Using Spanners to Describe Protein Structure

Leonidas Guibas, Daniel Russel

Stanford University

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Outline

• Goal• Spanners of proteins• Trajectories• Future work

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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

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Outline

• Goal• Spanners of proteins• Trajectories• Future work

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A Geometric Spanner

A graph spanner formsa compact encoding of allproximities among the points

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Protein Structure Descriptors

• Detect and capture – Proximities – Global conformation– Large-scale changes

during motion

3-spanner of BBA5

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Static Shapes: Alpha Helix

…1VDF

Protein sequence

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Mostly Beta

hairpins

1IHV

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Outline

• Goal• Spanners of proteins• Trajectories• Future work

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Folding Steps

• Matching successive frames– Easy correspondence– Small conformation changes

• We want stable descriptions

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Matching Edges

• Edge from atom i to atom j– Interval (i,j)

• Find best bipartite matching

Spanner edges from successive MD frames

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Folding Trajectory

End of helix stabilizes

helix and strand adopt

final conformation

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Spanners are Unstable

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Edges Come and Go

White is -Green is -Cyan is -

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Some Do Not Fold

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Outline

• Goal• Spanners of proteins• Trajectories• Future work

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Future Work

• Motif finding/Matching– Induced patterns as intervals– 1D matching problem– Gaps Hard

• Statistical models

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Simplified Distance Matrices

• Cover distance matrix with rectangles• For each rectangle, the two

subsequences are well separated, geometrically

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Simplified Distances: α-helix

Distance Matrix WSP decomposition

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Simplified Matrix: Mostly β-strands

Distance Matrix WSP decomposition

hairpin