Exploring Protein Folding Trajectories Using Geometric

SpannersDaniel Russel and Leonidas Guibas

Stanford University

Goals

● Existing work understanding protein structure– Classification– Comparison– Matching

● Qualitative understanding of proteins motions– We do not quite get there yet

Outline

● Overview of spanners● Properties of interest● Displaying trajectories● Future work

Geometric Spanners

● Vertexes are points● Edges have length● Expansion factor

– Ratio between graph distanceand Euclidean distance

Geometric Spanners

● Vertexes are points● Edges have length● Expansion factor

– Ratio between graph distanceand Euclidean distance

2CI2 Spanner

Expansion factor is 2Expansion factor is 2

backbone atom index

back

bon

e a

tom

ind

ex

spanner edges

2CI2 Spanner: Helix

helix

backbone atom index

back

bon

e a

tom

ind

ex

2CI2 Spanner: Strands

parallel strands

antiparallel strands

backbone atom index

back

bon

e a

tom

ind

ex

2CI2 Spanner: Tertiary Structure

tertiary interactions

backbone atom index

back

bon

e a

tom

ind

ex

Outline

● Overview of spanners● Properties of interest● Displaying trajectories● Future work

Combinatorializes Structure

● Picks a few pairs per set of adjacencies

Other Properties

● Regular structure of helices and strands

● Local meaning for edge

Family of Descriptors

● Vary expansion factor ● From distance matrix to sparse● Linear for many values

1.5 1.75 2.0 2.5 3.0

Outline

● Overview of spanners● Properties of interest● Displaying trajectories● Future work

Protein Folding Trajectories

● Input: Protein trajectories– Currently small proteins, later larger– Nanosecond timescales, later millisecond– Tens or hundreds of folding trajectories– Many non-folding

● Want to understand– Individual trajectories– Structure of conformation space

Processing Spanners

● Edges move slightly between frames● Tracked edge

initial frame edgesfinal frame edgesidentical edges

(identical edges colored grey or magenta)

some m

atching pairs

Another View

Initial edgesFinal edges

backbone index

Smoothing

● Birth and death events● Persistent edges● Gaps

time

pers

iste

nt

ed

ges

Tracked edge

BirthDeath

Strip Spanner Representation

start vertex

length

Strip Historyedge length

index along backbone

fram

e

brighten

Segmenting Trajectories

sample conformation

segmentdividers

Example: BBA5

Native state(1) (3)

(2)

Outline

● Overview of spanners● Properties of interest● Displaying trajectories● Future work

Sidechains

●

● Breaks the ordering of nodes● How many points?

Comparing Motions

● Compare pairs of trajectories– Partial matches

● Cluster conformations– Find bottlenecks

● Cluster trajectories– Which are similar– Which are unique

Deeper Understanding

● Proteins → spanner

● Graph → proteins

● Spanner distances vs. conformation distances