protein structure similarity

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Protein Structure Similarity

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Secondary Structure Elements:

helicesstrands/sheets & loops

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Structure Prediction/Determination

Computational tools• Homology, threading• Molecular dynamics

Experimental tools

NMR spectrometryX-ray crystallography

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X-ray diffraction crystallography

Protein Structure Determination (1)

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Protein Structure Determination (2)

Nuclear magnetic resonance spectroscopy

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Protein Data Bank

1990 250 new structures1999 2500 new structures2000 >20,000 structures total2004 ~30,000 structures total

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Protein Data Bank

1990 250 new structures1999 2500 new structures2000 >20,000 structures total2004 ~30,000 structures total

Only about 10% of structures have been determined for known protein sequences

Protein Structure Initiative (PSI)

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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

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Alignment of 1xis and 1nar (TIM-Barrels)

Alignment computed by DALI helix axes

1xis1nar

Sayle, R. RasMol. A protein visualization tool.http://www.umass.edu/microbio/rasmol/index2.htm.

ribbon format

backbone format

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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)

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12

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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

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Sequence Structure Function

sequencesimilarity

Why Comparing Protein Folded Structures?

Low sequence similarity may yield very similar structures Sometimes high sequence similarity yields different

structures

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Alignment of 1xis and 1nar (TIM-Barrels)

1xis and 1nar have only 7% sequenceidentity, but approximately 70% of the residues are structurally similar

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Sequence Structure Function

sequencesimilarity

structuresimilarity

Why Comparing Protein Folded Structures?

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

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Ill-Posed Problem Multiple Terminology

(Dis-)similarity analysis Structure comparison Alignment, superposition, matching Classification

Applications Definitions and issuesMethods

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A Few Web Sites Protein Data Bank (PDB):

http://www.rcsb.org/pdb/ Protein classification:

SCOP:http://scop.berkeley.edu/

CATHhttp://www.biochem.ucl.ac.uk/bsm/cath/

Protein alignment: DALI:

http://www.ebi.ac.uk/dali/ LOCK:

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

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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 C 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)

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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]

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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]

Increasing size of PDB Automatic classifiers: CATH [Orengo et al., 1997]; Pclass [Singh et al.]; FSSP [Holm and Sander]

Class: Similar secondary structure content

Fold: SSE’s in similar arrangement

Family: Clear evolutionary relationship

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Manuel vs. Automatic Classification

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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

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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

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Application #4: Find Pharmacophore

1TLP

4TMN

5TMN

6TMN

Inhibitors of thermolysin

Clusters of low-energy conformations of 1TLP

The 4 ligands overlappedwith their pharmacophorematched

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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):731-747, July 2000

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Applications Definitions and issuesMethods

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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…

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Matching of Structures

Two structures A and B match iff:

1.Correspondence: There is a one-to-one map between their elements

2.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 .

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Complete Match

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Alignment of 3adk and 1gky

But a complete match is rarely possible: The molecules have different sizes Their shapes are only locally similar

Both matching and non-matching secondary structure elements

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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

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Mathematical Relative

f

g

||f g||2

s

Over which support?

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Mathematical Relative

f

g

||f g||2

s

Over which support?

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Multiple Partial Matches

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Distributed Support

B

A

B

A

Gap

σ(A)

σ(B)

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What is Best?

B

A

B

A

Should gaps be penalized?

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What About This?

B

A

Sequence along backbone is not preserved

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Similarity measure is unlikely to satisfy triangular inequality for partial match

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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!

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Examples

2( )

max / 2( )

1

Ti T

i i

ANGAP

a T b

B

2( )

1min ( )

| ( ) |T i ii T

a T bT

RMSD dissimilarity measure emphasizes differences smaller support

STRUCTAL’s similarity measure emphasizes similarities larger support

Gap penalty

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Comparison of Similarity Measures

A.C.M. May. Toward more meaningful hierarchical classification of amino acids scoring functions. Protein Engineering, 12:707-712, 1999reviews 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

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Bottom LineFinding 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

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Computational Questions

Given a (dis)similarity measure and two proteins, compute the best match:

Which support? Which correspondence? Which alignment transform?

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Applications Definitions and issuesMethods

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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

Several possible outputsVariant of the Largest Common Point Set problem[Akutsu and Halldorsson, 1994]

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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]

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Some Existing SoftwareC 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 C atoms: VAST [Gibrat et al., 1996]

LOCK [Singh and Brutlag, 1996]

3dSEARCH [Singh and Brutlag, 1999]

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RMSD ≠ SimilarityBut matches and RMSD’s are not exactly what we need

In general, we need to computea similarity measure of the form maxT S(A,T(B)) where S is more complex than RMSD

Two-step approach: 1. Compute best matches using RMSD 2. Adjust transform to maximize similarity measure

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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 !!!

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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

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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]

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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]

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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]

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

[Horn, 87] quaternion

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Trial-and-Error Approach to Protein Structure

Comparison

Guess small correspondence set

Compute T

Update correspondence set (correspondence from proximity)

Apply T

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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)

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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

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How to get seed correspondences?