Seminar in structural bioinformatics - BioInfo3D Group

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1

Seminar in structural
bioinformatics

Pairwise Structural Alignment

Presented by: Dana Tsukerman

2

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

3

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.

4

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.

5

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).

6

Quick rehearsal
-

Basic terms

regular
hydrogen bond
patterns of
backbone atoms


7

3
D observation of proteins

Alpha
-
helix

Beta
-
sheet

Loop and Turn

Turn or coil

8

3
D 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.


9

Comparison in
3
D


Starting from an example:


A

B

E

C

D

B

A

E

D

C

back

10

Comparison in
3
D


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.

11

Why
3
D comparison?

12

Why
3
D comparison?

Wait a minute
-

isn

t sequence
comparison
enough?

13

Why
3
D 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.

14

Why
3
D 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.

15

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.


What

s

best

?

Most atoms
matched with the
lowest RMSD

16

Our goal

Find out the
correspondence
between the
structures


transformation T

17

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.

18

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.

19

RMSD
-

rehearsal


A tool we use to evaluate the correspondence we
found.


RMSD
-

R
oot
M
ean
S
quare
D
eviation



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).

20

RMSD
-

Example


21

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
3
D space to
2
D 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 sequence
-
order
-
dependent structural comparison methods,
like the
docking problem
.

22

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

23

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.

24

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.

25

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 nodes: triangles!

The triangle

s sides length
doesn

t change when we move
it or rotate it, and thus
invariant!

26




now please pay attention


27

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.

28

Motivation remainder

29

Step
1
-

Finding seed matches


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


Remember what we talked about in
geometric hashing?


How to represent?


Extensive search of the structures.


Most difficult and time consuming step.


Those will be referred as
seed matches
.

30

Finding seed matches

-

cont.


Seed

match

-

list of matching pairs of atoms.


Model & Target.


Assumption
: the structures to be compared
are described by sets of interest points and
their
3
-
D coordinates (for example: C
α

atoms)
.


Pair

-

correspondence between atoms from
different structures.

31

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

32

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
.

33

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)

c

b

a

34

Reference frames
-

clarification

Same
shape,
different
reference
frames

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

35

Preprocessing

How to store the

information
efficiently?

36

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.

37

Preprocessing Complexity
Discussion


The complexity is highly dependent on the
invariants we use for hashing.


Complexity:

O(n
3
)


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(n
2
).

38

Preprocessing example



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
).

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

39

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.

40

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).

41

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.

42

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(n
3
*k)



k indicates the # of triangles in each hash table
entry.


Can be of order O(n
2
) after optimizing
preprocessing.

43

Recognition example

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)?

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

44

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.

45

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.

46

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.


47

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:



m = # of seed matches to be clustered.

48

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.

49

Extending
-

intro.



:# 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
maximum distance of

50

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).


51

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.

52

Preprocessing Optimization


We can do better (complexity wise)!


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


Assumption:

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

53

Preprocessing Optimization
-

complexity discussion


Maximum allowed distance between the atoms
of the reference set: r
1
=
5
Å ( )


Maximum allowed distance between a third
point and the atoms of a reference set: r
2
=
20
Å


Theoretically, the complexity is now


Practically,


Example:
138
residues










13
,
359
triangles

r
1

r
1

r
2

54

Implementation
-

Examples



http://bioinfo
3
d.cs.tau.ac.il/

c_alpha_match/prog.html


6
LYZ vs.
2
LZM

Result
1

55

Implementation
-

Examples

1
pmy vs.
1
pza

1
pmy vs.
1
aaj

56

“Rasmol” example

57

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.

58

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

59

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.

60

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.

61

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(n
3
)

62

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
.

63

Lecture summary


3
D 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.

64

65

That’s it…