13. lecture ws 2005/06bioinformatics iii1 v13 – protein docking, fft, electron tomography fast...

13. Lecture WS 2005/06

Bioinformatics III 1

V13 – protein docking, FFT, electron tomography

Fast Fourier Transform

Electron Tomography



Prediction of Assemblies from Pairwise Docking

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

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.



Pairwise docking: Katchalski-Kazir algorithm; FTDOCK

Gabb et al. J. Mol. Biol. (1997)

Discretize proteins A and B on a grid.

Every node is assigned a value

Use FFT to compute correlation efficiently.

Output: solutions with best surface

complementarity.



Our docking strategy: FTDOCK + CHARMM

Flöck & Helms, PROTEINS (2002)



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)



(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.



(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.



(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 (Cayley‘s 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.



(2) Combinatorial assembly module:algorithm


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.



(2) Combinatorial assembly module:algorithm


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.



Flowchart

www.cs.tau.ac.il/~inbaryuv/combdoc/



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.



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.



Performance for known complexes




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.



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.



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.




after: Numerical Recipes

Discrete Fourier Transform of a function from a finite number of its sampled points.

Suppose that we have N consecutive sampled values

1,...,2,1,0,, Nkktthh kkk

so that the sampling interval is . Let us assume that N is even.

The discrete Fourier transform of the N points hk is

1

0

/2N

k

Niknkn ehH

The formula for the discrete inverse Fourier transform, which recovers the set

of hk‘s exactly from the Hn‘s is:

1

0

/21 N

n

Niknnk eH

Nh




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 numberNieW /2

Then we can write

1

0

N

kk

nkn hWH

The vector of hk‘s 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 Hn’s.

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.




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.



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.

ok

kek

N

jj

NikjkN

jj

Nikj

N

jj

NjikN

jj

Njik

N

jj

Nijkk

FWF

feWfe

fefe

feF

12

012

2/212

02

2/2

12

012

/12212

02

/22

1

0

/2

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




The wonderful property of the Danielson-Lanczos-Lemma is that it can be used

recursively.

Having reduced the problem of computing Fk to that of computing Fke and Fk

o ,

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 e‘s and o‘s, there is a one-point transform that is just

one of the input numbers fn

nfF noeeeoeeoeo

k somefor ...




The next trick is to figure out which value of n corresponds to which pattern of e‘s

and o‘s in noeeeoeeoeo

k fF ...

Answer: reverse the pattern of e‘s and o‘s, 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.




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.



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)



Identification of macromolecular complexes in cryoelectron tomograms of phantom cells

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

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.



Detection and idenfication strategy


The correlation of two functions is defined as

Correlation theorem for the transform pairs:

dhtghgCorr ,

fHfGhgCorr *,



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:

(i) Scan reconstructed volume by using small boxes of the size of the target

structure (real space method)

(ii) Paste template into a box of the size of the reconstructed volume (Fourier

space method). This method is much more efficient.



Correlation with Nonlinear Weighting

R

nn

R

nn

R

nnn

rRrxRx

rxRrxCC

1

22

1

22

1


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

rx and

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 >> R), nonlinear filtering has to be applied.

This filtering is done in the form of nonlinear weighting.



Raw data


Central x-y slices through the 3D reconstructions of ice-embedded phantom

cells filled with

(a) 20S proteasomes,

(b) thermosomes,

(c) and a mixture of both particles.

At low magnification, the macromolecules appear as small dots.



Correlation coefficients


(a) 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.

(b) 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.



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 1:1

ratio of both proteins. The algorithm

identifies 52% as thermosomes and

48% as 20S proteasomes.



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



Reconstruction of endoplasmatic reticulum

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

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



Reconstruction of actin 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).

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).




Science fiction


Reconstruct proteome of real biological cells.

Required steps:

(1) obtain EM maps of isolated (e.g. 6000 yeast) proteins

(2) enhance resolution of tomography

(3) speed up detection algorithm




Summary

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

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

13. lecture ws 2005/06bioinformatics iii1 v13 – protein docking, fft, electron tomography fast...

Documents

input graph

input n protein structures

input structural units

v13 protein docking

protein subunits

pair of proteins

docking strategy

candidate complex