A flexible approach for understanding protein stability

Protein Function Analysis through Machine Learning

Machine learning (ML) has been an important arsenal in computational biology used to elucidate protein function for decades. With the recent burgeoning of novel ML methods and applications, new ML approaches have been incorporated into many areas of computational biology dealing with protein function. We examine how ML has been integrated into a wide range of computational models to improve prediction accuracy and gain a better understanding of protein function. The applications discussed are protein structure prediction, protein engineering using sequence modifications to achieve stability and druggability characteristics, molecular docking in terms of protein-ligand binding, including allosteric effects, protein-protein interactions and protein-centric drug discovery. To quantify the mechanisms underlying protein function, a holistic approach that takes structure, flexibility, stability, and dynamics into account is required, as these aspects become inseparable through their interdependence. Another key component of protein function is conformational dynamics, which often manifest as protein kinetics. Computational methods that use ML to generate representative conformational ensembles and quantify differences in conformational ensembles important for function are included in this review. Future opportunities are highlighted for each of these topics.

In silico Prediction of Deleterious Single Nucleotide Polymorphism in S100A4 Metastatic Gene: Potential Early Diagnostic Marker

S100A4 protein overexpression has been reported in different types of cancer and plays a key role by interacting with the tumor suppressor protein Tp53. Single nucleotide polymorphisms (SNP) in S100A4 could directly influence the biomolecular interaction with the tumor suppressor protein Tp53 due to their aberrant conformations. Hence, the study was designed to predict the deleterious SNP and its effect on the S100A4 protein structure and function. Twenty-one SNP data sets were screened for nonsynonymous mutations and subsequently subjected to deleterious mutation prediction using different computational tools. The screened deleterious mutations were analyzed for their changes in functionality and their interaction with the tumor suppressor protein Tp53 by protein-protein docking analysis. The structural effects were studied using the 3DMissense mutation tool to estimate the solvation energy and torsion angle of the screened mutations on the predicted structures. In our study, 21 deleterious nonsynonymous mutations were screened, including F72V, E74G, L5P, D25E, N65S, A28V, A8D, S20L, L58P, and K26N were found to be remarkably conserved by exhibiting the interaction either with the EF-hand 1 or EF-hand 2 domain. The solvation and torsion values significantly deviated for the mutant-type structures with S20L, N65S, and F72L mutations and showed a marked reduction in their binding affinity with the Tp53 protein. Hence, these deleterious mutations might serve as prospective targets for diagnosing and developing personalized treatments for cancer and other related diseases.

Insights into the mechanism of inhibition of phospholipase A2 by resveratrol: An extensive molecular dynamics simulation and binding free energy calculation

Phospholipase A2 (PLA2) is one of the enzymes involved in the development of cardiovascular diseases, vascular inflammation, risk of heart attacks, and strokes. This enzyme is responsible for catalyzing the hydrolytic cleavage of ester bonds of phospholipids in the biological pathway of inflammation. To prevent the undesired hydrolysis of phospholipids, the catalytic activity of PLA2 needs to be blocked. Resveratrol is a plant-derived polyphenol inhibitor, proven to have anti-inflammatory properties. However, there is still substantial ambiguity about its inhibitory function. The present study uncovers a detailed molecular mechanism behind the resveratrol action in inhibition of PLA2, by applying and comparing two 200-ns molecular dynamics simulations. The results of structural analyses revealed that the binding of resveratrol to PLA2 reduces the content of β-sheets and increases a 5-helix to PLA2 structure, producing more folding and stability in protein. In the active site, the resveratrol is placed between the N-terminal α-helix and the newly formed 5-helix through the hydrophobic interactions with ILE19 and LEU3 residues, as well as the hydrogen bond interactions. These interactions play the role of a network at the entrance of the enzyme active site and prevent the penetration of water molecules into the PLA2 cavity. A high occupancy hydrogen bonding has been identified between SER23 of the protein and hydroxyl group of resveratrol. Furthermore, the estimation of binding free energy verified the binding affinity of resveratrol is thermodynamically sufficient to be stably bounded to PLA2. It also proved that the van der Waals interactions, particularly hydrophobic interactions, have the most significant role in PLA2-resveratrol binding and stability. Overall, our results provide useful information on the stepwise mechanism of the inhibition of PLA2 enzyme by resveratrol, as a target for improving the pharmacological applications.

Rigidity and flexibility characteristics of DD[E/D]-transposases Mos1 and Sleeping Beauty

DD[E/D]-transposases catalyze the multistep reaction of cut-and-paste DNA transposition. Structurally, several DD[E/D]-transposases have been characterized, revealing a multi-domain structure with the catalytic domain possessing the RNase H-like structural motif that brings three catalytic residues (D, D, and E or D) into close proximity for the catalysis. However, the dynamic behavior of DD[E/D]-transposases during transposition remains poorly understood. Here, we analyze the rigidity and flexibility characteristics of two representative DD[E/D]-transposases Mos1 and Sleeping Beauty (SB) using the minimal distance constraint model (mDCM). We find that the catalytic domain of both transposases is globally rigid, with the notable exception of the clamp loop being flexible in the DNA-unbound form. Within this globally rigid structure, the central β-sheet of the RNase H-like motif is much less rigid in comparison to its surrounding α-helices, forming a cage-like structure. The comparison of the original SB transposase to its hyperactive version SB100X reveals the region where the change in flexibility/rigidity correlates with increased activity. This region is found to be within the RNase H-like structural motif and comprise the loop leading from beta-strand B3 to helix H1, helices H1 and H2, which are located on the same side of the central beta-sheet, and the loop between helix H3 and beta-strand B5. We further identify the RKEN214-217DAVQ mutations of the set of hyperactive mutations within the catalytic domain of SB transposase to be the driving factor that induces change in residue-pair rigidity correlations within SB transposase. Given that a signature RNase H-like structural motif is found in DD[E/D]-transposases and, more broadly, in a large superfamily of polynucleotidyl transferases, our results are relevant to these proteins as well.

Multiscale weighted colored graphs for protein flexibility and rigidity analysis

Protein structural fluctuation, measured by Debye-Waller factors or B-factors, is known to correlate to protein flexibility and function. A variety of methods has been developed for protein Debye-Waller factor prediction and related applications to domain separation, docking pose ranking, entropy calculation, hinge detection, stability analysis, etc. Nevertheless, none of the current methodologies are able to deliver an accuracy of 0.7 in terms of the Pearson correlation coefficients averaged over a large set of proteins. In this work, we introduce a paradigm-shifting geometric graph model, multiscale weighted colored graph (MWCG), to provide a new generation of computational algorithms to significantly change the current status of protein structural fluctuation analysis. Our MWCG model divides a protein graph into multiple subgraphs based on interaction types between graph nodes and represents the protein rigidity by generalized centralities of subgraphs. MWCGs not only predict the B-factors of protein residues but also accurately analyze the flexibility of all atoms in a protein. The MWCG model is validated over a number of protein test sets and compared with many standard methods. An extensive numerical study indicates that the proposed MWCG offers an accuracy of over 0.8 and thus provides perhaps the first reliable method for estimating protein flexibility and B-factors. It also simultaneously predicts all-atom flexibility in a molecule.

Mutations in Antibody Fragments Modulate Allosteric Response Via Hydrogen-Bond Network Fluctuations

A mechanical perturbation method that locally restricts conformational entropy along the protein backbone is used to identify putative allosteric sites in a series of antibody fragments. The method is based on a distance constraint model that integrates mechanical and thermodynamic viewpoints of protein structure wherein mechanical clamps that mimic substrate or cosolute binding are introduced. Across a set of six single chain-Fv fragments of the anti-lymphotoxin-β receptor antibody, statistically significant responses are obtained by averaging over 10 representative structures sampled from a molecular dynamics simulation. As expected, the introduced clamps locally rigidify the protein, but long-ranged increases in both rigidity and flexibility are also frequently observed. Expanding our analysis to every molecular dynamics frame demonstrates that the allosteric responses are modulated by fluctuations within the hydrogen-bond network where the native ensemble is comprised of conformations that both are, and are not, affected by the perturbation in question. Population shifts induced by the mutations alter the allosteric response by adjusting which hydrogen-bond networks are the most probable. These effects are compared using response maps that track changes across each single chain-Fv fragment, thus providing valuable insight into how sensitive allosteric mechanisms are to mutations.

Multiscale Gaussian network model (mGNM) and multiscale anisotropic network model (mANM)

Gaussian network model (GNM) and anisotropic network model (ANM) are some of the most popular methods for the study of protein flexibility and related functions. In this work, we propose generalized GNM (gGNM) and ANM methods and show that the GNM Kirchhoff matrix can be built from the ideal low-pass filter, which is a special case of a wide class of correlation functions underpinning the linear scaling flexibility-rigidity index (FRI) method. Based on the mathematical structure of correlation functions, we propose a unified framework to construct generalized Kirchhoff matrices whose matrix inverse leads to gGNMs, whereas, the direct inverse of its diagonal elements gives rise to FRI method. With this connection, we further introduce two multiscale elastic network models, namely, multiscale GNM (mGNM) and multiscale ANM (mANM), which are able to incorporate different scales into the generalized Kirchhoff matrices or generalized Hessian matrices. We validate our new multiscale methods with extensive numerical experiments. We illustrate that gGNMs outperform the original GNM method in the B-factor prediction of a set of 364 proteins. We demonstrate that for a given correlation function, FRI and gGNM methods provide essentially identical B-factor predictions when the scale value in the correlation function is sufficiently large. More importantly, we reveal intrinsic multiscale behavior in protein structures. The proposed mGNM and mANM are able to capture this multiscale behavior and thus give rise to a significant improvement of more than 11% in B-factor predictions over the original GNM and ANM methods. We further demonstrate the benefits of our mGNM through the B-factor predictions of many proteins that fail the original GNM method. We show that the proposed mGNM can also be used to analyze protein domain separations. Finally, we showcase the ability of our mANM for the analysis of protein collective motions.

Flexibility-rigidity index for protein-nucleic acid flexibility and fluctuation analysis

Protein-nucleic acid complexes are important for many cellular processes including the most essential functions such as transcription and translation. For many protein-nucleic acid complexes, flexibility of both macromolecules has been shown to be critical for specificity and/or function. The flexibility-rigidity index (FRI) has been proposed as an accurate and efficient approach for protein flexibility analysis. In this article, we introduce FRI for the flexibility analysis of protein-nucleic acid complexes. We demonstrate that a multiscale strategy, which incorporates multiple kernels to capture various length scales in biomolecular collective motions, is able to significantly improve the state of art in the flexibility analysis of protein-nucleic acid complexes. We take the advantage of the high accuracy and O(N) computational complexity of our multiscale FRI method to investigate the flexibility of ribosomal subunits, which are difficult to analyze by alternative approaches. An anisotropic FRI approach, which involves localized Hessian matrices, is utilized to study the translocation dynamics in an RNA polymerase.

Novel Ricin Subunit Antigens With Enhanced Capacity to Elicit Toxin-Neutralizing Antibody Responses in Mice

RiVax is a candidate ricin toxin subunit vaccine antigen that has proven to be safe in human phase I clinical trials. In this study, we introduced double and triple cavity-filling point mutations into the RiVax antigen with the expectation that stability-enhancing modifications would have a beneficial effect on overall immunogenicity of the recombinant proteins. We demonstrate that 2 RiVax triple mutant derivatives, RB (V81L/C171L/V204I) and RC (V81I/C171L/V204I), when adsorbed to aluminum salts adjuvant and tested in a mouse prime-boost-boost regimen were 5- to 10-fold more effective than RiVax at eliciting toxin-neutralizing serum IgG antibody titers. Increased toxin neutralizing antibody values and seroconversion rates were evident at different antigen dosages and within 7 days after the first booster. Quantitative stability/flexibility relationships analysis revealed that the RB and RC mutations affect rigidification of regions spanning residues 98-103, which constitutes a known immunodominant neutralizing B-cell epitope. A more detailed understanding of the immunogenic nature of RB and RC may provide insight into the fundamental relationship between local protein stability and antibody reactivity.

Dancing through Life: Molecular Dynamics Simulations and Network-Centric Modeling of Allosteric Mechanisms in Hsp70 and Hsp110 Chaperone Proteins

Hsp70 and Hsp110 chaperones play an important role in regulating cellular processes that involve protein folding and stabilization, which are essential for the integrity of signaling networks. Although many aspects of allosteric regulatory mechanisms in Hsp70 and Hsp110 chaperones have been extensively studied and significantly advanced in recent experimental studies, the atomistic picture of signal propagation and energetics of dynamics-based communication still remain unresolved. In this work, we have combined molecular dynamics simulations and protein stability analysis of the chaperone structures with the network modeling of residue interaction networks to characterize molecular determinants of allosteric mechanisms. We have shown that allosteric mechanisms of Hsp70 and Hsp110 chaperones may be primarily determined by nucleotide-induced redistribution of local conformational ensembles in the inter-domain regions and the substrate binding domain. Conformational dynamics and energetics of the peptide substrate binding with the Hsp70 structures has been analyzed using free energy calculations, revealing allosteric hotspots that control negative cooperativity between regulatory sites. The results have indicated that cooperative interactions may promote a population-shift mechanism in Hsp70, in which functional residues are organized in a broad and robust allosteric network that can link the nucleotide-binding site and the substrate-binding regions. A smaller allosteric network in Hsp110 structures may elicit an entropy-driven allostery that occurs in the absence of global structural changes. We have found that global mediating residues with high network centrality may be organized in stable local communities that are indispensable for structural stability and efficient allosteric communications. The network-centric analysis of allosteric interactions has also established that centrality of functional residues could correlate with their sensitivity to mutations across diverse chaperone functions. This study reconciles a wide spectrum of structural and functional experiments by demonstrating how integration of molecular simulations and network-centric modeling may explain thermodynamic and mechanistic aspects of allosteric regulation in chaperones.

Rigidity Emerges during Antibody Evolution in Three Distinct Antibody Systems: Evidence from QSFR Analysis of Fab Fragments

The effects of somatic mutations that transform polyspecific germline (GL) antibodies to affinity mature (AM) antibodies with monospecificity are compared among three GL-AM Fab pairs. In particular, changes in conformational flexibility are assessed using a Distance Constraint Model (DCM). We have previously established that the DCM can be robustly applied across a series of antibody fragments (VL to Fab), and subsequently, the DCM was combined with molecular dynamics (MD) simulations to similarly characterize five thermostabilizing scFv mutants. The DCM is an ensemble based statistical mechanical approach that accounts for enthalpy/entropy compensation due to network rigidity, which has been quite successful in elucidating conformational flexibility and Quantitative Stability/Flexibility Relationships (QSFR) in proteins. Applied to three disparate antibody systems changes in QSFR quantities indicate that the VH domain is typically rigidified, whereas the VL domain and CDR L2 loop become more flexible during affinity maturation. The increase in CDR H3 loop rigidity is consistent with other studies in the literature. The redistribution of conformational flexibility is largely controlled by nonspecific changes in the H-bond network, although certain Arg to Asp salt bridges create highly localized rigidity increases. Taken together, these results reveal an intricate flexibility/rigidity response that accompanies affinity maturation.

Antibodies are protective proteins used by the immune system to recognize and neutralize foreign objects through interactions with a specific part of the target, called an antigen. Antibody structures are Y-shaped, contain multiple protein chains, and include two antigen-binding sites. The binding sites are located at the end of the Fab fragments, which are the upward facing arms of the Y-structure. The Fab fragments maintain binding affinity by themselves, and are thus often used as surrogates to student antibody-antigen interactions. High affinity antibodies are produced during the course of an immune response by successive mutations to germline gene-encoded antibodies. Germline antibodies are more likely to be polyspecific, whereas the affinity maturation process yields monoclonal antibodies that bind specifically to the target antigen. In this work, we use a computational Distance Constraint Model to characterize how mechanical properties change as three disparate germline antibodies are converted to affinity mature. Our results reveal a rich set of mechanical responses throughout the Fab structure. Nevertheless, increased rigidity in the VH domain is consistently observed, which is consistent with the transition from polyspecificity to monospecificity. That is, flexible antibody structures are able to recognize multiple antigens, while increased affinity and specificity is achieved—in part—by structural rigidification.

Flexibility Correlation between Active Site Regions Is Conserved across Four AmpC β-Lactamase Enzymes

β-lactamases are bacterial enzymes that confer resistance to β-lactam antibiotics, such as penicillins and cephalosporins. There are four classes of β-lactamase enzymes, each with characteristic sequence and structure properties. Enzymes from class A are the most common and have been well characterized across the family; however, less is known about how physicochemical properties vary across the C and D families. In this report, we compare the dynamical properties of four AmpC (class C) β-lactamases using our distance constraint model (DCM). The DCM reliably predicts thermodynamic and mechanical properties in an integrated way. As a consequence, quantitative stability/flexibility relationships (QSFR) can be determined and compared across the whole family. The DCM calculates a large number of QSFR metrics. Perhaps the most useful is the flexibility index (FI), which quantifies flexibility along the enzyme backbone. As typically observed in other systems, FI is well conserved across the four AmpC enzymes. Cooperativity correlation (CC), which quantifies intramolecular couplings within structure, is rarely conserved across protein families; however, it is in AmpC. In particular, the bulk of each structure is composed of a large rigid cluster, punctuated by three flexibly correlated regions located at the active site. These regions include several catalytic residues and the Ω-loop. This evolutionary conservation combined with active their site location strongly suggests that these coupled dynamical modes are important for proper functioning of the enzyme.

A virtual pebble game to ensemble average graph rigidity

Background

The body-bar Pebble Game (PG) algorithm is commonly used to calculate network rigidity properties in proteins and polymeric materials. To account for fluctuating interactions such as hydrogen bonds, an ensemble of constraint topologies are sampled, and average network properties are obtained by averaging PG characterizations. At a simpler level of sophistication, Maxwell constraint counting (MCC) provides a rigorous lower bound for the number of internal degrees of freedom (DOF) within a body-bar network, and it is commonly employed to test if a molecular structure is globally under-constrained or over-constrained. MCC is a mean field approximation (MFA) that ignores spatial fluctuations of distance constraints by replacing the actual molecular structure by an effective medium that has distance constraints globally distributed with perfect uniform density.

Results

The Virtual Pebble Game (VPG) algorithm is a MFA that retains spatial inhomogeneity in the density of constraints on all length scales. Network fluctuations due to distance constraints that may be present or absent based on binary random dynamic variables are suppressed by replacing all possible constraint topology realizations with the probabilities that distance constraints are present. The VPG algorithm is isomorphic to the PG algorithm, where integers for counting “pebbles” placed on vertices or edges in the PG map to real numbers representing the probability to find a pebble. In the VPG, edges are assigned pebble capacities, and pebble movements become a continuous flow of probability within the network. Comparisons between the VPG and average PG results over a test set of proteins and disordered lattices demonstrate the VPG quantitatively estimates the ensemble average PG results well.

Conclusions

The VPG performs about 20% faster than one PG, and it provides a pragmatic alternative to averaging PG rigidity characteristics over an ensemble of constraint topologies. The utility of the VPG falls in between the most accurate but slowest method of ensemble averaging over hundreds to thousands of independent PG runs, and the fastest but least accurate MCC.

Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis

Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Hamiltonian. Consequently, bypassing the matrix diagonalization, the original FRI has a computational complexity of O(N(2)). This work introduces a fast FRI (fFRI) algorithm for the flexibility analysis of large macromolecules. The proposed fFRI further reduces the computational complexity to O(N). Additionally, we propose anisotropic FRI (aFRI) algorithms for the analysis of protein collective dynamics. The aFRI algorithms permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that the FRI, particularly the fFRI, is orders of magnitude more efficient and about 10% more accurate overall than some of the most popular methods in the field. The proposed fFRI is able to predict B-factors for α-carbons of the HIV virus capsid (313 236 residues) in less than 30 seconds on a single processor using only one core. Finally, we demonstrate the application of FRI and aFRI to protein domain analysis.

Redistribution of flexibility in stabilizing antibody fragment mutants follows Le Châtelier's principle

Le Châtelier's principle is the cornerstone of our understanding of chemical equilibria. When a system at equilibrium undergoes a change in concentration or thermodynamic state (i.e., temperature, pressure, etc.), La Châtelier's principle states that an equilibrium shift will occur to offset the perturbation and a new equilibrium is established. We demonstrate that the effects of stabilizing mutations on the rigidity ⇔ flexibility equilibrium within the native state ensemble manifest themselves through enthalpy-entropy compensation as the protein structure adjusts to restore the global balance between the two. Specifically, we characterize the effects of mutation to single chain fragments of the anti-lymphotoxin-β receptor antibody using a computational Distance Constraint Model. Statistically significant changes in the distribution of both rigidity and flexibility within the molecular structure is typically observed, where the local perturbations often lead to distal shifts in flexibility and rigidity profiles. Nevertheless, the net gain or loss in flexibility of individual mutants can be skewed. Despite all mutants being exclusively stabilizing in this dataset, increased flexibility is slightly more common than increased rigidity. Mechanistically the redistribution of flexibility is largely controlled by changes in the H-bond network. For example, a stabilizing mutation can induce an increase in rigidity locally due to the formation of new H-bonds, and simultaneously break H-bonds elsewhere leading to increased flexibility distant from the mutation site via Le Châtelier. Increased flexibility within the VH β4/β5 loop is a noteworthy illustration of this long-range effect.

Towards comprehensive analysis of protein family quantitative stability-flexibility relationships using homology models

The Distance Constraint Model (DCM) is a computational modeling scheme that uniquely integrates thermodynamic and mechanical descriptions of protein structure. As such, quantitative stability-flexibility relationships (QSFR) that describe the interrelationships of thermodynamics and mechanics can be quickly computed. Using comparative QSFR analyses, we have previously investigated these relationships across a small number of protein orthologs, ranging from two to a dozen [1, 2]. However, our ultimate goal is provide a comprehensive analysis of whole protein families, which requires consideration of many more structures. To that end, we have developed homology modeling and assessment protocols so that we can robustly calculate QSFR properties for proteins without experimentally derived structures. The approach, which is presented here, starts from a large ensemble of potential homology models and uses a clustering algorithm to identify the best models, thus paving the way for a comprehensive QSFR analysis across hundreds of proteins in a protein family.

A case study comparing quantitative stability-flexibility relationships across five metallo-β-lactamases highlighting differences within NDM-1

The Distance Constraint Model (DCM) is an ensemble-based biophysical model that integrates thermodynamic and mechanical viewpoints of protein structure. The DCM outputs a large number of structural characterizations that collectively allow for Quantified Stability-Flexibility Relationships (QSFR) to be identified and compared across protein families. Using five metallo-β-lactamases (MBLs) as a representative set, we demonstrate how QSFR properties are both conserved and varied across protein families. Similar to our characterizations on other protein families, the backbone flexibility of the five MBLs are overall visually conserved, yet there are interesting specific quantitative differences. For example, the plasmid-encoded NDM-1 enzyme, which leads to a fast spreading drug-resistant version of Klebsiella pneumoniae, has several regions of significantly increased rigidity relative to the other four. In addition, the set of intramolecular couplings within NDM-1 are also atypical. While long-range couplings frequently vary significantly across protein families, NDM-1 is distinct because it has limited correlated flexibility, which is isolated within the active site S3/S4 and S11/H6 loops. These loops are flexibly correlated in the other members, suggesting it is important to function, but the others also have significant amounts of correlated flexibility throughout the rest of their structures.

Efficient and robust analysis of biomacromolecular flexibility using ensembles of network topologies based on fuzzy noncovalent constraints

We describe an approach (ENT(FNC)) for performing rigidity analyses of biomacromolecules on ensembles of network topologies (ENT) generated from a single input structure. The ENT is based on fuzzy noncovalent constraints, which considers thermal fluctuations of biomacromolecules without actually sampling conformations. Definitions for fuzzy noncovalent constraints were derived from persistency data from molecular dynamics (MD) simulations. A very good agreement between local flexibility and rigidity characteristics from ENT(FNC) and MD simulations-generated ensembles is found. Regarding global characteristics, convincing results were obtained when relative thermostabilities of citrate synthase and lipase A structures were computed. The ENT(FNC) approach significantly improves the robustness of rigidity analyses, is highly efficient, and does not require a protein-specific parameterization. Its low computational demand makes it especially valuable for the analysis of large data sets, e.g., for data-driven protein engineering.

Variations within class-A β-lactamase physiochemical properties reflect evolutionary and environmental patterns, but not antibiotic specificity

The bacterial enzyme β-lactamase hydrolyzes the β-lactam ring of penicillin and chemically related antibiotics, rendering them ineffective. Due to rampant antibiotic overuse, the enzyme is evolving new resistance activities at an alarming rate. Related, the enzyme's global physiochemical properties exhibit various amounts of conservation and variability across the family. To that end, we characterize the extent of property conservation within twelve different class-A β-lactamases, and conclusively establish that the systematic variations therein parallel their evolutionary history. Large and systematic differences within electrostatic potential maps and pairwise residue-to-residue couplings are observed across the protein, which robustly reflect phylogenetic outgroups. Other properties are more conserved (such as residue pKa values, electrostatic networks, and backbone flexibility), yet they also have systematic variations that parallel the phylogeny in a statistically significant way. Similarly, the above properties also parallel the environmental condition of the bacteria they are from in a statistically significant way. However, it is interesting and surprising that the only one of the global properties (protein charge) parallels the functional specificity patterns; meaning antibiotic resistance activities are not significantly constraining the global physiochemical properties. Rather, extended spectrum activities can emerge from the background of nearly any set of electrostatic and dynamic properties.

In silico screening and molecular dynamics simulation of disease-associated nsSNP in TYRP1 gene and its structural consequences in OCA3

Oculocutaneous albinism type III (OCA3), caused by mutations of TYRP1 gene, is an autosomal recessive disorder characterized by reduced biosynthesis of melanin pigment in the hair, skin, and eyes. The TYRP1 gene encodes a protein called tyrosinase-related protein-1 (Tyrp1). Tyrp1 is involved in maintaining the stability of tyrosinase protein and modulating its catalytic activity in eumelanin synthesis. Tyrp1 is also involved in maintenance of melanosome structure and affects melanocyte proliferation and cell death. In this work we implemented computational analysis to filter the most probable mutation that might be associated with OCA3. We found R326H and R356Q as most deleterious and disease associated by using PolyPhen 2.0, SIFT, PANTHER, I-mutant 3.0, PhD-SNP, SNP&GO, Pmut, and Mutpred tools. To understand the atomic arrangement in 3D space, the native and mutant (R326H and R356Q) structures were modelled. Finally the structural analyses of native and mutant Tyrp1 proteins were investigated using molecular dynamics simulation (MDS) approach. MDS results showed more flexibility in native Tyrp1 structure. Due to mutation in Tyrp1 protein, it became more rigid and might disturb the structural conformation and catalytic function of the structure and might also play a significant role in inducing OCA3. The results obtained from this study would facilitate wet-lab researches to develop a potent drug therapies against OCA3.

Global and local indices for characterizing biomolecular flexibility and rigidity

Understanding flexibility and rigidity characteristics of biomolecules is a prerequisite for understanding biomolecular structural stability and function. Computational methods have been implemented that directly characterize biomolecular flexibility and rigidity by constraint network analysis. For deriving maximal advantage from these analyses, their results need to be linked to biologically relevant characteristics of a structure. Such links are provided by global and local measures ("indices") of biomolecular flexibility and rigidity. To date, more than 14 indices are available with sometimes overlapping or only vague definitions. We present concise definitions of these indices, analyze the relation between, and the scope and limitations of them, and compare their informative value. For this, we probe the structural stability of the calcium binding protein α-lactalbumin as a showcase, both in the "ground state" and after perturbing the system by changing the network topology. In addition, we introduce three indices for the first time that extend the application domain of flexibility and rigidity analyses. The results allow us to provide guidelines for future studies suggesting which of these indices could best be used for analyzing, understanding, and quantifying structural features that are important for biomolecular stability and function. Finally, we make suggestions for proper index notations in future studies to prevent the misinterpretation and to facilitate the comparison of results obtained from flexibility and rigidity analyses.

A rigid network of long-range contacts increases thermostability in a mutant endoglucanase

Thermodynamic stability of a protein at elevated temperatures is a key factor for thermostable enzymes to catalyze their specific reactions. Yet our understanding of biological determinants of thermostability is far from complete. Many different atomistic factors have been suggested as possible means for such proteins to preserve their activity at high temperatures. Among these factors are specific local interatomic interactions or enrichment of specific amino acid types. The case of glycosyl hydrolase family endoglucanase of Trichoderma reesei defies current hypotheses for thermostability because a single mutation far from the active site (A35 V) converts this mesostable protein into a thermostable protein without significant change in the protein structure. This substantial change in enzymatic activity cannot be explained on the basis of local intramolecular interactions alone. Here we present a more global view of the induced thermostability and show that the A35 V mutation affects the underlying structural rigidity of the whole protein via a number of long-range, non-local interactions. Our analysis of this structure reveals a precisely tuned, rigid network of atomic interactions. This cooperative, allosteric effect promotes the transformation of this mesostable protein into a thermostable one.

Develop and test a solvent accessible surface area-based model in conformational entropy calculations

It is of great interest in modern drug design to accurately calculate the free energies of protein-ligand or nucleic acid-ligand binding. MM-PBSA (molecular mechanics Poisson-Boltzmann surface area) and MM-GBSA (molecular mechanics generalized Born surface area) have gained popularity in this field. For both methods, the conformational entropy, which is usually calculated through normal-mode analysis (NMA), is needed to calculate the absolute binding free energies. Unfortunately, NMA is computationally demanding and becomes a bottleneck of the MM-PB/GBSA-NMA methods. In this work, we have developed a fast approach to estimate the conformational entropy based upon solvent accessible surface area calculations. In our approach, the conformational entropy of a molecule, S, can be obtained by summing up the contributions of all atoms, no matter they are buried or exposed. Each atom has two types of surface areas, solvent accessible surface area (SAS) and buried SAS (BSAS). The two types of surface areas are weighted to estimate the contribution of an atom to S. Atoms having the same atom type share the same weight and a general parameter k is applied to balance the contributions of the two types of surface areas. This entropy model was parametrized using a large set of small molecules for which their conformational entropies were calculated at the B3LYP/6-31G* level taking the solvent effect into account. The weighted solvent accessible surface area (WSAS) model was extensively evaluated in three tests. For convenience, TS values, the product of temperature T and conformational entropy S, were calculated in those tests. T was always set to 298.15 K through the text. First of all, good correlations were achieved between WSAS TS and NMA TS for 44 protein or nucleic acid systems sampled with molecular dynamics simulations (10 snapshots were collected for postentropy calculations): the mean correlation coefficient squares (R²) was 0.56. As to the 20 complexes, the TS changes upon binding; TΔS values were also calculated, and the mean R² was 0.67 between NMA and WSAS. In the second test, TS values were calculated for 12 proteins decoy sets (each set has 31 conformations) generated by the Rosetta software package. Again, good correlations were achieved for all decoy sets: the mean, maximum, and minimum of R² were 0.73, 0.89, and 0.55, respectively. Finally, binding free energies were calculated for 6 protein systems (the numbers of inhibitors range from 4 to 18) using four scoring functions. Compared to the measured binding free energies, the mean R² of the six protein systems were 0.51, 0.47, 0.40, and 0.43 for MM-GBSA-WSAS, MM-GBSA-NMA, MM-PBSA-WSAS, and MM-PBSA-NMA, respectively. The mean rms errors of prediction were 1.19, 1.24, 1.41, 1.29 kcal/mol for the four scoring functions, correspondingly. Therefore, the two scoring functions employing WSAS achieved a comparable prediction performance to that of the scoring functions using NMA. It should be emphasized that no minimization was performed prior to the WSAS calculation in the last test. Although WSAS is not as rigorous as physical models such as quasi-harmonic analysis and thermodynamic integration (TI), it is computationally very efficient as only surface area calculation is involved and no structural minimization is required. Moreover, WSAS has achieved a comparable performance to normal-mode analysis. We expect that this model could find its applications in the fields like high throughput screening (HTS), molecular docking, and rational protein design. In those fields, efficiency is crucial since there are a large number of compounds, docking poses, or protein models to be evaluated. A list of acronyms and abbreviations used in this work is provided for quick reference.

Changes in Lysozyme Flexibility upon Mutation Are Frequent, Large and Long-Ranged

We investigate changes in human c-type lysozyme flexibility upon mutation via a Distance Constraint Model, which gives a statistical mechanical treatment of network rigidity. Specifically, two dynamical metrics are tracked. Changes in flexibility index quantify differences within backbone flexibility, whereas changes in the cooperativity correlation quantify differences within pairwise mechanical couplings. Regardless of metric, the same general conclusions are drawn. That is, small structural perturbations introduced by single point mutations have a frequent and pronounced affect on lysozyme flexibility that can extend over long distances. Specifically, an appreciable change occurs in backbone flexibility for 48% of the residues, and a change in cooperativity occurs in 42% of residue pairs. The average distance from mutation to a site with a change in flexibility is 17–20 Å. Interestingly, the frequency and scale of the changes within single point mutant structures are generally larger than those observed in the hen egg white lysozyme (HEWL) ortholog, which shares 61% sequence identity with human lysozyme. For example, point mutations often lead to substantial flexibility increases within the β-subdomain, which is consistent with experimental results indicating that it is the nucleation site for amyloid formation. However, β-subdomain flexibility within the human and HEWL orthologs is more similar despite the lowered sequence identity. These results suggest compensating mutations in HEWL reestablish desired properties.

The functional importance of protein dynamics is universally accepted, making the study of dynamical similarities and differences among proteins of the same function an intriguing problem. While some metrics are likely to be conserved across family, differences are also very common. In previous works we have used a Distance Constraint Model to quantify flexibility differences across sets of orthologous proteins, which reproduce this diversity. In the same manner, this work investigates changes occurring upon individual point mutations. Somewhat surprisingly, the small structural perturbations caused by mutation lead to changes throughout the protein. These changes can be quite large, actually surpassing the scale for differences between ortholog pairs. Moreover, changes in flexibility frequently occur at sites far from the mutation site. These results underscore the sensitivity of protein dynamics in connection with allostery, and help explain why differences across protein families are so common.

Calculating ensemble averaged descriptions of protein rigidity without sampling

Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the integer body-bar Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations to accurately account for fluctuations in network topology. We have developed a mean field Virtual Pebble Game (VPG) that represents the ensemble of networks by a single effective network. That is, all possible number of distance constraints (or bars) that can form between a pair of rigid bodies is replaced by the average number. The resulting effective network is viewed as having weighted edges, where the weight of an edge quantifies its capacity to absorb degrees of freedom. The VPG is interpreted as a flow problem on this effective network, which eliminates the need to sample. Across a nonredundant dataset of 272 protein structures, we apply the VPG to proteins for the first time. Our results show numerically and visually that the rigidity characterizations of the VPG accurately reflect the ensemble averaged properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability.

Electrostatics of hydrogen exchange for analyzing protein flexibility

Electrostatic interactions at the protein-aqueous interface modulate the reactivity of solvent-exposed backbone amides by a factor of at least a billion fold. The brief (∼10 ps) lifetime of the peptide anion formed during the hydroxide-catalyzed exchange reaction helps enable the experimental rates to be robustly predictable by continuum dielectric methods. Since this ability to predict the structural dependence of exchange reactivity also applies to the protein amide hydrogens that are only rarely exposed to the bulk solvent phase, electrostatic analysis of the experimental exchange rates provides an effective assessment of whether a given model ensemble is consistent with the properly weighted Boltzmann conformational distribution of the protein native state.

Flexibility analysis of biomacromolecules with application to computer-aided drug design

Flexibility characteristics of biomacromolecules can be efficiently determined down to the atomic level by a graph-theoretical technique as implemented in the FIRST (Floppy Inclusion and Rigid Substructure Topology) and ProFlex software packages. The method has been successfully applied to a series of protein and nucleic acid structures. Here, we describe practical guidelines for setting up and performing a flexibility analysis, discuss current bottlenecks of the approach, and provide sample applications as to how this technique can support computer-aided drug design approaches.

Ensemble properties of network rigidity reveal allosteric mechanisms

The distance constraint model (DCM) is a unique computational modeling paradigm that integrates mechanical and thermodynamic descriptions of macromolecular structure. That is, network rigidity calculations are used to account for nonadditivity within entropy components, thus restoring the utility of free-energy decomposition. The DCM outputs a large number of structural characterizations that collectively allow for quantified stability-flexibility relationships (QSFR) to be identified. In this review, we describe the theoretical underpinnings of the DCM and introduce several common QSFR metrics. Application of the DCM across protein families highlights the sensitivity within the set of protein structure residue-to-residue couplings. Further, we have developed a perturbation method to identify putative allosteric sites, where large changes in QSFR upon rigidification (mimicking ligand-binding) detect sites likely to invoke allosteric changes.

Characterizing protein motions from structure

To clarify the extent structure plays in determining protein dynamics, a comparative study is made using three models that characterize native state dynamics of single domain proteins starting from known structures taken from four distinct SCOP classifications. A geometrical simulation using the framework rigidity optimized dynamics algorithm (FRODA) based on rigid cluster decomposition is compared to the commonly employed elastic network model (specifically the Anisotropic Network Model ANM) and molecular dynamics (MD) simulation. The essential dynamics are quantified by a mode subspace constructed from ANM and a principal component analysis (PCA) on FRODA and MD trajectories. Aggregate conformational ensembles are constructed to provide a basis for quantitative comparisons between FRODA runs using different parameter settings to critically assess how the predictions of essential dynamics depend on a priori arbitrary user-defined distance constraint rules. We established a range of physicality for these parameters. Surprisingly, FRODA maintains greater intra-consistent results than obtained from MD trajectories, comparable to ANM. Additionally, a mode subspace is constructed from PCA on an exemplar set of myoglobin structures from the Protein Data Bank. Significant overlap across the three model subspaces and the experimentally derived subspace is found. While FRODA provides the most robust sampling and characterization of the native basin, all three models give similar dynamical information of a native state, further demonstrating that structure is the key determinant of dynamics.

Nonadditivity in conformational entropy upon molecular rigidification reveals a universal mechanism affecting folding cooperativity

Previously, we employed a Maxwell counting distance constraint model (McDCM) to describe α-helix formation in polypeptides. Unlike classical helix-coil transition theories, the folding mechanism derives from nonadditivity in conformational entropy caused by rigidification of molecular structure as intramolecular cross-linking interactions form along the backbone. For example, when a hydrogen bond forms within a flexible region, both energy and conformational entropy decrease. However, no conformational entropy is lost when the region is already rigid because atomic motions are not constrained further. Unlike classical zipper models, the same mechanism also describes a coil-to-β-hairpin transition. Special topological features of the helix and hairpin structures allow the McDCM to be solved exactly. Taking full advantage of the fact that Maxwell constraint counting is a mean field approximation applied to the distribution of cross-linking interactions, we present an exact transfer matrix method that does not require any special topological feature. Upon application of the model to proteins, cooperativity within the folding transition is yet again appropriately described. Notwithstanding other contributing factors such as the hydrophobic effect, this simple model identifies a universal mechanism for cooperativity within polypeptide and protein-folding transitions, and it elucidates scaling laws describing hydrogen-bond patterns observed in secondary structure. In particular, the native state should have roughly twice as many constraints as there are degrees of freedom in the coil state to ensure high fidelity in two-state folding cooperativity, which is empirically observed.

Nonadditivity in the alpha-helix to coil transition

The Lifson-Roig Model (LRM) and all its variants describe the α-helix to coil transition in terms of additive component-free energies within a free energy decomposition scheme, and these contributions are interpreted through sequence-context dependent nucleation and propagation parameters. Although this phenomenological approach is able to adequately fit experimental data on helix content and heat capacity, the number of required parameters increases dramatically with additional sequence variation. Moreover, due to nonadditive competing microscopic effects that are difficult to disentangle within a LRM, large uncertainties within the parameters emerge. We offer an alternative view that removes the need for sequence-context parameterization by focusing on individual microsopic interactions within a free energy decomposition and explicitly account for nonadditivity in conformational entropy through network rigidity using a Distance Constraint Model (DCM). We apply a LRM and a DCM to previously published experimental heat capacity and helix content data for a series of heterogeneous polypeptides. Both models describe the experimental data well, and the parameters from both models are consistent with prior work. However, the number of DCM parameters is independent of sequence-variability, the parameter values exhibit better transferability, and the helix nucleation is predicted by the DCM explicitly through the nonadditive nature of conformational entropy. The importance of these results is that the DCM offers a system-independent approach for modeling stability within polypeptides and proteins, where the demonstrated accuracy for the α-helix to coil transition over a series of heterogeneous polypeptides described here is one case in point.

Allosteric response is both conserved and variable across three CheY orthologs

A computational method to identify residues likely to initiate allosteric signals has been developed. The method is based on differences within stability and flexibility profiles between wild-type and perturbed structures as computed by a distance constraint model. Application of the approach to three bacterial chemotaxis protein Y (CheY) orthologs provides a comparison of allosteric response across protein family divergence. Interestingly, we observe a rich mixture of both conservation and variability within the identified allosteric sites. While similarity within the overall response parallels the evolutionary relationships, >50% of the best scoring putative sites are only identified in a single ortholog. These results suggest that detailed descriptions of intraprotein communication are substantially more variable than structure and function, yet do maintain some evolutionary relationships. Finally, structural clusters of large response identify four allosteric hotspots, including the β4/α4 loop known to be critical to relaying the CheY phosphorylation signal.

A signal processing method to explore similarity in protein flexibility

Understanding mechanisms of protein flexibility is of great importance to structural biology. The ability to detect similarities between proteins and their patterns is vital in discovering new information about unknown protein functions. A Distance Constraint Model (DCM) provides a means to generate a variety of flexibility measures based on a given protein structure. Although information about mechanical properties of flexibility is critical for understanding protein function for a given protein, the question of whether certain characteristics are shared across homologous proteins is difficult to assess. For a proper assessment, a quantified measure of similarity is necessary. This paper begins to explore image processing techniques to quantify similarities in signals and images that characterize protein flexibility. The dataset considered here consists of three different families of proteins, with three proteins in each family. The similarities and differences found within flexibility measures across homologous proteins do not align with sequence-based evolutionary methods.

Refining efficacy: allosterism and bias in G protein-coupled receptor signaling

Receptors on the surface of cells function as conduits for information flowing between the external environment and the cell interior. Since signal transduction is based on the physical interaction of receptors with both extracellular ligands and intracellular effectors, ligand binding must produce conformational changes in the receptor that can be transmitted to the intracellular domains accessible to G proteins and other effectors. Classical models of G protein-coupled receptor (GPCR) signaling envision receptor conformations as highly constrained, wherein receptors exist in equilibrium between single "off" and "on" states distinguished by their ability to activate effectors, and ligands act by perturbing this equilibrium. In such models, ligands can be classified based upon two simple parameters; affinity and efficacy, and ligand activity is independent of the assay used to detect the response. However, it is clear that GPCRs assume multiple conformations, any number of which may be capable of interacting with a discrete subset of possible effectors. Both orthosteric ligands, molecules that occupy the natural ligand-binding pocket, and allosteric modulators, small molecules or proteins that contact receptors distant from the site of ligand binding, have the ability to alter the conformational equilibrium of a receptor in ways that affect its signaling output both qualitatively and quantitatively. In this context, efficacy becomes pluridimensional and ligand classification becomes assay dependent. A more complete description of ligand-receptor interaction requires the use of multiplexed assays of receptor activation and screening assays may need to be tailored to detect specific efficacy profiles.

Refining efficacy: exploiting functional selectivity for drug discovery

Early models of G protein-coupled receptor (GPCR) activation envisioned the receptor in equilibrium between unique "off" and "on" states, wherein ligand binding affected signaling by increasing or decreasing the fraction of receptors in the active conformation. It is now apparent that GPCRs spontaneously sample multiple conformations, any number of which may couple to one or more downstream effectors. Such "multistate" models imply that the receptor-ligand complex, not the receptor alone, defines which active receptor conformations predominate. "Functional selectivity" refers to the ability of a ligand to activate only a subset of its receptor's signaling repertoire. There are now numerous examples of ligands that "bias" receptor coupling between different G protein pools and non-G protein effectors such as arrestins. The type 1 parathyroid hormone receptor (PTH(1)R) is a particularly informative example, not only because of the range of biased effects that have been produced, but also because the actions of many of these ligands have been characterized in vivo. Biased PTH(1)R ligands can selectively couple the PTH(1)R to G(s) or G(q/11) pathways, with or without activating arrestin-dependent receptor desensitization and signaling. These reagents have provided insight into the contribution of different signaling pathways to PTH action in vivo and suggest it may be possible to exploit ligand bias to uncouple the anabolic effects of PTH(1)R from its catabolic and calcitropic effects. Whereas conventional agonists and antagonists only modulate the quantity of efficacy, functionally selective ligands qualitatively change GPCR signaling, offering the prospect of drugs with improved therapeutic efficacy or reduced side effects.

Ensemble-based methods for describing protein dynamics

Molecular dynamics (MD) simulation is a natural approach for studying protein dynamics, and coupled with the ideas of multiscale modeling, MD proves to be the gold standard in computational biology to investigate mechanistic details related to protein function. In principle, if MD trajectories are long enough, the ensemble of protein conformations generated allows thermodynamic and kinetic properties to be predicted. We know from experiments that proteins exhibit a high degree of fidelity in function, and that empirical kinetic models are successful in describing kinetics, suggesting that the ensemble of conformations cluster into well-defined thermodynamic states, which are frequently metastable. The experimental evidence suggest that more efficient computational models that retain only essential properties of the protein can be constructed to faithfully reproduce the relatively few observed thermodynamic states, and perhaps describe transition states if the model is sufficiently detailed. Indeed, there are many so-called ensemble-based methods that attempt to generate more complete ensembles than MD can provide by focusing on the most important driving forces through simplified representations of how elements within the protein interact. Although coarse-graining is employed in MD and other approaches, such as in elastic network models, the key distinguishing factor of ensemble-based methods is that they are meant to efficiently generate a large ensemble of conformations without solving explicit equations of motion. This review highlights three types of ensemble-based methods, illustrated by 'COREX' and the Wako-Saito-Munoz-Eaton (WSME) model, the Framework Rigidity Optimized Dynamic Algorithm (FRODA) and the distance constraint model (DCM).

Seven transmembrane receptors as shapeshifting proteins: the impact of allosteric modulation and functional selectivity on new drug discovery

It is useful to consider seven transmembrane receptors (7TMRs) as disordered proteins able to allosterically respond to a number of binding partners. Considering 7TMRs as allosteric systems, affinity and efficacy can be thought of in terms of energy flow between a modulator, conduit (the receptor protein), and a number of guests. These guests can be other molecules, receptors, membrane-bound proteins, or signaling proteins in the cytosol. These vectorial flows of energy can yield standard canonical guest allostery (allosteric modification of drug effect), effects along the plane of the cell membrane (receptor oligomerization), or effects directed into the cytosol (differential signaling as functional selectivity). This review discusses these apparently diverse pharmacological effects in terms of molecular dynamics and protein ensemble theory, which tends to unify 7TMR behavior toward cells. Special consideration will be given to functional selectivity (biased agonism and biased antagonism) in terms of mechanism of action and potential therapeutic application. The explosion of technology that has enabled observation of diverse 7TMR behavior has also shown how drugs can have multiple (pluridimensional) efficacies and how this can cause paradoxical drug classification and nomenclatures.

Helix/coil nucleation: a local response to global demands

A complete description of protein structure and function must include a proper treatment of mechanisms that lead to cooperativity. The helix/coil transition serves as a simple example of a cooperative folding process, commonly described by a nucleation-propagation mechanism. The prevalent view is that coil structure must first form a short segment of helix in a localized region despite paying a free energy cost (nucleation). Afterward, helical structure propagates outward from the nucleation site. Both processes entail enthalpy-entropy compensation that derives from the loss in conformational entropy on helix formation with concomitant gain in favorable interactions. Nucleation-propagation models inherently assume that cooperativity arises from a sequential series of local events. An alternative distance constraint model asserts there is a direct link between available degrees of freedom and cooperativity through the nonadditivity in conformational entropy. That is, helix nucleation is a concerted manifestation of rigidity propagating through atomic structure. The link between network rigidity and nonadditivity of conformational entropy is shown in this study by solving the distance constraint model using a simple global constraint counting approximation. Cooperativity arises from competition between excess and deficiency in available degrees of freedom in the coil and helix states respectively.

Thermostability in rubredoxin and its relationship to mechanical rigidity

The source of increased stability in proteins from organisms that thrive in extreme thermal environments is not well understood. Previous experimental and theoretical studies have suggested many different features possibly responsible for such thermostability. Many of these thermostabilizing mechanisms can be accounted for in terms of structural rigidity. Thus a plausible hypothesis accounting for this remarkable stability in thermophilic enzymes states that these enzymes have enhanced conformational rigidity at temperatures below their native, functioning temperature. Experimental evidence exists to both support and contradict this supposition. We computationally investigate the relationship between thermostability and rigidity using rubredoxin as a case study. The mechanical rigidity is calculated using atomic models of homologous rubredoxin structures from the hyperthermophile Pyrococcus furiosus and mesophile Clostridium pasteurianum using the FIRST software. A global increase in structural rigidity (equivalently a decrease in flexibility) corresponds to an increase in thermostability. Locally, rigidity differences (between mesophilic and thermophilic structures) agree with differences in protection factors.

Peptide conformer acidity analysis of protein flexibility monitored by hydrogen exchange

The amide hydrogens that are exposed to solvent in the high-resolution X-ray structures of ubiquitin, FK506-binding protein, chymotrypsin inhibitor 2, and rubredoxin span a billion-fold range in hydroxide-catalyzed exchange rates which are predictable by continuum dielectric methods. To facilitate analysis of transiently accessible amides, the hydroxide-catalyzed rate constants for every backbone amide of ubiquitin were determined under near physiological conditions. With the previously reported NMR-restrained molecular dynamics ensembles of ubiquitin (PDB codes 2NR2 and 2K39) used as representations of the Boltzmann-weighted conformational distribution, nearly all of the exchange rates for the highly exposed amides were more accurately predicted than by use of the high-resolution X-ray structure. More strikingly, predictions for the amide hydrogens of the NMR relaxation-restrained ensemble that become exposed to solvent in more than one but less than half of the 144 protein conformations in this ensemble were almost as accurate. In marked contrast, the exchange rates for many of the analogous amides in the residual dipolar coupling-restrained ubiquitin ensemble are substantially overestimated, as was particularly evident for the Ile 44 to Lys 48 segment which constitutes the primary interaction site for the proteasome targeting enzymes involved in polyubiquitylation. For both ensembles, “excited state” conformers in this active site region having markedly elevated peptide acidities are represented at a population level that is 10² to 10³ above what can exist in the Boltzmann distribution of protein conformations. These results indicate how a chemically consistent interpretation of amide hydrogen exchange can provide insight into both the population and the detailed structure of transient protein conformations.

Unifying mechanical and thermodynamic descriptions across the thioredoxin protein family

We compare various predicted mechanical and thermodynamic properties of nine oxidized thioredoxins (TRX) using a Distance Constraint Model (DCM). The DCM is based on a nonadditive free energy decomposition scheme, where entropic contributions are determined from rigidity and flexibility of structure based on distance constraints. We perform averages over an ensemble of constraint topologies to calculate several thermodynamic and mechanical response functions that together yield quantitative stability/flexibility relationships (QSFR). Applied to the TRX protein family, QSFR metrics display a rich variety of similarities and differences. In particular, backbone flexibility is well conserved across the family, whereas cooperativity correlation describing mechanical and thermodynamic couplings between the residue pairs exhibit distinctive features that readily standout. The diversity in predicted QSFR metrics that describe cooperativity correlation between pairs of residues is largely explained by a global flexibility order parameter describing the amount of intrinsic flexibility within the protein. A free energy landscape is calculated as a function of the flexibility order parameter, and key values are determined where the native-state, transition-state, and unfolded-state are located. Another key value identifies a mechanical transition where the global nature of the protein changes from flexible to rigid. The key values of the flexibility order parameter help characterize how mechanical and thermodynamic response is linked. Variation in QSFR metrics and key characteristics of global flexibility are related to the native state X-ray crystal structure primarily through the hydrogen bond network. Furthermore, comparison of three TRX redox pairs reveals differences in thermodynamic response (i.e., relative melting point) and mechanical properties (i.e., backbone flexibility and cooperativity correlation) that are consistent with experimental data on thermal stabilities and NMR dynamical profiles. The results taken together demonstrate that small-scale structural variations are amplified into discernible global differences by propagating mechanical couplings through the H-bond network.