Protein Structure Similarity

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Protein Structure Similarity
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Secondary Structure Elements a helices, b
strands/sheets, loops
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Structure Prediction/Determination

• Computational tools
• Homology, threading
• Molecular dynamics
• Experimental tools

X-ray crystallography
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Protein Structure Determination (1)

• X-ray diffraction crystallography

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Protein Structure Determination (2)

• Nuclear magnetic resonance spectroscopy

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Protein Data Bank
1990 ? 250 new structures 1999 ? 2500 new
structures 2000 ? gt20,000 structures total 2004 ?
30,000 structures total
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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 ? gt20,000 structures total 2004 ?
30,000 structures total
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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)
Sayle, R. RasMol. A protein visualization
tool. http//www.umass.edu/microbio/rasmol/index2.
htm.
ribbon format
1xis 1nar
backbone format
Alignment computed by DALI
a helix axes
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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 (15 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 (15 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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Why Comparing Protein Folded Structures?
Sequence
Structure
Function

• 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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Why Comparing Protein Folded Structures?
Sequence
Structure
Function

• 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 issues
• Methods

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A Few Web Sites

• Protein Data Bank (PDB)http//www.rcsb.org/pdb/
• Protein classification
• SCOPhttp//scop.berkeley.edu/
• CATHhttp//www.biochem.ucl.ac.uk/bsm/cath/
• Protein alignment
• DALIhttp//www.ebi.ac.uk/dali/
• LOCKhttp//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 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)

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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
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 SSEs 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
  100s) 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
Inhibitors of thermolysin
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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 issues
• Methods

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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
• Correspondence There is a one-to-one map
  between their elements
• AlignmentThere 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.

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Complete Match
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Alignment of 3adk and 1gky

• Both matching and non-matching secondary
  structure elements

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Partial Match

• Notion of support s of the match the match is
  between s(A) and s(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
g
f
s
f - g2
Over which support?
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Mathematical Relative
g
f
s
f - g2
Over which support?
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Multiple Partial Matches
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Distributed Support
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What is Best?
Should gaps be penalized?
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What About This?
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 s 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
RMSD dissimilarity measure ? emphasizes
differences ? smaller support
STRUCTALs similarity measure? emphasizes
similarities ? larger support
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Comparison of Similarity Measures

• A.C.M. May. Toward more meaningful hierarchical
  classification of amino acids scoring functions.
  Protein Engineering, 12707-712, 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 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

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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 issues
• Methods

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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 e
• Several possible outputs

Variant of the Largest Common Point Set
problemAkutsu 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
  igti, are possible correspondence pairs iff
  jgtjE.g., sequence along backbone

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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

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RMSD ? Similarity
But matches and RMSDs 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 !!!

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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 (Si1,...,nbi)/n mean of the bis
• 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 ais

Arun, Huang, and Blostein, 87
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O(n) SVD-Based Algorithm

• A3?n a1-a, ..., an-a B3?n b1-b, ...,
  bn-b
• Compute SVD decomposition of 33 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
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? 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)

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? 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

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• How to get seed correspondences?