QSD Quadratic Shape Descriptors

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QSD Quadratic Shape Descriptors

• Surface Matching and Molecular Docking Using
  Quadratic Shape Descriptors

Goldman BB, Wipke WT. Quadratic Shape
Descriptors. 1. Rapid Superposition of Dissimilar
Molecules Using Geometrically Invariant Surface
Descriptors. J.Chem. Inf. Comput. Sci., 40 (3),
644 -658, 2000
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QSD idea
Define a geometrical invariant representation of
small surface sections (if two molecules have a
similar surface region then its small parts are
also similar) . In case a geometrical invariant
allows to define a reference frame then the
number of all superpositions is nm. n (m) -
number of invariants in the first (second)
molecule Principle curvature and principle
directions provide an elegant formalism that
captures these notions.
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Reminder curvature properties
k1 gt k2 gt k3 0
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knormal curvature - curvature of normal section
at p Principal Curvatures kmax , kmin -
normal curvatures with maximal-minimal
values Principal Directions ? max , ? min -
tangent vectors associated with principal
curvatures. kmax ? kmin ? ? max - ? min
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Molecular Surface Calculation

• The preprocessing stage of the algorithm computes
  the molecular surface of a molecule by using the
  original Connolly MS program.

Critical Points Calculation

• The critical points of the surface as defined by
  Lin et al.40 are calculated.
• These critical points are the center of gravity
  of each face of the Connolly surface projected
  back onto the surface.

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Critical Points

• To reduce the number of the critical points used
  to describe a molecule, the critical points
  associated with the toroidal sections (light
  purple) of the surface are not used.

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S p1, ..., pn, where p (v, n) is composed
of the surface point location v in
three-dimensional space and n is the unit vector
normal to the surface at p.v
C c1, ..., cm - set of critical points, where
ci in S
Surface neighborhood around c
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N is transformed s.t. c.v (0,0,0) c.n
(0,0,1)
Redefine points N
Hessian matrix (second fundamental form)
Local principal curvatures and directions are
eigenvalues and eigenvectors, respectively, of
the II matrix.
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Calculate matrix II by fitting the points of N to
the second order part of the Taylor expansion of
w
w(u,v)
Notice w(0,0)0 and so the first derivatives.
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Finally, two right-handed orthogonal coordinate
systems can be constructed from the local
principal curvature directions
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Principal curvature directions are in cyan.
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Shape Index

• (? min, ? min) and (? max, ? max) represent
  the local principal curvatures and directions of
  the surface patch.The shape index represents the
  degree of concavity of a local surface section
  and is defined by

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Shape Index Similarity

• The shape index provides a convenient mechanism
  for determining the similarity between two
  section of surface.
• The Similarity measure for two surface patches
  with shape indexes S1 and S2 is

1.0 shapes are identical 0.0 shapes are
exactly opposite
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Total Shape Similarity Score Y

• The score is simply a summation of the individual
  similarity scores for each pair of matching
  descriptors.ML ml0,,mln, where ml (ri,lj)
  indicates that ith QSD on the receptor matchs the
  jth QSD on the ligand.S(ml.x) represent the
  value of the shape index S for the match list QSD
  ml.x.

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QSD Preprocessing Algorithm.
Input M Coordinates of Molecule ? Distance
parameter Variables A Alignment Matrix
S Shape Index Algorithm Create molecular
surface for molecule M the Connolly algorithm.
Calculate critical points C c1,,cm of
surface using Lins method. for each c ? C
(c,S,A) ? Create QSD at point c with distance
range ? store (c,S,A) end
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Surface matching phase

• This phase of the algorithm commences with the
  input of the ligand and proteins atomic
  coordinates along with the set of quadratic shape
  descriptors approximating threir molecular
  surface.
• The surface of the active site has been inverted,
  and shape complementary between the ligand and
  receptor surfaces is referred to as shape
  similarity.
• An additional input parameter, the shape filter
  ?S, is used as a filter to determine the extant
  of similarity between two surface sections.

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• Surface matching phase
• InputML,MR Coordinates of Ligand and
  receptorQL,QR QSD set describing Ligand and
  receptor?S Shape Filter
• Algorithmfor each ql ? QL for each qr ? QR
• if (ql.S qr.S) ? ?S) Dock QL to QR
  as dictated by alignment of ql to qr if
  (sufficient QSDs from QR superimpose on QSD from
  QL) Dock ML onto MR as dictated by
  alignment of ql onto qr if
  (acceptable steric clash between MR and
  transformed ML) store docking end
  if end if end if end forend for

Steric collisions are quickly evaluated usinga
three-dimensional grid-based procedure.
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Scoring

• The scoring module uses three types of scoring
  routines to prioritize the computed dockings
• Empirical estimate of ?gbinding (using Bohms
  algorithm).
• Measure of shape similarity ?.
• Clustering algorithm.

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Matching Scoring Phase Complexity

• Let n,m represent the number of QSDs used to
  describe the shape of the target molecule and the
  moving molecule.
• The total number of the dockings calculated
  O(mn).
• For each docking calculated, all of the QSDs in
  the moving set are transformed, matched with QSDs
  in the target set and then the surface similarity
  score assessed.
• The total complexity of the matching phase is
  thus O(nm2).

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High level flow chart for QSD docking algorithm
Create Molecular Surface for Ligand and Receptor
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High level flow chart for QSD docking algorithm
Create Molecular Surface for Ligand and Receptor
Calculate Molecular Surface Critical Points
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High level flow chart for QSD docking algorithm
Create Molecular Surface for Ligand and Receptor
Calculate Molecular Surface Critical Points
Preprocessing
Calculate Quadratic Shape Descriptors
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High level flow chart for QSD docking algorithm
Create Molecular Surface for Ligand and Receptor
Calculate Molecular Surface Critical Points
Preprocessing
Calculate Quadratic Shape Descriptors
Dock Ligands To Receptor Using QSD
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High level flow chart for QSD docking algorithm
Create Molecular Surface for Ligand and Receptor
Calculate Molecular Surface Critical Points
Preprocessing
Calculate Quadratic Shape Descriptors
Dock Ligands To Receptor Using QSD
Object Recognition
Score Successful Dockings
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Preprocessing Times
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Crystallographic Scores
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QSD Matching Results
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QSD Docking Results on Ligand Into Protein and
Comparison With Cocrystalized Structure Position
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Comparison of QSDock a Times to DOCK2 and
Geometric Hashing (GH)
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Conclusion

• QSDock is capable of reproducing the
  crystallographically determined orientations
  using only shape.
• QSD for shape-based docking dretically reduces
  the computational complexity of the docking
  problem.

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Preprocessing

• The preprocessing algorithm accepts as input the
  three-dimensional coordinates of a molecule and
  calculate the set of QSDs describing its surface
  shape.
• The preprocessing is done only once for each
  molecule.

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Shape Descriptors Calculation

• A QSD is a macroscopic interpretation of the
  classical differential geometric surface
  properties of principal curvatures and principal
  directions.
• A QSD is calculated by least-squares fitting of a
  quadratic surface to a 2.0 Å circular patch of
  molecular surface surrounding a critical point.
• After the least-squares fitting procedure, the
  principal curvatures and directions of the
  surface at p are calculated.