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Pebble game extensions- Detecting relevant regions,
protein hinge motions, allostery …
Adnan Sljoka, York University
Work with Walter Whiteley
Applying Relevant Region Algorithm to Detect Hinge Locations
Inorganic pyrophosphatase PDB ID: 1k23
Froda (Framework Rigidity Optimized Dynamic Algorithm)
Immunoglobulin
Initial motivation for relevant/irrelevant regions
Froda (Framework Rigidity Optimized Dynamic Algorithm)
Immunoglobulin
Core
Gc
Decomposing G into Relevant and Irrelevant regions with respect to (core) Gc G
Core has 7 free pebbles:
6 trivial DOF +
1 internal (extra) DOF
Core
Gc
Intuition: Which region (subgraph) of the multigraph G outside of the core can we remove so that the maximum number of free pebbles on the core remains unchanged? Call it irrelevant region.
Which region (when removed) would increase the number of free pebbles on the core? Call it relevant region.
Decomposing G into Relevant and Irrelevant regions with respect to (core) Gc G
Decomposing G into Relevant and Irrelevant region with respect to Gc (core)
Intuition …
Imagine we remove this edge
Core
Decomposing G into Relevant and Irrelevant region with respect to Gc (core)
Returning pebble off that edge to this vertex (body)
Core
Decomposing G into Relevant and Irrelevant region with respect to Gc (core)
No outgoing edges here, so removal of that edge has no effect on max. number of free pebbles on the core (it is still 7)
Core
Decomposing G into Relevant and Irrelevant region with respect to Gc (core)
Core
Removing this entire dangling end does not change the answer on the core (this subgraph is irrelevant).
Decomposing G into Relevant and Irrelevant region with respect to Gc (core)
Core
Removing this entire dangling end does not change the answer on the core (this subgraph is irrelevant).
Similarly (no outgoing arrows)
Decomposing G into Relevant and Irrelevant region with respect to Gc (core)
Core
Removing edges in this subgraph does change the answer (relevant)!
1. Run the pebble game algorithm on G
2. Draw back maximum number of free pebbles to the core (region of interest)
3. If there are any out-going edges from the core, look for a (capped) failed search region to identify the relevant region
Algorithm
Core
Relevant and Irrelevant regions
Gc
1. Run the pebble game algorithm on G
2. Draw back maximum number of free pebbles to the core (region of interest)
3. If there are any out-going edges from the core, look for a (capped) failed search region to identify the relevant region
This pebble is frozen, it is part of Gc (core).
Algorithm Relevant and Irrelevant regions
1. Run the pebble game algorithm on G
2. Draw back maximum number of free pebbles to the core (region of interest)
3. If there are any out-going edges from the core, look for a (capped) failed search region to identify the relevant region
Relevant
Region
This outgoing edge indicates that the relevant region has removed one pebble (DOF) from the core
Algorithm Relevant and Irrelevant regions
CoreGc
IrrelevantRegion
Relevant + Core
Removing the entire irrelevant region does not change the maximum number of free pebbles (7) (DOF) on the core.
Relevant and Irrelevant regions
For test cases we selected proteins from Hinge Atlas Gold (HAG) dataset - a set of proteins with manually annotated hinge locations available at MOLMOV
Mark Gerstein Lab
Finding Relevant Regions as a Hinge Detection Algorithm
Run flexibility analysis with FIRST
Select two largest (appropriate)
rigid clusters from the Rigid Cluster Decomposition
Inorganic pyrophosphatase PDB ID: 1k23
Define two rigid clusters as Core
PDB ID: 1k23
Find the Relevant region of Core,
which is the predictedhinge location
PDB ID: 1k23
HAG residues 189-190
PDB ID: 1k23
Hinge prediction by finding Relevant regions: residues 189-190
Free Pebbles on Core = 9 = (6+3) Hinge of 3DOF
We can also extract DOF count for hinge.
Hinge detections …
pdb: 2lao (LAO Binding Protein)
Multiple hinge regions can be detected.
pdb: 2lao (LAO Binding Protein)
Not all paths between 2 rigid clusters are relevant (i.e. some connections are too flexible)
Which clusters and energy cutoff to select?
pdb: 3cln (Calmodulin)
pdb: 3cln (Calmodulin)
Exact Match with Hinge Atlas Gold (HAG) prediction residues 80, 81
Free Pebbles = 7 = (6+1) Hinge of 1DOF
Toy model of allostery: finding relevant/irrelevant region
• Remove an edge from R1, extra free pebble appears on R2 Number of free pebbles on R2 will increase from 8 to 9 (1 extra DOF)
• Add an edge in R1 a free pebble from R2 will be removed
• The possible transmission paths should occur only occur over the relevant region
Toy model of allostery: finding relevant/irrelevant region
Application: Allostery (on/off switch in proteins)
We suspect, that allosteric communication occurs over the relevant region, and not irrelevant!
Hinge with 3 DOF (9 free pebbles – 6)
R1, R2 rigid regions (i.e. draw back max. 6 free pebbles)
Hinge with 1 DOF (7 free pebbles – 6)
Theorem: Let G = (V, E) be a multigraph (edge multiplicity at most six) and Gc G. The maximum number of free pebbles we can draw back to Gc is invariant under different plays of the pebble game on G.
i.e. Greedy property is preserved for subgraphs
Core
Gc