative interactions: comparison and characterization of cooperativity

 In the frame of a Markov chain model for cooperative interactions in proteins, previously introduced by us, we deal here with estimation of unknown parameters from protein energy data. One of these parameters characterizes the cooperativity of a protein; we propose to measure it also by the so-called approximate entropy. By our computations the approximate entropy turns out to be a decreasing function of the cooperativity. We analyse both simulated data of the Markov chain, and protein energy data obtained by molecular dynamics simulation. Moreover, we compare two rubredoxin proteins at different temperatures, according to their degrees of cooperativity. Received: 2 October 2000 / Revised version: 4 April 2001 / Published online: 14 March 2002     

Kinetic and thermodynamic characterization of HIV-1 protease inhibitors

Interaction kinetic and thermodynamic analyses provide information beyond that obtained in general inhibition studies, and may contribute to the design of improved inhibitors and increased understanding of molecular interactions. Thus, a biosensor-based method was used to characterize the interactions between HIV-1 protease and seven inhibitors, revealing distinguishing kinetic and thermodynamic characteristics for the inhibitors. Lopinavir had fast association and the highest affinity of the tested compounds, and the interaction kinetics were less temperature-dependent as compared with the other inhibitors. Amprenavir, indinavir and ritonavir showed non-linear temperature dependencies of the kinetics. The free energy, enthalpy and entropy (DeltaG, DeltaH, DeltaS) were determined, and the energetics of complex association (DeltaG(on), DeltaH(on), DeltaS(on)) and dissociation (DeltaG(off), DeltaH(off), DeltaS(off)) were resolved. In general, the energetics for the studied inhibitors was in the same range, with the negative free energy change (DeltaG < 0) due primarily to increased entropy (DeltaS > 0). Thus, the driving force of the interaction was increased degrees of freedom in the system (entropy) rather than the formation of bonds between the enzyme and inhibitor (enthalpy). Although the DeltaG(on) and DeltaG(off) were in the same range for all inhibitors, the enthalpy and entropy terms contributed differently to association and dissociation, distinguishing these phases energetically. Dissociation was accompanied by positive enthalpy (DeltaH(off) > 0) and negative entropy (DeltaS(off) < 0) changes, whereas association for all inhibitors except lopinavir had positive entropy changes (DeltaS(on) > 0), demonstrating unique energetic characteristics for lopinavir. This study indicates that this type of data will be useful for the characterization of target-ligand interactions and the development of new inhibitors of HIV-1 protease.     

Joint Entropy for Space and Spatial Frequency Domains Estimated from Psychometric Functions of Achromatic Discrimination

We used psychometric functions to estimate the joint entropy for space discrimination and spatial frequency discrimination. Space discrimination was taken as discrimination of spatial extent. Seven subjects were tested. Gábor functions comprising unidimensionalsinusoidal gratings (0.4, 2, and 10 cpd) and bidimensionalGaussian envelopes (1°) were used as reference stimuli. The experiment comprised the comparison between reference and test stimulithat differed in grating''s spatial frequency or envelope''s standard deviation. We tested 21 different envelope''s standard deviations around the reference standard deviation to study spatial extent discrimination and 19 different grating''s spatial frequencies around the reference spatial frequency to study spatial frequency discrimination. Two series of psychometric functions were obtained for 2%, 5%, 10%, and 100% stimulus contrast. The psychometric function data points for spatial extent discrimination or spatial frequency discrimination were fitted with Gaussian functions using the least square method, and the spatial extent and spatial frequency entropies were estimated from the standard deviation of these Gaussian functions. Then, joint entropy was obtained by multiplying the square root of space extent entropy times the spatial frequency entropy. We compared our results to the theoretical minimum for unidimensional Gábor functions, 1/4π or 0.0796. At low and intermediate spatial frequencies and high contrasts, joint entropy reached levels below the theoretical minimum, suggesting non-linear interactions between two or more visual mechanisms. We concluded that non-linear interactions of visual pathways, such as the M and P pathways, could explain joint entropy values below the theoretical minimum at low and intermediate spatial frequencies and high contrasts. These non-linear interactions might be at work at intermediate and high contrasts at all spatial frequencies once there was a substantial decrease in joint entropy for these stimulus conditions when contrast was raised.     

Change in entropy of the free polypeptide chain during formation of hydrogen bonds]

Formation probabilities of different hydrogen bonds between carbonyl oxygen and amide hydrogen were determined by Monte Carlo simulations using a computer model in the space of sterically allowable conformations of alanine and glycine oligopeptides, and the corresponding entropy losses for the peptide backbone, T delta S, were calculated. The model was studied at different criteria of steric interactions. Comparison with the data of other authors showed the values of T delta S to be mainly determined by overall extent and type of the state space and to be only slightly dependent on its energy profile. Both short-range and long-range steric interactions were shown to prevent hydrogen bonding, especially in alanine peptides. In the model studied, the initiation of alpha(R)-helices is associated with T delta S = 8-10 kT, and prior formation of a 3/10-turn or one three-center H-bond does not appreciably decrease this entropy barrier. Elongation of the alpha(R)-helix by one residue leads to T delta S = 3.0-3.7 kT, the helices begin to stabilize after at least three sequential H-bonds are formed. The difference in the probability of insertion of Ala and Gly into the helix is lower than it follows from comparison of their mobility. The results could be explained assuming that factors different from helical H-bonds take part in the stabilization of the helices. One may suppose upon modeling of folding that even three sequential H-bonds are unable to fix the structure of a flexible peptide loop, while the elongation of alpha(R)-helices in the supersecondary helix-loop-helix structure is favorable as long as the loop conformation remains nearly optimal.