1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Nov 22, 2013: Topological methods for exploring low-density states in biomolecular
folding pathways.
Fall 2013 course offered through the
University of Iowa Division of Continuing Education
Isabel K. Darcy, Department of Mathematics
Applied Mathematical and Computational Sciences, University of Iowa

2. You can join live lecture Wednesday and Friday either by going to
or joining via regular classroom.
NOTE: to ask questions, you need to joing via regular classroom.

3. IMA Annual Program Year Workshop, December 9-13, 2013
Topological Structures in Computational Biology
Tuesday December 10, 2013
11:30am-12:20pm Pek Lum (Ayasdi, Inc.)
Friday December 13, 2013
9:00am-9:50am Monica Nicolau (Stanford University)

5. Example: Point cloud data representing a hand.
Data Set:
Example: Point cloud data
representing a hand.
Function f : Data Set  R
Example: x-coordinate
f : (x, y, z)  x

6. Put data into overlapping bins. Example: f-1(ai, bi)
( ( ) ( ) ( ) ( ) ( ) )
Function f : Data Set  R
Ex 1: x-coordinate
f : (x, y, z)  x

7. Vertex = a cluster of a bin. Edge = nonempty intersection
D) Cluster each bin
& create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters
Need covering Resolution multiscale

8. Vertex = a cluster of a bin. Edge = nonempty intersection
D) Cluster each bin
& create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters
Need covering Resolution multiscale

9. Vertex = a cluster of a bin. Edge = nonempty intersection
D) Cluster each bin
& create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters
Need covering Resolution multiscale

10. Example: Point cloud data representing a hand.
A) Data Set
Example: Point cloud data
representing a hand.
B) Function f : Data Set  R
Example: x-coordinate
f : (x, y, z)  x
Put data into overlapping bins.
Example: f-1(ai, bi)
Cluster each bin & create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters

11. Chose filter

12. Chose filter

13. http://scitation. aip. org/content/aip/journal/jcp/130/14/10. 1063/1

14. Data: Contact maps from 2,800 Serial Replica Exchange Molecular Dynamics (SREMD) simulations of the GCAA tetraloop on the distributed computing platform.
760 trajectories with a complete unfolding event
550 trajectories with a complete refolding event.
Goal: To determine secondary structure pathways between folded and unfolded state

15. Problem: Many more folded and unfolded conformations than intermediate conformations
How to distinguish intermediate conformations from noise?
Solution
Choose f: space of conformations  R
f(conformation) = density

17. 550 trajectories with a complete refolding event
2952 configurations

18. Distance = Hamming distance

19. 550 trajectories with a complete refolding event
2952 configurations

21. 760 trajectories with a complete refolding event
4330 configurations

22. An eQTL biological data visualization challenge and approaches from the visualization community,
Bartlett et al.
BMC Bioinformatics
2012, 13(Suppl 8):S8
Mapper applied
to SNP data:

23. Monday December 09, 2013
9:00am-9:50am
Visualizing and Exploring Molecular Simulation Data via Energy Landscape Metaphor
Yusu Wang (The Ohio State University)

24. E(conformation) = energy of the conformation
Motivation:
Let S = set of conformations of the survivin protein
Energy landscape E: S  R
E(conformation) = energy of the conformation

25. Data from:
20,000 conformations obtained via replica exchange molecular dynamics.
The backbone = 46 alpha-carbon atoms
= 1035 dimensional vector of pairwise distances
describing the protein shape.
Intrinsic dimensionality of the conformational manifold has been estimated at around 20.

26. level set = f-1(r) = { x in M | f(x) = r }
contour tree
level set = f-1(r) = { x in M | f(x) = r }
A contour = a connected component of a level set.
Let Cq = the contour in M that is collapsed to q
Let TopoComp(edge) = U Cq
1940’s
Reeb graph
How do you embed the tree?
q in edge

27. Given f: Md  R,
Find g: R2  R such that
f and g share same contour tree
(2) the area of TopoComp(edge) of g is the same as the volumes of the corresponding TopoComp(edge) of f for each edge in the contour tree.
Expands upon Weber’s Topological Landscapes, 2007

28. f: M^2  R

31. Figure 8: (a) Slice-and-dice and (b) Voronoi treemap layouts of terrains in Figure 6.