chapter 9 superposition and dynamic programming 1 chapter 9 superposition and dynamic programming...

Chapter 9 Superposition and Dynamic Programming 1

Chapter 9 Superposition and dynamic programming

Most methods for comparing structures use some sorts of superposition and dynamic programming

We have two structures A and B with elements in a given order

A = A1 A2 ... Am and B = B1 B2 ... Bn

An equivalence for (A, B) is a set of pairs of elements

(Ai1, Bj1) (Ai2, Bj2) ... (Air, Bjr)

If i1<i2<...<ir and j1<j2<...<jr then the set of pairs is an alignment

We shall look at methods for finding ”good”

- equivalences

- alignments


Superposition

• Determine how ”good” an equivalence is:• Put the structures on top of each other, and see how good the

equivalent pairs fit each other in the space, when translation and rotation are allowed

• If the geometry of the structures are not changed, this is called rigid body superposition

• Let B be fixed in space• Transform A with a transformation T over B• We can measure how good T is by giving a score to the resulting

superposition by measuring the root mean square deviation (RMSD) between the equivalent pairs of (transformed ) A and B

• Low RMSD values are best, zero indicates exact equality between the (sub)structures


Root Mean Square Deviation

Two different measures are mainly used for scoring the transformation (superposition)

• Coordinate RMSD

• Distance RMSD


Coordinate RMSD

• Let (1,1) ,..., (r,r) be the coordinate set (three values each) of the equivalent elements of the equivalence E

• The problem is then to find a transformation T for A which minimizes the coordinate root mean square deviation, that is

where wi are weights corresponding to each pair (i,i), and often set to one.

• A transformation can be performed by – A translation (three distances)– A rotation (three angles, around the x-, y-, z-axis)– The rotation can also be performed around one axis, the direction of

which has to be calculated for each rotation

r

iiiir

ii

TC Tw

wE

1

2

1

)(1

min)(RMSD


Coordinate RMSD, cont’

• A transformation for the minimum RMSD can be found by 1. Shift the centroids (geometrical centres) of each structure to the origin

of a common coordinate system

2. Find the rotation of A that minimizes the RMSDC

• A rotation around the origin can be described by an orthogonal matrix R3,3 (3D space) with determinant equal to 1

– There exist equations describing the connections between the angles (3) and the values of the matrix (9)

• A matrix is orthogonal if – The scalar product of any two different columns is 0

– The scalar product of any column with itself is 1

• The deteminant of a 3 x 3 matrix R with elements {rij} is calculated as

r11r22r33 + r12r23r31 + r13r21r32

- r11r23r32 - r12r21r33 - r13r22r31



• The orthogonal requirement is for assuring that the distances between the points are not changed (ridig body)

• The determinant requirement is for not ”reflecting” (”mirrorring”) the structure


Example

• A point (1, -1, 1) are to be rotated by an orthogonal matrix with deteminant equal to one

• Show that the matrix is orthogonal• Show that the determinant is one• Find the new coordinates

}

1

1

1

}{

0

{

21

23

43

41

23

43

3

43

2

21

1

ccc



• The formula can therefore be described by a rotation matrix R and a translation vector t, and we search for a pair (R,t) which minimizes the expression (assuming wi=1)

• Since t is found by moving to common centroids, the problem can be formulated as finding the orthogonal matrix, with determinant one, that minimizes the function

where (1,1) ,..., (r,r) are now the coordinates after the structures are moved to common origin

• Algorithms exist for finding such a matrix

r

iii tR

1

2)(

r

iiiR

1

2)(


Distance RMSD

• The distance score method measures how equal corresponding pairwise distances in the two structures are

• It alleviates the need for finding a translation and rotation of one of the structures, and is therefore faster

where ijA is the spatial distance between the elements of A in pairs i and

j of the equivalence

• However, it has a (serious) weakness, it is invariant under reflection

• The two measures are experimentally shown to have a close to linear relation

r

i

r

j

Bij

AijrD E

1 1

21 )()(RMSD


Using RMSD as scoring of structure similarities

• The problem of pairwise structure comparison is often to find equivalences with low RMSD values

• Several quite different equivalences with similar scores might be found, which one is ”correct” is often not easy to determine

• However, always consider the number of elements in the equivalences– for random comparisons the expected RMSD values seems to be

proportional to the square root of the number of equivalent residues

• Different measures can then be used for evaluating how well two structures can be superposed


Using RMSD as scoring of structure similarities


Comparison methods using superpositon and dynamic programming

• Dynamic programming cannot be directly used for structure comparison

• Two different approaches are used for combining superposition and dynamic programming– Alternating superposition and dynamic programming

– ”Contemporary” superposition and dynamic programming


Alternating superposition and dynamic programming

p:=0, Eo := an initial equivalence (sometimes subalignment required)iter

find the transformation Tp for min RMSD, using the r pairs in Ep

use Tp on the whole structure A, giving A*

calculate a scoring matrix Rij as function of the distance between residue i in A* and residue j in B

find the best alignment P with use of R (use dynamic programming)

p:=p+1

Ep:= the r pairs in P with shortest distances in R

until Ep=Ep-1 or p<pmax

P now contains a found equivalence (or alignment)


Alternating superposition and dynamic programming, cont’

The scoring matrix can have more components, in addition to the distance

• Sequence component, how similar are the amino acids?• How similar are the local environments?

Number of pairs in each equivalence• Instead of a fix number for r, r can vary with the number of

”overlapping” pairs

chapter 9 superposition and dynamic programming 1 chapter 9 superposition and dynamic programming...

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