Role of Rigid Components in Protein Structure

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Role of Rigid Components in Protein Structure

• Pramod Abraham Kurian

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Objectives

• Finding and Maintaining Rigid components in a
  Protein
• Rigidity in Higher Dimensions
• Identifying Over constrained regions
• Identifying Under constrained regions
• FIRST (Floppy Inclusion and Rigid Substructure
  Topography)

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Finding and Maintaining Rigid components in a
Protein

• Rigidity in Planar bar-and-joint framework
• 1) Leman Graph
• A graph is generically
  minimally rigid in 2D if and only
• if it has 2n-3 edges and no
  subgraph of k vertices has
• more than 2k-3 edges.
• 2) Pebble Game (2D)
• Each node is assigned 2 pebbles 2n
  pebbles in total.
• An edge is covered by having one
  pebble placed on either of its ends .
• A pebble covering is an assignment
  of pebbles so that all edges
• in graph are covered

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• Rigidity in Higher Dimensions
• 1) D dimension body-and-bar framework
• N rigid bodies connected by
  rigid bars
• 2) (k, l) Sparse Graph
• Generalized (k,l)-Pebble Game

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Definitions

• (k, l) Sparse Graph
• A multi-graph on n vertices is
  (k, l) sparse if every subset of n lt n
  vertices spans at most kn - l edges, 0 lt l lt
  2K
• k arborescence
• A multi-graph is a k-arborescence
  if it is the union of k ,
• edge-disjoint spanning trees
• (k, a)-arborescence
• A multi-graph is a (k, a)-arborescence if
  the addition of any a edges results in a
  k-arborescence.

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D dimension body-and-bar framework

• A body-and-bar framework is a structure built
  from n rigid bodies connected by rigid bars
• Induces a Graph/Mapped to a graph (How ??)
• Vertex in graph associates to each body and an
  edge to each bar.
• Tay Theorem
• states that the structure is
  (generically) rigid in dimension d, iff the
  associated graph is a k-arborescence, for k
  (d1)
• 
  2
• ( Illustration - figure 1)
• If bars are removed from a rigid structure (and
  edges
• from the corresponding graph), the structure
  becomes
• flexible.
• Some Parts may still be connected together in a
  rigid fashion .Such rigid sub-substructures are
  called maximal sub-arborescence.

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Continued.
Figure 1
The corresponding graph decomposes into 6
edge-disjoint spanning trees.
Generic minimally rigid body-hinge-and- bar
framework in 3d four rigid bodies joined along
three hinges and three bars.
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• Fundamental Problems with Graph Rigidity
• 1) Decision Problem
• asks if G is minimally rigid
• 2) Extraction Problem
• asks for a maximal, minimally rigid
  subgraph of G.
• 3) Optimization Problem
• When weights are given for the edges of
  G, the Optimization problem asks for the maximum
  weight, minimally rigid subgraph of G.
• 4) Component Problem
• Given a graph with some fexibility, the
  Components problem asks for G's maximal rigid
  subgraphs, or components.

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Double Banana Problem

• http//www.cs.concordia.ca/cccg/papers/42.pdf

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Quick Reminder

• If a subgraph has more edges than necessary, some
  edges are redundant.
• Non-redundant edges are independent.
• Each independent edge removes a degree of
  freedom.
• Therefore, 2n-3 independent edges guarantee
  rigidity.

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Pebble Game

• 3D Pebble Game
• 1) 3 pebbles, representing 3 translation
  degrees of freedom is assigned to each vertex
  in the graph.
• 2) Free Pebbles Pebbles associated with
  vertices and they represents independent degrees
  of freedom remaining in the network.
• 3) Covering Rule Once an independent
  constrain is covered by a pebble, it must always
  be covered by any pebble associated with the
  incident sites.
• 4) Rearrangement is possible, without
  violating covering rule.

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3D Pebble Game(Conti)

• 5) Essential feature of 3D pebble game is
  that each distance constrain associated with
  vertices V1 and V2 must have associated with it a
  second nearest neighbor constrain around both V1
  and V2.
• 6) For each new distance constrain,
  pebbles are rearranged to test if it is an
  independent distance constrain or not.
• 7) If it is independent, then it is
  covered by a pebble
• else
• it is uncovered

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3D Pebble Game Algorithm

• Place a distance constrain between V1 and V2.
• Rearrange pebbles covering to collect 3 pebbles
  on vertex V1.
• Rearrange the pebble covering to get maximum
  number of pebbles in vertex V2, while holding 3
  pebbles in vertex v1
• If the number of pebbles on vertex V2 is 2, then
  the distance constrain is redundant.
• OTHERWISE, 3 pebbles reside at vertices V1
  and V2
• 1) Hold the 3 pebbles on both V1 and
  V2.

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Algorithm (Conti)

• 2) For each neighbor of vertex V2, attempt
  to collect a pebble.
• 3) If for any neighbor of vertex V2 a
  pebble cannot be obtained, then that distance
  constraint is redundant.
• If the distance constrain is not redundant, cover
  it with a pebble from vertex V2.

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Demo (3,3)-Pebble Game
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Generalized (k,a)-Pebble Game

• Given a graph G(V,E)
• Place k pebbles on each vertex
• Add edges one at a time, in arbitrary order
• Acceptance condition for an edge
• Collect a 1 pebbles on the endpoints.
• If edge is accepted
• as directed edge
• Cover with a pebble from the source endpoint
• Successful termination
• All edges have been inserted successfully
• Well-constrained
• No more edges can possibly be inserted

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Identifying Over constrained Regions

• Redundant constrains are identified by failed
  pebble search
• A failed pebble search physically means a length
  mismatch between pairs of vertices.
• This means the bond length and the angle within
  this region will become distorted and internally
  stressed.
• These areas are identified as Over constrained
  regions.
• So as distance constrained are added to the
  network more over constrained regions can be
  found.
• As this continues at one subtle point, stress can
  propagate from 1 floppy region to another
  (Feature not in 2D).

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Identifying Under Constrained Regions

• Rigid Clusters
• 1) Bulk Vertex
• When all its neighbor vertices
  belong to the same rigid cluster
• 2) Surface Vertex
• When atleast one neighbor belong
  to a different rigid class
• So a labeling scheme for vertices in rigid
  clusters, Bulk Vertices will have a unique
  cluster label and surface verteces will have
  different cluster label.
• To identify under constrained regions find hinge
  joint.
• To identify hinge joint, check the 2 incident
  vertices associated with distance constrain
• If the vertices have different cluster label,
  then dihedral rotation is possible and joint is
  hinge joint , else dihedral angle is locked.

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Reference

• Flexibility and Rigidity in proteins, Donald
  Jacob Michael Thorpe
• An Algorithm for Rigidity Percolation Pebble
  Game, Donald Jacob
• Pebble Game Algorithm for (k, l) Sparse Graph,
  Audrey Lee
• http//www.cs.concordia.ca/cccg/papers/42.pdf