Finding Largest Wellpredicted Subset of Protein Structure Models

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Finding Largest Well-predicted Subset of Protein
Structure Models
Shuai Cheng Li, Dongbo Bu, Jinbo Xu and Ming
Li University of Waterloo
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8th Community Wide Experiment on theCritical
Assessment of Techniques for Protein Structure
Prediction
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Life
Phenotype
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Protein Structure Model Assessment

• A number of protein structure prediction methods
  are available.
• each method produces a large number of models
• Evaluation of the quality of these models is a
  difficult and fundamental subject
• Root Means Square Deviation (RMSD) is the most
  popular measuring method.
• RMSD fails to identify the quality of a model
  when only a substructure is predicted correctly.
• RMSD is also not equivalent between targets of
  different lengths.
• the quality of a model of 10 residues with an
  RMSD of 3A is considered bad, while
• the quality of a model of 100 residues with an
  RMSD of 3A is considered an accurate model.
• Intensive studies have been conducted
• MaxSub
• Global Distance Test (GDT)
• Local/Global Alignment(LGA)
• TMScore, etc.
• Most of the proposed methods are heuristic and do
  not have theoretical performance bound
• Min I sqrt (? i1 n ai - I(bi)2)/n.

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Largest Well-Predicted Subset

• Largest Well-Predicted Subset (LWPS) measure is
  proposed
• LWPS is elegant and intuitive
• It was believed to be NP-complete, and heuristic
  approaches have been proposed
• We found that it was actually solvable in O(n7)
  by techniques from computation geometry.

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Outline

• Preliminaries
• A Discretization of the Rigid Transformation
• An Randomized Algorithm for Globular Protein
  Structure
• Approximating the Bottleneck Distance

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Problem Definition

• A protein structure A consists of an ordered set
  of n 3D-points
• A(a1, a2,, an).
• Each point represents a Ca atom.
• A is bounded within a sphere of radius RA.
• RAO(n), for general proteins
• RAcn1/3, for globular proteins
• Distance Constraints
• between two non-consecutive points is no less
  than 4A.
• between any two consecutive points is about
  3.8\AA.
• the maximum number of points that can be
  encapsulated in a given sphere with radius r is
  proportional to the volume of the sphere

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Problem Definition

• The predicted model B of the protein consists of
  an ordered set of n points,
• Bb1, b2, , bn.
• B has the geometry properties of protein
  structure.
• Given a threshold d and a rigid transformation I,
  
• if ai-I (bi)ltd, we say ai matches bi or bi
  fits into ai under I.
• The match set of B is the set of points of B
  matches the corresponding point of A.
• When the context is clear, we simply refer it as
  match set.

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Problem Definition

• Given a protein structure A, a model B and a
  threshold d, the largest well-predicted subset
  problem (denote as LWPS(A, B, d), is to identify
• a maximum set Mopt? 1, 2, , n, and
• a corresponding rigid transformation Iopt (a
  rotation and translation) (See MaxSub, Lan03).
• d is called the bottleneck distance.
• Denote AoptAMopt and BoptBMopt.
• LWPS(A, B, d) is solvable in O(n7).

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Approximation Versions

• Distance Approximation for LWPS(A, B, d)
• Find a transformation T to bring a subset B' ?
  B, such that
• B' is of size at least Bopt, and
• ?bi?B', ai-T(bi) lt(1e)d, for some constant
  e.
• Bottleneck Distance Approximation.
• Find a transformation T such that
• ?bi?B, ai-T(bi) lt(1e)dopt

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Outline

• Preliminaries
• A Discretization of the Rigid Transformation
• An Randomized Algorithm for Globular Protein
  Structure
• Approximating the Bottleneck Distance

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Rigid Transformation

• Rigid transformation T on point set P
• consists of a rotation and a translation.
• Decompose Rigid Transformation T into two steps
• (1) An transformation T
• that transforms an arbitrary chosen radial pair
  ltp1, p2gt of P into their positions under T,
• i.e T(p1)T (p1) and T(p2)T (p2)
• (2) A rotation R around the axis along vector
• T(p1)?T(p2) ? p? P, R(T(p))T(p).
• Approximating Rigid Transformation T on P
• If transformation T moves radial pair ltp1, p2gt
  near enough (e) to T(p1), and T(p2), then there
  exist a rotation R which rotates point set P with
  axis T(p1)?T(p2), such that every point p of P
  is near enough (3e) to T(p).
• Proof omitted

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Matching a Radial Pair

• Assume we know a radial pair ltbi, bjgt of the
  matching set.
• bi is in the d-sphere of ai
• bj is in the d-sphere of aj
• Discretize d-sphere of ai with grid size ?d
• Use grid points as possible positions of T(bi)
• Totally, there are O(1/e3) possible positions for
  T(bi)
• Keep bi fixed, the possible position of T(bj) is
  a sphere
• Centered at T(bi)
• Radius is bi-bi
• Discretize the sphere surface with grid size ?d,
• there are O(1/e2) possible positions for T(bj)
• Totally, there are O(1/e5) possible positions for
  ltbi, bjgt

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Exact Algorithm for Given Rotation Axis

• If B is rotated around a given axis, an angle
  q?0, 2p), such that
• the number of matched pairs is maximized.
• Represent 0, 2p) as a unit circle.
• the angle that moves bi into the d-sphere of ai
  is an arc.
• With a plane sweep approach, we can find a point
  on the circle (which corresponding to an angle)
  which contained by maximum number of circles in
  O(nlogn)
• Approximating the LWPS problem
• For each pair of B,
• Try to match the pair to the corresponding pair
  of B approximately, O(1/?5) choices for each
  pair.
• Rotation B along the pair to find a maximum
  match
• The running time is O(n3logn/e5)

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Outline

• Preliminaries
• A Discretization of the Rigid Transformation
• An Randomized Algorithm for Globular Protein
  Structure
• Approximating the Bottleneck Distance

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Basic Idea

• Algorithm LWPS is inefficient.
• Time complexity O(n3 log(n) )
• Observation
• Given a radial pair of Bopt, we can solve LWPS
  in O(n log(n) / ?5 ).
• If the probability of a pair to be a radial pair
  is high, we can solve the problem with high
  probability by trying only a few pairs instead of
  trying all the possible pairs.

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Two Key Facts

• Pseudo radial pair
• A pair ltbi, bjgt is pseudo radial pair if b1-b2
  gt (1/2 ? n)1/3
• a is some constant.
• Fact 1 Pseudo radial pair will introduce small
  error.
• By creating smaller grid, the constant ? can be
  absorbed.
• Fact 2 There exist enough pseudo radial pairs.
• BMopt contains at least ½ M2opt pseudo radial
  pairs.
• Proof omitted.

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Randomized Algorithm

• We randomly sample 1/?2 log(n) pairs from B.
• There are at least ½ (?n)2 pseudo radial pairs,
• the probability of no pseudo radial pair is
  1-O(1/n).
• Time complexity
• Each pair takes time O( nlogn/ ?5 )
• O(logn) pairs tried
• Total time O(n (logn)2 / ?5 )

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Outline

• Preliminaries
• A Discretization of the Rigid Transformation
• An Randomized Algorithm for Globular Protein
  Structure
• Approximating the Bottleneck Distance

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Approximating the Bottleneck Distance

• To compute the minimum distance d
• such that each point bi in B can fit into
  d-sphere of ai.
• A binary search approach is used here.

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Bound the Bottleneck Distance

• Let RMSD(A, B) to be D, then
• Dltdopt,ltsqrt(n)D
• Proof ommited
• RMSD can be computed in O(n)
• Given D, sqrt(n)D
• Subdivide interval it into subintervals D, 2D,
  (2D, 4D, (4D, 8D, , (2kD, 2k1D,
• Totally there are O(loglog(n)) sub intervals
• Use binary search to bound for dopt in one of the
  subintervals.
• Assume dopt?(2kD, 2k1D for some k.
• Subdivide (2kD, 2k1D into subintervals of size
  0.5?2kD?
• Totally there are O(?) intervals.
• Apply a binary search again to further bound dopt
  into one of the subinterval
• Totally time complexity is O(n(loglognlog1/e)
  /?5)
• Each binary search operation takes time O(n/?5)
• Since we want to match all the point in B, a
  radial pair can be pre-computed

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Experimental Results
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• TMscore and MaxSub are two popular package for
  protein structure comparison.
• We compare ApproxSub against MaxSub and TMscore
  in terms of finding the number of matched pairs.
• Testing set includes 1fc2, 1enh, 2gb1, 2cro,
  1ctf, and 4icb. These six proteins are commonly
  used for testing protein structure prediction
  methods.

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• X the ratio of matched pair by MaxSub (TMscore)
• Y the ratio of matched pair by ApproxSub
• Observation
• ApproxSub can find more matched pairs than MaxSub
  and TMscore
• MaxSub/TMscore are poor to find matched pairs
  when ratiogt0.60.
• This is due to the heuristic nature of these two
  methods, and they extend matches by superimposing
  a local fragment of A and B first. This may trap
  them at a local minimum.

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• Running Time
• X number of residues of protein
• Y CPU time (sec)
• Observation
• ApproxSub is much faster than TMscore.

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• Q and A