Download Conformational Space

Conformational Space
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
Conformational Space
qj
qi
qN-1
q2
qN
q1
Conformational Space
q0
q1
qn
q4
q3
Relation to Robotics/Graphics
q0
q1
q2
qn
t(t)
q4
Configuration space
q3
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)?
Metric in Conformational Space
 A metric over conformational space C is a
function:
d: c,c’  C  d(c,c’)  +{0}
such that:
 d(c,c’) = 0  c = c’
 d(c,c’) = d(c’,c)
 d(c,c’) + d(c’,c”)  d(c,c”)
(non-degeneracy)
(symmetry)
(triangle inequality)
But not all metrics are “good”
 Euclidean metric:
d(c,c’) =
Si=1,...,n(|fi-fi’|2+ |yi-yi’|2)
Metric in Conformational Space
 A “good” metric should measure how
well the atoms in two conformations can
be aligned
 Usual metrics: cRMSD, dRMSD
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)Si=1,…,n||ai-bi||2]1/2
where ||ai-bi|| denotes the Euclidean
distance between ai and bi in 3
 RMSD(A,B) = 0 iff ai = bi for all i
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)Si=1,…,n||ai(c) – T(ai(c’))||2]1/2
where the minimization is over all
possible rigid-body transform T
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
protein’s backbone
Representation Restricted
to Ca Atoms
Protein 1tph
- The positions of AA residue centers (Cα atoms)
mainly determine the structure of a protein.
- In structural comparison, people usually work
only on the backbone of Cα atoms, and neglect
the other atoms.
Possible project: Design a method for
efficiently finding nearest neighbors in
a sampled conformation space of a
protein, using the cRMSD metric.
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))Si=1,…,n-1Sj=i+1,…,n(dij(c) – dij(c’))2]1/2
 {dij} is usually restricted to a subset of atoms,
e.g., the Ca atoms on a protein’s backbone
Intra-Molecular Distance Matrix
Distances between Ca pairs of a protein with 142 residues.
Darker squares represent shorter distances.
Intra-Molecular Distance Matrix
45
40
85
1
Distances between Ca pairs of a protein with 142 residues.
Darker squares represent shorter distances.
Intra-Molecular Distance Matrix
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))Si=1,…,n-1Sj=i+1,…,n(dij(c) – dij(c’))2]1/2
 {dij} is usually restricted to a subset of atoms,
e.g., the Ca atoms on a protein’s backbone
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))Si=1,…,n-1Sj=i+1,…,n(dij(c) – dij(c’))2]1/2
 {dij} is usually restricted to a subset of atoms,
e.g., the Ca atoms on a protein’s backbone
 Advantage: No aligning transform
 Drawback: Takes quadratic time to compute
Is dRMSD a metric?
 dRMSD(c, c’) =
[(2/n(n-1))Si=1,…,n-1Sj=i+1,…,n(dij(c) – dij(c’))2]1/2
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
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
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 10,000’s to millions!!!
Can be improved by:
1. Reducing L
2. More efficient algorithm (e.g., kd-tree)
kd-Tree
In a d-dimensional space, where d>2, range searching for a point takes O(dn1-1/d)
k-Nearest-Neighbors Problem
Idea: simplify protein’s description
Assume that each conformation is described by
the coordinates of the n Ca atoms
cRMSD  O(n) time
dRMSD  O(n2) time
This representation is highly
redundant
 Proximity along the chain entails spatial
proximity
d  3l
 Atoms can’t bunch up, hence far away
atoms along the chain are on average
spatially distant
ci
cj
 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
Evaluation: Test Sets
[Lotan 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)
Running Times
Further Reduction for dRMSD
1) Stack m-averaged distance matrices
as vectors of a matrix A
N
r
1  n  n 

r        1
2 m  m

A
Vector ai of elements of
distance matrix of
ith conformation (i = 1 to N)
1
dRMSDm (c,c
ai -aj
i j )=
r
2
Further Reduction for dRMSD
1) Stack m-averaged distance matrices
as vectors of a matrix A
2) Compute the SVD A = UDVT
SVD Decomposition
N
r
A
(rxN)
=
Vector aj of elements of
distance matrix of
jth conformation (j = 1 to N)
U
(rxr)
D
(rxr)
Diagonal matrix
Orthonormal
(rotation) matrix
VT
(rxN)
SVD Decomposition
N
r
A
(rxN)
=
Vector aj of elements of
distance matrix of
jth conformation (j = 1 to N)
U
(rxr)
s1
s2 0
0
sr
VT
(rxN)
Diagonal matrix
s1  s2  ...  sr  0
(singular values)
Orthonormal
(rotation) matrix
SVD Decomposition
N
r
A
(rxN)
=
Vector aj of elements of
distance matrix of
jth conformation (j = 1 to N)
U
(rxr)
VT
D
(rxr)
(rxN)
vjT
vkT
Diagonal matrix
Orthonormal
(rotation) matrix
Matrix with
orthonormal rows
vi and vj are orthogonal
unit Nx1 vectors
SVD Decomposition
N
A
(rxN)
r
=
U
(rxr)
D
(rxr)
y
X
VT
(rxN)
Representation of
A in space (X,Y)
1
dRMSDm (c,c
)=
ai -aj
i j
r
Y
does not depend on the
coordinate system!
r-dimensional space r  1  n    n   1 
2  m    m 

x
2
SVD Decomposition
N
r
s1
s2
A
(rxN)
=
U
(rxr)
v1 T
v2 T
s3
sr
D
(rxr)
VT
(rxN)
||s1v1||  ||s2v2|| ...
SVD Decomposition
N
r
s1
s2
A
(rxN)
=
U
(rxr)
v1 T
v2 T
s3
sr
vpT
D
(rxr)
VT
(rxN)
p principal
components
SVD Decomposition
N
r
s1
A
(rxN)
=
U
(rxr)
v1 T
v2 T
s2
sp
vpT
0
D
(rxr)
VT
(rxN)
p principal
components
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
Correlation
between dRMSD and dRMSD4PC
dRMSD is
PC
4
reduced to
summing up 12 to
20 terms
(instead of ~ 80 to
200, since the
proteins have 54 to
76 amino acids)
Complexity of SVD
 SVD of rxN matrix, where N > r, takes
O(r2N) time
 Here r ~ (n/m)2
 So, time complexity is O(n4N)
 Would be too costly without m-averaging
Evaluation for 1CTF Decoy Sets
[Lotan and Schwarzer, 2003]

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
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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