Computing the Amino Acid Specificity of Fluctuations in Biomolecular Systems

Anisotropic Network Analysis of Open/Closed HCN4 Channel Advocates Asymmetric Subunit Cooperativity in cAMP Modulation of Gating

Hyperpolarization-activated cyclic nucleotide-modulated (HCN) channels are opened in an allosteric manner by membrane hyperpolarization and cyclic nucleotides such as cAMP. Because of conflicting reports from experimental studies on whether cAMP binding to the four available binding sites in the channel tetramer operates cooperatively in gating, we employ here a computational approach as a promising route to examine ligand-induced conformational changes after binding to individual sites. By combining an elastic network model (ENM) with linear response theory (LRT) for modeling the apo-holo transition of the cyclic nucleotide-binding domain (CNBD) in HCN channels, we observe a distinct pattern of cooperativity matching the "positive-negative-positive" cooperativity reported from functional studies. This cooperativity pattern is highly conserved among HCN subtypes (HCN4, HCN1), but only to a lesser extent visible in structurally related channels, which are only gated by voltage (KAT1) or cyclic nucleotides (TAX4). This suggests an inherent cooperativity between subunits in HCN channels as part of a ligand-triggered gating mechanism in these channels.

Insights into Activation Dynamics and Functional Sites of Inwardly Rectifying Potassium Channel Kir3.2 by an Elastic Network Model Combined with Perturbation Methods

The inwardly rectifying potassium channel Kir3.2, a member of the inward rectifier potassium (Kir) channel family, exerts important biological functions through transporting potassium ions outside of the cell, during which a large-scale synergistic movement occurs among its different domains. Currently, it is not fully understood how the binding of the ligand to the Kir3.2 channel leads to the structural changes and which key residues are responsible for the channel gating and allosteric dynamics. Here, we construct the Gaussian network model (GNM) of the Kir3.2 channel with the secondary structure and covalent interaction information considered (sscGNM), which shows a better performance in reproducing the channel's flexibility compared with the traditional GNM. In addition, the sscANM-based perturbation method is used to simulate the channel's conformational transition caused by the activator PIP2's binding. By applying certain forces to the PIP2 binding pocket, the coarse-grained calculations generate the similar conformational changes to the experimental observation, suggesting that the topology structure as well as PIP2 binding are crucial to the allosteric activation of the Kir3.2 channel. We also utilize the sscGNM-based thermodynamic cycle method developed by us to identify the key residues whose mutations significantly alter the channel's binding free energy with PIP2. We identify not only the residues important for the specific binding but also the ones critical for the allosteric transition coupled with PIP2 binding. This study is helpful for understanding the working mechanism of Kir3.2 channels and can provide important information for related drug design.

Biotite: new tools for a versatile Python bioinformatics library

BACKGROUND

Biotite is a program library for sequence and structural bioinformatics written for the Python programming language. It implements widely used computational methods into a consistent and accessible package. This allows for easy combination of various data analysis, modeling and simulation methods.

RESULTS

This article presents major functionalities introduced into Biotite since its original publication. The fields of application are shown using concrete examples. We show that the computational performance of Biotite for bioinformatics tasks is comparable to individual, special purpose software systems specifically developed for the respective single task.

CONCLUSIONS

The results show that Biotite can be used as program library to either answer specific bioinformatics questions and simultaneously allow the user to write entire, self-contained software applications with sufficient performance for general application.

A generic force field for simulating native protein structures using dissipative particle dynamics

A coarse-grained force field for molecular dynamics simulations of the native structures of proteins in a dissipative particle dynamics (DPD) framework is developed. The parameters for bonded interactions are derived by mapping the bonds and angles for 20 amino acids onto target distributions obtained from fully atomistic simulations in explicit solvent. A dual-basin potential is introduced for stabilizing backbone angles, to cover a wide spectrum of protein secondary structures. The backbone dihedral potential enables folding of the protein from an unfolded initial state to the folded native structure. The proposed force field is validated by evaluating the structural properties of several model peptides and proteins including the SARS-CoV-2 fusion peptide, consisting of α-helices, β-sheets, loops and turns. Detailed comparisons with fully atomistic simulations are carried out to assess the ability of the proposed force field to stabilize the different secondary structures present in proteins. The compact conformations of the native states were evident from the radius of gyration and the high intensity peaks of the root mean square deviation histograms, which were found to be within 0.4 nm. The Ramachandran-like energy landscape on the phase space of backbone angles (θ) and dihedrals (ϕ) effectively captured the conformational phase space of α-helices at ∼(ϕ = 50°,θ = 90°) and β-strands at ∼(ϕ = ±180°,θ = 90-120°). Furthermore, the residue-residue native contacts were also well reproduced by the proposed DPD model. The applicability of the model to multidomain complexes was assessed using lysozyme and a large α-helical bacterial pore-forming toxin, cytolysin A. Our study illustrates that the proposed force field is generic, and can potentially be extended for efficient in silico investigations of membrane bound polypeptides and proteins using DPD simulations.

Equally Weighted Multiscale Elastic Network Model and Its Comparison with Traditional and Parameter-Free Models

Dynamical properties of proteins play an essential role in their function exertion. The elastic network model (ENM) is an effective and efficient tool in characterizing the intrinsic dynamical properties encoded in biomacromolecule structures. The Gaussian network model (GNM) and anisotropic network model (ANM) are the two often-used ENM models. Here, we introduce an equally weighted multiscale ENM (equally weighted mENM) based on the original mENM (denoted as mENM), in which fitting weights of Kirchhoff/Hessian matrixes in mENM are removed since they neglect the details of pairwise interactions. Then, we perform its comparison with the mENM, traditional ENM, and parameter-free ENM (pfENM) in reproducing dynamical properties for the six representative proteins whose molecular dynamics (MD) trajectories are available in http://mmb.pcb.ub.es/MoDEL/. In the results, for B-factor prediction, mENM performs best, while the equally weighted mENM performs also well, better than the traditional ENM and pfENM models. As to the dynamical cross-correlation map calculation, mENM performs worst, while the results produced from the equally weighted mENM and pfENM models are close to those from MD trajectories with the latter a little better than the former. Furthermore, encouragingly, the equally weighted mANM displays the best performance in capturing the functional motional modes, followed by pfANM and traditional ANM models, while the mANM fails in all the cases. This work is helpful for strengthening the understanding of the elastic network model and provides a valuable guide for researchers to utilize the model to explore protein dynamics.

Inferring functional units in ion channel pores via relative entropy

Coarse-grained protein models approximate the first-principle physical potentials. Among those modeling approaches, the relative entropy framework yields promising and physically sound results, in which a mapping from the target protein structure and dynamics to a model is defined and subsequently adjusted by an entropy minimization of the model parameters. Minimization of the relative entropy is equivalent to maximization of the likelihood of reproduction of (configurational ensemble) observations by the model. In this study, we extend the relative entropy minimization procedure beyond parameter fitting by a second optimization level, which identifies the optimal mapping to a (dimension-reduced) topology. We consider anisotropic network models of a diverse set of ion channels and assess our findings by comparison to experimental results.

Advances in coarse-grained modeling of macromolecular complexes

Recent progress in coarse-grained (CG) molecular modeling and simulation has facilitated an influx of computational studies on biological macromolecules and their complexes. Given the large separation of length-scales and time-scales that dictate macromolecular biophysics, CG modeling and simulation are well-suited to bridge the microscopic and mesoscopic or macroscopic details observed from all-atom molecular simulations and experiments, respectively. In this review, we first summarize recent innovations in the development of CG models, which broadly include structure-based, knowledge-based, and dynamics-based approaches. We then discuss recent applications of different classes of CG models to explore various macromolecular complexes. Finally, we conclude with an outlook for the future in this ever-growing field of biomolecular modeling.

Mechanical transduction of cytoplasmic-to-transmembrane-domain movements in a hyperpolarization-activated cyclic nucleotide-gated cation channel

Hyperpolarization-activated cyclic nucleotide–gated cation (HCN) channels play a critical role in the control of pacemaking in the heart and repetitive firing in neurons. In HCN channels, the intracellular cyclic nucleotide–binding domain (CNBD) is connected to the transmembrane portion of the channel (TMPC) through a helical domain, the C-linker. Although this domain is critical for mechanical signal transduction, the conformational dynamics in the C-linker that transmit the nucleotide-binding signal to the HCN channel pore are unknown. Here, we use linear response theory to analyze conformational changes in the C-linker of the human HCN1 protein, which couple cAMP binding in the CNBD with gating in the TMPC. By applying a force to the tip of the so-called “elbow” of the C-linker, the coarse-grained calculations recapitulate the same conformational changes triggered by cAMP binding in experimental studies. Furthermore, in our simulations, a displacement of the C-linker parallel to the membrane plane (i.e. horizontally) induced a rotational movement resulting in a distinct tilting of the transmembrane helices. This movement, in turn, increased the distance between the voltage-sensing S4 domain and the surrounding transmembrane domains and led to a widening of the intracellular channel gate. In conclusion, our computational approach, combined with experimental data, thus provides a more detailed understanding of how cAMP binding is mechanically coupled over long distances to promote voltage-dependent opening of HCN channels.

The maximum penalty criterion for ridge regression: application to the calibration of the force constant in elastic network models

Tikhonov regularization, or ridge regression, is a popular technique to deal with collinearity in multivariate regression. We unveil a formal analogy between ridge regression and statistical mechanics, where the objective function is comparable to a free energy, and the ridge parameter plays the role of temperature. This analogy suggests two novel criteria for selecting a suitable ridge parameter: specific-heat (C_v) and maximum penalty (MP). We apply these fits to evaluate the relative contributions of rigid-body and internal fluctuations, which are typically highly collinear, to crystallographic B-factors. This issue is particularly important for computational models of protein dynamics, such as the elastic network model (ENM), since the amplitude of the predicted internal motion is commonly calibrated using B-factor data. After validation on simulated datasets, our results indicate that rigid-body motions account on average for more than 80% of the amplitude of B-factors. Furthermore, we evaluate the ability of different fits to reproduce the amplitudes of internal fluctuations in X-ray ensembles from the B-factors in the corresponding single X-ray structures. The new ridge criteria are shown to be markedly superior to the commonly used two-parameter fit that neglects rigid-body rotations and to the full fits regularized under generalized cross-validation. In conclusion, the proposed fits ensure a more robust calibration of the ENM force constant and should prove valuable in other applications.

A reduced mechanical model for cAMP-modulated gating in HCN channels

We developed an in silico mechanical model to analyze the process of cAMP-induced conformational modulations in hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which conduct cations across the membrane of mammalian heart and brain cells. The structural analysis reveals a quaternary twist in the cytosolic parts of the four subunits in the channel tetramer. This motion augments the intrinsic dynamics of the very same protein structure. The pronounced differences between the cAMP bound and unbound form include a mutual interaction between the C-linker of the cyclic nucleotide binding domain (CNBD) and the linker between the S4 and S5 transmembrane domain of the channel. This allows a mechanistic annotation of the twisting motion in relation to the allosteric modulation of voltage-dependent gating of this channel by cAMP.

Drug-resistant HIV-1 protease regains functional dynamics through cleavage site coevolution

Drug resistance is caused by mutations that change the balance of recognition favoring substrate cleavage over inhibitor binding. Here, a structural dynamics perspective of the regained wild-type functioning in mutant HIV-1 proteases with coevolution of the natural substrates is provided. The collective dynamics of mutant structures of the protease bound to p1-p6 and NC-p1 substrates are assessed using the Anisotropic Network Model (ANM). The drug-induced protease mutations perturb the mechanistically crucial hinge axes that involve key sites for substrate binding and dimerization and mainly coordinate the intrinsic dynamics. Yet with substrate coevolution, while the wild-type dynamic behavior is restored in both p1-p6 ((LP) (1'F)p1-p6D30N/N88D) and NC-p1 ((AP) (2) (V)NC-p1V82A) bound proteases, the dynamic behavior of the NC-p1 bound protease variants (NC-p1V82A and (AP) (2) (V)NC-p1V82A) rather resemble those of the proteases bound to the other substrates, which is consistent with experimental studies. The orientational variations of residue fluctuations along the hinge axes in mutant structures justify the existence of coevolution in p1-p6 and NC-p1 substrates, that is, the dynamic behavior of hinge residues should contribute to the interdependent nature of substrate recognition. Overall, this study aids in the understanding of the structural dynamics basis of drug resistance and evolutionary optimization in the HIV-1 protease system.

Effective harmonic potentials: insights into the internal cooperativity and sequence-specificity of protein dynamics

The proper biological functioning of proteins often relies on the occurrence of coordinated fluctuations around their native structure, or on their ability to perform wider and sometimes highly elaborated motions. Hence, there is considerable interest in the definition of accurate coarse-grained descriptions of protein dynamics, as an alternative to more computationally expensive approaches. In particular, the elastic network model, in which residue motions are subjected to pairwise harmonic potentials, is known to capture essential aspects of conformational dynamics in proteins, but has so far remained mostly phenomenological, and unable to account for the chemical specificities of amino acids. We propose, for the first time, a method to derive residue- and distance-specific effective harmonic potentials from the statistical analysis of an extensive dataset of NMR conformational ensembles. These potentials constitute dynamical counterparts to the mean-force statistical potentials commonly used for static analyses of protein structures. In the context of the elastic network model, they yield a strongly improved description of the cooperative aspects of residue motions, and give the opportunity to systematically explore the influence of sequence details on protein dynamics.

Free energy of contact formation in proteins: efficient computation in the elastic network approximation

Biomolecular simulations have become a major tool in understanding biomolecules and their complexes. However, one can typically only investigate a few mutants or scenarios due to the severe computational demands of such simulations, leading to a great interest in method development to overcome this restriction. One way to achieve this is to reduce the complexity of the systems by an approximation of the forces acting upon the constituents of the molecule. The harmonic approximation used in elastic network models simplifies the physical complexity to the most reduced dynamics of these molecular systems. The reduced polymer modeled this way is typically comprised of mass points representing coarse-grained versions of, e.g., amino acids. In this work, we show how the computation of free energy contributions of contacts between two residues within the molecule can be reduced to a simple lookup operation in a precomputable matrix. Being able to compute such contributions is of great importance: protein design or molecular evolution changes introduce perturbations to these pair interactions, so we need to understand their impact. Perturbation to the interactions occurs due to randomized and fixated changes (in molecular evolution) or designed modifications of the protein structures (in bioengineering). These perturbations are modifications in the topology and the strength of the interactions modeled by the elastic network models. We apply the new algorithm to (1) the bovine trypsin inhibitor, a well-known enzyme in biomedicine, and show the connection to folding properties and the hydrophobic collapse hypothesis and (2) the serine proteinase inhibitor CI-2 and show the correlation to Φ values to characterize folding importance. Furthermore, we discuss the computational complexity and show empirical results for the average case, sampled over a library of 77 structurally diverse proteins. We found a relative speedup of up to 10 000-fold for large proteins with respect to repeated application of the initial model.

Structural model of the gas vesicle protein GvpA and analysis of GvpA mutants in vivo

Gas vesicles are gas-filled protein structures increasing the buoyancy of cells. The gas vesicle envelope is mainly constituted by the 8 kDa protein GvpA forming a wall with a water excluding inner surface. A structure of GvpA is not available; recent solid-state NMR results suggest a coil-α-β-β-α-coil fold. We obtained a first structural model of GvpA by high-performance de novo modelling. Attenuated total reflection (ATR)-Fourier transform infrared spectroscopy (FTIR) supported this structure. A dimer of GvpA was derived that could explain the formation of the protein monolayer in the gas vesicle wall. The hydrophobic inner surface is mainly constituted by anti-parallel β-strands. The proposed structure allows the pinpointing of contact sites that were mutated and tested for the ability to form gas vesicles in haloarchaea. Mutations in α-helix I and α-helix II, but also in the β-turn affected the gas vesicle formation, whereas other alterations had no effect. All mutants supported the structural features deduced from the model. The proposed GvpA dimers allow the formation of a monolayer protein wall, also consistent with protease treatments of isolated gas vesicles.

Membrane anchoring and interaction between transmembrane domains are crucial for K+ channel function

The small viral channel Kcv is a Kir-like K(+) channel of only 94 amino acids. With this simple structure, the tetramer of Kcv represents the pore module of all complex K(+) channels. To examine the structural contribution of the transmembrane domains (TMDs) to channel function, we performed Ala scanning mutagenesis of the two domains and tested the functionality of the mutants in a yeast complementation assay. The data reveal, in combination with computational models, that the upper halves of both TMDs, which face toward the external medium, are rather rigid, whereas the inner parts are more flexible. The rigidity of the outer TMD is conferred by a number of essential aromatic amino acids that face the membrane and probably anchor this domain in the bilayer. The inner TMD is intimately connected with the rigid part of the outer TMD via π···π interactions between a pair of aromatic amino acids. This structural principle is conserved within the viral K(+) channels and also present in Kir2.2, implying a general importance of this architecture for K(+) channel function.

Efficient quantification of the importance of contacts for the dynamical stability of proteins

Understanding the stability of the native state and the dynamics of a protein is of great importance for all areas of biomolecular design. The efficient estimation of the influence of individual contacts between amino acids in a protein structure is a first step in the reengineering of a particular protein for technological or pharmacological purposes. At the same time, the functional annotation of molecular evolution can be facilitated by such insight. Here, we use a recently suggested, information theoretical measure in biomolecular design - the Kullback-Leibler-divergence - to quantify and therefore rank residue-residue contacts within proteins according to their overall contribution to the molecular mechanics. We implement this protocol on the basis of a reduced molecular model, which allows us to use a well-known lemma of linear algebra to speed up the computation. The increase in computational performance is around 10(1)- to 10(4)-fold. We applied the method to two proteins to illustrate the protocol and its results. We found that our method can reliably identify key residues in the molecular mechanics and the protein fold in comparison to well-known properties in the serine protease inhibitor. We found significant correlations to experimental results, e.g., dissociation constants and Φ values.

Rationale for more diverse inhibitors in competition with substrates in HIV-1 protease

The structural fluctuations of HIV-1 protease in interaction with its substrates versus inhibitors were analyzed using the anisotropic network model. The directions of fluctuations in the most cooperative functional modes differ mainly around the dynamically key regions, i.e., the hinge axes, which appear to be more flexible in substrate complexes. The flexibility of HIV-1 protease is likely optimized for the substrates' turnover, resulting in substrate complexes being dynamic. In contrast, in an inhibitor complex, the inhibitor should bind and lock down to inactivate the active site. Protease and ligands are not independent. Substrates are also more flexible than inhibitors and have the potential to meet the dynamic distributions that are inherent in the protease. This may suggest a rationale and guidelines for designing inhibitors that can better fit the ensemble of binding sites that are dynamically accessible to the protease.

Minimalist models for proteins: a comparative analysis

The last decade has witnessed a renewed interest in the coarse-grained (CG) models for biopolymers, also stimulated by the needs of modern molecular biology, dealing with nano- to micro-sized bio-molecular systems and larger than microsecond timescale. This combination of size and timescale is, in fact, hard to access by atomic-based simulations. Coarse graining the system is a route to be followed to overcome these limits, but the ways of practically implementing it are many and different, making the landscape of CG models very vast and complex. In this paper, the CG models are reviewed and their features, applications and performances compared. This analysis, restricted to proteins, focuses on the minimalist models, namely those reducing at minimum the number of degrees of freedom without losing the possibility of explicitly describing the secondary structures. This class includes models using a single or a few interacting centers (beads) for each amino acid. From this analysis several issues emerge. The difficulty in building these models resides in the need for combining transferability/predictive power with the capability of accurately reproducing the structures. It is shown that these aspects could be optimized by accurately choosing the force field (FF) terms and functional forms, and combining different parameterization procedures. In addition, in spite of the variety of the minimalist models, regularities can be found in the parameters values and in FF terms. These are outlined and schematically presented with the aid of a generic phase diagram of the polypeptide in the parameter space and, hopefully, could serve as guidelines for the development of minimalist models incorporating the maximum possible level of predictive power and structural accuracy.

BioPhysConnectoR: Connecting sequence information and biophysical models

Background

One of the most challenging aspects of biomolecular systems is the understanding of the coevolution in and among the molecule(s).

A complete, theoretical picture of the selective advantage, and thus a functional annotation, of (co-)mutations is still lacking. Using sequence-based and information theoretical inspired methods we can identify coevolving residues in proteins without understanding the underlying biophysical properties giving rise to such coevolutionary dynamics. Detailed (atomistic) simulations are prohibitively expensive. At the same time reduced molecular models are an efficient way to determine the reduced dynamics around the native state. The combination of sequence based approaches with such reduced models is therefore a promising approach to annotate evolutionary sequence changes.

Results

With the R package BioPhysConnectoR we provide a framework to connect the information theoretical domain of biomolecular sequences to biophysical properties of the encoded molecules - derived from reduced molecular models. To this end we have integrated several fragmented ideas into one single package ready to be used in connection with additional statistical routines in R. Additionally, the package leverages the power of modern multi-core architectures to reduce turn-around times in evolutionary and biomolecular design studies. Our package is a first step to achieve the above mentioned annotation of coevolution by reduced dynamics around the native state of proteins.

Conclusions

BioPhysConnectoR is implemented as an R package and distributed under GPL 2 license. It allows for efficient and perfectly parallelized functional annotation of coevolution found at the sequence level.

Multiscale modeling of proteins

The activity within a living cell is based on a complex network of interactions among biomolecules, exchanging information and energy through biochemical processes. These events occur on different scales, from the nano- to the macroscale, spanning about 10 orders of magnitude in the space domain and 15 orders of magnitude in the time domain. Consequently, many different modeling techniques, each proper for a particular time or space scale, are commonly used. In addition, a single process often spans more than a single time or space scale. Thus, the necessity arises for combining the modeling techniques in multiscale approaches. In this Account, I first review the different modeling methods for bio-systems, from quantum mechanics to the coarse-grained and continuum-like descriptions, passing through the atomistic force field simulations. Special attention is devoted to their combination in different possible multiscale approaches and to the questions and problems related to their coherent matching in the space and time domains. These aspects are often considered secondary, but in fact, they have primary relevance when the aim is the coherent and complete description of bioprocesses. Subsequently, applications are illustrated by means of two paradigmatic examples: (i) the green fluorescent protein (GFP) family and (ii) the proteins involved in the human immunodeficiency virus (HIV) replication cycle. The GFPs are currently one of the most frequently used markers for monitoring protein trafficking within living cells; nanobiotechnology and cell biology strongly rely on their use in fluorescence microscopy techniques. A detailed knowledge of the actions of the virus-specific enzymes of HIV (specifically HIV protease and integrase) is necessary to study novel therapeutic strategies against this disease. Thus, the insight accumulated over years of intense study is an excellent framework for this Account. The foremost relevance of these two biomolecular systems was recently confirmed by the assignment of two of the Nobel prizes in 2008: in chemistry for the discovery of GFP and in medicine for the discovery of HIV. Accordingly, these proteins were studied with essentially all of the available modeling techniques, making them ideal examples for studying the details of multiscale approaches in protein modeling.

Relating sequence evolution of HIV1-protease to its underlying molecular mechanics

We investigate the connection between sequence evolution of the human immunodeficiency virus (HIV) type 1 protease under neutral selection or selective pressure induced by protease inhibitors and the functional and molecular-stability characteristics of the molecule in the physical domain. To this end we analyze sequence data on more than 45,000 patients with bioinformatical tools, namely mutual information between residue pairings. In addition we perform extensive computations on the molecular mechanics of the molecule subject to artificial mutations. The changes in the mechanics and dynamics of the molecule in three-dimensional space upon perturbation are then related to the sequence stability as described by the mutual information. We distinguish physical interactions by their evolutionary background and give hints for potential new drug targets. In addition we discuss how such targets can be efficiently chosen to give the HI virus less opportunity to develop resistance towards such drugs while maintaining the protease function at the same time. The interactions between residue no. 28 and 23' in different chains as well as the interaction between residue no. 92 and 94 within one chain were identified as particular crucial. In addition we find interactions in the beta-sheet-dimerization interface to be important for conserving the protein function and stability while these are at the same time evolutionary conserved - implications of and comparisons to experimental results are finally discussed.

Ensemble refinement of protein crystal structures: validation and application

X-ray crystallography typically uses a single set of coordinates and B factors to describe macromolecular conformations. Refinement of multiple copies of the entire structure has been previously used in specific cases as an alternative means of representing structural flexibility. Here, we systematically validate this method by using simulated diffraction data, and we find that ensemble refinement produces better representations of the distributions of atomic positions in the simulated structures than single-conformer refinements. Comparison of principal components calculated from the refined ensembles and simulations shows that concerted motions are captured locally, but that correlations dissipate over long distances. Ensemble refinement is also used on 50 experimental structures of varying resolution and leads to decreases in R(free) values, implying that improvements in the representation of flexibility observed for the simulated structures may apply to real structures. These gains are essentially independent of resolution or data-to-parameter ratio, suggesting that even structures at moderate resolution can benefit from ensemble refinement.

Anisotropic fluctuations of amino acids in protein structures: insights from X-ray crystallography and elastic network models

MOTIVATION

A common practice in X-ray crystallographic structure refinement has been to model atomic displacements or thermal fluctuations as isotropic motions. Recent high-resolution data reveal, however, significant departures from isotropy, described by anisotropic displacement parameters (ADPs) modeled for individual atoms. Yet, ADPs are currently reported for a limited set of structures, only.

RESULTS

We present a comparative analysis of the experimentally reported ADPs and those theoretically predicted by the anisotropic network model (ANM) for a representative set of structures. The relative sizes of fluctuations along different directions are shown to agree well between experiments and theory, while the cross-correlations between the (x-, y- and z-) components of the fluctuations show considerable deviations. Secondary structure elements and protein cores exhibit more robust anisotropic characteristics compared to disordered or flexible regions. The deviations between experimental and theoretical data are comparable to those between sets of experimental ADPs reported for the same protein in different crystal forms. These results draw attention to the effects of crystal form and refinement procedure on experimental ADPs and highlight the potential utility of ANM calculations for consolidating experimental data or assessing ADPs in the absence of experimental data.

AVAILABILITY

The ANM server at http://www.ccbb.pitt.edu/anm is upgraded to permit users to compute and visualize the theoretical ADPs for any PDB structure, thus providing insights into the anisotropic motions intrinsically preferred by equilibrium structures.

SUPPLEMENTARY INFORMATION

Two Supplementary Material files can be accessed at the journal website. The first presents the tabulated results from computations (Pearson correlations and KL distances with respect to experimental ADPs) reported for each of the 93 proteins in Set I (the averages over all proteins are presented above in Table 3). The second file consists of three sections: (A) detailed derivation of Equation (7), (B) analysis of the effect of ANM parameters on computed ADPs and identification of parameters that achieve optimal correlation with experiments and (C) description of the method for computing the tangential and radial components of equilibrium fluctuations.

Protein structural variation in computational models and crystallographic data

Normal mode analysis offers an efficient way of modeling the conformational flexibility of protein structures. We use anisotropic displacement parameters from crystallography to test the quality of prediction of both the magnitude and directionality of conformational flexibility. Normal modes from four simple elastic network model potentials and from the CHARMM force field are calculated for a data set of 83 diverse, ultrahigh-resolution crystal structures. While all five potentials provide good predictions of the magnitude of flexibility, all-atom potentials have a clear edge at prediction of directionality, and the CHARMM potential has the highest prediction quality. The low-frequency modes from different potentials are similar, but those computed from the CHARMM potential show the greatest difference from the elastic network models. The comprehensive evaluation demonstrates the costs and benefits of using normal mode potentials of varying complexity.