Transcript Seminar in structural bioinformatics

Seminar in structural
bioinformatics
Pairwise Structural Alignment
Presented by: Dana Tsukerman
1
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
2
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.
3
Definitions
• Sequence alignment (remainder from last
lecture), unambiguously distinguishes only
between protein pairs of similar structure and nonsimilar 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.
4
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).
5
Quick rehearsal - Basic terms
regular
hydrogen bond
patterns of
backbone atoms
6
3D observation of proteins
Turn or coil
Alpha-helix
Beta-sheet
Loop and Turn
7
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.”
8
Comparison in 3D
• Starting from an example:
A
B
A
B
C
E
back
D
C
E
D
9
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.
10
Why 3D comparison?
11
Why 3D comparison?
Wait a minute isn’t sequence
comparison
enough?
12
Why 3D comparison?
• Structures are more conserved than
sequences.
• Detection of distant evolutionary
relationships.
• Structural alignment can imply a functional
similarity that isn’t detectable from a
sequence alignment.
• The protein docking problem.
• Structure based drug design.
• Applications and implications to the
protein folding problem.
13
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.
14
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.
Most atoms
• NP-Hard.
matched with the
What’s
“best”?
lowest RMSD
15
Our goal
Find out the
correspondence
between the
structures
transformation T
16
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.
17
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.
x  Rx  T
x
3
• 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
 || x  y ||
i
RMSD(n; x, y ) 
2
i
i
Where,
n
• 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.:
RMSD(n; T *( x), y )  min T
2
||
T
*(
x
)

y
||

i
i
i
n
• We know how to do that in O(n).
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RMSD - Example
X  (x , x , x )
1
i
1
i1
1
i2
1
i3
i
j
X  (x , x , x )
2
j
2
j1
2
j2
2
j3
20
Other methods for structural
alignment
• Dynamic programming - building a score matrix,
with a score for each pair of residues. (n5 ) or (n 4 )
• 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
couldn’t be approached previously by sequenceorder-dependent structural comparison methods,
like the docking problem.
21
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 couldn’t be found similar by a
progression rule based method (sequence dependent).
Example
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PDB - Protein Data Bank
http://www.rcsb.org/pdb/index.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.
23
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 system’s
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 won’t change.
HOW? Building triangles!
for n nodes: n3 triangles!
The triangle’s sides length
doesn’t change when we move
it or rotate it, and thus
invariant!
25
now please pay attention…
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The algorithm – major steps
1. Find (relatively small) subsets of the
structures that form an initial match;
2. Find clusters in initial matches that
represent similar transformations;
3. Extend the clusters to contain additional
matching pairs of residues.
27
Motivation remainder
28
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?
29
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: Cα
atoms).
• Model & Target.
30
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
31
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.
32
Preprocessing - intro.
• We won’t 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)
b
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
34
Preprocessing
How to store the
information
efficiently?
35
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.
36
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).
37
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).
38
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.
39
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).
40
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(n3*k)
• k indicates the # of triangles in each hash table
entry.
• Can be of order O(n2) after optimizing
preprocessing.
42
Recognition
example
Note: the example is in the 2-D case (basic ideas the same as the 3-D case)
For instance, let’s
look on point #f, it’s
coordinates are (0, 4)
and so this is the key
to H. H(0,4) contains
the reference frame
(1,3), thus it’s 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.
44
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 it’s transformation.
45
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 don’t 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: o(m 2 )
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.
48
Extending - intro.
• N it :# of iterations to extend each seed
match (small constant).
• ε - maximum allowed distance.
• At iteration i we extend the match to
contain pairs of atoms that lie at a
i
maximum distance of  
N it
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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
i

N it
• Extend the match by heuristic matching
algorithm (given a threshold value).
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Extending - cont.
• After N it iterations, repeat the first 3 steps to
refine the last matching.
• Complexity: as the heuristic matching algorithm
( O ( n) or O(n3 ) )
• 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.
52
Preprocessing Optimization complexity discussion
• Maximum allowed distance between the atoms
of the reference set: r1 = 5Å ( 1Å = 1010 m )
• Maximum allowed distance between a third
point and the atoms of a reference set: r2 = 20Å
• Theoretically, the complexity is now o ( n )
2
• Practically, o( n )
• Example: 138 residues
r1
r1
r2
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 isn’t
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.
57
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 isn’t 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
58
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.
59
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.
60
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)
61
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.
62
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 it’s use in the algorithm.
• The algorithm and it’s implementation.
• Extensions and additional features - a look ahead.
63
64
That’s it…
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