Role of Rigid Components in Protein Structure Pramod Abraham Kurian

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)

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

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

Definitions (k, l) Sparse Graph A multi-graph on n vertices is (k, l) sparse if every subset of n’ < =n vertices spans at most kn’ - l edges, 0 <= l < 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.

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 = (d+1) 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.

Continued…. Generic minimally rigid body-hinge-and- bar framework in 3d: four rigid bodies joined along three hinges and three bars. The corresponding graph decomposes into 6 edge- disjoint spanning trees. Figure 1

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.

Double Banana Problem s/42.pdf

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.

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.

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

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.

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.

Demo: (3,3)-Pebble Game

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

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).

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.

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