EECS 730Introduction to Bioinformatics

Structure Comparison

Luke HuanElectrical Engineering and Computer Science

http://people.eecs.ku.edu/~jhuan/

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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The State of the Strucutre Space

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) 2007: ~ 34,000 structures in PDB

~ 1,000 different folds (1:34 ratio)

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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 Compute Structure Similarity?

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 Compute 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

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Ill-Posed Problem Multiple Terminology

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

Definitions Applications Methods Issues

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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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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 StructuresTwo structures A and B match if:

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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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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Definitions Applications Methods Issues

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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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Compute Structure Similarity

Structure presentation Similarity measurement Computational solution

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Structure presentation

Element based representation A structure is broken down to a list of structure

elements We represent a protein structure by its geometry,

topology, and attributes: Geometry: the coordinates of the elements Topology: the physical and chemical interaction of

elements Attributes: the physical and chemical attributes of the

elements

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Structure Representation

There are three major groups of structure presentation Point list: treat protein as a list of points in a 3D

space Point set: treat protein as a set of points in a 3D

space Graphs: treat protein as a graph

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Comparing two point sets

Similarity measure:

Given two point set P = {p1, p2, …, pn} and Q = {q1, q2, …, qm}, (n≤ m), find a Euclidian transformation T (rotation + translation), and a 1-1 mapping f from P to Q such that

S (P, Q) = sqrt( id2(pi, T(f(pi)) ) is minimized .

S is called the RMSD (root-mean-spared-distance) between the two structures

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Comparing two point sets

If m = n, there is a close-form solution to find the exact solution to the problem of comparing the two point sets

If m ≠n, the problem is much harder

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Common Point Subset Problem

Find the largest common point subset Given two point set P = {p1, p2, …, pn} and Q = {q1,

q2, …, qm}, (n≤ m), find a Euclidian transformation T (rotation + translation), and a 1-1 partial mapping f with maximal cardinality from P to Q such that

d (pi, T(f(pi)) ) < t for all i defined in f Also a harder problem (but not a NP-hard problem)

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Geometric Hashing

Originally used for automatic visual recognition of geometric figures

The principle We have two geometric figures

model A with m points (can have several models) quary B with n points

Discover similar subfigures in A and B invariant under placement, rotation (and often size)

Let the figures be described by points Try to find the largest set of points from (A, B) with

coinciding points

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Coinciding points Example from 2

dimension Find six overlapping

pairs (1,a)(2,d)(3,c)(4,e)

(6,f)(7,g) The coinciding pairs

are independent of the labeling

Note that the figures can be translated and rotated

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Reference frames The points of the figures are specified in coordinate

systems or reference frames A reference frame can in 2D be defined by two points Choose two points from A (ai,ak) and two from B (bj, bl),

called basises, and define the reference frames (RF) from the basises Example: origin in ai and the x-axis along the line ai,ak, or origin

at the middle of ai,ak

Find the positions in RF of all the other points, called reference frame system, RFS

”Overlap” (the x,y-axes) RFSA and RFSB, and count the number of coinciding points

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Reference frame system, example

Model (1,3) [(0,0)(6,2)(8,0)(9,4)(6,10)(3,8)(-1,6)]

four coinciding points Query (a,c) [(0,0)(3,-2) (8,0)(6,2)

(10,4)(3,8)(0,6)] only the origins coincidies Model (3,5) [(0,0)(1,8)(2,2)(4,-2)

(10,0)(8,3)(8,7)]

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Comparison of (Reference) Frame Systems

The number of coinciding points depends on the basises

Should therefore try all possible pairs as basises

This would result in m(m-1)n(n-1) comparison of reference frame systems, but many of those comparisons are redundant

Geometric hashing is used for efficiently performing ”simultaneously” many comparisons

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Hashing

Compare simultaneously a query frame system to all model frame systems

Assume a 2D hashing table H, a simple hashing function One bucket for each square of the frame system, identified

by (p,q) Let (u,v) H(p,q) mean that the frame system with basis (u,v)

has a point in the square (p,q) (a very simple hash function) H is filled in a preprocessing of the model

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Hashing preprocessing example

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Recognition Compare the query with the model (several frame

systems) Select a basis in the query and define reference frame Find the positions in the reference frame to all the other

points For each point r in the query reference system do

Calculate the position (x,y) in H that r hashes to Vote one for each model reference system in

H(x,y) End Recognize the model reference systems with highest votes

Repeat for more query reference systems, if not enough coinciding points are found

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Example recognition

query (a,c) [(0,0)

(3,-2) (8,0)(6,2)(10,4)(3,8)(0,6)]

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Use of several models

Can have several models in the same hashtable Must then identify model and reference system in

the hash table Example: Have a database of structures, stored in

a hashtable

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Geometric hashing for structure comparison Need methods invariant under translation and rotation Use geometric hashing to find subsets and coincident residues

(points), residues that superpose well1. Define referance frames

Three atoms can be used: ai, ak, ar. Example Origin in ai The x-axis along ai,ak The y-axis in the plane defined of ai, ak, ar in

counterclockwise The z-axis orthogonal to the plane

2. The residues may have labels (attributes)1. Implement the labels explicit: in the hastable2. Implement the labels implicit: in the hashing