Normal modes as refinement parameters for the F-actin model.

The slow normal modes of G-actin were used as structural parameters to refine the F-actin model against 8-A resolution x-ray fiber diffraction data. The slowest frequency normal modes of G-actin pertain to collective rearrangements of domains, motions that are characterized by correlation lengths on the order of the resolution of the fiber diffraction data. Using a small number of normal mode degrees of freedom (< or = 12) improved the fit to the data significantly. The refined model of F-actin shows that the nucleotide binding cleft has narrowed and that the DNase I binding loop has twisted to a lower radius, consistent with other refinement techniques and electron microscopy data. The methodology of a normal mode refinement is described, and the results, as applied to actin, are detailed.     

Normal mode refinement: crystallographic refinement of protein dynamic structure. I. Theory and test by simulated diffraction data.

A dynamic structure refinement method for X-ray crystallography, referred to as the normal mode refinement, is proposed. The Debye-Waller factor is expanded in terms of the low-frequency normal modes whose amplitudes and eigenvectors are experimentally optimized in the process of the crystallographic refinement. In this model, the atomic fluctuations are treated as anisotropic and concerted. The normal modes of the external motion (TLS model) are also introduced to cover the factors other than the internal fluctuations, such as the lattice disorder and diffusion. A program for the normal mode refinement (NM-REF) has been developed. The method has first been tested against simulated diffraction data for human lysozyme calculated by a Monte Carlo simulation. Applications of the method have demonstrated that the normal mode refinement has: (1) improved the fitting to the diffraction data, even with fewer adjustable parameters; (2) distinguished internal fluctuations from external ones; (3) determined anisotropic thermal factors; and (4) identified concerted fluctuations in the protein molecule.     

Normal-mode refinement of anisotropic thermal parameters for potassium channel KcsA at 3.2 A crystallographic resolution

We report a normal-mode method for anisotropic refinement of membrane-protein structures, based on a hypothesis that the global near-native-state disordering of membrane proteins in crystals follows low-frequency normal modes. Thus, a small set of modes is sufficient to represent the anisotropic thermal motions in X-ray crystallographic refinement. By applying the method to potassium channel KcsA at 3.2 A, we obtained a structural model with an improved fit with the diffraction data. Moreover, the improved electron density maps allowed for large structural adjustments for 12 residues in each subunit, including the rebuilding of 3 missing side chains. Overall, the anisotropic KcsA structure at 3.2 A was systematically closer to a 2.0 A KcsA structure, especially in the selectivity filter. Furthermore, the anisotropic thermal ellipsoids from the refinement revealed functionally relevant structural flexibility. We expect this method to be a valuable tool for structural refinement of many membrane proteins with moderate-resolution diffraction data.     

Normal mode refinement: crystallographic refinement of protein dynamic structure applied to human lysozyme.

A new method of dynamic structure refinement of protein x-ray crystallography, normal mode refinement, is developed. In this method the Debye-Waller factor is expanded in terms of the low-frequency normal modes and external normal modes, whose amplitudes and couplings are optimized in the process of crystallographic refinement. By this method, internal and external contributions to the atomic fluctuations can be separated. Also, anisotropic atomic fluctuations and their interatomic correlations can be determined experimentally even with a relatively small number of adjustable parameters. The method is applied to the analysis of experimental data of human lysozyme to reveal its dynamic structure.     

Constitutive modeling of mouse carotid arteries using experimentally measured microstructural parameters

Changes in the local mechanical environment and tissue mechanical properties affect the biological activity of cells and play a key role in a variety of diseases, such as cancer, arthritis, nephropathy, and cardiovascular disease. Constitutive relations have long been used to predict the local mechanical environment within biological tissues and to investigate the relationship between biological responses and mechanical stimuli. Recent constitutive relations for soft tissues consider both material and structural properties by incorporating parameters that describe microstructural organization, such as fiber distributions, fiber angles, fiber crimping, and constituent volume fractions. The recently developed technique of imaging the microstructure of a single artery as it undergoes multiple deformations provides quantitative structural data that can reduce the number of estimated parameters by using parameters that are truly experimentally intractable. Here, we employed nonlinear multiphoton microscopy to quantify collagen fiber organization in mouse carotid arteries and incorporated measured fiber distribution data into structurally motivated constitutive relations. Microscopy results demonstrate that collagen fibers deform in an affine manner over physiologically relevant deformations. The incorporation of measured fiber angle distributions into constitutive relations improves the model's predictive accuracy and does not significantly reduce the goodness of fit. The use of measured structural parameters rather than estimated structural parameters promises to improve the predictive capabilities of the local mechanical environment, and to extend the utility of intravital imaging methods for estimating the mechanical behavior of tissues using in situ structural information.     

Simulation of planar soft tissues using a structural constitutive model: Finite element implementation and validation

Computational implementation of physical and physiologically realistic constitutive models is critical for numerical simulation of soft biological tissues in a variety of biomedical applications. It is well established that the highly nonlinear and anisotropic mechanical behaviors of soft tissues are an emergent behavior of the underlying tissue microstructure. In the present study, we have implemented a structural constitutive model into a finite element framework specialized for membrane tissues. We noted that starting with a single element subjected to uniaxial tension, the non-fibrous tissue matrix must be present to prevent unrealistic tissue deformations. Flexural simulations were used to set the non-fibrous matrix modulus because fibers have little effects on tissue deformation under three-point bending. Multiple deformation modes were simulated, including strip biaxial, planar biaxial with two attachment methods, and membrane inflation. Detailed comparisons with experimental data were undertaken to insure faithful simulations of both the macro-level stress–strain insights into adaptations of the fiber architecture under stress, such as fiber reorientation and fiber recruitment. Results indicated a high degree of fidelity and demonstrated interesting microstructural adaptions to stress and the important role of the underlying tissue matrix. Moreover, we apparently resolve a discrepancy in our 1997 study (Billiar and Sacks, 1997. J. Biomech. 30 (7), 753–756) where we observed that under strip biaxial stretch the simulated fiber splay responses were not in good agreement with the experimental results, suggesting non-affine deformations may have occurred. However, by correctly accounting for the isotropic phase of the measured fiber splay, good agreement was obtained. While not the final word, these simulations suggest that affine fiber kinematics for planar collagenous tissues is a reasonable assumption at the macro level. Simulation tools such as these are imperative in the design and simulation of native and engineered tissues.     

Homology‐based hydrogen bond information improves crystallographic structures in the PDB

The Protein Data Bank (PDB) is the global archive for structural information on macromolecules, and a popular resource for researchers, teachers, and students, amassing more than one million unique users each year. Crystallographic structure models in the PDB (more than 100,000 entries) are optimized against the crystal diffraction data and geometrical restraints. This process of crystallographic refinement typically ignored hydrogen bond (H‐bond) distances as a source of information. However, H‐bond restraints can improve structures at low resolution where diffraction data are limited. To improve low‐resolution structure refinement, we present methods for deriving H‐bond information either globally from well‐refined high‐resolution structures from the PDB‐REDO databank, or specifically from on‐the‐fly constructed sets of homologous high‐resolution structures. Refinement incorporating HOmology DErived Restraints (HODER), improves geometrical quality and the fit to the diffraction data for many low‐resolution structures. To make these improvements readily available to the general public, we applied our new algorithms to all crystallographic structures in the PDB: using massively parallel computing, we constructed a new instance of the PDB‐REDO databank ( https://pdb-redo.eu ). This resource is useful for researchers to gain insight on individual structures, on specific protein families (as we demonstrate with examples), and on general features of protein structure using data mining approaches on a uniformly treated dataset.     

Atomic Resolution Structure of a Protein Prepared by Non-Enzymatic His-Tag Removal. Crystallographic and NMR Study of GmSPI-2 Inhibitor

Purification of suitable quantity of homogenous protein is very often the bottleneck in protein structural studies. Overexpression of a desired gene and attachment of enzymatically cleavable affinity tags to the protein of interest made a breakthrough in this field. Here we describe the structure of Galleria mellonella silk proteinase inhibitor 2 (GmSPI-2) determined both by X-ray diffraction and NMR spectroscopy methods. GmSPI-2 was purified using a new method consisting in non-enzymatic His-tag removal based on a highly specific peptide bond cleavage reaction assisted by Ni(II) ions. The X-ray crystal structure of GmSPI-2 was refined against diffraction data extending to 0.98 Å resolution measured at 100 K using synchrotron radiation. Anisotropic refinement with the removal of stereochemical restraints for the well-ordered parts of the structure converged with R factor of 10.57% and R _free of 12.91%. The 3D structure of GmSPI-2 protein in solution was solved on the basis of 503 distance constraints, 10 hydrogen bonds and 26 torsion angle restraints. It exhibits good geometry and side-chain packing parameters. The models of the protein structure obtained by X-ray diffraction and NMR spectroscopy are very similar to each other and reveal the same β₂αβ fold characteristic for Kazal-family serine proteinase inhibitors.     

Real space refinement of neutron diffraction data from sperm whale carbonmonoxymyoglobin

Neutron diffraction data from crystals of sperm whale carbonmonoxymyoglobin have been refined by the real space refinement technique. Estimates of the neutron occupancies at the end of the refinement show that the mean for each atom type (including hydrogen and deuterium) is close to the expected value and has a standard deviation from the mean of about 5%. Mean neutron occupancies of main-chain atoms involved in deuterium bonds versus those not involved in deuterium bonds demonstrate that the hydrogen/deuterium exchange of the latter group is higher. The oxygen and deuterium co-ordinates for 40 water molecules have been determined: 27 of these water molecules were involved in bridges between protein atoms, and nine were involved in deuterium bonds with main-chain atoms. The deuterium-bond angles in helical regions show significant deviations from linearity. The mean ND … O angle was 154(3) °² and the mean CO … D angle was 145(3) °.     

Rigid-body refinement and conformation of α-chitin

The structure of α-chitin has been refined using the rigid body refinement technique. Polysaccharide chitin composed of repeating chitibiose units has been treated as a rigid body. The refinement has been done using group intensity data. Using the Full Matrix Least Square rigid body refinement procedure the structure has been refined to an R-factor of 40.7%. For the refined structure the agreement between the calculated and the observed intensities is reasonably good. The refinement leads to two possible orientations of the CH₂OH group and in both orientations it is not involved in hydrogen bonding. One of the conformations of CH₂OH group in the refined structure is close to that proposed earlier by Carlström (J. Biophys. Biochem, Cytol., 3 (1957) 669) from X-ray and optical data. The other parts of the chain and the packing are very similar to Carlström's proposition. The structure has been examined from a stereochemical point of view and from hydrogen bond energy considerations. The refined structure is found to be free from all short contracts and also to be well stabilised by one intrachain hydrogen bond of the O-H…O type and one interchain hydrogen bond of the N-H…O type. The strength of the hydrogen bond as compared with the best possible ones are discussed. It is believed that the structure obtained is possibly the limit of refinement that can be reached with visually estimated fibre intensity data.     

Flexibility of alpha-helices: results of a statistical analysis of database protein structures

Alpha-helices stand out as common and relatively invariant secondary structural elements of proteins. However, alpha-helices are not rigid bodies and their deformations can be significant in protein function (e.g. coiled coils). To quantify the flexibility of alpha-helices we have performed a structural principal-component analysis of helices of different lengths from a representative set of protein folds in the Protein Data Bank. We find three dominant modes of flexibility: two degenerate bend modes and one twist mode. The data are consistent with independent Gaussian distributions for each mode. The mode eigenvalues, which measure flexibility, follow simple scaling forms as a function of helix length. The dominant bend and twist modes and their harmonics are reproduced by a simple spring model, which incorporates hydrogen-bonding and excluded volume. As an application, we examine the amount of bend and twist in helices making up all coiled-coil proteins in SCOP. Incorporation of alpha-helix flexibility into structure refinement and design is discussed.     

Protein NMR Structures Refined without NOE Data

The refinement of low-quality structures is an important challenge in protein structure prediction. Many studies have been conducted on protein structure refinement; the refinement of structures derived from NMR spectroscopy has been especially intensively studied. In this study, we generated flat-bottom distance potential instead of NOE data because NOE data have ambiguity and uncertainty. The potential was derived from distance information from given structures and prevented structural dislocation during the refinement process. A simulated annealing protocol was used to minimize the potential energy of the structure. The protocol was tested on 134 NMR structures in the Protein Data Bank (PDB) that also have X-ray structures. Among them, 50 structures were used as a training set to find the optimal “width” parameter in the flat-bottom distance potential functions. In the validation set (the other 84 structures), most of the 12 quality assessment scores of the refined structures were significantly improved (total score increased from 1.215 to 2.044). Moreover, the secondary structure similarity of the refined structure was improved over that of the original structure. Finally, we demonstrate that the combination of two energy potentials, statistical torsion angle potential (STAP) and the flat-bottom distance potential, can drive the refinement of NMR structures.     

Bending, hydration and interstitial energies quantitatively account for the hexagonal-lamellar-hexagonal reentrant phase transition in dioleoylphosphatidylethanolamine.

We have accounted for the unusual structural change wherein dioleoylphosphatidylethanolamine undergoes a hexagonal-lamellar-hexagonal transition sequence as the water content is reduced systematically. We describe the role played by the energies of bending, hydration, voids in hexagonal interstices, and van der Waals interaction in this transition sequence. We have used the X-ray diffraction and osmotic stress experiments on the two phases to derive the structural parameters and all of the force constants defining the energetics of the hexagonal and lamellar phases. We have calculated the chemical potentials of lipid and water in both phases and derived the phase diagram of the lipid with no free, adjustable parameters. The calculated temperature/osmotic stress and temperature/composition diagrams quantitatively agree with experiment. The reentrant transition appears to be driven by a delicate balance between the hydration energy in the lamellar phase and bending energy in the hexagonal phase, whereas the energy of voids in hexagonal interstices defines its energy scale and temperature range. Van der Waals attraction between the bilayers in the lamellar phase does not appear to be important in this transition.     

Calculated compression,bending, shearing,torsion, and base-tilting force constants of B- and A-form poly(dG)·poly(dC)

Force constants associated with large-scale motions of the DNA double-helical homopolymer poly(dG)·poly(dC) are projected out of a long-range atom–atom Coulomb force field. Force constants for both B- and A-conformations have been calculated. Using the Coulomb interaction as previously incorporated into our DNA normal mode calculations, we obtain the forces determined by this long-range interaction. These include compression, bending, shearing, torsion, and base tilting. We show, quantitatively, how the shearing and torsional transverse interactions fall off with intercell distance more rapidly than the compressional (longitudinal) interactions. Reasonable values for the elastic moduli of B- and A-DNA are calculated. An important prediction of the present interaction theory is that although the single adjustable strength parameter in the Coulomb force field was chosen so as to reproduce an experimental value for the longitudinal sound velocity, the transverse torsional potential resulting from this fitted force field is in excellent agreement with results reported from supercoiling data and fluorescence measurements.     

A three-dimensional constitutive model for the stress relaxation of articular ligaments

A new nonlinear constitutive model for the three-dimensional stress relaxation of articular ligaments is proposed. The model accounts for finite strains, anisotropy, and strain-dependent stress relaxation behavior exhibited by these ligaments. The model parameters are identified using published uniaxial stress–stretch and stress relaxation data on human medial collateral ligaments (MCLs) subjected to tensile tests in the fiber and transverse to the fiber directions (Quapp and Weiss in J Biomech Eng Trans ASME 120:757–763, 1998; Bonifasi-Lista et al. in J Orthop Res 23(1):67–76, 2005). The constitutive equation is then used to predict the nonlinear elastic and stress relaxation response of ligaments subjected to shear deformations in the fiber direction and transverse to the fiber direction, and an equibiaxial extension. A direct comparison with stress relaxation data collected by subjecting human MCLs to shear deformation in the fiber direction is presented in order to demonstrate the predictive capabilities of the model.     

Gating mechanisms of mechanosensitive channels of large conductance, II: systematic study of conformational transitions

Part II of this study is based on the continuum mechanics-based molecular dynamics-decorated finite element method (MDeFEM) framework established in Part I. In Part II, the gating pathways of Escherichia coli-MscL channels under various basic deformation modes are simulated. Upon equibiaxial tension (which is verified to be the most effective mode for gating), the MDeFEM results agree well with both experiments and all-atom simulations in literature, as well as the analytical continuum models and elastic network models developed in Part I. Different levels of model sophistication and effects of structural motifs are explored in detail, where the importance of mechanical roles of transmembrane helices, cytoplasmic helices, and loops are discussed. The conformation transitions under complex membrane deformations are predicted, including bending, torsion, cooperativity, patch clamp, and indentation. Compared to atom-based molecular dynamics simulations and elastic network models, the MDeFEM framework is unusually well-suited for simulating complex deformations at large length scales. The versatile hierarchical framework can be further applied to simulate the gating transition of other mechanosensitive channels and other biological processes where mechanical perturbation is important.     

An analysis of core deformations in protein superfamilies

An analysis is presented on how structural cores modify their shape across homologous proteins, and whether or not a relationship exists between these structural changes and the vibrational normal modes that proteins experience as a result of the topological constraints imposed by the fold. A set of 35 representative, well-populated protein families is studied. The evolutionary directions of deformation are obtained by using multiple structural alignments to superimpose the structures and extract a conserved core, together with principal components analysis to extract the main deformation modes from the three-dimensional superimposition. In parallel, a low-resolution normal mode analysis technique is employed to study the properties of the mechanical core plasticity of these same families. We show that the evolutionary deformations span a low dimensional space of 4-5 dimensions on average. A statistically significant correspondence exists between these principal deformations and the approximately 20 slowest vibrational modes accessible to a particular topology. We conclude that, to a significant extent, the structural response of a protein topology to sequence changes takes place by means of collective deformations along combinations of a small number of low-frequency modes. The findings have implications in structure prediction by homology modeling.     

Deriving correlated motions in proteins from X-ray structure refinement by using TLS parameters

Dynamic information in proteins may provide valuable information for understanding allosteric regulation of protein complexes or long-range effects of the mutations on enzyme activity. Experimental data such as X-ray B-factors or NMR order parameters provide a convenient estimate of atomic fluctuations (or atomic auto-correlated motions) in proteins. However, it is not as straightforward to obtain atomic cross-correlated motions in proteins — one usually resorts to more sophisticated computational methods such as Molecular Dynamics, normal mode analysis or atomic network models. In this report, we show that atomic cross-correlations can be reliably obtained directly from protein structure using X-ray refinement data. We have derived an analytic form of atomic correlated motions in terms of the original TLS parameters used to refine the B-factors of X-ray structures. The correlated maps computed using this equation are well correlated with those of the method based on a mechanical model (the correlation coefficient is 0.75) for a non-homologous dataset comprising 100 structures. We have developed an approach to compute atomic cross-correlations directly from X-ray protein structure. Being in analytic form, it is fast and provides a feasible way to compute correlated motions in proteins in a high throughput way. In addition, avoiding sophisticated computational operations; it provides a quick, reliable way, especially for non-computational biologists, to obtain dynamics information directly from protein structure relevant to its function.     

A computational study of invariant I5 in a nearly incompressible transversely isotropic model for white matter

The aligned axonal fiber bundles in white matter make it suitable to be modeled as a transversely isotropic material. Recent experimental studies have shown that a minimal form, nearly incompressible transversely isotropic (MITI) material model, is capable of describing mechanical anisotropy of white matter. Here, we used a finite element (FE) computational approach to demonstrate the significance of the fifth invariant (I₅) when modeling the anisotropic behavior of white matter in the large-strain regime. We first implemented and validated the MITI model in an FE simulation framework for large deformations. Next, we applied the model to a plate-hole structural problem to highlight the significance of the invariant I₅ by comparing with the standard fiber reinforcement (SFR) model. We also compared the two models by fitting the experiment data of asymmetric indentation, shear test, and uniaxial stretch of white matter. Our results demonstrated the significance of I₅ in describing shear deformation/anisotropy, and illustrated the potential of the MITI model to characterize transversely isotropic white matter tissues in the large-strain regime.     

DNA bending in the mycobacterial plasmid pAL5000 origin-RepB complex

Plasmid pAL5000 represents a family of relatively newly discovered cryptic plasmids in gram-positive Actinomycetes bacteria. The replication regions of these plasmids comprise a bicistronic operon, repA-repB, encoding two replication proteins. Located upstream is a cis-acting element that functions as the origin of replication. It comprises an ~200-bp segment spanning two binding sites for the replication protein RepB, a low-affinity (L) site and a high-affinity (H) site separated by an ~40-bp spacer sequence. The trajectory of the DNA in the RepB-origin complex has been investigated, and it has been found that the origin undergoes significant bending movements upon RepB binding. RepB binding not only led to local bending effects but also caused a long-range polar curvature which affected the DNA sequences 3′ to the H site. These movements appear to be essential for the in-phase alignment of the L and H sites that leads to the formation of a looped structure. A novel property of RepB unearthed in this study is its ability to form multimers. This property may be an important factor that determines the overall trajectory of the DNA in the RepB-origin complex. The results presented in this study suggest that the origins of replication of pAL5000 and related plasmids are highly flexible and that multimeric, RepB-like initiator proteins bind the origin and induce local deformations and long-range curvatures which are probably necessary for the proper functioning of the origin.