V13

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V13 protein docking, FFT, electron tomography

Fast Fourier Transform Electron Tomography
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Prediction of Assemblies from Pairwise Docking
CombDock first fully automated approach for
predicting hetero multimolecular assembly only
based on structural models of its protein
subunits. Problem appears more difficult than
the pairwise docking problem it is
NP-hard. Idea exploit additional geometric
constraints embraced in the combinatorial
problem. Input a set of protein structural
models. Unlike a 3D puzzle, where two connected
pieces in the puzzle solution match perfectly,
we would like to tolerate some extent of
penetration, due to the flexible nature of the
proteins.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Pairwise docking Katchalski-Kazir algorithm
FTDOCK
Discretize proteins A and B on a grid. Every node
is assigned a value
Use FFT to compute correlation efficiently. Outpu
t solutions with best surface complementarity.
Gabb et al. J. Mol. Biol. (1997)
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Our docking strategy FTDOCK CHARMM
Flöck Helms, PROTEINS (2002)
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Protein-protein docking of cyt c552 and COX
Exercise on model system Complex of yeast
Cytochrome c Peroxidase with iso-1-Cytochrome c
X-ray structure (Kraut et al. 1992) Heme
positions of crystal complex and 19 best docked
and energy-minimized complexes. Crystal complex
has lowest energy. Docked complex with
second-best energy has RMSD of only 2.0 Å.
Flöck, Helms, PROTEINS (2002)
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(1) All pairs docking module
Module gets as its input N protein structures ?
predict pairwise interactions. Perform pairwise
docking for each of the N (N - 1) /2 pairs of
proteins. Keep K best solutions for each pair of
proteins. Since pairwise-docking is a difficult
problem, K should be set reasonably high. Here,
K was varied from dozens to hundreds.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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(2) Combinatorial assembly module
Input N subunits and N (N - 1) /2 sets of K
scored transformations. These are the candidate
interactions. Reduction to a spanning tree Build
weighted graph representing the input each
structural unit vertex each transformation
edge connecting the corresponding vertices edge
weight score of the transformation ? Since
the input contains K transformations for each
pair of subunits, we have a complete graph with K
parallel edges between each pair of vertices.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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(2) Combinatorial assembly module
For two subunits, each candidate complex is
represented by an edge and the two vertices. In
the case of N structural units a candidate
complex is represented by a spanning tree a
subgraph of the input graph that connects all
vertices and has no circles. Each spanning tree
of the input graph represents a complex of all
the input structural units. The problem of
finding complexes is equivalent to finding
spanning trees. The number of spanning trees in
a complete graph with no parallel edges is NN-2
(Cayleys formula). Since the input graph has K
parallel edges between each pair of vertices, the
number of spanning trees is NN-2 KN-1 . ?
Exhaustive searches are infeasible.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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(2) Combinatorial assembly modulealgorithm
Algorithm uses 2 basic principles (1)
hierarchical construction of the spanning
tree (2) greedy selection of subtrees Different
trees share common trees ? generate trees with n
vertices by connecting two trees of smaller size
that were previously generated with an input
edge. Thus, the common parts of different trees
are generated only once. When connecting
subtrees, validate only the inter-subtree
constraints. ? need to check whether there are
severe penetrations in the complex only between
pairs of subunits, where each is represented by a
different subtree.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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(2) Combinatorial assembly modulealgorithm
Stage 1 algorithm constructs trees of size 1.
Each tree contains a single vertex that
represents a subunit. Stage i the tree
complexes that consist of exactly i vertices
(subunits) are generated by connecting two trees
generated at a lower stage with an input edge
transformation. Tree complexes that fulfil the
penetration constraint are kept for the next
stages. Because it is impractical to search all
valid spanning trees, the algorithm performs a
greedy selection of subtrees. For each subset of
vertices, the algorithm keeps only the D
best-scoring valid trees that connect them. The
tree score is the sum of its edge weights.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Flowchart

www.cs.tau.ac.il/inbaryuv/combdoc/
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Example
The construction of the third-best scoring
solution of arp2/3 complex (RMSD 1.2 Å ). The
combinatorial assembly algorithm is hierarchical
at the first stage, each complex consists of a
single subunit. At the ith stage it constructs
complexes that consist of i subunits by
connecting complexes of smaller size using one of
the input candidate transformations. The arp2/3
complex consists of seven subunits shown at the
top. In this Figure we present only the complexes
of the different stages that are relevant to the
construction of the third-best scoring solution
(at the bottom of the Figure). Along with each
complex is its corresponding subgraph, where the
vertices represent the subunits and the edges
represent the pairwise interactions that were
used to construct the complex. In each graph, the
red edge represents the transformation of the
current stage, while blue edges represent
transformations of previous stages.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Final scoring
The geometric score evaluates the shape
complementarity between the subunits check
distances between surface points on adjacent
subunits. Close surface points increase
score, Penetrating surface points decrease
score. Physico-chemical component of the final
score counts the surface points that belong to
non-polar atoms gives an estimate of the
hydrophobic effect. Clustering of solutions (1)
compute contact maps between subunits array of N
( N 1 ) bins. If two subunits are in contact
within the complex, set the corresponding bit to
1, and to 0 otherwise. (2) superimpose complexes
that have the same contact map and compute
RMSD between C? atoms. If this distance is less
than a threshold, consider complexes as members
of a cluster. For each cluster, keep only the
complex with the highest score.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Performance for known complexes

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Method works with different contact topologies.
The near-native solutions for two complexes with
different contact topologies. Left CombDock
solution, Right solution superposed on the
crystal structure (gray thiner lines). (a) the
sixth-best scoring solution for the IkBa/NF-kB
complex of an unbound input, RMSD 1.9 Å. The p65
subunit was extracted from a homodimer structure
(PDB 1BFT). The structure used for the IkBa
subunit was generated by MODELLER6 v2 using bcl-3
(PDB 1k1b) as the template structure (b) the
second-best scoring solution of
VHL/elonginC/elonginB complex (PDB 1vcb), with an
RMSD of 0.5 Å . Each complex consists of three
subunits but, while in the IkBa/NF-kB complex all
the subunits are in contact with each other, in
the VHL/elonginC/ elonginB complex the elonginC
is the core of the complex (in yellow) and VHL
(in blue) and elonginB (in red) are not in
contact. The algorithm was able to predict a
near-native solution for both complexes
regardless of their contact topologies.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Examples of large complexes
Left CombDock solution, Right solution
superposed on the crystal structure (gray thinner
lines). The solutions are (a) the thirdbest
scoring assembly of the seven subunits of the
arp2/3 complex, RMSD 1.2 Å (b) the bestranked
complex of the ten subunits of RNA polymerase II,
RMSD 1.4 Å.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Discussion of CombDock
For the five different targets, CombDock
predicted at least one near-native solution and
ranked it in the top ten for both bound and
unbound cases. Problem in evaluating
performance full sets of unbound structures
are not available for complexes with a higher
number of subunits. It is unlikely that this
version of the algorithm (using rigid protein
conformations) will be able to correctly assemble
such complexes if the input subunits involve
significant conformational changes. ? future
version should include hinge-bending movements of
protein subunits.

Inbar et al., J. Mol. Biol. 349, 435 (2005)
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Fast Fourier Transform
Discrete Fourier Transform of a function from a
finite number of its sampled points. Suppose that
we have N consecutive sampled values

so that the sampling interval is ?. Let us assume
that N is even. The discrete Fourier transform
of the N points hk is
The formula for the discrete inverse Fourier
transform, which recovers the set of hks exactly
from the Hns is
after Numerical Recipes
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Fast Fourier Transform
How much computation is involved in computing the
discrete Fourier transform of N points? Until
the mid-1960s, the standard answer was
this Define W as the complex number

Then we can write
The vector of hks is multiplied by a matrix
whose (n,k)th element is the constant W to the
power n ? k. The matrix multiplication produces a
vector result whose components are the
Hns. This matrix multiplication requires N2
complex multiplications, plus a smaller number
of operations to generate the required powers of
W. So, the discrete Fourier transform appears to
be an O(N2) process.
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Fast Fourier Transform
However, the discrete Fourier transform (in 1
dimension) can be computed in O(N log2 N)
operations by an algorithm called the Fast
Fourier Transform. With N 106, the difference
between O(N2) and O(N log2 N) is 30 CPU seconds
against 2 CPU weeks! The FFT algorithm became
generally known in the mid-1960s from the work of
J.W. Cooley and J.W. Tukey. In fact, efficient
methods to compute discrete Fourier transforms
had been independently discovered many times,
starting with Gauss in 1805.

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FFT by Danielson and Lanczos (1942)
D. and L. showed that a discrete Fourier
transform of length N can be rewritten as the sum
of two discrete Fourier transforms, each of
length N/2. One of the two is formed from the
even-numbered points of the original N, the other
from the odd-numbered points.

W is the same constant as before. Fke k-th
component of the Fourier transform of length N/2
formed from the even components of the original
fj s Fko k-th component of the Fourier
transform of length N/2 formed from the odd
components of the original fj s
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FFT by Danielson and Lanczos (1942)
The wonderful property of the Danielson-Lanczos-Le
mma is that it can be used recursively. Having
reduced the problem of computing Fk to that of
computing Fke and Fko , we can do the same
reduction of Fke to the problem of computing the
transform of its N/4 even-numbered input data and
N/4 odd-numbered data. We can continue applying
the DL-Lemma until we have subdivided the data
all the way down to transforms of length 1. What
is the Fourier transform of length one? It is
just the identity operation that copies its one
input number into its one output slot. For every
pattern of log2N es and os, there is a
one-point transform that is just one of the input
numbers fn

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FFT by Danielson and Lanczos (1942)
The next trick is to figure out which value of n
corresponds to which pattern of es and os in

Answer reverse the pattern of es and os, then
let e 0 and o 1, and you will have, in
binary the value of n. Idea this works because
the successive subdividisions of the data into
even and odd are tests of successive low-order
(least significant) bits of n. This idea of bit
reversal can be exploited in a very clever way
which, along with the DL-Lemma, makes FFT
practical Suppose we take the original vector
of data fj and rearrange it into bit-reversed
order, so that the individual numbers are in the
order not of j, but of the number obtained by
bit-reversing j.
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FFT by Danielson and Lanczos (1942)
Reordering an array (here of length 8) by bit
reversal, (a) between two arrays, versus (b) in
place. The points as given are the one-point
transforms. We combine adjacent pairs to get
two-point transforms, then combine adjacent pairs
of pairs to get 4-point transforms, and so on
until the first and second halves of the whole
data set are combined into the final
transform. Each combination takes of order N
operations, and there are log2N
combinations. This, then, is the structure of an
FFT algorithm.

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New challenge Electron Tomography
Method overview a) The electron beam of an EM
microscope is scattered by the central object and
the scattered electrons are detected on the black
plate. By tilting the object in small steps,
collect electrons scattered at different
angles. b) reconstruction in the
computer. Back-projection (Fourier method) of the
scatter-information at different angles. The
superposition generates a three-dimensional
tomogrom.
Sali et al. Nature 422, 216 (2003)
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Identification of macromolecular complexes in
cryoelectron tomograms of phantom cells

Idea construct model system with well-defined
properties. Prepare phantom cells (ca. 400 nm
diameter) with well-defined contents liposomes
filled with thermosomes and 20S
proteasomes. Thermosome 933 kD, 16 nm diameter,
15 nm height, subunits assemble into toroidal
structure with 8-fold symmetry. 20S proteasome
721 kD, 11.5 nm diameter, 15 nm height, subunits
assemble into toroidal structure with 7-fold
symmetry. Collect Cryo-EM pictures of phantom
cells for a tilt series from -70º until 70º with
1.5º increments. Aim identify and map the 2
types of proteins in the phantom cell. This is a
problem of matching a template, ideally derived
from a high-resolution structure, to an image
feature, the target structure.
Frangakis et al., PNAS 99, 14153 (2002)
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Detection and idenfication strategy

The correlation of two functions is defined
as Correlation theorem for the transform pairs
Frangakis et al., PNAS 99, 14153 (2002)
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Search strategy

• Adjust pixel size of templates to the pixel size
  of the EM 3D reconstruction.
• The gray value of a voxel (volume element)
  containing ca. 30 atoms is obtained by summation
  of the atomic number of all atoms positioned in
  it.
• Possible search strategies
• Scan reconstructed volume by using small boxes of
  the size of the target structure (real space
  method)
• Paste template into a box of the size of the
  reconstructed volume (Fourier space method). This
  method is much more efficient.

Frangakis et al., PNAS 99, 14153 (2002)
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Correlation with Nonlinear Weighting
The correlation coefficient CC is a measure of
similarity of two features e.g. a signal x
(image) and a template r both with the same size
R. Expressed in one dimension
are the mean values of the subimage and the
template. The denominators are the variances
To derive the local-normalized cross correlation
function or, equivalently, the correlation
coefficients in a defined region R around each
voxel k, which belongs to a large volume N
(whereby N gtgt R), nonlinear filtering has to be
applied. This filtering is done in the form of
nonlinear weighting.
Frangakis et al., PNAS 99, 14153 (2002)
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Raw data

• Central x-y slices through the 3D reconstructions
  of ice-embedded phantom cells filled with
• 20S proteasomes,
• thermosomes,
• and a mixture of both particles.
• At low magnification, the macromolecules appear
  as small dots.

Frangakis et al., PNAS 99, 14153 (2002)
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Correlation coefficients

• Histogram of the correlation coefficients of the
  particles found in the proteasome-containing
  phantom cell scanned with the "correct"
  proteasome and the "false" thermosome template.
  Of the 104 detected particles, 100 were
  identified correctly. The most probable
  correlation coefficient is 0.21 for the
  proteasome template and 0.12 for the thermosome
  template.
• Histogram of the correlation coefficients of the
  particles found in the thermosome-containing
  phantom cell. Of the 88 detected particles,
  77 were identified correctly. The most probable
  correlation value is 0.21 for the thermosome
  template and 0.16 for the proteasome template.
• Detection in (a) works well, but is somehow
  problematic in (b) because (correct) thermosome
  and proteasome are not well separated.

Frangakis et al., PNAS 99, 14153 (2002)
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Reconstruction of phantom cell
Volume-rendered representation of a reconstructed
ice-embedded phantom cell containing a mixture of
thermosomes and 20S proteasomes. After applying
the template-matching algorithm, the protein
species were identified according to the maximal
correlation coefficient. The molecules are
represented by their averages thermosomes are
shown in blue, the 20S proteasomes in yellow.

The phantom cell contained a 11 ratio of both
proteins. The algorithm identifies 52 as
thermosomes and 48 as 20S proteasomes.
Frangakis et al., PNAS 99, 14153 (2002)
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Electron tomography

• Method has very high computational cost.
• Observation biological cells are not packed so
  densely as expected,
• allowing the identification of single proteins
  and protein complexes
• Problem for real cells molecular crowding.
• Potential difficulties to identify spots.
• - need to increase spatial resolution of tomograms

Frangakis et al., PNAS 99, 14153 (2002)
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Reconstruction of endoplasmatic reticulum
Picture rights shows rough endoplasmatic
reticulum (membrane network in eukaryotic cells
that generates proteins and new membranes) coated
with ribosomes. The picture is taken from an
intact cell. Membranes are shown in blue, the
ribosomes in green-yellow.

http//science.orf.at/science/news/61666 Dept. of
Structural Biology, Martinsried
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Reconstruction of actin filaments
Actin filaments are structural proteins they
form filaments which span the entire cell. They
stabilize the cellular shape, are required for
motion, and are involved in important cellular
transport processes (molecular motors like
kinesin walk along these filaments).

Shown is the cytoskeleton of Dictyostelium.
Apparently, filaments cross and bridge each other
at different angles, and are connected to the
cell membrane (right picture). Actin filaments
are shown in brown. The cell segment left has a
size of 815 x 870 x 97 nm3. Middle single actin
filaments connected at different angles. Right
actin filaments (brown) binding to the cell
membrane (blue).
http//science.orf.at/science/news/61666 Dept. of
Structural Biology, Martinsried
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Science fiction

• Reconstruct proteome of real biological cells.
• Required steps
• obtain EM maps of isolated (e.g. 6000 yeast)
  proteins
• enhance resolution of tomography
• speed up detection algorithm

http//science.orf.at/science/news/61666 Dept. of
Structural Biology, Martinsried
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Summary

• The structural characterization of large
  multi-protein complexes and the resolution of
  cellular architectures will likely be achieved by
  a combination of methods in structural biology
• X-ray crystallography and NMR for
  high-resolution structures of single proteins and
  pieces of protein complexes
• (Cryo) Electron Microscopy to determine
  medium-resolution structures of entire protein
  complexes
• Stained EM for still pictures at
  medium-resolution of cellular organells
• (Cryo) Electron Tomography to for 3-dimensional
  reconstructions of biological cells and for
  identification of the individual components.
• Mapping and idenfication steps require heavy
  computation.
• Employ protein-protein docking as a help to
  identify complexes?
• - Sali Baumeister
• - Russell Böttcher
• - Wriggers J. Frank and others

Botstein Risch, Nature Gen. 33, 228 (2003)