1. 24. Lecture WS 2008/09Bioinformatics III1 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)

2. 24. Lecture WS 2008/09Bioinformatics III2 Correlation-based fitting Wriggers, Chacon, Structure 9, 779 (2001) Correlation-mapping can also be used to position small fragments into large templates. It can also be adapted to accomodate molecular flexibility during fitting.

3. 24. Lecture WS 2008/09Bioinformatics III3 Aim: Accelerated Correlation-Based Fitting with FFT Wriggers, Chacon, Structure 9, 779 (2001) 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.

4. 24. Lecture WS 2008/09Bioinformatics III4 Matching densities 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 T x,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 ( , ,  ). 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?

5. 24. Lecture WS 2008/09Bioinformatics III5 Situs package: Automated low-resolution fitting Chacon, Wriggers J Mol Biol 317, 375 (2002) 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 b l,m,n with a Gaussian function g. The complexity of computing this correlation for all translations in direct space is O(N 6 ): O(N 3 ) for every value of x,y,z. The total complexity of this algorithm is therefore O(N 6 )  number of rotations

6. 24. Lecture WS 2008/09Bioinformatics III6 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 a ijk and  2 a ijk 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.

7. 24. Lecture WS 2008/09Bioinformatics III7 Schematic view of a Laplacian filter a i-1jk, a ijk, and a i+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 a i+1jk + a i-1jk -2 a ijk.

8. 24. Lecture WS 2008/09Bioinformatics III8 Effect of Laplacian filter Chacon, Wriggers J Mol Biol 317, 375 (2002) 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.

9. 24. Lecture WS 2008/09Bioinformatics III9 Efficient evaluation of correlation by FFT Chacon, Wriggers J Mol Biol 317, 375 (2002) Geometric match between two molecules A and B can be measured by the Laplacian cross-correlation: 6D rigid-body search has complexity N 6. Common problem in protein-ligand and in protein-protein docking. Efficient solution (Katchalski-Kazir algorithm): use FFT because FFT has complexity N 3 logN 3 

10. 24. Lecture WS 2008/09Bioinformatics III10 Situs package: success case Chacon, Wriggers J Mol Biol 317, 375 (2002) 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.).

11. 24. Lecture WS 2008/09Bioinformatics III11 Core-weighted fitting + Grid-threading Monte-Carlo Wu, Milne,.., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) 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 a ijk and  2 a ijk represent the density and the Laplacian filtered density at grid point (i,j,k).

12. 24. Lecture WS 2008/09Bioinformatics III12 Core-weighted fitting I core index Wu, Milne,.., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) Define core index, which describes the depth of a grid point located within this core: where f ijk is the core index of grid point (i,j,k), a c is a cutoff density min[f i  1jk, f ij  1k,f ijk  1 ] represents the minimum core index of the neighboring grid points around grid point (i,j,k).

13. 24. Lecture WS 2008/09Bioinformatics III13 Core-weighted fitting I core index Wu, Milne,.., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) 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  2 a ijk  0 and a ijk > a c. Use this iterative procedure for calculating the core incex: (a)initialize core index so that all core indices are 1 except the grid points at the boundary (b)loop over all grid points (c)repeat (b) until all grid points satisfy equation on p.31.

14. 24. Lecture WS 2008/09Bioinformatics III14 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)

15. 24. Lecture WS 2008/09Bioinformatics III15 Core-weighted correlation function Wu, Milne,.., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) 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)

16. 24. Lecture WS 2008/09Bioinformatics III16 Core-weighted fitting I core index - properties Wu, Milne,.., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) We expect the following features when we consider the match between the map of an individual component and the map of a multicomponent assembly: 1.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. 2.If the surfaces match, the distribution property of this surface region should not change appreciably for the correct fit. 3.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. 4.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.

17. 24. Lecture WS 2008/09Bioinformatics III17 Core-weighted fitting I core index - algorithm Wu, Milne,.., Subramaniam, Brooks, J Struct Biol 141, 63 (2003)  one needs to minimize the contribution from scenario 3 in the correlation function calculation. Can be achieved by „down-weighting“ such matches. Use where w mn is 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

18. 24. Lecture WS 2008/09Bioinformatics III18 Core-weighted fitting I core index - algorithm Wu, Milne,.., Subramaniam, Brooks, J Struct Biol 141, 63 (2003) 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.

19. 24. Lecture WS 2008/09Bioinformatics III19 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)

20. 24. Lecture WS 2008/09Bioinformatics III20 Algorithm (1)For a protein component, divide 6D search space to provide initial conformational states covering the whole space: n x  n y  n z for translational sampling n   n   n  for rotational sampling (2)Perform MC search starting from each grid point over N MC steps. At each ‚ move ‘ the component is translated along a random vector (x r, y r, z r ) and then rotated around x,y,z axes for random angles (  r,  r,  r ). A ‚ trial move ‘ is accepted if and rejected otherwise. T is a reduced temperature. Wu et al. J Struct Biol 141, 63 (2003)

21. 24. Lecture WS 2008/09Bioinformatics III21 Algorithm (3)Nonoverlapping local maxima are stored in sorted, linked list. Step (2) is repeated until all grid points are searched (4)Identify global maximum from linked list and assign to component. (5)Repeat steps (1) to (4) until all components have been fitted into the density map. Wu et al. J Struct Biol 141, 63 (2003)

22. 24. Lecture WS 2008/09Bioinformatics III22 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. Wu et al, J Struct Biol 141, 63 (2003) Observation: DC identifies wrong global maximum for this 15 Å map. Other methods are more stable at lower resolutions (see table).

23. 24. Lecture WS 2008/09Bioinformatics III23 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 (n 6 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)

24. 24. Lecture WS 2008/09Bioinformatics III24 Performance of grid search and Monte Carlo The core-weighted density correlation function as before during Monte Carlo searches starting from each of the 2 6 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. Wu et al. J Struct Biol 141, 63 (2003) 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.

25. 24. Lecture WS 2008/09Bioinformatics III25 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)

26. 24. Lecture WS 2008/09Bioinformatics III26 Success case (a)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)

27. 24. Lecture WS 2008/09Bioinformatics III27 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. (a)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 4 6 grid, N mc =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)

28. 24. Lecture WS 2008/09Bioinformatics III28 SOM: surface overlap maximization Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) 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.

29. 24. Lecture WS 2008/09Bioinformatics III29 SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) 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.

30. 24. Lecture WS 2008/09Bioinformatics III30 SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004) 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 v b, has to be searched In practice, the vector v a, 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.

31. 24. Lecture WS 2008/09Bioinformatics III31 SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004)

32. 24. Lecture WS 2008/09Bioinformatics III32 SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004)

33. 24. Lecture WS 2008/09Bioinformatics III33 SOM (II) fast fitting round Ceulemans, Russell J. Mol. Biol. 338, 783 (2004)

34. 24. Lecture WS 2008/09Bioinformatics III34 Mod-EM Topf,..., Sali J. Mol. Biol. 357, 1655 (2006) 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. Z s : (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.

35. 24. Lecture WS 2008/09Bioinformatics III35 Mod-EM Topf,..., Sali J. Mol. Biol. 357, 1655 (2006)

36. 24. Lecture WS 2008/09Bioinformatics III36 Mod-EM Topf,..., Sali J. Mol. Biol. 357, 1655 (2006)

37. 24. Lecture WS 2008/09Bioinformatics III37 Mod-EM Topf,..., Sali J. Mol. Biol. 357, 1655 (2006)

38. 24. Lecture WS 2008/09Bioinformatics III38 Mod-EM Topf,..., Sali J. Mol. Biol. 357, 1655 (2006)

39. 24. Lecture WS 2008/09Bioinformatics III39 Mod-EM Topf,..., Sali J. Mol. Biol. 357, 1655 (2006)

40. 24. Lecture WS 2008/09Bioinformatics III40 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.