v24 hybrid-methods for macromolecular complexes
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
24. Lecture WS 2008/09
Bioinformatics III 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)
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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.
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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.
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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 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 (, , ).
2,,,,),,,,,( BAzyxR zyx RT
N
l
N
m
N
nnmlzyxnmlzyx baC
1 1 1,,,,,,,,,, RT
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?
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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 bl,m,n with a
Gaussian function g.
N
l
N
m
N
nznymxlnmlzyx bgaC
1 1 1,,,,,,
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
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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:2
2
2
2
2
22
z
f
y
f
x
ff
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:
ijkijkijkkijkijjkijki
ijkijkijkijk
ijkkijijkkij
ijkjkiijkjkiijk
aaaaaaa
aaaa
aaaa
aaaaa
6111111
11
11
112
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.
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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.
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Effect of Laplacian filter
111111,,,,2 6 lmnlmnnlmnlmmnlmnlnmlnml aaaaaaaa
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.
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Efficient evaluation of correlation by FFT
N
l
N
m
N
nznymxlnmlzyx bgaC
1 1 1,,
2,,
2,,
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 N6.
Common problem in protein-ligand and in protein-protein docking.
Efficient solution (Katchalski-Kazir algorithm):
use FFT because FFT has complexity N3logN3
nmlnmlzyx bgFFTaFFTFFTC ,,2*
,,21
,,
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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.).
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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 aijk and 2aijk represent the density and the Laplacian filtered density at grid
point (i,j,k).
ijkijkijkkijkijjkijkiijk aaaaaaaa 61111112
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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 fijk is the core index of grid point (i,j,k),
ac is a cutoff density
min[fi1jk, fij1k ,fijk1] represents the minimum core index of the neighboring grid
points around grid point (i,j,k).
otherwise1min
0minand00
0minand0
1,1,1
1,1,12
1,1,1
ijkkijjki
ijkkijjkiijk
ijkkijjkicijk
ijk
fff
fffa
fffaa
f
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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 2aijk 0 and aijk > ac.
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.
otherwise1
or1oror1oror10 zyxijk
nkknjjniif
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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)
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Core-weighted correlation function
nm
nmnmmn aa
aaaaDC
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,
x y zn
i
n
j
n
kzyx
kjiannn
a ,,1
and 22 aaa
represent the average and fluctuation of the density fluctuation.
Alternatively, one can use the Laplacian correlation (LC)
nm
nmnmmn aa
aaaaLC
22
2222
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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.
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Core-weighted fitting I core index - algorithm
cbff
fw
an
am
am
mn
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 wmn 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:
nwmw
wnwmwnmmn XX
XXXXXCW
kjimn
kjimn
w kjiw
kjiXkjiw
X
,,
,,
,,
,,,,
wX
and 22www XXX
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Core-weighted fitting I core index - algorithm
nwmw
wnwmwnmmn aa
aaaaCWDC
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)
nwmw
wnwmwnmmn aa
aaaaCWLC
22
2222
The core-weighted correlation functions are designed to down-weight the regions
overlapping with other components, while emphasizing the regions with no
overlap.
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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)
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Algorithm
(1) 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
(2) 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 ‚trial move‘ is accepted if
and rejected otherwise.
T is a reduced temperature.
Wu et al. J Struct Biol 141, 63 (2003)
T
CC oldnewexp
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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)
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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).
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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)
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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.
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.
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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)
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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)
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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 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)
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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.
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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.
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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 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.
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SOM (II) fast fitting round
Ceulemans, Russell J. Mol. Biol. 338, 783 (2004)
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SOM (II) fast fitting round
Ceulemans, Russell J. Mol. Biol. 338, 783 (2004)
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SOM (II) fast fitting round
Ceulemans, Russell J. Mol. Biol. 338, 783 (2004)
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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.
csZwZwF21
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.
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Mod-EM
Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006)
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Mod-EM
Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006)
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Mod-EM
Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006)
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Mod-EM
Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006)
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Mod-EM
Topf, ..., Sali J. Mol. Biol. 357, 1655 (2006)
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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.