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V24 Hybrid-methods for macromolecular complexes

V24 Hybrid-methods for macromolecular complexes. Structural Bioinformatics (a) Integration of structures of various protein components into one large complex. What to do if density is too small or too large?. Sali et al. Nature 422, 216 (2003). Correlation-based fitting.

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V24 Hybrid-methods for macromolecular complexes

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  1. V24 Hybrid-methods for macromolecular complexes Structural Bioinformatics (a) Integration of structures of various protein components into one large complex. What to do if density is too small or too large? Sali et al. Nature 422, 216 (2003) Bioinformatics III

  2. Correlation-based fitting Correlation-mapping can also be used to position small fragments into large templates. It can also be adapted to accomodate molecular flexibility during fitting. Wriggers, Chacon, Structure 9, 779 (2001) Bioinformatics III

  3. Aim: Accelerated Correlation-Based Fitting with FFT The initial data sets are a low-resolution map (target) and an atomic structure (probe), corresponding to direct space densities em(r) and atomic(r), respectively (blue box). The probe molecule is subject to a rotation matrix R (red box) that can be constructed from the three Euler angles. After lowering the resolution of the atomic structure (by direct space convolution with a Gaussian g) to that of the target map, the rotated probe molecule corresponds to the simulated density calc(r). An optional filter e (e.g., a Laplacian) can be applied to both em (r) and calc(r) before the structure factors are computed (f denotes the FFT and the asterisk denotes the complex conjugate). The definition of a direct space convolution of a density function b(r) with a kernel a(r) is given in the green box. The definition of the direct space correlation C as a function of a translational displacement T is given in the orange box. By virtue of the Fourier correlation theorem, C can be computed for all T from the inverse Fourier transform of the previously calculated structure factors. Wriggers, Chacon, Structure 9, 779 (2001) Bioinformatics III

  4. Matching densities Intuitively, we want to compute the overlap of the two densities after placing the two lattices on top of each other. But what means 'on top of each other' in mathematical terms? Orienting the two lattices can be done with respect to 6 degrees of freedom, 3 for translation along x, y, and z, and 3 for rotation around the angles , , and . Among all these possibilities, one wishes to identify the relative orientation x, y, z, , ,  that minimizes the sum of least squares Here, R,,is a three-dimensional rotation matrix and Tx,y,z is a translation operator that translates molecule B to the position x, y, z. Minimizing the sum of squared errors is equivalent to maximizing the linear cross-correlation of A and B, for a given translation vector (x,y,z) and rotation (, , ). Bioinformatics III

  5. Situs package: Automated low-resolution fitting • The data sets need to be compared at comparable resolution • project atomic structure B on the cubic lattice of the EM data A by tri-linear interpolation, and convolute each lattice points bl,m,n with a Gaussian function g. The complexity of computing this correlation for all translations in direct space is O(N6): O(N3) for every value of x,y,z. The total complexity of this algorithm is therefore O(N6)  number of rotations Chacon, Wriggers J Mol Biol 317, 375 (2002) Bioinformatics III

  6. Laplacian filter for edge enhancement In the absence of hard boundaries, the contour of a low-resolution object is contained in the 3D edge information instead of a 2D surface. A simple and computationally cheap filter for 3D edge enhancement is the Laplacian filter: that approximates the Laplace operator of the second derivative. Applied to the density gradient on a grid, the Laplacian filtered density can be quickly computed by a finite difference scheme: where aijk and 2aijk represent the density and the Laplacian filtered density at grid point (i,j,k). The expression compares the values at grid points +1 and -1 along all three directions to the value of the grid point ijk. Bioinformatics III

  7. Schematic view of a Laplacian filter ai-1jk, aijk, and ai+1jk are the density values at three neighboring grid points in one direction. The grey lines denote the difference between the central point and the values to the left and to the right. These are finite difference approximations of the first derivative left and right of the grid point ijk. The dotted line and dotted arrow illustrate how the two first derivatives are combined to obtain an approximation of the second derivative at grid point ijk by finite difference as ai+1jk + ai-1jk-2 aijk. Bioinformatics III

  8. Effect of Laplacian filter Include „surface“ information in the volume docking procedure. Laplacian filter: Effect of Laplacian filter: Left: cross-section of 15Å simulated density of RecA hexameric structure. Right: same density after application of Laplacian filter. Secondary derivatives are maximal here because signal increases in various directions. Chacon, Wriggers J Mol Biol 317, 375 (2002) Bioinformatics III

  9. Efficient evaluation of correlation by FFT Geometric match between two molecules A and B can be measured by the Laplacian cross-correlation: 6D rigid-body search has complexity N6. Common problem in protein-ligand and in protein-protein docking. Efficient solution (Katchalski-Kazir algorithm): use FFT because FFT has complexity N3logN3  Chacon, Wriggers J Mol Biol 317, 375 (2002) Bioinformatics III

  10. Situs package: success case Fitting of tubulin components to an experimental 20Å resolution map of microtubuli. Without any a priori consideration about the relative orientation of  and  tubulins, the atomic structure of the -tubulin dimer could be reconstructed to within 2Å of the known dimer X-ray structure (labeled by Nogales et al.). Chacon, Wriggers J Mol Biol 317, 375 (2002) Bioinformatics III

  11. Core-weighted fitting + Grid-threading Monte-Carlo Idea: define „core“ region of a structure as the part whose density distribution is unlikely to be altered by the presence of adjacent components. „Surface“ region: is accessible/may interact with other components. Use again Laplacian filter defined by a finite difference approximation to define the boundary of the surface: where aijk and 2aijk represent the density and the Laplacian filtered density at grid point (i,j,k). Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  12. Core-weighted fitting I core index Define core index, which describes the depth of a grid point located within this core: where fijk is the core index of grid point (i,j,k), ac is a cutoff density min[fi1jk, fij1k ,fijk1] represents the minimum core index of the neighboring grid points around grid point (i,j,k). Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  13. Core-weighted fitting I core index The core index is zero for grid points outside the core and increases progressively for grid points located deeper in the core. A grid point outside the core region must neighbor at least one grid point that is also outside the core. A grid point within the core cannot neighbor a grid point outside the core unless it satisfies the condition 2aijk 0 and aijk > ac. • Use this iterative procedure for calculating the core incex: • initialize core index so that all core indices are 1 except the grid points at the boundary • loop over all grid points • repeat (b) until all grid points satisfy equation on p.31. Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  14. Core indices for 2 proteins and their complex Grid points labelled by value of core index. Regions of protein density are colored grey. For both proteins, the core index is 0 outside the domains, 1 at the outer edge and becomes larger inside the proteins. Bold numbers indicate the core indices of proteins A and B that change upon formation of the AB complex. Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  15. Core-weighted correlation function The match in density between two maps is again described by a density correlation function (DC): m and n refer to the two maps being compared, and represent the average and fluctuation of the density fluctuation. Alternatively, one can use the Laplacian correlation (LC) Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  16. Core-weighted fitting I core index - properties • We expect the following features when we consider the match between the map of an individual component and the map of a multicomponent assembly: • If the core region of an individual component matches the core region of the complex, the distribution property of this core region should not change appreciably for the correct fit. • If the surfaces match, the distribution property of this surface region should not change appreciably for the correct fit. • If the surface (low core index) of an individual component matches the core (high core index) of the complex, the distribution property of the surface region should change significantly for the correct fit. • If the core (high core index) of an individual component matches the surface (low core index) of the complex, it cannot be a correct fit. • A correlation function works fine for scenarios 1, 2, and 4 to distinguish the correct fit from wrong fits. Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  17. Core-weighted fitting I core index - algorithm • one needs to minimize the contribution from scenario 3 in the correlation function calculation. Can be achieved by „down-weighting“ such matches. Use where wmnis the core-weighting function for the individual component m to the complex n. a, b, c are suitable parameters. • core-weighted correlation function where represents a core-weighted average of property X: and Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  18. Core-weighted fitting I core index - algorithm If we choose densities for the calculation, we obtain the core-weighted density correlation (CWDC) and if we choose to apply the Laplacian filter, we obtain the core-weighted Laplacian correlation (CWLC) The core-weighted correlation functions are designed to down-weight the regions overlapping with other components, while emphasizing the regions with no overlap. Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  19. Grid-threading Monte-Carlo Shown on the right is a grid-threading Monte Carlo search in 2D. It is a combination of a grid search and a Monte Carlo sampling. The conformational space is divided into a 3×3 grid. From each of the 9 grid points, short MC searches (shown as purple curves) are performed to locate a nearby local maximum. The global maximum is identified from among these local maxima. Only conformations along the 9 Monte Carlo paths are searched. Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  20. Algorithm • For a protein component, divide 6D search space to provide initial conformational states covering the whole space: • nx ny  nz for translational sampling • n n nfor rotational sampling • Perform MC search starting from each grid point over NMC steps. At each ‚move‘ the component is translated along a random vector (xr, yr, zr) and then rotated around x,y,z axes for random angles (r,r,r). • A ‚trialmove‘ is accepted if • and rejected otherwise. • T is a reduced temperature. Wu et al. J Struct Biol 141, 63 (2003) Bioinformatics III

  21. Algorithm • Nonoverlapping local maxima are stored in • sorted, linked list. Step (2) is repeated until • all grid points are searched • Identify global maximum from linked list and • assign to component. • Repeat steps (1) to (4) until all components • have been fitted into the density map. Wu et al. J Struct Biol 141, 63 (2003) Bioinformatics III

  22. Test of Core-weighting method (a) The X-ray structure of TCR variable domain (PDB code: 1A7N) and a 15 Å map generated from the structure using pdblur from Situs. (b) The  -chain (red) at the maximum density correlation position. The -chain is at its X-ray position for reference. Observation: DC identifies wrong global maximum for this 15 Å map. Other methods are more stable at lower resolutions (see table). Wu et al, J Struct Biol 141, 63 (2003) Bioinformatics III

  23. Performance of systematic sampling • The maximum core-weighted density correlations between the map of TCR • -chain and the map of the TCR  complex identified from grid searches of the 6D conformational space (n6 grid points). 15 Å resolution maps. • Black dashed line: correlation value for the X-ray coordinates. • An exponential increase in grid sampling size is required to improve the correlation values. • grid searches are computationally inefficient. Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  24. Performance of grid search and Monte Carlo The core-weighted density correlation function as before during Monte Carlo searches starting from each of the 26 grid points. The Monte Carlo searches were performed with max=15 Å, max=30°, and T=0.01. Each line represents one Monte Carlo search procedure. The ability to converge to the correct fit and the speed of convergence depend significantly on the starting position. Useful strategy: identify best local fit by short MC search. Select global fit among these candidates. This is the basis of the grid-threading MC search. Wu et al. J Struct Biol 141, 63 (2003) Bioinformatics III

  25. Performance of different correlation functions The rms deviations of the best fits from the X-ray structure using different correlation functions. RMSD > 20 Å indicates that search converged to a far maximum. MC with DC alone does not converge to the correct fit. This is due to the fact that map resolutions were 15 Å or worse where DC does not work. Laplacian correlation works until 15 Å, Core-weighted density correlation until 20 Å and core-weighted Laplacian correlation even at 30 Å. Wu et al. J Struct Biol 141, 63 (2003) Bioinformatics III

  26. Success case • Surface representation of the experimental map (at 14 Å resolution) of the icosahedral complex formed from 60 copies of the E2 catalytic domain of the pyruvate dehydrogenase. • (b) The X-ray structure of the same complex (PDB code: 1B5S). Wu, Milne, .., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Bioinformatics III

  27. Success case continued • Comparison of the location of the E2 catalytic domain obtained using a GTMC search (green) with that of the corresponding domain from the X-ray structure (red). The experimental EM map is shown in blue. • The best fit obtained, RMS=2.13 Å; • (b). The worst fit obtained, RMS=6.52 Å. The grid-threading Monte Carlo search was conducted with a 46 grid, Nmc=5000, max=30 Å, max=30°, and T=0.01. • The core-weighted Laplacian correlation function was used. The average RMSD of the C backbone (averaged over all 60 copies) between the X-ray structure and the fitted coordinates is 3.73 Å. Wu et al. J Struct Biol 141, 63 (2003) Bioinformatics III

  28. SOM: surface overlap maximization I preprocessing: all voxels with density < cut-off are set to ‚false‘ all remaining voxels to ‚true‘  ‚template volume‘ ‚target volume‘ (atomic structure in PDB format): placed in a 3D grid with voxel size equal to that of the above density map. For grid voxel i, i  [1,3N] for all atoms in voxel i sum #electrons end store estimate of electron density in voxel i end smoothen model to the resolution of the density map. Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) Bioinformatics III

  29. SOM (II) fast fitting round • Score goodness-of-fit by surface overlap: fraction of surface voxels of the transformed target that are superimposed on template surface. • Determine all combinations of translations and rotations (around origin) that project at least one surface voxel of the target onto the template surface. • Effort? • target surface voxel a and template surface voxel b find set of transformations that superimpose a onto b. Each such transformation can be decomposed into the unique translation of a to b and a rotation about b. Expectation: rotations need to be searched exhaustively. Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) Bioinformatics III

  30. SOM (II) fast fitting round • Interestingly, many rotations about b need not to be explored. • If a really is the counterpart of b, the optimal transformation will superimposed the plane tangent to the target surface in a onto the plane tanget to the template surface in b. • only 1 rotational degree of freedom, around vb, has to be searched In practice, the vector va,is estimated: a and its 26 spatial neighbors are interpreted as vectors. Subtract all neighbors of a that score ‚true‘ in the volume matrix, from a. Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) Bioinformatics III

  31. SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) Bioinformatics III

  32. SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) Bioinformatics III

  33. SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) Bioinformatics III

  34. Mod-EM Task: Comparative (homology) modelling is imprecise at sequence identity levels of 10 %  x 30 %, the so-called „twilight zone“. Idea: use different homology models, combine with experimental EM density. Select model with best combined fitness function. Zs : (statistical potential score – mean  ) / standard deviation  The statistical potential score of a model is the sum of the solvent accessibility terms for all C atoms and distance-dependent terms for all pairs of C and C atoms. The solvent-accessibility term for a C atom depends on its residue type and the number of other C atoms within 10Å; the non-bonded terms depend on the atom and residue types spanning the distance, the distance itself, and the number of residues separating the distance-spanning atoms in the sequence. These potential terms reflect the statistical preferences observed in 760 non-redundant proteins of known structure. The density-fitting Z c-score is the maximized cross-correlation coefficient between the cryoEM density map and the probe (model) density calculated with Mod-EM. The normalization relies on the mean and standard deviation obtained from a population of ca. 7500 alignments constructed in 25 iterations of the Moulder program with the original fitness function that depends only on the statistical potential. When the fit is good, the density-fitting Z-score is positive; it usually ranges from -10 to 10. Five protocols of Moulder-EM were tested, corresponding to different weights ([w1,w2]) of [1,0], [1,1], [1,2], [1,8], and [0,1] for the statistical potential Z-score and the density-fitting Z-score in the fitness function, respectively. Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006) Bioinformatics III

  35. Mod-EM Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006) Bioinformatics III

  36. Mod-EM Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006) Bioinformatics III

  37. Mod-EM Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006) Bioinformatics III

  38. Mod-EM Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006) Bioinformatics III

  39. Mod-EM Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006) Bioinformatics III

  40. Summary Fitting objects into densities has become a standard area of structural bioinformatics. Main technique: compute the correlation of two densities. This can be efficiently done after Fourier transformation of the densities. Laplace filtering of the densities enhances the contrast. SOM: attempts matching of surface details (fast speed due to reduction of search space). Mod-EM: employs structure fitting as tool to support homology modelling in the twilight zone. Bioinformatics III

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