Using Spanners to Describe Protein Structure Leonidas Guibas, Daniel Russel Stanford University

2 Outline Goal Spanners of proteins Trajectories Future work

3 Folding Processes for Macromolecules Folding is a continuous process state of the fold is discrete –proximities between groups of atoms –residue exposure to solvent We want to combinatorialize the folding process

4 Outline Goal Spanners of proteins Trajectories Future work

5 A Geometric Spanner A graph spanner forms a compact encoding of all proximities among the points

6 Protein Structure Descriptors Detect and capture –Proximities –Global conformation –Large-scale changes during motion 3-spanner of BBA5

7 Static Shapes: Alpha Helix … 1VDF Protein sequence

8 Mostly Beta hairpins 1IHV

9 Outline Goal Spanners of proteins Trajectories Future work

10 Folding Steps Matching successive frames –Easy correspondence –Small conformation changes We want stable descriptions

11 Matching Edges Edge from atom i to atom j – Interval (i,j) Find best bipartite matching Spanner edges from successive MD frames

12 Folding Trajectory End of  helix stabilizes  helix and  strand adopt final conformation

13 Spanners are Unstable

14 Edges Come and Go White is  -  Green is  -  Cyan is  - 

15 Some Do Not Fold

16 Outline Goal Spanners of proteins Trajectories Future work

17 Future Work Motif finding/Matching –Induced patterns as intervals –1D matching problem –Gaps Hard Statistical models

18 Simplified Distance Matrices Cover distance matrix with rectangles For each rectangle, the two subsequences are well separated, geometrically

19 Simplified Distances: α-helix Distance MatrixWSP decomposition

20 Simplified Matrix: Mostly β-strands Distance MatrixWSP decomposition  hairpin