1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Nov 20, 2013: Intro to RNA & Topological Landscapes for Visualization of Scalar-Valued Functions.
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

3. Transfer ribonucleic acid (tRNA)

5. Messenger RNA (mRNA)

6. Ribosomal ribonucleic acid (rRNA)

7. Circular arcplots of E. cuniculi, E. coli and H
Circular arcplots of E. cuniculi, E. coli and H. volcanii 16S secondary structures.
Circular arcplots of E. cuniculi, E. coli and H. volcanii 16S secondary structures. Sequence is drawn as a circle, and each arc denotes a base pair. Column (a) shows an overlay of MFE (blue) and native (green) structures with common base pairs in red. Column (b) shows the MFE structure with base pairs annotated by the fraction of data-directed structures also containing that pair, as indicated by the color bar at the bottom. Column (c) show the native structure likewise annotated.
Sükösd Z et al. Nucl. Acids Res. 2013;41:
© The Author(s) Published by Oxford University Press.

8. Pseudoknot

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

12. 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

15. 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.

16. Figure 9: An energy landscape produced by PCA
Figure 9: An energy landscape produced by PCA. (a) Terrain produced from Delaunay triangulation of first two principal components of high-dimensional protein conformation point cloud. (b) Closeup view of the landscape emphasizing the jagged distortion around the global minimum.

17. (g) Side view of the terrain model generated using PCA embedding of Figure 9.

18. contour tree http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf
1940’s
Reeb graph
How do you embed the tree?
contour tree

19. Not simply connected:

20. level set = f-1(r) = { x in M | f(x) = r }
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

21. 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

22. f: M^2  R

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