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.

39. FIG. 2. Color a NMR structure of the GCAA tetraloop. b Contact map
for the native state. Bases are numbered from 1 to 12 and native basepair
contacts dotted lines are numbered 1–4.

40. FIG. 3. Color online Graphical representation of pathways by MAPPER
FIG. 3. Color online Graphical representation of pathways by MAPPER. a Unfolding pathway. b Folding pathway. In both cases, the top row graphs are
the outputs from MAPPER, while the bottom row depicts the mean contact maps of the corresponding clusters. For clarity in mean contact maps, we drop those
mean contacts lower than 0.4. The node colors from red to blue indicate the density from high to low, and the labels e.g., 100% show the percentage of
configurations of the same level included in the cluster corresponding to the node. We dropped all the clusters of size smaller than 3% of the level size. a
shows that unfolding has a single dominant pathway characterized by unzipping from the end base pair. b shows that folding process has two dominant
pathways, passing through either the formation of the closing base pair or the end base pair. A noisy cluster consisting 3% of the level size was also shown
in b, which accounts for reptation, i.e., sliding of the two strands of the stem.

42. FIG. 4. Color online Transition probability from two intermediate states.
Lag time is 2 ps. The left four nodes as extended structures Fig. 3b are
merged into node U, and the right three nodes as folded structures are
collected in node F. The two intermediate states on pathways are denoted by
I1 and I2, respectively. The transition probability from I1 and I2 to other
states are noted as numbers on the arrows. One can see that I1 and I2 are
kinetically separated.

43. FIG. 5. Color online The number of end base-pair clusters found by
K-means. Here k ranges from 2 to 80 with step 2. For each k, 20 experiments
are repeated with K-means clustering. The number of clusters with
end base pair formed are recorded. The star is the median of such numbers
and the bar delimits the distribution range from 10% to 90%. Starting from
around k=25, such clusters appear with at least 1/2 probability. Around k
=55, such clusters begin to split. The instability of K-means clusters is
increasing as k grows, indicated by the expanding ranges.

44. FIG. 6. Color online K-means clustering fails to capture the low-density
intermediate states with one end base pair formed. The illustration here
chooses k=30 for K-means clustering. a shows how end base-pair formed
structures are distributed in different k-means clusters; b illustrates the
mean structures of the top eight K-means clusters gray which contain
base-pair formed structures. K-means splits the MAPPER cluster and lumps
them with densest clusters.

54. 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
Put data into overlapping bins.
Example: f-1(ai, bi)

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