1. Protein Structure Similarity

2. Secondary Structure Elements: a helices, b strands/sheets, & loops

3. Structure Prediction/Determination
Computational tools
Homology, threading
Molecular dynamics
Experimental tools
NMR spectrometry
X-ray crystallography

4. Protein Structure Determination (1)
X-ray diffraction crystallography

5. Protein Structure Determination (2)
Nuclear magnetic resonance spectroscopy

6. Protein Data Bank 1990  250 new structures 1999  2500 new structures
2000  >20,000 structures total
2004  ~30,000 structures total

7. Protein Data Bank
Only about 10% of structures have been determined for known protein sequences
 Protein Structure Initiative (PSI)
1990  250 new structures
1999  2500 new structures
2000  >20,000 structures total
2004  ~30,000 structures total

8. Structure Similarity
Refers to how well (or poorly) 3D folded structures of proteins can be aligned
Expected to reflect functional similarities (interaction with other molecules)
Proteins in the TIM barrel fold family

9. Alignment of 1xis and 1nar (TIM-Barrels)
ribbon format
Sayle, R. RasMol. A protein visualization tool.
1xis
1nar
backbone format
Alignment computed by DALI
a helix axes

10. Structure Similarity
Refers to how well (or poorly) 3D folded structures of proteins can be aligned
Is expected to reflect functional similarities (interaction with other molecules)
2000: ~ 20,000 structures in PDB ~ 4,000 different folds (1:5 ratio)

13. Structure Similarity
Refers to how well (or poorly) 3D folded structures of proteins can be aligned
Is expected to reflect functional similarities (interaction with other molecules)
2000: ~ 20,000 structures in PDB ~ 4,000 different folds (1:5 ratio)
Three possible reasons: - evolution, - physical constraints (e.g., few ways to maximize hydrophobic interactions), - limits in techniques used for structure determination
Given a new structure, the probability is high that it is similar to an existing one

14. Why Comparing Protein Folded Structures?
sequence similarity
Sequence
Structure
Function
Low sequence similarity may yield very similar structures
Sometimes high sequence similarity yields different structures

15. Alignment of 1xis and 1nar (TIM-Barrels)
1xis and 1nar have only 7% sequence identity, but approximately 70% of the residues are structurally similar

16. Why Comparing Protein Folded Structures?
sequence similarity
Sequence
Structure
Function
structure similarity
Low sequence similarity may yield very similar structures
Sometimes high sequence similarity yields different structures
Structure comparison is expected to provide more pertinent information about functional (dis-)similarity among proteins, especially with non-evolutionary relationships or non-detectable evolutionary relationships

17. Ill-Posed Problem  Multiple Terminology
(Dis-)similarity analysis
Structure comparison
Alignment, superposition, matching
Classification
Applications
Definitions and issues
Methods

18. A Few Web Sites Protein Data Bank (PDB): http://www.rcsb.org/pdb/
Protein classification:
SCOP:
CATH
Protein alignment:
DALI:
LOCK:

19. Application #1: Find Global Similarities Among Protein Structures
Given two protein structures, find the largest similar substructures
For example, a substructure is a subset of Ca atoms or a subset of secondary structure elements in each molecule
Several possible similarity measures
Variants: 1-to-1, 1-to-many, many-to-many (PDB)
Must be automatic (and fast)

20. Application #2: Classify Proteins
Many proteins, but relatively few distinct fold families [Chotia, 1992; Holm and Sander, 1996; Brenner et al. 1997]
Hierarchical classification
Insight into functions and structure stabilization
Basis for homology and threading
Manual classification  SCOP [Murzin et al., 1995]

21. Application #2: Classify Proteins
Class: Similar secondary structure content
Many proteins, but relatively few distinct fold families [Chotia, 1992; Holm and Sander, 1996; Brenner et al. 1997]
Hierarchical classification
Insight into functions and structure stabilization
Basis for homology and threading
Manual classification  SCOP [Murzin et al., 1995]
Increasing size of PDB  Automatic classifiers: CATH [Orengo et al., 1997]; Pclass [Singh et al.]; FSSP [Holm and Sander]
Fold: SSE’s in similar arrangement
Family: Clear evolutionary relationship

22. Manuel vs. Automatic Classification

23. Application #3: Find Motif in Protein Structure
Given a protein structure and a motif (e.g., a small collection of atoms corresponding to a binding site)
Find whether the motif matches a substructure of the protein
Variant: One motif against many proteins
Active sites of 1PIP and 5PAD. Only 3 amino-acids participate in the motif

24. Application #4: Find Pharmacophore
Given:
Small collection (5-10) of small flexible ligands with similar activity (hence, assumed to bind at same protein site)
Low-energy conformations (several dozens to few 100’s) for each ligand
Find substructure (pharmacophore) that occurs in at least one conformation of each ligand
Key problem in drug design when binding site is unknown

25. Application #4: Find Pharmacophore
1TLP
4TMN
5TMN
6TMN
Clusters of low-energy
conformations of 1TLP
The 4 ligands overlapped with their pharmacophore matched
Inhibitors of thermolysin

26. Application #5: Search for Ligands Containing a Pharmacophore
Given:
Database containing several 100,000, or more, small ligands
A pharmacophore P
Find all ligands that have a low-energy conformation containing P
Data mining of pharmaceutical databases (lead generation)
S.M. LaValle, P.W. Finn, L.E. Kavraki, and J.C. Latombe. A Randomized Kinematics-Based Approach to Pharmacophore-Constrained Conformational Search and Database Screening. J. of Computational Chemistry, 21(9): , July 2000

27. Applications
Definitions and issues
Methods

28. 3D Molecular Structure
Collection of (possibly typed) atoms or groups of atoms in some given 3D relative placement
The placement of a group of atoms is defined by the position of a reference point (e.g., the center of an atom) and the orientation of a reference direction
The type can be the atom ID, the amino-acid ID, etc…

29. Matching of Structures
Two structures A and B match iff:
Correspondence: There is a one-to-one map between their elements
Alignment: There exists a rigid-body transform T such that the RMSD between the elements in A and those in T(B) is less than some threshold e.

30. Complete Match

31. Alignment of 3adk and 1gky
But a complete match is rarely possible:
The molecules have different sizes
Their shapes are only locally similar
Alignment of 3adk and 1gky
Both matching and non-matching secondary structure elements

32. Partial Match
Notion of support σ of the match: the match is between σ(A) and σ(B)
 Dual problem: What is the support? What is the transform?
Often several (many) possible supports
Small supports  motifs

33. Mathematical Relativeg f s
||f - g||2
Over which support?

34. Mathematical Relativeg f s
||f - g||2
Over which support?

35. Multiple Partial Matches

36. Distributed Support B A B A
σ(A)
σ(B)
Gap

37. What is Best? B A B A
Should gaps be penalized?

38. What About This? B A
Sequence along backbone is not preserved

39. Similarity measure is unlikely to satisfy triangular inequality for partial match

40. Scoring Issues Trade-off between size of σ and RMSD
How should gaps be counted?
Is there a “quality” of the correspondence?
[The correspondence may, or may not, satisfy type and/or backbone sequence preferences]
Should accessible surface be given more importance?
 Similarity measure may be different from the inverse of RSMD (though no consensus on best measure!)
But RMSD is computationally very convenient!

41. Examples
RMSD dissimilarity measure  emphasizes differences  smaller support
STRUCTAL’s similarity measure  emphasizes similarities
 larger support
Gap penalty

42. Comparison of Similarity Measures
A.C.M. May. Toward more meaningful hierarchical classification of amino acids scoring functions. Protein Engineering, 12: , 1999 reviews 37 protein structure similarity measures
The difficulty of defining a similarity score is probably due to the facts that structure comparison is an ill-posed problem and has multiple solutions

43. Bottom Line Finding an optimal partial match is NP-hard:
No fast algorithm is guaranteed to give an optimal answer for any given measure [Godzik, 1996]
 Heuristic/approximate algorithms
 Probably not a single solution, but application- dependent solutions
 But there exist general algorithmic principles

44. Computational Questions
Given a (dis)similarity measure and two proteins, compute the best match:
Which support?
Which correspondence?
Which alignment transform?

45. Applications
Definitions and issues
Methods

46. Find Global Similarities Among Protein Structures
Input: Two sets of features (atoms or groups of atoms) {a1,…,an} and {b1,…,bm} belonging to two different proteins A and B
Output: - Maximal correspondence set C of pairs (ai,bj), where all ai and all bj are distinct - Alignment transform T such that the RMSD of the pairs (ai,T(bj)) is less than a given e
Several possible outputs
Variant of the Largest Common Point Set problem [Akutsu and Halldorsson, 1994]

47. Possible Correspondence Constraints
Typed features: (ai,bj) is a possible correspondence pair iff Type(ai) = Type(bj)
Ordered features: (ai,bj) and (ai’,bj’), where i’>i, are possible correspondence pairs iff j’>j [E.g., sequence along backbone]

48. Some Existing Software
Ca atoms:
DALI [Holm and Sander, 1993]
STRUCTAL [Gerstein and Levitt, 1996]
MINAREA [Falicov and Cohen, 1996]
CE [Shindyalov and Bourne, 1998]
ProtDex [Aung,Fu and Tan, 2003]
Secondary structure elements and Ca atoms:
VAST [Gibrat et al., 1996]
LOCK [Singh and Brutlag, 1996]
3dSEARCH [Singh and Brutlag, 1999]

49. RMSD ≠ Similarity But matches and RMSD’s are not exactly what we need
In general, we need to compute a similarity measure of the form maxT S(A,T(B)) where S is more complex than
RMSD
Two-step approach: Compute best matches using RMSD Adjust transform to maximize similarity measure

50. 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 !!!

51. 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

52. 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]

53. 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]
[Arun, Huang, and Blostein, 87]

54. O(n) SVD-Based Algorithm
A3n = [a1-a, ..., an-a] B3n = [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]

55. [Arun, Huang, and Blostein, 87]
 rotation matrix
[Horn, 87]  quaternion

56.  Trial-and-Error Approach to Protein Structure Comparison
Guess small correspondence set
Compute T
Update correspondence set (correspondence from proximity)
Apply T

57.  Trial-and-Error Approach to Protein Structure Comparison
Set CS to a seed correspondence set (small set sufficient to generate an alignment transform)
Compute the alignment transform T for CS and apply T to the second protein B
Update CS to include all pairs of features that are close apart
If CS has changed, then return to Step 2 else return (CS,T)

58.  Trial-and-Error Approach to Protein Structure Comparison
- result = nil
- Iterate N times:
Set CS to a seed correspondence set (small set sufficient to generate an alignment transform)
Compute the alignment transform T for CS and apply T to the second protein B
Update CS to include all pairs of features that are close apart
If CS has changed, then return to Step 2 else result  result  {(CS,T)}
- Return result

59. How to get seed correspondences?