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 GcGc Decomposing G into Relevant and Irrelevant regions with respect to (core) G c G Core has 7 free pebbles: 6 trivial DOF + 1 internal (extra) DOF

Core GcGc 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) G c G

Decomposing G into Relevant and Irrelevant region with respect to G c (core) Intuition … Imagine we remove this edge Core

Decomposing G into Relevant and Irrelevant region with respect to G c (core) Returning pebble off that edge to this vertex (body) Core

Decomposing G into Relevant and Irrelevant region with respect to G c (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 G c (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 G c (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 G c (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 In the algorithm we keep track of these vertices Core Relevant and Irrelevant regions GcGc

In the algorithm we keep track of these vertices 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 G c (core). Algorithm Relevant and Irrelevant regions

In the algorithm we keep track of these vertices 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 Core GcGc

Irrelevant Region 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 predicted hinge location PDB ID: 1k23

HAG residues PDB ID: 1k23 Hinge prediction by finding Relevant regions: residues 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)

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 R 1, extra free pebble appears on R 2  Number of free pebbles on R 2 will increase from 8 to 9 (1 extra DOF) Add an edge in R 1  a free pebble from R 2 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) R 1, R 2 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 G c G. The maximum number of free pebbles we can draw back to G c is invariant under different plays of the pebble game on G. i.e. Greedy property is preserved for subgraphs Core GcGc