1

Protein Structure Similarity

2

Computation of Best Matches

Two “simultaneous” subproblems • Find maximal correspondence set C• Find alignment transform T

Chicken-and-egg issue: Each subproblem is relatively simple:

– If we knew C, we could compute T– If we knew T, we could get C by proximity

But the combination is hard !!!

3

Computation of Best Matches

Two “simultaneous” subproblems • Find maximal correspondence set C• Find alignment transform T

Chicken-and-egg issue: Each subproblem is relatively simple:

– If we knew C, we could compute T– If we knew T, we could get C by proximity

But the combination is hard !!!

Only requires computing 6 parameters

4

Find Alignment Transform

Two sets of points A= {a1,…,an} and B = {b1,…,bn}

Correspondence pairs (ai, bi) Find T = arg minT RMSD(A,T(B)) O(n) closed-form solution

[Arun, Huang, and Blostein, 87] [Horn, 87][Horn, Hilden, and Negahdaripour, 88]

5

O(n) SVD-Based Algorithm T combines translation t and rotation R,

such that T(bi) = t + R(bi)

b = (Σi=1,...,nbi)/n [mean of the bi’s] Place the origin of coordinate system at b

minT RMSD(A,T(B)) simplifies to (up to some constants):

t and R can be computed separately

t = a [mean of the ai’s]

n n

2

i i it,Ri=1 i=1

min a-t -2 a,R(b)

[Arun, Huang, and Blostein, 87]

6

O(n) SVD-Based Algorithm A3n = [a1-a, ..., an-a] B3n = [b1-b, ..., bn-

b]

Compute SVD decomposition of 3×3 correlation matrix BAT: BAT = UDVT where D is a diagonal matrices with decreasing non-negative entries (singular values) along the diagonal

If det(U)det(V) = 1 then S = I, else S = diag(1,1,-1)

R = USVT[Arun, Huang, and Blostein, 87]

7

O(n) SVD-Based Algorithm A3n = [a1-a, ..., an-a] B3n = [b1-b, ..., bn-

b]

Compute SVD decomposition of 3×3 correlation matrix BAT: BAT = UDVT where D is a diagonal matrices with decreasing non-negative entries (singular values) along the diagonal

If det(U)det(V) = 1 then S = I, else S = diag(1,1,-1)

R = USVT[Arun, Huang, and Blostein, 87]

8

[Arun, Huang, and Blostein, 87] rotation matrix

[Horn, 87] quaternion

9

Trial-and-Error Approach to Protein Structure

Comparison

Guess small correspondence set

Compute T

Update correspondence set (correspondence from proximity)

Apply T

10

Trial-and-Error Approach to Protein Structure

Comparison

1. Set CS to a seed correspondence set (small set sufficient to generate an alignment transform)

2. Compute the alignment transform T for CS and apply T to the second protein B

3. Update CS to include all pairs of features that are close apart

4. If CS has changed, then return to Step 2 else return (CS,T)

11

Trial-and-Error Approach to Protein Structure

Comparison- result = nil- Iterate N times:

1. Set CS to a seed correspondence set (small set sufficient to generate an alignment transform)

2. Compute the alignment transform T for CS and apply T to the second protein B

3. Update CS to include all pairs of features that are close apart

4. If CS has changed, then return to Step 2 else result result {(CS,T)}

- Return result

12

How to get seed correspondences?

13

Seed Generation from Fragment

1. From distance matricesE.g., DALI [Holm and Sander, 1996]

14

Using Distance Matrices (DALI)

Distances are invariant to rigid-body transformations DALI [Holm and Sander, 1996] looks for similar

hexapeptides by searching for similar 7x7 C-C distance matrices

1

40

85

45

15

Seed Generation from Fragment

1. From distance matricesE.g., DALI [Holm and Sander, 1996]

2. From secondary structure elements (SSE’s)E.g., LOCK [Singh and Brutlag, 1996]

3. From voting scheme (using geometric hashing)E.g., 3dSEARCH [Singh and Brutlag, 2000]

16

LOCK

A.P. Singh and D.L. Brutlag. Hierarchical Protein Structure Superposition Using Both Secondary and Atomic Representations. Proc. ISMB, pp. 284-293, 1997.

LOCK2:J. Shapiro and D.L. Brutlag. FoldMiner: Structural Motif Discovery Using an Improved Superposition Algorithm. Protein Science, 13:278-294, 2004.

http://motif.stanford.edu/lock2/

17

LOCK Two levels of features: SSEs and C

atoms Stage 1 (SSE alignment): Initial alignment

is computed using SSEs represented as vectors

Stage 2 (atom alignment): Alignment is refined using C atoms represented as points

18

Rationale for LOCK Using types of features is an effective way to

reduce combinatorial explosion and computation

SSEs, which are responsible for most of the stability and functionality of the proteins, are more meaningful and better conserved than types of atoms and amino-acids

If 2 structures are similar, some of their SSEs should form similar substructures

Drawback: It narrows down the set of possible applications, e.g., can’t find small motifs at atomic level

19

Vector-Based Representation

-helices

-strands

loops

One vector per SSE (helix, strand, loop)

20

Vector-Based Representation

DSSP [Kabsch and Sander, 1983] classifies residues into helices/strands

For -helix starting at residue i:Xorigin= (0.74Xi + Xi+1 + Xi+2 + 0.74Xi+3)/3.48where Xi is the position of the C atom of residue i(angle between two consecutive residues is 100dg factor 0.74)

Similar computation for Xend and for -strand

21

Position-independent differences: |angle(i,k)-angle(p,r)| |angle(i,j)-angle(p,q)| |angle(j,k)-angle(q,r)| |distance(i,k)-distance(p,r)| |length(k)-length(r)|

Position-dependent differences: angle(k,r) distance(k,r)

Scores are additive

Assume that i and p have been aligned. What is the score of the alignment of k and r?

S(di) = 1+(di/di0)2

2Mi - Mi

Score = S(di)

Scoring Similarity

Maximal score

Value of di forwhich score is 0

22

Stage 1: SSE Alignment1. For every pair of SSE vectors of protein A,

find all pairs of vectors in B that align well using orientation-independent scores seed correspondence sets

2. For each correspondence set: Find alignment transform and apply it to B Find correspondence set with maximal score

(record transform T and correspondence set CS that yields maximal score)

E.g., using start, middle, and end points of vectors

23

Stage 1: SSE Alignment A = (i, j, k, l, m) B = (p, q, r, s, t) Seed correspondence {(i,p),(j,q)}

(m,t)

(i,p), (j,q)

(k,r)

(l,t)

(l,r)(k,t) (m,r)(k,s)

(l,s) (m,s) (m,t) (m,s) (m,t)(l,t) (m,t)

• Simultaneous gaps in both structures are not allowed (not in SCOP2) • Terminate a path when score of new correspondence is negative • Re-compute new transform with each new correspondence (?)

24

Stage 2: Atom (Core) Alignment

1. Construct correspondence pairs of atoms : Atom i of A corresponds to atom j of T(B) iff i is

the closest atom in A to j and j is the closest atom in T(B) to i

The distance between i and T(j) is (3Å)

2. Prune correspondence set to largest subset of correspondence pairs that follow backbone alignment constraint

3. Re-compute T to be the transform that minimizes the RMSD of the atoms in the correspondence set

4. Iterate 1-2-3 until RSMD converges

25

Experimental Results

685 protein structures from PDB such that each pair has less than 25% sequence identity

3 families of folds (based on SCOP classification): - myoglobins (11 structures) – ~20% amino acid identity- TIM barrels (50 structures)- immunoglobulins (38 structures)

Goal: Given one query protein in each family, find the other members of the family (3×685 = 2055 alignments)

Method: For each query, sort the 685 structures by score (computed by LOCK). Select the top k proteins. Count members of family (true positives) and non-members (false positives)

26

# True positives

# False positives

11 0

Myoglobins (11)

# True positives

# False positives

40 0

45 1

50 5

TIM-barrels (50)

# True positives

# False positives

20 0

25 1

30 2

35 11

38 383

Immunoglobulins (38)

27

Alignment of 11 Myoglobins

28

Alignment of 50 TIM barrels

-helices in red-strands in yellow

29

Alignments of 31 Immunoglobulins

Only -strands are shown

30

ROC Curves

31

Running Time

~ 1ms per seed correspondence

~ 1h to search 10,000 protein structures

~ 100s of days to compare all pairs of proteins in PDB

Geometric hashing to speedup stage 1

32

Seed Generation from Fragment

1. From distance matricesE.g., DALI [Holm and Sander, 1996]

2. From secondary structure elements (SSE’s)E.g., LOCK [Singh and Brutlag, 1996]

3. From voting scheme (using geometric hashing)E.g., 3dSEARCH [Singh and Brutlag, 2000]

33

Voting Scheme with Hash Table

Many-to-many comparison requires a better organization of computation to avoid repeating the same computation again and again

Pre-computation: Index proteins in hash table Query phase: Voting scheme using hash table

Several variants on this theme 3d-Lookup [Holm and Sander, 1995]

3dSEARCH [Singh 2002]

34

Voting Scheme with Hash Table

Many-to-many comparison requires a better organization of computation to avoid repeting the same computation again and again

Pre-computation: Index proteins in hash table Query phase: Voting scheme using hash table

Several variants on this theme 3d-Lookup [Holm and Sander, 1995]

3dSEARCH [Singh 2002]

35

Indexing Target Structures in Hash Table (3dSEARCH [Singh

2002]) Hash table: 3-D regular grid of cubic bins (~2Å)

For each target structureFor each pair of vectors (i,j)1. Compute a coordinate system2. Place an entry for each other vector

k into the bin containing the coordinates of the midpoint of the vector (or average of coordinates of origin, middle, and end points). Store ID of coordinate system + k’s orientation and type ( or ) in the entry.

36

u

uv

v

Grid is same for all coordinate systems

37

uu

v

v

Grid is same for all coordinate systems

38

Indexing Target Structures in Hash Table (3dSEARCH [Singh

2002]) Hash table: 3-D regular grid of cubic bins (~2Å)

For each target structureFor each pair of vectors (i,j)1. Compute a coordinate system2. Place an entry for each other vector

k into the bin containing the coordinates of the midpoint of the vector (or average of coordinates of origin, middle, and end points). Store ID of coordinate system + k’s orientation and type ( or ) in the entry.

Grid is sparsely occupied hash table A structure with n SSEs contributes n(n-1)(n-2)

entries. Each vector is represented (n-1)(n-2) times 10,000 structures with 10 SSEs each yield ~7M

entries

39

Voting Using Hash TableGiven a query structure For each pair of vectors (i,j)

1. Compute a coordinate system2. For each other vector k

a. Retrieve the bin accessed by this vector and the neighboring bins

b. For every entry (vector) in those bins that has the same orientation and type as k, add a vote for the coordinate system stored in the entry

Sort target structures based on max number of votes received by any of its coordinate systems

Small number of target structures. Use LOCK for better alignment

Hours of pure LOCK are reduced to seconds

40

Advantages of Voting System

Very efficient in practice for many-to-many comparisons

Can establish correspondence between partial, disconnected substructures

Parallel implementation is straightforward Independent of the order in which vectors

are considered

Drawback (?): May establish correspondences that do not satisfy the backbone sequence constraint

41

Problem #4: Find Pharmacophore in

Ligands Given: Collection of N (= 5 to 10) small

flexible ligands with similar activity (binding at same sites)

Benzamidine binding to beta-Trypsin (3ptb)

Inhibitor binding to HIV protease

42

43

Problem #4: Find Pharmacophore in

Ligands Given: Collection of N (= 5 to 10) small

flexible ligands with similar activity (binding at same sites)

A set of low-energy conformations (dozens to few hundreds) for each ligand

44

Problem #4: Find Pharmacophore in

Ligands Given: Collection of N (= 5 to 10) small

flexible ligands with similar activity (binding at same sites)

A set of low-energy conformations (dozens to few hundreds) for each ligand

Find a substructure (pharmacophore) that has a match in at least one conformation of each ligand

45

46

O

O

OH

47

O

O

OH

48

O

O

OH

O

O

OHpharmacophore

49

Pharmacophore and Rational Drug Design

Pharmacophore identification is a form of “reverse engineering” to get a model of a binding site

A pharmacophore can be used to modify ligands into more potent drugs and/or to screen large databases of ligands for “leads”

50

Three Simultaneous Problems

Conformations? Correspondence? Transform?

But ligands are small molecules

51

Software

DISCO [Martin et al., 1993] DISCOtech and GASP [Tripos, Inc.] CATALYST and HIPHOP [Accelrys et al.;

Green et al., 1994; Barnum et al., 1996] RAPID

P.W. Finn, L.E. Kavraki, J.C. Latombe, R. Motwani, C. Shelton, S. Venkatasubramanian, and A. Yao. RAPID: Randomized Pharmacophore Identification for Drug Design. Computational Geometry: Theory and Applications, 10, pp. 263-272, 1998

52

M2M1 M3

53

Pairwise Comparison

Multi-Probe({M1,…,MN})

1) Extract invariants from M1 and M2 by calling Pair-Probe(P1,P2) on every pair of conformations of the two ligands

2) Test each candidate invariant S obtained at Step 1 against every ligand Mi, i = 3,…,N by calling Pair-Probe(S,P) on S and each conformation P of Mi

54

Pair-Proben: smallest number of atoms/features in a ligand: given constant (0 < ≤ 1) P1 and P2: Conformations of two distinct ligands (or candidate invariant)

Pair-Probe(P1,P2)Perform s times:

1) Pick a triplet of atoms at random from P1

2) Determine three atoms in P2 congruent to this triplet; compute the alignment transform T

3) Iterate: Apply T to P2; determine the atoms in P1 matching those in P2; update T

4) If the number of matching atoms exceed n, then return this atom set as a candidate invariant S

55

Magnitude of s

Pr[picking 3 atoms in invariant] 3

Pr[failing to find invariant] (1 3)s We want: (13)s

( is acceptable probability of failure) s ln()/ln(13) Since x < ln(1-x) for 0 < x < 1, we get:

s ln(1/)/3

For = 10-2 and = 0.3, we get s 180

56

1TLP 4TMN 5TMN 6TMN

Some Results

63 to 69 atoms with 10 to 15 torsional degrees of freedom

Feature: every non-H atom ~30 features of 6 types(atom types)

Invariant in active conformations: 7-atom pharmacophore + 7-atom scaffolding

11 800 44 20 10 5 2 1 0 0 1 0 0#conf t(s) #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14

57

Fuel for Thoughts

58

Idea: Many-to-many correspondence may be more

robust

Example: Hausdorf distance

[Huttenlocher et al., 1993]

59

Hausdorf Distance

Two sets of points A = {a1,...,an} and B = {b1,...,bm} in k

dH(A,B) = maxaA minbB ||a-b||

DH(A,B) = max {dH(A,B), dH(B,A)} Variation for shape similarity:

ΔH(A,B) = minT DH(A,T(B)) But efficient algorithms only exist for

planar sets of points

B

A

dH(A,B)

60

Other Idea: Minimize cost of transforming A into B

Old idea: Graphics: Morphing distance

Computer vision: Earth Mover’s distance[Rubner, Tomasi, and Guibas, 1998]

Protein similarity: Isotopic distance [Erdmann, 2004]

61

Structure Alignment Isotopies

Two curves are isotopic if one can be deformed into the other without self-collision

Example: Polygonal curve with n vertices

One may think of structure alignment as an isotopy deforming one structure into the other

Two structures are similar if the isotopy is “small”M.A. Erdmann. Protein Similarity from Knot Theory: GeometricConvolution and Line Weavings, CMU Tech. Rep. CMU-CS-04-138.

62

“Small” Isotopy Model a structure as a set of polygonal lines (e.g.,

vertices are C atoms) Two structures A and B are (T,δ)-isotopic if there

exists an isotopy deforming A into T(B) in such a way that no vertices of A moves further away than some δ from its initial or final location

[Erdmann 2004]

63

Similarity Measure

dT(A,B) = inf {δ | A is (T,δ)-isotopic to B}

d(A,B) = infT dT(A,B) d is computable [Erdmann,2004]

But as complex as path planning, hence exponential in the number of degrees of freedom

Possibility of approximating d using probabilistic roadmaps?

64

Topology of Line Weavings

helix axes

1xis1nar

M.A. Erdmann. Protein Similarity from Knot Theory: GeometricConvolution and Line Weavings, CMU Tech. Rep. CMU-CS-04-138.

65

66

2 topologically equivalent line weavings

3 equivalent classes for 4 lines[Erdmann 2004]

67

68

Another (incorrect) alignment of 1xis and 1nar

69

2 non-equivalent line weavings

2 equivalent classes for 3 lines

+

70

Why topology is interesting?

Two conformations may be geometrically close (small RMSD) may require a long continuous deformation to map one into the other (without steric clashes)

71

Conclusion Automatic computation of structure similarity is

essential due to the rapid growth of the PDB and other molecule (e.g., ligand) libraries

As the growth of new protein structures outpaces that of new folds, detecting structural similarity will have to be much more fine-grained than it is today

Biological discoveries will likely lie in local, possibly rare structure similarities, rather than in global fold-level classification

Need for better understanding of applications and radically new approaches

Still a lot of work ...