Modeling Tricks and Fitting Techniques for Multiresolution Structures

Abstract

The spectacular developments of recent years toward solving atomic structures of ever-increasing complexity underscore the importance of relating 3D structures of multicomponent cellular machines to molecular mechanisms. To understand the workings of these machines [1Alberts B. The cell as a collection of protein machines preparing the next generation of molecular biologists.Cell. 1998; 92: 291-294Abstract Full Text Full Text PDF PubMed Scopus (934) Google Scholar], we face many challenges, one of which is to describe the structure and functional mechanisms at the atomic level. The atomic structure of a molecular machine in a particular state of processing can be inferred by building it from its components, i.e., by combining multiresolution data from a variety of biophysical sources. This hybrid modeling approach holds much promise, provided that the docking procedure is reproducible and incorporates the constraints of molecular interactions and architecture. In the following, we present an overview of state-of-the-art computational fitting techniques and, wherever possible, we put them to a stringent test to discuss their advantages and limitations. The assessment of the scope and validity of individual methods will hopefully serve as a “consumer guide” that allows the reader to identify the most suitable docking criterion given a specific fitting problem. As a computational research area, multiresolution modeling is still in its infancy, but it will become increasingly attractive in the near future, when more and more low-resolution structures of large complexes and atomic structures of their components become available. This development is prompted in part by the advent of structural genomics, which promises to bring the sequences of most single-domain proteins within homology-modeling distance of a known structure [2Šali A. Kuriyan J. 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By combining data from high-throughput cryo-EM and structural genomics, multiresolution modeling will produce approximate but reasonably accurate atomic models of macromolecular assemblies. These models of large assemblies will be created routinely, often years before a comparable crystallographic structure can be solved by the more laborious, traditional X-ray or electron crystallography methods. In the past decade, significant progress was made by combining cryo-EM data with high-resolution structures determined by NMR spectroscopy or X-ray crystallography. The first EM maps into which atomic structures were fitted included actomyosin filaments [9Schröder R.R. Spudich J.A. et al.Three-dimensional atomic model of F-actin decorated with Dictyostelium myosin S1.Nature. 1993; 364: 171-174Crossref PubMed Scopus (265) Google Scholar, 10Rayment I. Milligan R.A. et al.Structure of the actin-myosin complex and its implications for muscle contraction.Science. 1993; 261: 58-65Crossref PubMed Scopus (1415) Google Scholar] and icosahedral viruses [11Olson N.H. Rossmann M.G. et al.Structure of a human rhinovirus complexed with its receptor molecule.Proc. Natl. Acad. Sci. USA. 1993; 90: 507-511Crossref PubMed Scopus (268) Google Scholar, 12Smith T.J. Baker T.S. et al.Structure of human rhinovirus complexed with Fab fragments from a neutralizing antibody.J. Virol. 1993; 67: 1148-1158Crossref PubMed Google Scholar, 13Stewart P.L. Fuller S.D. Burnett R.M. Difference imaging of adenovirus bridging the resolution gap between X-ray crystallography and electron microscopy.EMBO J. 1993; 12: 2589-2599Crossref PubMed Scopus (294) Google Scholar]. The most common of these hybrid strategies involves the visual docking of atomic structures into envelopes derived from low-resolution data [14Agrawal R.K. Heagle A.B. Penczek P. Grassucci R.A. Frank J. EF-G-dependent GTP hydrolysis induces translocation accompanied by large conformational changes in the 70S ribosome.Nature Struct. Biol. 1999; 6: 643-647Crossref PubMed Scopus (272) Google Scholar, 15Moores C.A. Keep N.H. Kendrick-Jones J. Structure of the utrophin actin-binding domain bound to F-actin reveals binding by an induced fit mechanism.J. Mol. Biol. 2000; 297: 465-480Crossref PubMed Scopus (53) Google Scholar]. The successful construction of such hybrid models for virus capsids and cytoskeletal motor-filament complexes constitutes a clear indication of the value of the visual approach [5Baker T.S. Johnson J.E. Low resolution meets high towards a resolution continuum from cells to atoms.Curr. Opin. Struct. Biol. 1996; 6: 585-594Crossref PubMed Scopus (74) Google Scholar, 7Baumeister W. Steven A.C. Macromolecular electron microscopy in the era of structural genomics.Trends Biochem. Sci. 2000; 25: 624-631Abstract Full Text Full Text PDF PubMed Scopus (127) Google Scholar, 16Stowell M.H.B. Miyazawa A. Unwin N. Macromolecular structure determination by electron microscopy new advances and recent results.Curr. Opinion Struct. Biol. 1998; 8: 595-600Crossref PubMed Scopus (25) Google Scholar]. More recently, several groups have recognized the need for computational tools to perform the fitting in a reliable and reproducible manner (Figure 1). A low-resolution image reconstruction of a macromolecular assembly from electron micrographs can be viewed as a convolution of an atomic structure of the assembly with a smoothing kernel (point-spread function). In general, the point-spread function depends on the electron optics of the microscope [4Frank J. Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Academic Press, San Diego1996Crossref Google Scholar]. Computationally, it is straightforward to lower the resolution of an atomic structure, e.g., by convolution with a Gaussian [17Belnap D.M. Kumar A. Folk J.T. Smith T.J. Baker T.S. Low-resolution density maps from atomic models how stepping “back” can be a step “forward.”.J. Struct. Biol. 1999; 125: 166-175Crossref PubMed Scopus (23) Google Scholar] that approximates the point-spread function. The reverse problem, then, is the deconvolution of the low-resolution density map utilizing the atomic structure of components. This multiresolution deconvolution, i.e., docking, poses a challenging computational problem that is the subject of this review. A variety of computational docking algorithms have recently become available (Figure 1). Some algorithms have been adopted by individual laboratories for their own use, while others are openly disseminated within the EM community. It is not possible in this review to do justice to all existing algorithms, since many laboratories devise a mix of individual docking techniques for particular practical applications [18Rossmann M.G. Fitting atomic models into electron-microscopy maps.Acta Crystallogr. D. 2000; 56: 1341-1349Crossref PubMed Scopus (164) Google Scholar]. Also, we omit specialized methods that impose a particular symmetry on the refined data, in particular, icosahedral symmetry in the case of virus capsids [19Baker T.S. Olson N.H. Fuller S.D. Adding the third dimension to virus life cycles three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs.Microbiol. Mol. Biol. Rev. 1999; 63: 862-922Crossref PubMed Google Scholar]. Instead, we focus here on describing the central principles underlying the most commonly used fitting methods, and we refer to the original articles for the detailed implementation. Following a discussion of differences and similarities between Fourier space and direct space fitting criteria, we sketch the advantages and limitations of data reduction techniques that reduce the complexity of direct space data for interactive and flexible fitting applications. We describe the use of crosscorrelation and convolution methods and outline how their viable resolution range can be extended by modifying the underlying correlation criterion. We conclude by outlining some outstanding problems that put forth tasks for future research such as the systematic evaluation of fitting methods and their use as database query tools. It comes as no surprise that some of the earliest fitting tools employed by low-resolution modelers were based on well-established methods for X-ray crystallographic refinement [19Baker T.S. Olson N.H. Fuller S.D. 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The problem of rigid-body fitting can be formulated as the minimization of the discrepancy between observed and calculated structure factors in Fourier space (Figure 1c) with respect to the rotational and translational parameters of the model and the scale factor λ. If the linear discrepancy (n = 1 in Figure 1c) and amplitude differences (R1, n = 1 in Figure 1c) are considered, this corresponds to the well-known crystallographic R factor [23Drenth J. Principles of Protein X-Ray Crystallography. Second Edition. Springer Verlag, New York1999Crossref Google Scholar]. Similarly, linear vector discrepancies (R2, n = 1) correspond to the vector R factor [24Brünger A.T. X-PLOR, Version 3.1. Yale University Press, New Haven, CT1992Google Scholar]. Alternatively, it is also possible to minimize a quadratic misfit (n = 2) of amplitudes or vectors [25Huber R. Schneider M. A group refinement procedure in protein crystallography using Fourier transforms.J. Appl. 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Fourier-based methods (R1) are the only recourse if diffraction amplitudes are the sole source of information, as in X-ray fiber diffraction [20Holmes K.C. Popp D. Gebhard W. Kabsch W. Atomic model of the actin filament.Nature. 1990; 347: 44-49Crossref PubMed Scopus (1286) Google Scholar]. In EM, the phases of the structure factors are known and, in principle, one can construct a 3D model of the density in direct space. Nevertheless, Fourier-based methods are valuable in EM if one wants to avoid the numerical complications of the transformation to direct space. For example, in helical-image analysis of EM micrographs, a standard reconstruction method relies on layer-line (helical-diffraction) data in reciprocal space [28DeRosier D.J. Klug A. Reconstruction of three dimensional structures from electron micrographs.Nature. 1968; 217: 130-134Crossref PubMed Scopus (911) Google Scholar, 29Stewart M. 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Biophysical Electron Microscopy. Academic Press, London1990Google Scholar]. If the fitting is performed directly in Fourier space, problems with overlapping layer lines and the laborious accounting of the entire diffraction pattern are avoided. The R value criterion, R1, is not very suitable for the docking of EM densities, since the additional phase information available from EM image reconstructions is ignored. To maximize the amount of information actually used for the docking, the vector discrepancies R2 should be minimized. Even if phase information is included [32Mendelson R. Morris E.P. The structure of acto-myosin subfragment 1 complex Results of searches using data from electron microscopy and x-ray crystallography.Proc. Natl. Acad. Sci. USA. 1997; 94: 8533-8538Crossref PubMed Scopus (79) Google Scholar], deviations between computed structure factors in Fourier space do not correspond to localized positional or orientational changes in direct space. In particular, it is difficult to refine the internal degrees of freedom of atomic structures against this data [32Mendelson R. Morris E.P. The structure of acto-myosin subfragment 1 complex Results of searches using data from electron microscopy and x-ray crystallography.Proc. Natl. Acad. Sci. USA. 1997; 94: 8533-8538Crossref PubMed Scopus (79) Google Scholar]. Overfitting is less of a problem in rigid-body docking, where one has only six degrees of freedom. However, problems with the “delocalized” Fourier space fitting are exemplified by a refinement of the actin monomer structure against low-resolution X-ray fiber diffraction data [21Lorenz M. Popp D. Holmes K.C. Refinement of the F-actin model against X-ray fiber diffraction data by the use of a directed mutation algorithm.J. Mol. Biol. 1993; 234: 826-836Crossref PubMed Scopus (441) Google Scholar] that resulted in a reduced stereochemical quality of the fitted model compared to the crystal structure [33Kabsch W. Mannherz H.G. 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It is straightforward, in direct space, to combine EM-based refinement with geometric constraints from biochemical experiments and with molecular force fields that govern the physical interactions of the atoms [35Mueller F. Brimacombe R. et al.The 3D arrangement of the 23 S and 5 S rRNA in the Escherichia coli 50S ribosomal subunit based on a cryo-electron microscopic reconstruction at 7.5 Å resolution.J. Mol. Biol. 2000; 298: 35-59Crossref PubMed Scopus (89) Google Scholar]. One can count the number of independent pieces of information available for the fitting of a model in direct space by dividing the volume of the structure by the volume of a resolution element, i.e., a cube whose length corresponds to the spatial resolution. For medium-resolution (∼10–30 Å) EM maps of single molecules, this number is surprisingly small, ranging from the lower single digits in the cases of actin [36Galkin V.E. Orlova A. Lukoyanova N. Wriggers W. Egelman E.H. 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Clearly, it would be beneficial for direct space fitting and modeling if we could represent this small number of shape-defining fiducials in a reliable and reproducible fashion. Clustering techniques have been used since the 1950s for digital signal compression in engineering applications such as digital speech and image processing [40Makhoul J. Roucos S. Gish H. Vector quantization in speech coding.Proc. IEEE. 1985; 73: 1551-1588Crossref Scopus (545) Google Scholar]. Such “lossy” data compression methods seek to represent a complex signal by a reduced number of vectors that identify the signal cluster centers. Certain clustering methods are rooted also in neural computing, where the goal is to find “faithful” neighborhood-preserving maps from an input space of sensory signals to a discrete network of neurons in the cortex [41van Hulle M.M. Faithful Representations and Topographic Maps. John Wiley and Sons, New York2000Google Scholar]. From both fields, algorithms emerged that essentially perform density estimation, yielding discrete estimates of data manifolds. Electron microscopists may be familiar with such methods in the context of classification of images during the data acquisition and management phase to reduce signal loss and to achieve improvements in resolution [4Frank J. Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Academic Press, San Diego1996Crossref Google Scholar]. Recently, it was proposed to utilize a clustering technique, termed vector quantization, for a reduced representation of 3D data that allows EM data to be matched with atomic structures [42Wriggers W. Milligan R.A. Schulten K. McCammon J.A. Self-organizing neural networks bridge the biomolecular resolution gap.J. Mol. Biol. 1998; 284: 1247-1254Crossref PubMed Scopus (66) Google Scholar, 43Wriggers W. Milligan R.A. McCammon J.A. Situs a package for docking crystal structures into low-resolution maps from electron microscopy.J. Struct. Biol. 1999; 125: 185-195Crossref PubMed Scopus (439) Google Scholar, 44Wriggers W. Birmanns S. Using Situs for flexible and rigid-body fitting of multi-resolution single molecule data.J. Struct. Biol. 2001; 133: 193-202Crossref PubMed Scopus (192) Google Scholar]. In vector quantization, a single-molecule data set is represented by k so-called codebook vectors (Figure 1b): wcalcj (corresponding to high-resolution data) or wemi (corresponding to low-resolution data; i,j = 1,…,k). An index map I : j → i defines the k pairs of corresponding vectors. There are two applications of this technology; in rigid-body fitting, I is not known a priori, and an exhaustive search of the k! possible permutations (I(1),…,I(k)) is carried out. The fits are then ranked by the residual rms deviation V (Figure 1c) after least-squares fitting of the vectors wcalcj to the wemIj. In flexible fitting, rigid-body docking alone gives poor alignment of the crystal structure and the low-resolution data set. For example, the structure of the ribosomal protein EF-G exhibits a striking “induced fit” conformational change on the 70S ribosome involving three protruding domains [14Agrawal R.K. Heagle A.B. Penczek P. Grassucci R.A. Frank J. EF-G-dependent GTP hydrolysis induces translocation accompanied by large conformational changes in the 70S ribosome.Nature Struct. Biol. 1999; 6: 643-647Crossref PubMed Scopus (272) Google Scholar, 39Wriggers W. Agrawal R.K. Drew D.L. McCammon J.A. Frank J. Domain motions of EF-G bound to the 70S ribosome insights from a hand-shaking between multi-resolution structures.Biophys. J. 2000; 79: 1670-1678Abstract Full Text Full Text PDF PubMed Scopus (68) Google Scholar]. In such situations I is known, and the deviating atomic structure can be brought into register with the EM density, effectively by forcing V (Figure 1c) to vanish. This is done in a molecular dynamics refinement of the atomic structure where a quantity equivalent to V forms a penalty that is imposed by distance constraints (see [44Wriggers W. Birmanns S. Using Situs for flexible and rigid-body fitting of multi-resolution single molecule data.J. Struct. Biol. 2001; 133: 193-202Crossref PubMed Scopus (192) Google Scholar] for details). The major advantage of vector quantization, apart from its obvious value for flexible fitting, is computational speed. Both quantization and docking by the reduced representation can be carried out within seconds of compute time. In contrast, the exhaustive search of all rigid-body degrees of freedom for the full data sets can take many hours (Figure 1c). Despite the fact that multiresolution structural data is docked indirectly (by means of the vector quantization), the accuracy that can be achieved in flexible and rigid-body docking with simulated (noise-free) data is one order of magnitude above the nominal resolution of the EM map, or better [44Wriggers W. Birmanns S. Using Situs for flexible and rigid-body fitting of multi-resolution single molecule data.J. Struct. Biol. 2001; 133: 193-202Crossref PubMed Scopus (192) Google Scholar]. The identification of spatial features by vector quantization is sensitive to noise that might originate from experimental limitations. One of the open questions in flexible docking is how to maintain the stereochemical quality of a fitted structure [45Rice W.J. Young H.S. Martin D.W. Sachs J.R. Stokes D.L. Structure of Na+, K+-ATPase at 11-Å resolution comparison with Ca2+-ATPase in E1 and E2 states.Biophys. J. 2001; 80: 2187-2197Abstract Full Text Full Text PDF PubMed Scopus (84) Google Scholar], since any overfitting to noise-induced vector displacements would compromise the quality of the atomic model. In a recent approach, intervector distances along the connected polypetide chain are constrained (Figure 2). The resulting vector skeletons (distance-constrained vectors) eliminate the longitudinal degrees of freedom that are deemed inessential for the flexible docking while permitting lateral flexibility. Thereby, the skeleton-based fitting approach provides additional robustness against the effects of noise and experimental uncertainty [44Wriggers W. Birmanns S. Using Situs for flexible and rigid-body fitting of multi-resolution single molecule data.J. Struct. Biol. 2001; 133: 193-202Crossref PubMed Scopus (192) Google Scholar]. Figures 2a–2e illustrate one limitation of vector quantization and a possible remedy: all density should be accounted for by the atomic structure. If there are extraneous densities, e.g., due to sequence insertions or neighboring structures as in the case of RNA polymerase, they should be identified and subtracted by discrepancy mapping [46Volkmann N. Hanein D. Quantitive fitting of atomic models into observed densities derived by electron microscopy.J. Struct. Biol. 1999; 125: 176-184Crossref PubMed Scopus (161) Google Scholar] prior to any docking. Discrepancy mapping can involve an iterative strategy in which the structure is first docked in a course manner, and then the refinement is done at a later stage once the extraneous densities have converged [S.A. Darst, N. Opalka, P.C., A. Polyakov, C. Richter, G. Zhang, and W.W., submitted]. The quantitative docking methods discussed so far involve symmetrical systems or systems where the subunit to be docked can be isolated. The methods presented in this section are capable—to varying degrees—of docking components into larger densities present in biomolecular assemblies. Naturally, these exhaustive search methods are computationally demanding and, at present, are limited to rigid-body docking. For the first time, we have evaluated the docking performance of three state-of-the-art criteria on simulated low-resolution data generated from known atomic structures. Perhaps the oldest fitting criterion is the (globally normalized) crosscorrelation coefficient C1 (Figure 1c). The method has been adopted by a large number of authors [13Stewart P.L. Fuller S.D. Burnett R.M. Difference imaging of adenovirus bridging the resolution gap between X-ray crystallography and electron microscopy.EMBO J. 1993; 12: 2589-2599Crossref PubMed Scopus (294) Google Scholar, 46Volkmann N. Hanein D. Quantitive fitting of atomic models into observed densities derived by electron microscopy.J. Struct. Biol. 1999; 125: 176-184Crossref PubMed Scopus (161) Google Scholar, 47Zhang X. Freemont P.S. et al.Structure of the AAA ATPase p97.Mol. 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