Protein flexibility predictions using graph theory

Techniques from graph theory are applied to analyze the bond networks in proteins and identify the flexible and rigid regions. The bond network consists of distance constraints defined by the covalent and hydrogen bonds and salt bridges in the protein, identified by geometric and energetic criteria. We use an algorithm that counts the degrees of freedom within this constraint network and that identifies all the rigid and flexible substructures in the protein, including overconstrained regions (with more crosslinking bonds than are needed to rigidify the region) and underconstrained or flexible regions, in which dihedral bond rotations can occur. The number of extra constraints or remaining degrees of bond-rotational freedom within a substructure quantifies its relative rigidity/flexibility and provides a flexibility index for each bond in the structure. This novel computational procedure, first used in the analysis of glassy materials, is approximately a million times faster than molecular dynamics simulations and captures the essential conformational flexibility of the protein main and side-chains from analysis of a single, static three-dimensional structure. This approach is demonstrated by comparison with experimental measures of flexibility for three proteins in which hinge and loop motion are essential for biological function: HIV protease, adenylate kinase, and dihydrofolate reductase.     

Stability and rigidity/flexibility—Two sides of the same coin?

Protein molecules require both flexibility and rigidity for functioning. The fast and accurate prediction of protein rigidity/flexibility is one of the important problems in protein science. We have determined flexible regions for four homologous pairs from thermophilic and mesophilic organisms by two methods: the fast FoldUnfold which uses amino acid sequence and the time consuming MDFirst which uses three-dimensional structures. We demonstrate that both methods allow determining flexible regions in protein structure. For three of the four thermophile–mesophile pairs of proteins, FoldUnfold predicts practically the same flexible regions which have been found by the MD/First method. As expected, molecular dynamics simulations show that thermophilic proteins are more rigid in comparison to their mesophilic homologues. Analysis of rigid clusters and their decomposition provides new insights into protein stability. It has been found that the local networks of salt bridges and hydrogen bonds in thermophiles render their structure more stable with respect to fluctuations of individual contacts. Such network includes salt bridge triads Agr-Glu-Lys and Arg-Glu-Arg, or salt bridges (such as Arg-Glu) connected with hydrogen bonds. This ionic network connects alpha helices and rigidifies the structure. Mesophiles can be characterized by stand alone salt bridges and hydrogen bonds or small ionic clusters. Such difference in the network of salt bridges results in different flexibility of homologous proteins. Combining both approaches allows characterizing structural features in atomic detail that determine the rigidity/flexibility of a protein structure. This article is a part of a Special Issue entitled: The emerging dynamic view of proteins: Protein plasticity in allostery, evolution and self-assembly.     

Recognition of native structure from complete enumeration of low-resolution models with constraints

Complete sets of low-resolution conformations are generated for eight small proteins by rotating the C_α-C_α virtual bonds at selected flexible regions, while the remaining structural elements are assumed to move in rigid blocks. Several filtering criteria are used to reduce the ensemble size and to ensure the sampling of well-constructed conformations. These filters, based on structure and energy constraints deduced from knowledge-based studies, include the excluded volume requirement, the radius of gyration constraint, and the occurrence of sufficiently strong attractive inter-residue potentials to stabilize compact forms. About 8,000 well-constructed decoys or “probable folds” (PFs) are constructed for each protein. A correlation between root-mean-square (rms) deviations from X-ray structure and total energies is observed, revealing a decrease in energy as the rms deviation decreases. The conformation with the lowest energy exhibits an rms deviation smaller than 3.0 Å, in most of the proteins considered. The results are highly sensitive to the choice of flexible regions. A strong tendency to assume native state rotational angles is revealed for some flexible bonds from the analysis of the distributions of dihedral angles in the PFs, suggesting the formation of foldons near these locally stable regions at early folding pathway. Proteins 32:211–222, 1998. © 1998 Wiley-Liss, Inc.     

Dynamics of cAPK type IIbeta activation revealed by enhanced amide H/2H exchange mass spectrometry (DXMS)

cAMP-dependent protein kinase (cAPK) is a key component in numerous cell signaling pathways. The cAPK regulatory (R) subunit maintains the kinase in an inactive state until cAMP saturation of the R-subunit leads to activation of the enzyme. To delineate the conformational changes associated with cAPK activation, the amide hydrogen/deuterium exchange in the cAPK type IIbeta R-subunit was probed by electrospray mass spectrometry. Three states of the R-subunit, cAMP-bound, catalytic (C)-subunit bound, and apo, were incubated in deuterated water for various lengths of time and then, prior to mass spectrometry analysis, subjected to digestion by pepsin to localize the deuterium incorporation. High sequence coverage (>99%) by the pepsin-digested fragments enables us to monitor the dynamics of the whole protein. The effects of cAMP binding on RIIbeta amide hydrogen exchange are restricted to the cAMP-binding pockets, while the effects of C-subunit binding are evident across both cAMP-binding domains and the linker region. The decreased amide hydrogen exchange for residues 253-268 within cAMP binding domain A and for residues 102-115, which include the pseudosubstrate inhibitory site, support the prediction that these two regions represent the conserved primary and peripheral C-subunit binding sites. An increase in amide hydrogen exchange for a broad area within cAMP-binding domain B and a narrow area within cAMP-binding domain A (residues 222-232) suggest that C-subunit binding transmits long-distance conformational changes throughout the protein.     

Electron paramagnetic resonance analysis of the vimentin tail domain reveals points of order in a largely disordered region and conformational adaptation upon filament assembly

Very little data have been reported that describe the structure of the tail domain of any cytoplasmic intermediate filament (IF) protein. We report here the results of studies using site directed spin labeling and electron paramagnetic resonance (SDSL‐EPR) to explore the structure and dynamics of the tail domain of human vimentin in tetramers (protofilaments) and filaments. The data demonstrate that in contrast to the vimentin head and rod domains, the tail domains are not closely apposed in protofilaments. However, upon assembly into intact IFs, several sites, including positions 445, 446, 451, and 452, the conserved “beta‐site,” become closely apposed, indicating dynamic changes in tail domain structure that accompany filament elongation. No evidence is seen for coiled‐coil structure within the region studied, in either protofilaments or assembled filaments. EPR analysis also establishes that more than half of the tail domain is very flexible in both the assembly intermediate and the intact IF. However, by positioning the spin label at distinct sites, EPR is able to identify both the rod proximal region and sites flanking the beta‐site motif as rigid locations within the tail. The rod proximal region is well assembled at the tetramer stage with only slight changes occurring during filament elongation. In contrast, at the beta site, the polypeptide backbone transitions from flexible in the assembly intermediate to much more rigid in the intact IF. These data support a model in which the distal tail domain structure undergoes significant conformational change during filament elongation and final assembly.     

Hierarchical domain‐motion analysis of conformational changes in sarcoplasmic reticulum Ca2+‐ATPase

Sarco(endo)plasmic reticulum Ca²⁺‐ATPase transports two Ca²⁺ per ATP‐hydrolyzed across biological membranes against a large concentration gradient by undergoing large conformational changes. Structural studies with X‐ray crystallography revealed functional roles of coupled motions between the cytoplasmic domains and the transmembrane helices in individual reaction steps. Here, we employed “Motion Tree (MT),” a tree diagram that describes a conformational change between two structures, and applied it to representative Ca²⁺‐ATPase structures. MT provides information of coupled rigid‐body motions of the ATPase in individual reaction steps. Fourteen rigid structural units, “common rigid domains (CRDs)” are identified from seven MTs throughout the whole enzymatic reaction cycle. CRDs likely act as not only the structural units, but also the functional units. Some of the functional importance has been newly revealed by the analysis. Stability of each CRD is examined on the morphing trajectories that cover seven conformational transitions. We confirmed that the large conformational changes are realized by the motions only in the flexible regions that connect CRDs. The Ca²⁺‐ATPase efficiently utilizes its intrinsic flexibility and rigidity to response different switches like ligand binding/dissociation or ATP hydrolysis. The analysis detects functional motions without extensive biological knowledge of experts, suggesting its general applicability to domain movements in other membrane proteins to deepen the understanding of protein structure and function. Proteins 2015; 83:746–756. © 2015 Wiley Periodicals, Inc.     

Functional flexibility of human cyclin-dependent kinase-2 and its evolutionary conservation

Cyclin-dependent kinase 2 (CDK2) is the most thoroughly studied of the cyclin-dependent kinases that regulate essential cellular processes, including the cell cycle, and it has become a model for studies of regulatory mechanisms at the molecular level. This contribution identifies flexible and rigid regions of CDK2 based on temperature B-factors acquired from both X-ray data and molecular dynamics simulations. In addition, the biological relevance of the identified flexible regions and their motions is explored using information from the essential dynamics analysis related to conformational changes of CDK2 and knowledge of its biological function(s). The conserved regions of CMGC protein kinases' primary sequences are located in the most rigid regions identified in our analyses, with the sole exception of the absolutely conserved G13 in the tip of the glycine-rich loop. The conserved rigid regions are important for nucleotide binding, catalysis, and substrate recognition. In contrast, the most flexible regions correlate with those where large conformational changes occur during CDK2 regulation processes. The rigid regions flank and form a rigid skeleton for the flexible regions, which appear to provide the plasticity required for CDK2 regulation. Unlike the rigid regions (which as mentioned are highly conserved) no evidence of evolutionary conservation was found for the flexible regions.     

BuildBeta—A system for automatically constructing beta sheets

We describe a method that can thoroughly sample a protein conformational space given the protein primary sequence of amino acids and secondary structure predictions. Specifically, we target proteins with β‐sheets because they are particularly challenging for ab initio protein structure prediction because of the complexity of sampling long‐range strand pairings. Using some basic packing principles, inverse kinematics (IK), and β‐pairing scores, this method creates all possible β‐sheet arrangements including those that have the correct packing of β‐strands. It uses the IK algorithms of ProteinShop to move α‐helices and β‐strands as rigid bodies by rotating the dihedral angles in the coil regions. Our results show that our approach produces structures that are within 4–6 Å RMSD of the native one regardless of the protein size and β‐sheet topology although this number may increase if the protein has long loops or complex α‐helical regions. Proteins 2010. © Published 2009 Wiley‐Liss, Inc.     

Is there nascent structure in the intrinsically disordered region of troponin I?

In striated muscle, the binding of calcium to troponin C (TnC) results in the removal of the C‐terminal region of the inhibitory protein troponin I (TnI) from actin. While structural studies of the muscle system have been successful in determining the overall organization of most of the components involved in force generation at the atomic level, the structure and dynamics of the C‐terminal region of TnI remains controversial. This domain of TnI is highly flexible, and it has been proposed that this intrinsically disordered region (IDR) regulates contraction via a “fly‐casting” mechanism. Different structures have been presented for this region using different methodologies: a single α‐helix, a “mobile domain” containing a small β‐sheet, an unstructured region, and a two helix segment. To investigate whether this IDR has in fact any nascent structure, we have constructed a skeletal TnC‐TnI chimera that contains the N‐domain of TnC (1–90), a short linker (GGAGG), and the C‐terminal region of TnI (97–182) and have acquired ¹⁵N NMR relaxation data for this chimera. We compare the experimental relaxation parameters with those calculated from molecular dynamic simulations using four models based upon the structural studies. Our experimental results suggest that the C‐terminal region of TnI does not contain any defined secondary structure, supporting the “fly‐casting” mechanism. We interpret the presence of a “plateau” in the ¹⁵N NMR relaxation data as being an intrinsic property of IDRs. We also identified a more rigid adjacent region of TnI that has implications for muscle performance under ischemic conditions. Proteins 2011. © 2011 Wiley‐Liss, Inc.     

Method for identification of rigid domains and hinge residues in proteins based on exhaustive enumeration

Many proteins undergo large‐scale motions where relatively rigid domains move against each other. The identification of rigid domains, as well as the hinge residues important for their relative movements, is important for various applications including flexible docking simulations. In this work, we develop a method for protein rigid domain identification based on an exhaustive enumeration of maximal rigid domains, the rigid domains not fully contained within other domains. The computation is performed by mapping the problem to that of finding maximal cliques in a graph. A minimal set of rigid domains are then selected, which cover most of the protein with minimal overlap. In contrast to the results of existing methods that partition a protein into non‐overlapping domains using approximate algorithms, the rigid domains obtained from exact enumeration naturally contain overlapping regions, which correspond to the hinges of the inter‐domain bending motion. The performance of the algorithm is demonstrated on several proteins. Proteins 2015; 83:1054–1067. © 2015 Wiley Periodicals, Inc.     

Molecular dynamics simulations of apocupredoxins: insights into the formation and stabilization of copper sites under entatic control

Cupredoxins perform copper-mediated long-range electron transfer (ET) in biological systems. Their copper-binding sites have evolved to force copper ions into ET-competent systems with decreased reorganization energy, increased reduction potential, and a distinct electronic structure compared with those of non-ET-competent copper complexes. The entatic or rack-induced state hypothesis explains these special properties in terms of the strain that the protein matrix exerts on the metal ions. This idea is supported by X-ray structures of apocupredoxins displaying “closed” arrangements of the copper ligands like those observed in the holoproteins; however, it implies completely buried copper-binding atoms, conflicting with the notion that they must be exposed for copper loading. On the other hand, a recent work based on NMR showed that the copper-binding regions of apocupredoxins are flexible in solution. We have explored five cupredoxins in their “closed” apo forms through molecular dynamics simulations. We observed that prearranged ligand conformations are not stable as the X-ray data suggest, although they do form part of the dynamic landscape of the apoproteins. This translates into variable flexibility of the copper-binding regions within a rigid fold, accompanied by fluctuations of the hydrogen bonds around the copper ligands. Major conformations with solvent-exposed copper-binding atoms could allow initial binding of the copper ions. An eventual subsequent incursion to the closed state would result in binding of the remaining ligands, trapping the closed conformation thanks to the additional binding energy and the fastening of noncovalent interactions that make up the rack.     

How different are structurally flexible and rigid binding sites? Sequence and structural features discriminating proteins that do and do not undergo conformational change upon ligand binding

Proteins are dynamic molecules and often undergo conformational change upon ligand binding. It is widely accepted that flexible loop regions have a critical functional role in enzymes. Lack of consideration of binding site flexibility has led to failures in predicting protein functions and in successfully docking ligands with protein receptors. Here we address the question: which sequence and structural features distinguish the structurally flexible and rigid binding sites? We analyze high-resolution crystal structures of ligand bound (holo) and free (apo) forms of 41 proteins where no conformational change takes place upon ligand binding, 35 examples with moderate conformational change, and 22 cases where a large conformational change has been observed. We find that the number of residue-residue contacts observed per-residue (contact density) does not distinguish flexible and rigid binding sites, suggesting a role for specific interactions and amino acids in modulating the conformational changes. Examination of hydrogen bonding and hydrophobic interactions reveals that cases that do not undergo conformational change have high polar interactions constituting the binding pockets. Intriguingly, the large, aromatic amino acid tryptophan has a high propensity to occur at the binding sites of examples where a large conformational change has been noted. Further, in large conformational change examples, hydrophobic-hydrophobic, aromatic-aromatic, and hydrophobic-polar residue pair interactions are dominant. Further analysis of the Ramachandran dihedral angles (phi, psi) reveals that the residues adopting disallowed conformations are found in both rigid and flexible cases. More importantly, the binding site residues adopting disallowed conformations clustered narrowly into two specific regions of the L-Ala Ramachandran map. Examination of the dihedral angles changes upon ligand binding shows that the magnitude of phi, psi changes are in general minimal, although some large changes particularly between right-handed alpha-helical and extended conformations are seen. Our work further provides an account of conformational changes in the dihedral angles space. The findings reported here are expected to assist in providing a framework for predicting protein-ligand complexes and for template-based prediction of protein function.     

Determinants of human glucokinase activation and implications for small molecule allosteric control

Glucokinase (GK) is an enzyme that catalyzes the ATP-dependent phosphorylation of glucose to form glucose-6-phosphate, and it is a tightly regulated checkpoint in glucose homeostasis. GK is known to undergo substantial conformational changes upon glucose binding. The monomeric enzyme possesses a highly exotic kinetic activity profile with an unusual sigmoidal dependence on glucose concentration. In this interdisciplinary study, which draws on small angle X-ray scattering (SAXS) integrated with 250?ns of atomistic molecular dynamics (MD) simulations and experimental glucose binding thermodynamics, we reveal that the critical regulation of this glucose sensor is due to a solvent controlled “switch”. We demonstrate that the “solvent switch” is driven by specific protein structural dynamics, which leads to an enzyme structure that has a much more favorable solvation energy than most of the protein ensemble. These findings uncover the physical workings of an agile and flexible protein scaffold, which derives its long-range allosteric control through specific regions with favorable solvation energy. The physiological framework presented herein provides insights that have direct implications for the design of small molecule GK activators as anti-diabetes therapeutics as well as for understanding how proteins can be designed to have built-in regulatory functions via solvation energy dynamics.     

Building a foundation for structure‐based cellulosome design for cellulosic ethanol: Insight into cohesin‐dockerin complexation from computer simulation

The organization and assembly of the cellulosome, an extracellular multienzyme complex produced by anaerobic bacteria, is mediated by the high‐affinity interaction of cohesin domains from scaffolding proteins with dockerins of cellulosomal enzymes. We have performed molecular dynamics simulations and free energy calculations on both the wild type (WT) and D39N mutant of the C. thermocellum Type I cohesin‐dockerin complex in aqueous solution. The D39N mutation has been experimentally demonstrated to disrupt cohesin‐dockerin binding. The present MD simulations indicate that the substitution triggers significant protein flexibility and causes a major change of the hydrogen‐bonding network in the recognition strips—the conserved loop regions previously proposed to be involved in binding—through electrostatic and salt‐bridge interactions between β‐strands 3 and 5 of the cohesin and α‐helix 3 of the dockerin. The mutation‐induced subtle disturbance in the local hydrogen‐bond network is accompanied by conformational rearrangements of the protein side chains and bound water molecules. Additional free energy perturbation calculations of the D39N mutation provide differences in the cohesin‐dockerin binding energy, thus offering a direct, quantitative comparison with experiments. The underlying molecular mechanism of cohesin‐dockerin complexation is further investigated through the free energy profile, that is, potential of mean force (PMF) calculations of WT cohesin‐dockerin complex. The PMF shows a high‐free energy barrier against the dissociation and reveals a stepwise pattern involving both the central β‐sheet interface and its adjacent solvent‐exposed loop/turn regions clustered at both ends of the β‐barrel structure.     

Active and Inactive Cdc42 Differ in Their Insert Region Conformational Dynamics

Cell division control protein 42 homolog (Cdc42) protein, a Ras superfamily GTPase, regulates cellular activities, including cancer progression. Using all-atom molecular dynamics (MD) simulations and essential dynamic analysis, we investigated the structure and dynamics of the catalytic domains of GDP-bound (inactive) and GTP-bound (active) Cdc42 in solution. We discovered substantial differences in the dynamics of the inactive and active forms, particularly in the “insert region” (residues 122–135), which plays a role in Cdc42 activation and binding to effectors. The insert region has larger conformational flexibility in the GDP-bound Cdc42 than in the GTP-bound Cdc42. The G2 loop and switch I at the effector lobe of the catalytic domain exhibit large conformational changes in both the GDP- and the GTP-bound systems, but in the GTP-bound Cdc42, the switch I interactions with GTP are retained. Oncogenic mutations were identified in the Ras superfamily. In Cdc42, the G12V and Q61L mutations decrease the GTPase activity. We simulated these mutations in both GDP- and GTP-bound Cdc42. Although the overall structural organization is quite similar between the wild type and the mutants, there are small differences in the conformational dynamics, especially in the two switch regions. Taken together, the G12V and Q61L mutations may play a role similar to their K-Ras counterparts in nucleotide binding and activation. The conformational differences, which are mainly in the insert region and, to a lesser extent, in the switch regions flanking the nucleotide binding site, can shed light on binding and activation. We propose that the differences are due to a network of hydrogen bonds that gets disrupted when Cdc42 is bound to GDP, a disruption that does not exist in other Rho GTPases. The differences in the dynamics between the two Cdc42 states suggest that the inactive conformation has reduced ability to bind to effectors.     

hdANM: a new comprehensive dynamics model for protein hinges

Hinge motions are essential for many protein functions, and their dynamics are important to understand underlying biological mechanisms. The ways that these motions are represented by various computational methods differ significantly. By focusing on a specific class of motion, we have developed a new hinge-domain anisotropic network model (hdANM) that is based on the prior identification of flexible hinges and rigid domains in the protein structure and the subsequent generation of global hinge motions. This yields a set of motions in which the relative translations and rotations of the rigid domains are modulated and controlled by the deformation of the flexible hinges, leading to a more restricted, specific view of these motions. hdANM is the first model, to our knowledge, that combines information about protein hinges and domains to model the characteristic hinge motions of a protein. The motions predicted with this new elastic network model provide important conceptual advantages for understanding the underlying biological mechanisms. As a matter of fact, the generated hinge movements are found to resemble the expected mechanisms required for the biological functions of diverse proteins. Another advantage of this model is that the domain-level coarse graining makes it significantly more computationally efficient, enabling the generation of hinge motions within even the largest molecular assemblies, such as those from cryo-electron microscopy. hdANM is also comprehensive as it can perform in the same way as the well-known protein dynamics models (anisotropic network model, rotations-translations of blocks, and nonlinear rigid block normal mode analysis), depending on the definition of flexible and rigid parts in the protein structure and on whether the motions are extrapolated in a linear or nonlinear fashion. Furthermore, our results indicate that hdANM produces more realistic motions as compared to the anisotropic network model. hdANM is an open-source software, freely available, and hosted on a user-friendly website.     

Conformational dynamics of Peb4 exhibit “mother’s arms” chain model: a molecular dynamics study

Peb4 from Campylobacter jejuni is an intertwined dimeric, periplasmic holdase, which also exhibits peptidyl prolyl cis/trans isomerase (PPIase) activity. Peb4 gene deletion alters the outer membrane protein profile and impairs cellular adhesion and biofilm formation for C. jejuni. Earlier crystallographic study has proposed that the PPIase domains are flexible and might form a cradle for holding the substrate and these aspects of Peb4 were explored using sub-microsecond molecular dynamics simulations in solution environment. Our simulations have revealed that PPIase domains are highly flexible and undergo a large structural change where they move apart from each other by 8 nm starting at .5 nm. Further, this large conformational change renders Peb4 as a compact protein with crossed-over conformation, forms a central cavity, which can “cradle” the target substrate. As reported for other chaperone proteins, flexibility of linker region connecting the chaperone and PPIase domains is key to forming the “crossed-over” conformation. The conformational transition of the Peb4 protein from the X-ray structure to the crossed-over conformation follows the “mother’s arms” chain model proposed for the FkpA chaperone protein. Our results offer insights into how Peb4 and similar chaperones can use the conformational heterogeneity at their disposal to perform its much-revered biological function.     

Structures of mesophilic and extremophilic citrate synthases reveal rigidity and flexibility for function

Citrate synthase (CS) catalyses the entry of carbon into the citric acid cycle and is highly‐conserved structurally across the tree of life. Crystal structures of dimeric CSs are known in both “open” and “closed” forms, which differ by a substantial domain motion that closes the substrate‐binding clefts. We explore both the static rigidity and the dynamic flexibility of CS structures from mesophilic and extremophilic organisms from all three evolutionary domains. The computational expense of this wide‐ranging exploration is kept to a minimum by the use of rigidity analysis and rapid all‐atom simulations of flexible motion, combining geometric simulation and elastic network modeling. CS structures from thermophiles display increased structural rigidity compared with the mesophilic enzyme. A CS structure from a psychrophile, stabilized by strong ionic interactions, appears to display likewise increased rigidity in conventional rigidity analysis; however, a novel modified analysis, taking into account the weakening of the hydrophobic effect at low temperatures, shows a more appropriate decreased rigidity. These rigidity variations do not, however, affect the character of the flexible dynamics, which are well conserved across all the structures studied. Simulation trajectories not only duplicate the crystallographically observed symmetric open‐to‐closed transitions, but also identify motions describing a previously unidentified antisymmetric functional motion. This antisymmetric motion would not be directly observed in crystallography but is revealed as an intrinsic property of the CS structure by modeling of flexible motion. This suggests that the functional motion closing the binding clefts in CS may be independent rather than symmetric and cooperative. Proteins 2014; 82:2657–2670. © 2014 Wiley Periodicals, Inc.