Nonlinear Dimensionality Reduction Applied to Problems in Structural Bioinformatics

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Nonlinear Dimensionality Reduction Applied to
Problems in Structural Bioinformatics
Forbes Burkowski Shirley Hui School of
Computer Science University of Waterloo
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Dimensionality Reduction

• Introduction
• Dimensionality reduction is a technique that may
  be used to organize high dimensional data by
  discovering a more compact representation of the
  data.
• We regard the data as a set of points in a high
  dimensional space.
• This strategy is suitable for data that are
  generated by some process that situates the
  points in a lower dimensional manifold of that
  space.

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Data Set Example 1

• The swiss roll data set is a 2D manifold
  situated in a higher dimensional 3D space.
• The goal of a dimensionality reduction exercise
  would be the discovery of the 2D intrinsic
  coordinates that would allow us to see the data
  as it is displayed in figure (c).

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Data Set Example 2

• Suppose we have a camera that takes pictures of a
  face that is rotated through a continuous range
  of angles.
• If a picture is represented using 256 grey scale
  values in a 32 by 32 array then each picture may
  be represented as a point in a space of 1024
  dimensions.
• Since the angle of rotation is a single parameter
  the sequence of pictures corresponds to a
  sequence of points that trace out a 1D curve in
  the 1024D space.

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Motivation

• Why is this important?
• Visualization of high dimensional data
• Data compression
• Recognition of similarity in data sets
• It can be very useful to extract meaningful
  dimensions.
• We hope to determine an intrinsic coordinate
  system for the data.
• We want to unroll the manifold.
• Interpolation

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Working with the geodesic distance

• Often the distance between two points is more
  meaningful when calculated by going through
  adjacent points in the lower dimensional space
  rather than simply using the Euclidean distance
  in the high dimensional space.

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LLE Overview

• We assume that a point Xi in the high dimension
  space resides in a hyper plane that is determined
  by Xi and its k closest neighbouring points.
• Calculate the best weights Wij that would
  linearly reconstruct Xi from its neighbours.
• This is done by solving a constrained
  least-squares problem, minimizing the error
• Weights are zero if Xj does not belong to the
  neighbour set of Xi .
• We also need

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LLE Overview (cont.)

• Compute the low d-dimensional embedding vectors
  Yi that are best reconstructed by the weights
  Wij.
• The Yi minimize
• This is done by finding the smallest eigenvectors
  of the sparse symmetric matrix
• Reference
• Roweis, S.T. and Saul, L.K., Nonlinear
  Dimensionality Reduction by Locally Linear
  Embedding, Science, Vol. 29, Dec. 22, 2000, pp.
  2323-2326.

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LLE Overview (cont.)

• Summary of the 3 steps for LLE
• High dimensional space
• Low dimensional space

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Images of Lips Mapped Via LLE
Images were mapped into the embedding space
described by the first two coordinates of LLE.

• Saul, L.K. Roweis, S.T.
• An Introduction to Locally Linear Embedding
• http//www.cs.toronto.edu/roweis/lle/papers/llein
  tro.pdf

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Images of Faces Mapped Via LLE

• These images were mapped into the embedding space
  described by the first two coordinates of LLE.

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HIV Protease 2D Manifold
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Dimensionality Reduction and Protein Flexibility

• The study of protein motion is difficult because
  of the many degrees of freedom.
• A medium sized protein may have a few thousand
  degrees of freedom.
• Can dimensionality reduction be used to obtain a
  reduced basis representation of protein
  flexibility?
• Teodoro et al. answer in the affirmative
• Teodoro, M.L., Phillips, G.N.Jr., Kavraki, L.E.,
  A Dimensionality Reduction Approach to Modeling
  Protein Flexibility, RECOMB 02, Washington, Apr.
  2002, 299-308.

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Modeling Flexibility of HIV Protease

• Teodoro et al use a molecular dynamics program to
  generate 14,000 conformations of the protein.
• They get a vector set that is subjected to
  dimensionality reduction via PCA (Principle
  Component Analysis).
• The reduced basis representation retains the
  critical information about the directions of
  preferred motion of the protein.
• They have a convincing animation that shows the
  flap movement in 4HVP.

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Nonlinear Dimension Reduction for Proteins

• Can we use LLE in lieu of PCA?
• S. Hui and U. Shakeel (University of Waterloo,
  School of Computer Science graduate students)
  repeated the HIV protease study using LLE instead
  of PCA.
• very good results
• but quite computationally expensive
• Results will be presented in poster form at this
  years ISMB in Glasgow.

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Coordinate Systems for Molecules

• A point in the high dimensional space will have
  coordinates. Some possibilities are
• The 3N numbers representing the Cartesian
  coordinates of N atoms in the molecule.
• The phi/psi angles of the alpha carbons.
• The interatomic distances of particular atoms in
  the molecule (useful in drug design).
• We now look at the use of LLE for another
  application dealing with proteins structure
  alignment.

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Structure Alignment

• From a biological perspective demonstrating the
  similarity of two proteins by comparing their 3D
  structure is very important because protein
  functionality is strongly related to structure.
• From an evolutionary viewpoint Structure is more
  conserved than sequence.

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What is Structure Alignment?

• Objective
• We are given the 3D coordinates of all atoms for
  two proteins P and Q.
• So we know their 3D conformations.
• How do we find the translation and rotation
  operations that lead to the most significant
  amount of superimposition?
• If flexibility of the protein is to be allowed,
  then the transformation operations may include
  some local deformations of either P or Q or both.

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Variations of the Problem

• There are variations of the problem depending on
  whether we allow flexibility and on whether we
  have the same or different sequences.
• Flexible same sequence
• Trivial We simply assume that P and Q can adopt
  the same shape.
• Rigid same sequence
• We find the rotation and translation that
  minimizes the distance between corresponding
  alpha carbons in P and Q.
• Measure via RMSD.
• Rigid different sequence
• More difficult It is necessary to find an
  alignment between P and Q that provides a
  matching between a subset of the amino acids in P
  with those of Q.
• Then we calculate the rotation and translation
  that minimizes the RMSD measured only with
  respect to the matching alpha carbons.

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Variations of the Problem (cont.)

• Flexible and different sequence
• A similar approach to the last case complicated
  by protein deformations that should be made with
  due attention to energy considerations.
• Other concerns
• For both of the last two cases, the alignment
  (matching amino acids) may change as the
  rotation, translation, and deformation
  transformations are performed.
• So many of the algorithms used are heuristic and
  iterative.
• It is possible that a less than optimal alignment
  may lead to a structural alignment that is more
  biologically relevant.
• There is no universally accepted structural
  alignment definition.

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Prior Art

• In all currently available methodologies
  developed and applied to the comparisons of
  protein structures, the molecules are considered
  to be rigid objects
• Verbitsky, G., Nussinov, R., Wolfson, H.,
  Flexible Structural Comparison Allowing
  Hinge-Bending, Swiveling Motions. Proteins
  Structure, Function, and Genetics 34 1999, pp.
  232-254.
• Their paper goes on to describe an algorithm that
  does structure comparison allowing molecular
  parts such as domains, subdomains, and loops, to
  rotate around preselected point-hinges.

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Reverse Chiropraxis

• In our approach we wish to derive a structural
  comparison that involves automatically selected
  bonds chosen as the places where rotation takes
  place.
• So we wish to take the protein backbone from its
  native conformation to some nearby conformation
  that has small rotational alterations to its
  backbone.
• This to be done with due attention to potential
  energy considerations.
• Introducing subluxations of the protein
  backbone ?

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A Structural Alignment Heuristic (1)

• Each protein will have a set of conformations
  each conformation being a point in its space of
  conformations.
• Assuming we have an alignment algorithm that
  establishes the alpha carbons that are to be put
  into correspondence then we can create a
  coordinate system that is based on the positions
  of these atoms.
• In this way, the points representing
  conformations will have the same frames of
  reference for both proteins P and Q.

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A Structural Alignment Heuristic (2)

• If the two manifolds intersect then a point of
  intersection will specify conformations for P and
  Q that are the same (at least with respect to the
  alpha carbons in the alignment).
• It would remain to investigate whether the
  energies of these conformations of P and Q were
  acceptable.
• Ideally, a point of intersection is close to the
  points in the manifolds that represent the native
  conformations of P and Q.

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A Structural Alignment Heuristic (3)

• If the two manifolds do not intersect (or even
  when they do) we would want to find two points,
  one in the manifold for P and the other in the
  manifold for Q such that the points are close to
  one another and correspond to reasonably low
  energy levels.
• We essentially have a search problem in the low
  dimensional space.

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Interpolation (1)

• Calculating the intersection of two manifolds is
  based on the observation that the manifolds are
  defined by discrete sets of points corresponding
  to the conformations that were fed to the
  software doing the dimension reduction.
• A simpler heuristic is to assume that one protein
  (say P) is rigid and so its conformation space is
  really a single point corresponding to the native
  conformation.

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Interpolation (2)

• We find the line through P that is normal to the
  manifold at a point that is on a line between the
  point representing the native structure of Q and
  some neighbouring point.

Native Structure of P
Next interpolated conformation for Q
Conformational Neighbour of Native Structure of Q
Native Structure of Q
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Some Empirical Results

• My grad student Shirley Hui has done various
  experiments using PCA and LLE for dimensionality
  reduction applied to structural alignment.

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Experiments

• The goal of the following experiment was to align
  two protein chains where one is flexible and one
  is rigid.  
• Aligned fragments of the two proteins are
  obtained.
• The alignment of these fragments is improved upon
  by flexing the flexible protein.
• Same data as used in

Shindyalov IN, Bourne PE (1998) Protein
structure alignment by incremental combinatorial
extension (CE) of the optimal path.
Protein Engineering 11(9) 739-747.
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Method

• The flexible structural algorithm developed
  consists of 3 steps
• 1)  Perform a Rigid Alignment
• The initial rigid alignment provides an idea of
  which areas of the structures should be aligned.
  When the rigid alignment is complete, we know
  which areas of the structures are aligned and
  which areas are gaps in both of the chains.
• 2)  Achieve a Tighter Fit
• A tighter fit may be achieved if the flexible
  molecule is flexed in the aligned areas. We do
  not flex in the gap areas.
• 3)  Achieve a Feasible Fit
• Since the chain moves as a system, flexing the
  chain in individual areas in step two may distort
  the structure so much that although it is a
  better fit, it is not one that is energetically
  feasible. In order to achieve a feasible fit,
  the flexible proteins flexibility manifold is
  searched for a conformation that is as close as
  possible to the flexed chain generated from step
  2. The resulting conformation is one that is
  feasible since it lies on the manifold.

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Data Used

• The protein chains used are as follows Example
  One 1) Flexible Chain - C-phycocyanin L Chain
  (atoms 408 - 716)2) Rigid Chain Colicin
  A Chain (atoms 508 - 747)Example Two1)
  Flexible Chain - C-phycocyanin L Chain
  (atoms 2040 - 2315)2) Rigid Chain - Colicin A
  Chain (atoms 955 - 1372)

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Example 1 Results
Blue COL A (assumed rigid) Green CPC L
(assumed rigid) Red CPC L (assumed flexible)
Flexibility allowed more superposition with an
improvement in RMSD of 0.606 Angstrom.
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Example 2 Results
Blue COL A (assumed rigid) Green CPC L
(assumed rigid) Red CPC L (assumed flexible)
Flexibility allowed more superposition with an
improvement in RMSD of 0.489 Angstrom.