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 http://www.math.uiowa.edu/~idarcy/AppliedTopology.html

www.elmhurst.edu/~chm/vchembook/580DNA.html http://en.wikipedia.org/wiki/File:RNA-comparedto-DNA_thymineAndUracilCorrected.png

Transfer ribonucleic acid (tRNA) http://en.wikipedia.org/wiki/File:TRNA-Phe_yeast_en.svg

http://en.wikipedia.org/wiki/File:TRNA-Phe_yeast_1ehz.png

Messenger RNA (mRNA) http://en.wikipedia.org/wiki/File:MRNA-interaction.png

Ribosomal ribonucleic acid (rRNA) http://en.wikipedia.org/wiki/File:RF00177.jpg

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;41:2807-2816 © The Author(s) 2013. Published by Oxford University Press.

Pseudoknot http://en.wikipedia.org/wiki/File:Pseudoknot.svg

http://www.stamms-lab.net/pdfs/lectures/RNA%20introduction.pdf

http://www.stamms-lab.net/pdfs/lectures/RNA%20introduction.pdf

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

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 http://pubs.acs.org/doi/pdf/10.1021/jp911085d

https://parasol.tamu.edu/foldingserver/FAQ_Technique.php

https://parasol.tamu.edu/foldingserver/FAQ_Technique.php

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.

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. http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

(g) Side view of the terrain model generated using PCA embedding of Figure 9. http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

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

Not simply connected: http://en.wikipedia.org/wiki/File:3D-Leveltorus-Reebgraph.png

level set = f-1(r) = { x in M | f(x) = r } http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf 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

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

f: M^2  R http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

Figure 8: (a) Slice-and-dice and (b) Voronoi treemap layouts of terrains in Figure 6. http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf

http://www.cse.ohio-state.edu/~yusu/papers/Eurovis10.pdf