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Protein Structure Comparison

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Comparison based on 2D or 3D distance matrices (Holm, Sander, 96) ... Geometric Hashing, Indexing (Wolfson et al, Holm et al, Guerra et al ) ... – PowerPoint PPT presentation

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Title: Protein Structure Comparison


1
Protein Structure Comparison
2
Protein Structure Alignment
Human Hemoglobin alpha-chain pdb1jebA
Human Myoglobin pdb2mm1
Another example G-Proteins 1c1yA,
1kk1A6-200 Sequence id 18 Structural id 72
3
Protein Structure ComparisonMotivation
  • Understanding of the folding process.
  • Protein classification
  • Finding binding sites of the protein
  • Identifying structurally conserved regions in the
    protein

4
Comparison of proteins by sequence or shape?
  • Protein sequence comparison is simpler
  • 3D structures are available in a few percent of
    all proteins
  • Shape similarity is detectable even though the
    sequences may have changed in the course of
    evolution

5
Different Instances of Structure Comparison
  • All-to-All comparison
  • Classify all known structures
  • Search for a structural motif
  • Study interaction between structures and other
    molecules (Protein Docking)
  • Use known structures to predict structure from
    sequence (Protein Threading)

6
Sequence order dependence
  • Sequence order dependent alignment
  • an 1-D task.
  • Sequence order independent alignment
  • a real 3-D task.

7
Why Sequence order dependence
  • Substructures preserving sequence order might be
    biologically more meaningful
  • With the sequence order constraint the
    computational task is simpler

8
The problem
  • Given a pair of protein structures, find the
    correspondences between the Ca atoms of the
    backbone that best align the two structures
  • Tradeoffs between the number of corresponding
    atoms and the lowest distance

9
Finding Correspondences
  • Point-based approaches
  • Geometric Hashing, Indexing (Wolfson et al.,
    1998)
  • Comparison based on 2D or 3D distance matrices
    (Holm, Sander, 96)
  • Dynamic Programming (Gerstein, Levitt, 98)
  • Combinatorial Extension (Bourne et al, 96)

10
Algorithms for Structure Alignment
  • Distance based methods
  • DALI(Holm and Sander) Aligning scalar distance
    plots
  • STRUCTAL(Gerstein and Levitt) Dynamic
    programming using pairwise inter-molecular
    distances
  • SSAP(Orengo and Taylor) Dynamic programming
    using intra-molecular vector distance
  • Vector based methods
  • VAST (Bryant) Graph theory based secondary
    structure alignment
  • 3dSearch (Singh and Brutlag) Fast secondary
    structure index lookup
  • Both vector and distance based
  • LOCK (Singh and Brutlag) Hierarchically uses
    both secondary structures vectors and atomic
    distances

11
Hashing function
  • From an Object
  • To invariant Features
  • To t-ple of numbers
  • To indeces
  • Use the t indeces to access a t-dimensional hash
    table

12
Indexing Methodsfor Fast retrieval of 3D
patterns
  • Select a set of target proteins
  • Create and store a hash table indexed by
    invariant geometric properties of the selected
    folds
  • Update the databases as new structures are found
  • Use the table to identify the nearest fold for a
    target protein.

13
Reference Frame
  • A 3-D reference frame (r. f.) can be defined by
    three non collinear points
  • Invariant
  • the coordinates of any other point in the r.f.

14
Secondary Structures Representation
  • Secondary structures are represented as linear
    vectors (segments)
  • the axis for the alpha helix and the best fit
    segment for a beta strand
  • A SVD-based alignment algorithm is used to match
    an a helix segments with known axes to determine
    helix axis. Direct segment fits were made to fit
    b-sheet strands.

15
Visualization
  • Each segment associated to a secondary structure
    is displayed as a cylinder

16
Secondary Structure-based Approaches
  • Geometric Hashing, Indexing (Wolfson et al, Holm
    et al, Guerra et al )
  • Graph-based (Grindley et al)
  • Dynamic Programming (Singh, Brutlag)

17
Indexing techniques based on Secondary
Structures(Guerra et al)
  • Consider all the triplets of secondary structures
    and their associated segments
  • Construct a 3D table indexed by the angles
    relating three secondary structures.

18
Table Construction
  • For each triplet a1 , a2 , a3 of secondary
    structures of protein P
  • compute the angles between
  • (a1 , a2 ), (a1 , a3 ), and (a2 , a3 ),
  • and use them as indexes to an entry in the a-a-a
    Table where (P, (a1 , a2 , a 3)) is stored.
  • Each cell of the table at the end contains
    information about all triplets that hashed into
    it (including distances between secondary
    structures)

19
Table construction
  • Time Complexity O(s3n)
  • s is the of secondary structures in a
    protein
  • n the of proteins.

20
Searching the table
  • For a query protein, compute the same invariants
    used for the target proteins.
  • For each invariant and corresponding indeces,
    access the corresponding cell in the table where
    a vote is cast
  • List the target proteins according to their votes

21
Distribution of table entries(D. Platt, C.
Guerra, I. Rigoutsos, G. Zanotti, 2003)
  • There is a strong preference for triplets to fall
    into cells with indexes a,b, g satisfying
  • a b g
  • corresponding to segments lying on parallel planes

22
Analysis of Distribution of globally selected
secondary structures
  • Distributions show much stronger preference for
    alignment than expected for randomly uniform
    vectors.
  • There is a greater preference for alignment
    between any two secondary structure elements if a
    third structure element aligns with either of the
    first two -- the alignment angles are not
    independent variates.

23
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26
Geometric Indexing
PDB contains 27000 proteins
Hash Table
Query protein
Proteins Superposition
Search for Similarity
Hypotheses of Similarity
List of Similar Proteins
Pairwise atomic Superimposition with a selected
protein
Alignment by Dynamic Programming
27
Refinement of the matching procedure
  • Alignment
  • Find a collection of corresponding pairs of
    secondary structures (SS) which maximizes a
    given similarity measure
  • Dynamic programming

28
D(i,j) number of times SS i of protein A is
associated to SS j of protein B in a triplet of
equivalent SS
29
Problem
  • Find the increasing path in the D(i,j) matrix
    that maximizes the total similarity measure

30
Dynamic Programming
  • Let M be a 2D matrix such that M(i,j) is the
    similarity measure between s1 s2 .... si and
    t1t2 .... tj
  • Compute
  • M(i,j) max M(i-1,j) d(ti , f),
  • M(i-1,j-1) d(si tj),
    M(i,j-1) d(f, s j )
  • The solution
  • D(A,B)M(n,m)
  • Quadratic time complexity

31
Integration of matching strategies
  • Using different protein representations at
  • atomic level
  • secondary structure level
  • sequence level

32
Superposition
  • Find a rigid transformation which optimally
    superimposes the atoms of two proteins
  • Horn method

33
Accuracy vs coverage
  • Accuracy how many of the solutions found were
    correct?
  • A F intersection T /F
  • Coverage How many of the correct solutions were
    found?
  • C F intersection T /T

T correct sol.
Fsolutions found
34
Evaluating PROuST
  • Using as standard of truth SCOP
  • Against other existing servers

35
Accuracy of results for 1tim chain A
36
Another Algorithm DALI
  • Based on aligning 2-D intra-molecular distance
    matrices
  • Computes the best subset of corresponding
    residues from the two proteins such that
    similarity between the 2-D distance matrices is
    maximized.
  • Searches through all possible alignments of
    residues using Monte-Carlo algorithms

37
DALI
38
Distance matrix (2)
  • Advantages
  • - invariant with respect to rotation and
    translation
  • - can be used to compare proteins
  • Disadvantages
  • - the distance matrix is O(n2) for a protein
    with n residues
  • - comparing distance matrix is a hard problem
  • - insensitive to chirality

39
Distance matrix
5.9
2
4
8.1
3
6.0
1
40
DALI
  • DALI has been used to do an ALL vs. ALL
    comparison of proteins in the PDB, and to create
    a hierarchical clustering of families.
  • FSSPFold classification based on
    Structure-Structure alignment of Proteins
  • http//www.ebi.ac.uk/dali/fssp/fssp.html

41
VAST-Vector Alignment Search Tool
  • Aligns only secondary structure elements (SSE)
  • Represents each SSE as a vector
  • Finds all possible pairs of vectors from the two
    structures that are similar
  • Uses a graph theory algorithms to find maximal
    subset of similar vectors
  • Overall alignment scores is based on the number
    of similar pairs of vectors between the two
    structures.

42
VAST
  • VAST has been used to do an ALL vs. ALL
    comparison of proteins in the MMDB (NCBIs
    structure database), and to find structure
    neighbors for each structure.
  • MMDB provides service of searching structure
    neighbors using VAST.
  • http//www.ncbi.nlm.nih.gov/Structure/VAST/vast.sh
    tml

43
LOCK
  • Define local secondary structures
  • Find an initial superposition by using DP to
    align secondary structure vectors.
  • Use greedy algorithms to find nearest neighbors
    and minimize RMSD between the C-? atoms from
    query and target.
  • Find the core of aligned C-? atoms and minimize
    RMSD between them.

44
Comparison of methods
45
Execution times for comparing a query structure
to 27,000 target structures
46
Execution times for comparing a query structure
to 685 target structures
47
Data
  • 30,000 proteins extracted from PDB
  • Approx. 27,000 proteins inserted in the geometric
    database
  • 0 to 528 segments per protein
  • 13.5 segments on average
  • 48,000,000 triplets
  • 4x20x20x20 table

48
GEOMETRIC PATTERN MATCHING UNDER RIGID MOTION(C.
Guerra, V. Pascucci, 1999)
  • Problem 1. Find a transformation T, if it exists
    that brings A to within a given distance, say e,
    of B, i.e. H(T(A),B)
  • Problem 2. Find the minimum Hausdorff distance
    under a rigid motion
  • D(A, B) min t (t(A), B)
  • where t is a rigid motion

49
Hausdorff Distance
  • Let A a1, a2, ..., am B b1, b 2, ..., bn
    be sets of either points or segments.
  • Definition. (Hausdorff Distance)
  • H(A, B) max (h(A, B), h(B, A))
  • where the one-way Hausdorff distance is
  • h(A, B) maxa minb r (a, b)
  • where a (b) is a point of A (B) and r (a, b), is
    a metric.

50
Segment Hausdorff distance
  • HS(A, B) max (hS(A, B), hS(B, A))
  • where
  • hS(A, B) max ai (min bj H(ai,bj))

51
Oriented Segment Hausdorff distance
  • HOS(A, B) max (hOS(A, B), hOS(B, A)),
  • where
  • hOS(A, B) maxai
  • (minbj (max( d(ais,bjs),d(aie,bje)) ))
  • ais , aie are the endpoints of ai

52
Exact solution in 2D
  • This problem is generally solved as a problem of
    intersection of unions of disks in the
    transformation space.
  • Time complexity O( m3 n3 log2nm) in R2

53
The Matching algorithm
  • Find a rigid body transformation (translation
    plus rotation) that minimizes the Hausdorff
    distance between the segments of A and B.
  • Derive
  • T A?B
  • based on three representative segments of A and
    all
  • triplets of segments of B, and choose the best
    T.

54
Practical Approach
  • 1. Select three representative'' segments a,
    a' , a' of A as follows
  • 1.a Choose randomly one representative a for A.
  • 1.b Select a' to be the segment containing the
    point a'f farthest from the midpoint ac of a .

55
Practical approach (contd)
  • 1.3. Select a'' as the segment that contains the
    point at maximum distance from the line ac ,a'f.
  • 2. For each triplet b, b', b'' of elements of B
    determine the rotation and translation that maps
    a, a' , a'' into b, b' , b''.
  • 3. Choose the best transformation among the
    examined ones.

56
Segment Nearest-Neighbor
  • The nearest-neighbor among n segments in Rd is
    equivalent to a query among 2n points in R2d.
  • HSS(a, b) min(max(d(as,bs),d(ae,be)), max
    (d(as,be), d(a e,be)).
  • Approximate nearest neighbor of a point q in Rd
    (within a factor of (1e )) (Arya et al. )
  • Time complexity O(logn) with O(nlogn)
    preprocessing.

57
Complexity Analysis
  • Time complexity O(mn3log n)
  • Approximation error factor 8

58
Protein 1rpa
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Questions?
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