Seminar in structural bioinformatics

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Seminar in structural bioinformatics

• Pairwise Structural Alignment

Presented by Dana Tsukerman
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Outline

• Definitions.
• What is structural alignment?
• Why structural alignment?
• structural alignment vs. sequence alignment
• Problem definition
• Background
• preparing the ground for the algorithm.
• The algorithm

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Outline - cont.

• Implementation of the algorithm and an example of
  using a real software, based on the algorithm
  that will be presented.
• Method results.
• Method discussion
• Method summary.
• Extensions and additional features - a look
  ahead.
• Lecture summary.

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Definitions

• Sequence alignment (remainder from last lecture),
  unambiguously distinguishes only between protein
  pairs of similar structure and non-similar
  structures when the pairwise sequence identity is
  high.
• Structure alignment - the precise arrangement of
  the amino acid side chains in the three
  dimensional structure of the protein that
  dictates its function.

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Quick rehearsal - Basic terms

• Primary structure refers to the order (and
  sequence) of amino acids along one chain.
• Some regions form regular local structure
  (folding patterns)
• Alpha helices
• Beta strands
• Collectively called secondary structure elements
  (SSEs).
• Regions connecting SSEs are loops.
• Secondary structure is the description of the
  type and locations of the SSEs.
• Tertiary structure is the 3-D coordinates of the
  atoms in a chain.
• Quaternary structure describes the spatial
  packing of several folded chains (not all
  proteins have a quaternary structure).

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Quick rehearsal - Basic terms
regular hydrogen bond patterns of backbone atoms
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3D observation of proteins
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3D observation of proteins

• If one looks at the collection of protein
  structures, one is reminded of the works of an
  Origami artist Certain basic folding patterns
  are used over and over again and cleverly
  modified by minor adaptations to generate a wide
  variety of different protein structures. Where
  one such folding units is insufficient to
  generate the required complexity, multiple
  domains can be combined, such as in the camel or
  giraffe structure on this picture.

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Comparison in 3D

• Starting from an example

A
B
B
A
C
C
E
D
D
E
back
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Comparison in 3D

• Rotation and translation coordinates - 6 degrees
  of freedom.
• The method is independent of the amino acid
  sequence.
• What does it mean?
• This method is insensitive to insertions,
  deletions and displacements of equivalents
  substructures betweens the molecules being
  compared.
• Proteins with similar sequences adopt very
  similar structures.

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Why 3D comparison?
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Why 3D comparison?
Wait a minute - isnt sequence comparison enough?
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Why 3D comparison?

• Structures are more conserved than sequences.
• Detection of distant evolutionary relationships.
• Structural alignment can imply a functional
  similarity that isnt detectable from a sequence
  alignment.
• The protein docking problem.
• Structure based drug design.
• Applications and implications to the protein
  folding problem.

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Why 3D comparison? Cont.

• For homologous proteins, this provides the gold
  standard for sequence alignment.
• For nonhomologous proteins, it allows us to
  identify common substructures of interest.
• Allows us to classify proteins into clusters,
  based on structural similarity.
• Design and engineering of synthetic proteins.

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Problem Definition

• Input 3-D coordinate data of the structures to
  be compared.
• Output regions of structural similarity (more
  than one, if exists), that lead to the best
  alignment.
• NP-Hard.

Most atoms matched with the lowest RMSD
Whats best?
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Our goal
Find out the correspondence between the structures
transformation T
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Preparing the ground

• Transformation definition.
• How can we evaluate the match we found?
• RMSD rehearsal from the opening lecture.
• Other methods besides the one we will discuss
  and why our method is better.
• Progression rule definition.
• PDB functionality rehearsal.
• Geometric Hashing introduction.

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3-D Transformation

• Rotation - the movement of a body in such a way
  that any given point of that body remains at a
  constant distance from some other fixed point.
  Will be denoted by R.
• Translation - the transformation of moving every
  point by a fixed distance in the same direction
  (addition of a constant vector to every point).
  Will be denoted by T.
• What is preserved under translation and rotation?
  
• relative distances within an object (e.g.
  Shapes)
• In total, the 3-D transformation has 6 degrees of
  freedom 3 for rotation and 3 for translation.

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RMSD - rehearsal

• A tool we use to evaluate the correspondence we
  found.
• RMSD - Root Mean Square Deviation
• Where,
• n number of atoms
• x, y the proteins we want to compare
  (structures)
• We want to find 3-D transformation T, such that
  the RMSD will be minimal, i.e.
• We know how to do that in O(n).

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RMSD - Example
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Other methods for structural alignment

• Dynamic programming - building a score matrix,
  with a score for each pair of residues.
  or
• Other improvements of that method.
• Simplify the problem by moving from 3D space to
  2D space sacrificing the optimum result for the
  speed.
• Comparing secondary structure elements (SSE)
• Our method allows access to problems that
  couldnt be approached previously by
  sequence-order-dependent structural comparison
  methods, like the docking problem.

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Progression rule

• Rule definition for elements i and k from one
  sequence and elements j and l from the other
  sequence, if element i is matched to element j
  and element k is matched to element l, and if k
  is to the right of i in the first sequence, the l
  must also be to the right of j in the second
  sequence.
• For example, the structures we saw at the
  beginning couldnt be found similar by a
  progression rule based method (sequence
  -dependent).

Example
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PDB - Protein Data Bankhttp//www.rcsb.org/pdb/in
dex.html

• International structure database.
• Archive of experimentally determined 3-D
  structures of biological macromolecules, together
  with extensive annotation.
• Established at Brookhaven National Laboratories
  in 1971. in the beginning it held 7 structures.
• In 2003, 4,831 structures were deposited to the
  PDB archive.
• January 2004 snapshot 23,792 released atomic
  coordinate entries.

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Geometric hashing

• Introduced for model-based object recognition in
  computer vision.
• Goal identify and locate in an image all the
  instances of models which appear in the systems
  database.
• Represents the objects to be compared in a
  translational and rotational invariant fashion.
• On which the first step of the algorithm
  presented today is based.

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Geometric hashing - cont.

• We search for a way to represent object in a way
  we will be able to move them, and the
  representation wont change.
• HOW? Building triangles!
• for nodes triangles!

The triangles sides length doesnt change when
we move it or rotate it, and thus invariant!
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• now please pay attention

• Wake Up!

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The algorithm major steps

• Find (relatively small) subsets of the structures
  that form an initial match
• Find clusters in initial matches that represent
  similar transformations
• Extend the clusters to contain additional
  matching pairs of residues.

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Motivation remainder

• now let's jump into the water...

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Step 1 - Finding seed matches

• Goal search through the structures to find
  candidate initial matches.

• Those will be referred as seed matches.

• Most difficult and time consuming step.

• Extensive search of the structures.

• How to represent?

• Remember what we talked about in geometric
  hashing?

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Finding seed matches - cont.

• Seed match - list of matching pairs of atoms.

• Pair - correspondence between atoms from
  different structures.

• Assumption the structures to be compared are
  described by sets of interest points and their
  3-D coordinates (for example Ca atoms).

• Model Target.

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Finding seed matches - cont.

• Redefinition of the problem is there a rotated
  and translated subset of the interest points of
  the target which matches those of the model?

• Two phases
• preprocessing
• recognition

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Preprocessing - intro.

• Goal represent the information about the atoms
  of the model molecule in a rotation and
  translation invariant manner.
• Off-line. Why?
• This information will be later used in the
  recognition phase.
• 3 non-collinear atoms specify a unique
  orthonormal reference frame (unique coordinate
  system).
• This will be a full reference frame.

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Preprocessing - intro.

• We wont use a full reference frame only 2 atoms
  (not unique). Those 2 atoms will be called
  reference set.
• Each atom b in the molecule is represented by the
  triplet of distances of the sides of the triangle
  formed between b and the atoms of the reference
  set.

Reference set (c,a)
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Reference frames - clarification
Note the example is in the 2-D case (basic ideas
the same as the 3-D case)
Same shape, different reference frames
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Preprocessing
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Preprocessing

• Hash table
• representation of each model atom
• triplets of distances (from the atom to
  reference pair)
• the corresponding reference pair and the atom
  which obtained this key.
• Note
• each atom has a redundant representation in all
  possible reference sets.
• Many triangles can occupy the same hash table
  entry.

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Preprocessing Complexity Discussion

• The complexity is highly dependent on the
  invariants we use for hashing.
• Complexity O(n3)
• n is the of atoms in the model.
• But We can do better!
• we will later see an optimization that will
  reduce the complexity to O(n2).

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Preprocessing example
Note the example is in the 2-D case (basic ideas
the same as the 3-D case)

• Reference frame here is a pair of coordinates.
• For instance, in cell (3, 2) we find point 2,
  in both reference frames, and so we store those
  reference frames in the hash table H(3, 2).

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Recognition - intro.

• Goal discover candidate matching substructures
  in the target and model molecules.
• Reference set - pair of atoms.
• Each such matching substructure is based on a
  certain reference set, which appears both in the
  model and target molecules.

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Recognition algorithm

• For each reference set of the target
• Hold a vote counter for each reference set
  appearing in the hash table.
• any ideas what will it hold?
• Of course, it will hold the current number of
  matching atoms, and the list of matching pairs.
• We will call this list the vote list.
• In the beginning the list is initialized with
  null.
• Pick a target atom (take predefined threshold
  distance into consideration).

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Recognition - cont.

• Use the 3 sides of the triangle formed to compute
  their hash table key.
• Access the hash table in this key
• Extract all the model triangles in this entry.
• For each triangle
• Vote_counter
• Vote_list.add(current_triangle)
• Go back to picking another atom, until we
  considered all of them.

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Recognition - cont.

• Check the vote counters of all the entries and
  consider the ones with a large of votes.
• Verification.
• Choose another reference set in the target
  molecule and go back to the beginning.
• Complexity O(n3k)
• k indicates the of triangles in each hash table
  entry.
• Can be of order O(n2) after optimizing
  preprocessing.

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Recognition example
Note the example is in the 2-D case (basic ideas
the same as the 3-D case)
For instance, lets look on point f, its
coordinates are (0, 4) and so this is the key to
H. H(0,4) contains the reference frame (1,3),
thus its counter will be increased (a vote for
the base pairs in H) and the pair (7, f) will be
added to the matched list.
Why (7, f)?
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Step 2 - Clustering

• Goal clustering the seed matches that represent
  almost identical transformations.
• Why clustering? Many of the seed matches obtained
  in step 1 represent the same transformation (but
  contain different pairs of matching atoms).
• We use the lists of matching atoms to compute the
  3-D rotation and translation, which gives us the
  minimal least squares distance between the target
  and the model.

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Clustering - cont.

• The computed 3-D transformation has 6 parameters
  (3 for rotation (angles) and 3 for translation
  (distances)).
• Join similar transformations into new groups.
• What's similar?
• Small 6-D distance between the parameter vectors
  of the transformations.
• Clustering algorithm (iterative)
• At the beginning, each seed match forms a group
  represented by 6 parameters of its
  transformation.

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Clustering - cont.

• The pair of groups having the minimal distance
  between their transformations is chosen and a new
  group is formed by merging these two groups.
• Who will be the parameters of the new group?
• A threshold is defined to determine an end to the
  algorithm.
• What do we have so far?
• of groups, each represents one transformation
  obtained by averaging the individual
  transformations that were joined to the group.

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Clustering - cont.

• The seed match of a group is obtained by choosing
  matching pairs from the original seed matches
  that composed the group.
• But, we dont take the union of all pairs!
• Improve accuracy by choosing pairs that appear in
  at least certain percentage of the seed matches.
• The new correspondence lists are considered more
  reliable than in step 1.
• Complexity
• m of seed matches to be clustered.

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Step 3 - Extending

• Goal extend the correspondence lists from step 2
  to contain additional matching pairs.
• Remember! the transformation representing each
  group was computed by taking the average of the
  initial transformation.
• How can we find more matches?
• Compute again a transformation which gives the
  minimal least squares distance between the
  matched pairs.
• The pairs that survive the second transformation
  are candidate additional matches.

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Extending - intro.

• of iterations to extend each seed match
  (small constant).
• e - maximum allowed distance.
• At iteration i we extend the match to contain
  pairs of atoms that lie at a maximum distance of

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Extending - algorithm

• For iteration i
• Find the transformation of the current match
  using least squares procedure.
• Transform the target according to this
  transformation.
• Remove pairs from the current match that lie in a
  distance larger than
• Extend the match by heuristic matching algorithm
  (given a threshold value).

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Extending - cont.

• After iterations, repeat the first 3 steps
  to refine the last matching.
• Complexity as the heuristic matching algorithm
• ( or )
• Output the best extended matches.
• A remainder What is best?
• of matching pairs
• Minimal RMSD between the matching atoms.

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Preprocessing Optimization

• We can do better (complexity wise)!

• Assumption there is spatial proximity between
  the atoms of the relevant matching substructures.

• Conclusion the triangles we will consider are
  those composed of three atoms whose atom-to-atom
  distances are below certain threshold.

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Preprocessing Optimization - complexity discussion

• Maximum allowed distance between the atoms of the
  reference set r1 5Å ( )
• Maximum allowed distance between a third point
  and the atoms of a reference set r2 20Å
• Theoretically, the complexity is now
• Practically,
• Example 138 residues
• 13,359 triangles

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Implementation - Examples

• http//bioinfo3d.cs.tau.ac.il/
• c_alpha_match/prog.html
• 6LYZ vs. 2LZM
• Result 1

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Implementation - Examples
1pmy vs. 1pza
1pmy vs. 1aaj
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Rasmol example
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Results of the algorithm

• 3-D comparison method that isnt constrained by
  linear order of the amino acid chain.
• Self comparison - outputs the best match besides
  the trivial one. Could not be obtained in a
  sequence-dependent method.
• Successful on a wide range of protein comparison
  problems.

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Method discussion - cont.

• 2 factors in structural comparison (might be
  conflicting)
• Sequential order conservation.
• Geometric pattern conservation.
• Most of known methods strict constraint has been
  placed on the search - sequential order
  conservation.
• Much easier (structural alignment is NP-Hard).
• Linear order conservation isnt necessarily
  undesirable
• Comparing proteins whose evolutionary relatedness
  is certain
• But neither desirable
• If the exact evolutionary relationship between
  the structures is unknown
• Possible generic mutations could have occurred

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Method discussion

• Sequence independent
• Help find common 3-D folding units
• Dealing with the question of convergence to a
  similar structure or divergence from a common
  ancestor.
• Classical example TIM barrel proteins.
• Demonstrates that a strictly linear match is not
  the best geometrical match between
• two barrel structures.

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Method summary

• Based on the geometric hashing paradigm.
• Pure 3-D approach (sequence-independent).
• No a-priori knowledge of the motifs nor an
  initial alignment are required.
• Not sensitive to insertions, deletions, gaps or
  displacements of equivalent substructures between
  the molecules being compared.
• Efficient and fully automated.
• Seconds for typical pairwise comparisons.
• Successful on a wide range of protein comparison
  problems.

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Method summary - cont.

• In most of the examples, the best match
  corresponds to a linear alignment match.
• Provides a way to compare proteins without the
  bias of other methods (sequence dependent).
• Capable of discovering partial structural
  similarities.
• Sole criterion geometry!
• Complexity O(n3)

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Extensions and additional features - a look ahead

• The method can be extended to allow simultaneous
  and efficient comparison of a target structure
  with a data base of many model structure.
• Protein and amino acid properties can be
  exploited in the definition of the reference
  frame and thus taken into consideration in the
  algorithm.
• Different choices of interest points.
• Strategies to reduce the of triangles.
• Assigning weights to the matches according to
  certain factors (recognition phase change).
• Extending and adapting the technique to be used
  in the docking problem.

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Lecture summary

• 3D observation of proteins.
• Why structural alignment?
• Studies of catalogued motifs can aid in
  understanding the evolutionary relationship
  between the proteins.
• The method presented allows addressing the
  question of of protein structural classes found
  in nature.
• In particular, the availability of such a library
  is expected to aid in the investigation of the
  protein folding problem.
• Sequence alignment vs. structure alignment.
• Geometric hashing and its use in the algorithm.
• The algorithm and its implementation.
• Extensions and additional features - a look ahead.

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• Questions?

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Thats it