Conformational Space

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Conformational Space
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Conformational Space

• Conformation of a molecule specification of the
  relative positions of all atoms in 3D-space,
• Typical parameterizations
• List of coordinates of atom centers
• List of torsional angles (e.g., the f-y-c for a
  protein)
• Conformational space Space of all conformations

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Conformational Space
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Conformational Space
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Relation to Robotics/Graphics
Configuration space
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Need for a Metric

• Simulation and sampling techniques can produce
  millions of conformations
• Which conformations are similar?
• Which ones are close to the folded one?
• Do some conformations form small clusters (e.g.
  key intermediates while folding)?

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Metric in Conformational Space

• A metric over conformational space C is a
  function d c,c ? C ? d(c,c) ? ??0such
  that
• d(c,c) 0 ? c c (non-degeneracy)
• d(c,c) d(c,c) (symmetry)
• d(c,c) d(c,c) ? d(c,c) (triangle
  inequality)

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But not all metrics are good

• Euclidean metric
• d(c,c) Si1,...,n(fi-fi2 yi-yi2)

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Metric in Conformational Space

• A good metric should measure how well the atoms
  in two conformations can be aligned
• Usual metrics cRMSD, dRMSD

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RMSD

• Given two sets of n points in ?3 A a1,,an
  and B b1,,bn
• The RMSD between A and B is RMSD(A,B)
  (1/n)Si1,,nai-bi21/2
• where ai-bi denotes the Euclidean distance
  between ai and bi in ?3
• RMSD(A,B) 0 iff ai bi for all i

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cRMSD

• Molecule M with n atoms a1,,an
• Two conformations c and c of M
• ai(c) is position of ai when M is at c
• cRMSD(c,c) is the minimized RMSD between the two
  sets of atom centers minT(1/n)Si1,,nai(c)
  T(ai(c))21/2
• where the minimization is over all possible
  rigid-body transform T

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cRMSD

• cRMSD verifies triangle inequality
• cRMSD takes linear time to compute
• Often, cRMSD is restricted to a subset of atoms,
  e.g., the Ca atoms on a proteins backbone

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Representation Restricted to Ca Atoms
- The positions of AA residue centers (Ca atoms)
mainly determine the structure of a
protein. - In structural comparison, people
usually work only on the backbone of Ca
atoms, and neglect the other atoms.
Protein 1tph
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• Possible project Design a method for
  efficiently finding nearest neighbors in a
  sampled conformation space of a protein, using
  the cRMSD metric.

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dRMSD

• Molecule M with n atoms a1,,an
• Two conformations c and c of M
• dij(c) n?n symmetrical intra-molecular
  distance matrix in M at c
• dRMD(c, c) is (1/n(n-1))Si1,,n-1Sji1,,n(d
  ij(c) dij(c))21/2
• dij is usually restricted to a subset of atoms,
  e.g., the Ca atoms on a proteins backbone

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Intra-Molecular Distance Matrix
Distances between Ca pairs of a protein with 142
residues. Darker squares represent shorter
distances.
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Intra-Molecular Distance Matrix
Distances between Ca pairs of a protein with 142
residues. Darker squares represent shorter
distances.
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Intra-Molecular Distance Matrix
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dRMSD

• Molecule M with n atoms a1,,an
• Two conformations c and c of M
• dij(c) n?n symmetrical intra-molecular
  distance matrix in M at c
• dRMSD(c, c) (2/n(n-1))Si1,,n-1Sji1,,n(dij
  (c) dij(c))21/2
• dij is usually restricted to a subset of atoms,
  e.g., the Ca atoms on a proteins backbone

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dRMSD

• Molecule M with n atoms a1,,an
• Two conformations c and c of M
• dij(c) n?n symmetrical intra-molecular
  distance matrix in M at c
• dRMSD(c, c) (2/n(n-1))Si1,,n-1Sji1,,n(dij
  (c) dij(c))21/2
• dij is usually restricted to a subset of atoms,
  e.g., the Ca atoms on a proteins backbone
• Advantage No aligning transform
• Drawback Takes quadratic time to compute

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Is dRMSD a metric?

• dRMSD(c, c)
• (2/n(n-1))Si1,,n-1Sji1,,n(dij(c)
  dij(c))21/2
• is a metric in the n(n-1)/2-dimensional space,
  where a conformation c is represented by
  dij(c)
• But, in this representation, the same point
  represents both a conformation and its mirror
  image

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k-Nearest-Neighbors Problem
Given a set S of conformations of a protein and a
query conformation c, find the k conformations in
S most similar to c (w.r.t. cRMSD, dRMSD, other
metric)
Can be done in time O(N(log k L)) where
- N size of S- L time
to compare two conformations
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k-Nearest-Neighbors Problem
The total time needed to compute the k nearest
neighbors of every conformation in S is O(N2(log
k L)) Much too long for large datasets where
N ranges from 10,000s to millions!!!
Can be improved by 1. Reducing L 2. More
efficient algorithm (e.g., kd-tree)
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kd-Tree
In a d-dimensional space, where dgt2, range
searching for a point takes O(dn1-1/d)
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k-Nearest-Neighbors Problem
Idea simplify proteins description
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Assume that each conformation is described by
the coordinates of the n Ca atoms
cRMSD ? O(n) time dRMSD ? O(n2) time
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This representation is highly redundant

• Proximity along the chain entails spatial
  proximity
• Atoms cant bunch up, hence far away atoms along
  the chain are on average spatially distant

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? m-Averaged Approximation

• Cut the backbone into fragments of m Ca atoms
• Replace each fragment by the centroid of the m Ca
  atoms
• ? Simplified cRMSD and dRMSD

3n coordinates
3n/m coordinates
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Evaluation Test SetsLotan and Schwarzer, 2003

• 8 diverse proteins (54 -76 residues)
• Decoy sets of N 10,000 conformations from the
  Park-Levitt set Park et al, 1997

Correlation
m cRMSD dRMSD
3 0.99 0.96-0.98
4 0.98-0.99 0.94-0.97
6 0.92-0.99 0.78-0.93
9 0.81-0.98 0.65-0.96
12 0.54-0.92 0.52-0.69
Higher correlation for random sets (? greater
savings)
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Running Times
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Further Reduction for dRMSD

1. Stack m-averaged distance matrices as vectors of
   a matrix A

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N
A
r
Vector ai of elements of distance matrix of ith
conformation (i 1 to N)
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Further Reduction for dRMSD

• Stack m-averaged distance matrices as vectors of
  a matrix A
• Compute the SVD A UDVT

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SVD Decomposition
N
A(rxN)
U(rxr)
D(rxr)
VT(rxN)

r
Vector aj of elements of distance matrix of jth
conformation (j 1 to N)
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SVD Decomposition
N
s1 s2 sr
A(rxN)
U(rxr)
VT(rxN)
0

r
0
Vector aj of elements of distance matrix of jth
conformation (j 1 to N)
Diagonal matrix
s1 ? s2 ? ... ? sr ? 0 (singular values)
Orthonormal(rotation) matrix
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SVD Decomposition
N
A(rxN)
U(rxr)
D(rxr)
VT(rxN)

r
Vector aj of elements of distance matrix of jth
conformation (j 1 to N)
Diagonal matrix
Matrix withorthonormal rows
Orthonormal(rotation) matrix
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SVD Decomposition
N
A(rxN)
U(rxr)
D(rxr)
VT(rxN)

r
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SVD Decomposition
N
A(rxN)
U(rxr)
D(rxr)
VT(rxN)

r
s1v1 ? s2v2 ...
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SVD Decomposition
N
A(rxN)
U(rxr)
D(rxr)
VT(rxN)

r
p principal components
vpT
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SVD Decomposition
N
A(rxN)
U(rxr)
D(rxr)
VT(rxN)

r
s1 s2 sp
v1T
v2T
p principal components
vpT
0
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Further Reduction for dRMSD

1. Stack m-averaged distance matrices as vectors of
   a matrix A
2. Compute the SVD A UDVT
3. Project onto p principal components

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Correlation
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Complexity of SVD

• SVD of rxN matrix, where N gt r, takes O(r2N) time
• Here r (n/m)2
• So, time complexity is O(n4N)
• Would be too costly without m-averaging

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Evaluation for 1CTF Decoy SetsLotan and
Schwarzer, 2003

• N 100,000, k 100, 4-averaging, 16 PCs
• 70 correct, with furthest NN off by 20
• Brute-force
  84 h
• Brute-force m-averaging 4.8 h
• Brute-force m-averaging PC 41 min
• kD-tree m-averaging PC 19 min
• Speedup greater than x200
• 6k approximate NNs contain all true k NNs
• ? Use m-averaging and PC reduction as fast
  filters