Crystallographic structure refinement of Chromatium high potential iron protein at two Angstroms resolution.

The structure of Chromatium high potential iron protein (HiPIP) has been refined by semiautomatic Fo-Fc (observed minus calculated structure amplitude Fourier methods to a convential R index, R=sum of the absolute value of Fo-Fc divided by the sum of Fo, of 24.7% for a model in which bond distances and angles are constrained to standard values. Bond length and angle constraints were applied only intermittenly during the computations. At a late stage of the refinement, atomic parameters for only the Fe4S4 cluster plus the 4 associated cystein S-gamma atoms were adjusted by least squares methods and kept fixed during the rest of the refinement. The refined model consists of 625 of the 632 nonhydrogen atoms in the protein plus 75 water molecules. Seven side chain atoms could not be located in the final electron density map. A computer program rather than visual inspection was used wherever possible in the refinement: for locating water molecules, for removing water molecules that too closely approach other atoms, for deleting atoms that lay in regions of low electron density, and for evaluating the progress of refinement. Fo-Fc Fourier refinement is sufficiently economical to be applied routinely in protein crystal structure determinations. The complete HiPIP refinement required approximately 12 hours of CDC 3600 computer time and cost less than $3000 starting from a "trial structure," based upon multipe isomorphoous replacement phases, which gave an R of 43%...     

Modeling protein-induced configurational changes in DNA minicircles

A method is offered for obtaining minimum energy configurations of DNA minicircles constrained by one or more DNA-binding proteins. The minicircles are modeled as elastic rods, while the presence of bound protein is implied by rigidly fixing portions of these chains. The configurations of the geometrically constrained circular rods are sampled stochastically and optimized according to a simple elastic energy model of nicked DNA. The shapes of the minimum energy structures identified after a simulated annealing process are analyzed in terms of relative protein orientation and writhing number. The procedure is applied to minicircles 500 base pairs in length, bound to two evenly spaced DNA-wrapping proteins. The presence of histone octamers is suggested by rigidly fixing the two protein-bound portions of each minicircle as small superhelices similar in dimension to nucleosomal DNA. The folded minimum energy forms of sample chains with different degrees of protein wrapping are noteworthy in themselves in that they offer a new resolution to the well-known minichromosome linking number paradox and point to future minicircle simulations of possible import. © 1997 John Wiley & Sons, Inc.     

Improving ranking of models for protein complexes with side chain modeling and atomic potentials

An atomically detailed potential for docking pairs of proteins is derived using mathematical programming. A refinement algorithm that builds atomically detailed models of the complex and combines coarse grained and atomic scoring is introduced. The refinement step consists of remodeling the interface side chains of the top scoring decoys from rigid docking followed by a short energy minimization. The refined models are then re‐ranked using a combination of coarse grained and atomic potentials. The docking algorithm including the refinement and re‐ranking, is compared favorably to other leading docking packages like ZDOCK, Cluspro, and PATCHDOCK, on the ZLAB 3.0 Benchmark and a test set of 30 novel complexes. A detailed analysis shows that coarse grained potentials perform better than atomic potentials for realistic unbound docking (where the exact structures of the individual bound proteins are unknown), probably because atomic potentials are more sensitive to local errors. Nevertheless, the atomic potential captures a different signal from the residue potential and as a result a combination of the two scores provides a significantly better prediction than each of the approaches alone. Proteins 2013. © 2012 Wiley Periodicals, Inc.     

Real space refinement of acto-myosin structures from sectioned muscle

We have adapted a real space refinement protocol originally developed for high-resolution crystallographic analysis for use in fitting atomic models of actin filaments and myosin subfragment 1 (S1) to 3-D images of thin-sectioned, plastic-embedded whole muscle. The rationale for this effort is to obtain a refinement protocol that will optimize the fit of the model to the density obtained by electron microscopy and correct for poor geometry introduced during the manual fitting of a high-resolution atomic model into a lower resolution 3-D image. The starting atomic model consisted of a rigor acto-S1 model obtained by X-ray crystallography and helical reconstruction of electron micrographs. This model was rebuilt to fit 3-D images of rigor insect flight muscle at a resolution of 7 nm obtained by electron tomography and image averaging. Our highly constrained real space refinement resulted in modest improvements in the agreement of model and reconstruction but reduced the number of conflicting atomic contacts by 70% without loss of fit to the 3-D density. The methodology seems to be well suited to the derivation of stereochemically reasonable atomic models that are consistent with experimentally determined 3-D reconstructions computed from electron micrographs.     

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.     

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.     

Effects of ligand binding upon flexibility of proteins

Binding of a ligand on a protein changes the flexibility of certain parts of the protein, which directly affects its function. These changes are not the same at each point, some parts become more flexible and some others become stiffer. Here, an equation is derived that gives the stiffness map for proteins. The model is based on correlations of fluctuations of pairs of points in proteins, which may be evaluated at different levels of refinement, ranging from all atom molecular dynamics to general elastic network models, including the simplest case of isotropic Gaussian Network Model. The latter is used, as an example, to evaluate the changes of stiffness upon dimerization of ACK1. Proteins 2015; 83:805–808. © 2015 Wiley Periodicals, Inc.     

Computational approaches for automatic structural analysis of large biomolecular complexes

We present computational solutions to two problemsof macromolecular structure interpretation from reconstructedthree-dimensional electron microscopy (3D-EM) maps of largebio-molecular complexes at intermediate resolution (5A-15A). Thetwo problems addressed are: (a) 3D structural alignment (matching)between identified and segmented 3D maps of structure units(e.g. trimeric configuration of proteins), and (b) the secondarystructure identification of a segmented protein 3D map (i.e.locations of a-helices, b -sheets). For problem (a), we presentan efficient algorithm to correlate spatially (and structurally)two 3D maps of structure units. Besides providing a similarityscore between structure units, the algorithm yields an effectivetechnique for resolution refinement of repeated structure units,by 3D alignment and averaging. For problem (b), we present anefficient algorithm to compute eigenvalues and link eigenvectorsof a Gaussian convoluted structure tensor derived from theprotein 3D Map, thereby identifying and locating secondarystructural motifs of proteins. The efficiency and performanceof our approach is demonstrated on several experimentallyreconstructed 3D maps of virus capsid shells from single-particlecryo-EM, as well as computationally simulated protein structuredensity 3D maps generated from protein model entries in theProtein Data Bank.     

Mathematical aspects of protein structure determination with NMR orientational restraints

The field of structural biology is becoming increasingly important as new technological developments facilitate the collection of data on the atomic structures of proteins and nucleic acids. The solid-state NMR method is a relatively new biophysical technique that holds particular promise for determining the structures of peptides and proteins that are located within the cell membrane. This method provides information on the orientation of the peptide planes relative to an external magnetic field. In this article, we discuss some of the mathematical methods and tools that are useful in deriving the atomic structure from these orientational data. We first discuss how the data are viewed as tensors, and how these tensors can be used to construct an initial atomic model, assuming ideal stereochemistry. We then discuss methods for refining the models using global optimization, with stereochemistry constraints treated as penalty functions. These two processes, initial model building followed by refinement, are the two crucial steps between data collection and the final atomic model.     

On contribution of known atomic partial charges of protein backbone in electrostatic potential density maps

Partial charges of atoms in a molecule and electrostatic potential (ESP) density for that molecule are known to bear a strong correlation. In order to generate a set of point‐field force field parameters for molecular dynamics, Kollman and coworkers have extracted atomic partial charges for each of all 20 amino acids using restrained partial charge‐fitting procedures from theoretical ESP density obtained from condensed‐state quantum mechanics. The magnitude of atomic partial charges for neutral peptide backbone they have obtained is similar to that of partial atomic charges for ionized carboxylate side chain atoms. In this study, the effect of these known atomic partial charges on ESP is examined using computer simulations and compared with the experimental ESP density recently obtained for proteins using electron microscopy. It is found that the observed ESP density maps are most consistent with the simulations that include atomic partial charges of protein backbone. Therefore, atomic partial charges are integral part of atomic properties in protein molecules and should be included in model refinement.     

Study of the structural dynamics of the E coli 70S ribosome using real-space refinement

Cryo-EM density maps showing the 70S ribosome of E. coli in two different functional states related by a ratchet-like motion were analyzed using real-space refinement. Comparison of the two resulting atomic models shows that the ribosome changes from a compact structure to a looser one, coupled with the rearrangement of many of the proteins. Furthermore, in contrast to the unchanged inter-subunit bridges formed wholly by RNA, the bridges involving proteins undergo large conformational changes following the ratchet-like motion, suggesting an important role of ribosomal proteins in facilitating the dynamics of translation.     

Effect of anisotropy and anharmonicity on protein crystallographic refinement. An evaluation by molecular dynamics

Molecular dynamics simulations are employed to determine the errors introduced by anharmonicity and anisotropy in the structure and temperature factors obtained for proteins by refinement of X-ray diffraction data. Simulations (25 ps and 300 ps) of metmyoglobin are used to generate time-averaged diffraction data at 1.5 A resolution. The crystallographic restrained-parameter least-squares refinement program PROLSQ is used to refine models against these simulated data. The resulting atomic positions and isotropic temperature factors are compared with the average structure and fluctuations calculated directly from the simulations. It is found that significant errors in the atomic positions and fluctuations are introduced by the refinement, and that the errors increase with the magnitude of the atomic fluctuations. Of particular interest is the fact that the refinement generally underestimates the atomic motions. Moreover, while the actual fluctuations go up to a mean-square value of about 5 A2, the X-ray results never go above approximately 2 A2. This systematic deviation in the motional parameters appears to be due to the use of a single-site isotropic model for the atomic fluctuations. Many atoms have multiple peaks in their probability distribution functions. For some atoms, the multiple peaks are seen in difference electron density maps and it is possible to include these in the refinement as disordered residues. However, for most atoms the refinement fits only one peak and neglects the rest, leading to the observed errors in position and temperature factor. The use of strict stereochemical restraints is inconsistent with the average dynamical structure; nevertheless, refinement with tight restraints results in structures that are comparable to those obtained with loose restraints and better than those obtained with no restraints. The results support the use of tight stereochemical restraints, but indicate that restraints on the variation of temperature factors are too restrictive.     

Improving the physical realism and structural accuracy of protein models by a two-step atomic-level energy minimization

Most protein structural prediction algorithms assemble structures as reduced models that represent amino acids by a reduced number of atoms to speed up the conformational search. Building accurate full-atom models from these reduced models is a necessary step toward a detailed function analysis. However, it is difficult to ensure that the atomic models retain the desired global topology while maintaining a sound local atomic geometry because the reduced models often have unphysical local distortions. To address this issue, we developed a new program, called ModRefiner, to construct and refine protein structures from Cα traces based on a two-step, atomic-level energy minimization. The main-chain structures are first constructed from initial Cα traces and the side-chain rotamers are then refined together with the backbone atoms with the use of a composite physics- and knowledge-based force field. We tested the method by performing an atomic structure refinement of 261 proteins with the initial models constructed from both ab initio and template-based structure assemblies. Compared with other state-of-art programs, ModRefiner shows improvements in both global and local structures, which have more accurate side-chain positions, better hydrogen-bonding networks, and fewer atomic overlaps. ModRefiner is freely available at http://zhanglab.ccmb.med.umich.edu/ModRefiner.     

Structure prediction of domain insertion proteins from structures of individual domains

Multidomain proteins continue to be a major challenge in protein structure prediction. Here we present a Monte Carlo (MC) algorithm, implemented within Rosetta, to predict the structure of proteins in which one domain is inserted into another. Three MC moves combine rigid-body and loop movements to search the constrained conformation by structure disruption and subsequent repair of chain breaks. Local searches find that the algorithm samples and recovers near-native structures consistently. Further global searches produced top-ranked structures within 5 A in 31 of 50 cases in low-resolution mode, and refinement of top-ranked low-resolution structures produced models within 2 A in 21 of 50 cases. Rigid-body orientations were often correctly recovered despite errors in linker conformation. The algorithm is broadly applicable to de novo structure prediction of both naturally occurring and engineered domain insertion proteins.     

REMO: A new protocol to refine full atomic protein models from C‐alpha traces by optimizing hydrogen‐bonding networks

Protein structure prediction approaches usually perform modeling simulations based on reduced representation of protein structures. For biological utilizations, it is an important step to construct full atomic models from the reduced structure decoys. Most of the current full atomic model reconstruction procedures have defects which either could not completely remove the steric clashes among backbone atoms or generate final atomic models with worse topology similarity relative to the native structures than the reduced models. In this work, we develop a new protocol, called REMO, to generate full atomic protein models by optimizing the hydrogen‐bonding network with basic fragments matched from a newly constructed backbone isomer library of solved protein structures. The algorithm is benchmarked on 230 nonhomologous proteins with reduced structure decoys generated by I‐TASSER simulations. The results show that REMO has a significant ability to remove steric clashes, and meanwhile retains good topology of the reduced model. The hydrogen‐bonding network of the final models is dramatically improved during the procedure. The REMO algorithm has been exploited in the recent CASP8 experiment which demonstrated significant improvements of the I‐TASSER models in both atomic‐level structural refinement and hydrogen‐bonding network construction. Proteins 2009. © 2009 Wiley‐Liss, Inc.     

Coarse-grained normal mode analysis in structural biology

The realization that experimentally observed functional motions of proteins can be predicted by coarse-grained normal mode analysis has renewed interest in applications to structural biology. Notable applications include the prediction of biologically relevant motions of proteins and supramolecular structures driven by their structure-encoded collective dynamics; the refinement of low-resolution structures, including those determined by cryo-electron microscopy; and the identification of conserved dynamic patterns and mechanically key regions within protein families. Additionally, hybrid methods that couple atomic simulations with deformations derived from coarse-grained normal mode analysis are able to sample collective motions beyond the range of conventional molecular dynamics simulations. Such applications have provided great insight into the underlying principles linking protein structures to their dynamics and their dynamics to their functions.     

Refinement of protein termini in template-based modeling using conformational space annealing

The rapid increase in the number of experimentally determined protein structures in recent years enables us to obtain more reliable protein tertiary structure models than ever by template-based modeling. However, refinement of template-based models beyond the limit available from the best templates is still needed for understanding protein function in atomic detail. In this work, we develop a new method for protein terminus modeling that can be applied to refinement of models with unreliable terminus structures. The energy function for terminus modeling consists of both physics-based and knowledge-based potential terms with carefully optimized relative weights. Effective sampling of both the framework and terminus is performed using the conformational space annealing technique. This method has been tested on a set of termini derived from a nonredundant structure database and two sets of termini from the CASP8 targets. The performance of the terminus modeling method is significantly improved over our previous method that does not employ terminus refinement. It is also comparable or superior to the best server methods tested in CASP8. The success of the current approach suggests that similar strategy may be applied to other types of refinement problems such as loop modeling or secondary structure rearrangement.     

Real-space refinement with DireX: from global fitting to side-chain improvements

Single-particle cryo-electron microscopy (cryo-EM) has become an important tool to determine the structure of large biomolecules and assemblies thereof. However, the achievable resolution varies considerably over a wide range of about 3.5-20 ?. The interpretation of these intermediate- to low-resolution density maps in terms of atomic models is a big challenge and an area of active research. Here, we present our real-space structure refinement program DireX, which was developed primarily for cryo-EM-derived density maps. The basic principle and its main features are described. DireX employs Deformable Elastic Network (DEN) restraints to reduce overfitting by decreasing the effective number of degrees of freedom used in the refinement. Missing or reduced density due to flexible parts of the protein can lead to artifacts in the structure refinement, which is addressed through the concept of restrained grouped occupancy refinement. Furthermore, we describe the performance of DireX in the 2010 Cryo-EM Modeling Challenge, where we chose six density maps of four different proteins provided by the Modeling Challenge exemplifying typical refinement results at a large resolution range from 3 to 23 ?.     

Myoglobin and CO: structure, energetics, and disorder

 The physiological significance of steric inhibition of CO binding to heme proteins is a fundamentally quantitative issue, since it has meaning only relative to the enhancement of O₂ binding by the protein. Previously, difficulties in reconciling structural and energetic data have hindered a quantitative assessment of the energetic cost of distorting the bound CO. Recent progress on both fronts suggests that the energetic cost of CO distortion in myoglobin is quite small. However, distortion of the surrounding protein to accommodate the heme-CO complex may contribute significantly to the free energy of CO binding. Polarized IR measurements on single crystals of MbCO not only yield precise structural information on the CO geometry, but also address the limitations of conventional structural refinement by characterizing conformational disorder in the crystalline state. Received, accepted: 23 May 1997     

Can local energy minimization refine the PDB structures of different resolution universally?

Local energy minimization was statistically tested as the refinement strategy for PDB structure pairs of different resolution. The 13 pairs of structures with the only difference being the resolution were extracted from PDB and represented structures of 11 identical proteins obtained with different x-ray diffraction techniques. The rmsd distribution was calculated for these pairs before and after local energy minimization of each structure. MMFF94 was used for energy calculations and the quasi-Newtonian method was used for local energy minimization. By comparison of these two rmsd distributions, the local energy minimization was proved to statistically increase the structural differences in pairs, so it cannot be used for refinement purposes. To explore the prospects of complex refinement strategies based on energy minimization, randomized structures were obtained by moving the initial PDB structures as far as the minimized structures had been moved in the multidimensional space of atomic coordinates. For these randomized structures the rmsd distribution was calculated and compared with the one for minimized structures. The significant differences in their mean values proved the energy surface of the protein to have only few minima near the conformations of different resolution obtained by x-ray analysis for PDB. Some other results we obtained exploring the energy surface near these conformations are also presented. These results are expected to be useful for the development of new protein refinement strategies based on energy minimization.