1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Nov 4, 2013
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. DNA trivia: Who are the authors of the 1953 paper on DNA with the following quotes:
“DNA is a helical structure”
with “two co-axial molecules.”
“period is 34 Å”
“one repeating unit contains ten
nucleotides on each of two co-axial molecules.'’
“The phosphate groups lie on the outside of the structural unit, on a helix of diameter about 20 Å”
“the sugar and base groups must accordingly be turned inwards towards the helical axis.”

4. Also published in the same issue of Nature:

5.
JAMA. 2013;309(20): doi: /jama
“IN MARCH, THE PUBLICATION OF THE complete genome sequence of cancer cells from a Maryland woman who died in 1951 ignited an ethical firestorm. These cells, called HeLa because they were derived from the cervical tumor of Henrietta Lacks, have been widely cultured in laboratories and used in research.”

6.
1951 Biopsy of Henrietta Lacks’ tumour collected without her knowledge or consent. HeLa cell line soon established.
1971 The journal Obstetrics and Gynecology names Henrietta Lacks as HeLa source; word later spreads in Nature, Science and mainstream press.
1973 Lacks family members learn about HeLa cells. Scientists later collect their blood to map HeLa genes, without proper informed consent.
1996 Lacks family honoured at the first annual HeLa Cancer Control Symposium, organized by former student of scientist who isolated HeLa cells.
2013 HeLa genome published without knowledge of the family, which later endorses restricted access to HeLa genome data.

7. out the contamination problem.
…HeLa cells could float on dust particles in the air & travel on unwashed hands and contaminate other cultures. It became an enormous controversy. In the midst of that, one group of scientists tracked down Henrietta’s relatives to take some samples with hopes that they could use the family’s DNA to make a map of Henrietta’s genes so they could tell which cell cultures were HeLa and which weren’t, to begin straightening
out the contamination problem.

9. In the USA, Under GINA (2009), it's (mostly) illegal for
In the USA, Under GINA (2009), it's (mostly) illegal for
an employer to fire someone based on his genes, and it's illegal for health insurers to raise rates or to deny coverage because of someone's genetic code.
But the law has a loophole: It only applies to health insurance. It doesn't say anything about companies
that sell life insurance, disability insurance or
long-term-care insurance.”
people who discover they have the gene, ApoE4, associated with Alzheimer’s are 5 times more likely
than the average person to go out and buy
long-term-care insurance.

11. The boy took the saliva sample late and sent it off to an online genealogy DNA-testing service.
boy's genetic father had never supplied his DNA
2 men on the database matched his Y-chromosome
The two men did not know each other, but shared a surname, albeit with a different spelling.
genetic similarity of their Y-chromosomes suggested a 50% chance that the 2 men & the boy shared the same father, grandfather or great-grandfather.
bought information on everyone born in the same place and on the same date as his father.

12. Science 18 January 2013: Vol. 339 no. 6117 p. 262
NEWS & ANALYSIS GENETICS
Genealogy Databases Enable Naming of Anonymous DNA Donors by John Bohannon
Science 18 January 2013: Vol. 339 no pp
Identifying Personal Genomes by Surname Inference
Melissa Gymrek, Amy L. McGuire, David Golan, Eran Halperin, Yaniv Erlich
able to expose the identity of 50 individuals whose DNA was donated anonymously for scientific study through consortiums such as the 1000 Genomes Project.

13. Stability

14. Discrete & Computational Geometry
January 2007, Volume 37, Issue 1, pp
Stability of Persistence Diagrams
David Cohen-Steiner, Herbert Edelsbrunner, John Harer

15. || (x1,…,xn) – (y1,…,yn) ||∞ = max{|x1 – y1|,…,|xn - yn|}
Given sets X, Y and bijection g: X  Y,
Bottleneck Distance: dB(X, Y) = inf sup || x – g(x) ||∞
g x

16. dB(D(f), D(g)) ≤ || f − g ||∞ = sup{|f(x) – g(x)|}
Stability theorem:
Let X be a triangulable space with continuous
tame functions f, g : X  R. Then the persistence diagrams satisfy
dB(D(f), D(g)) ≤ || f − g ||∞ = sup{|f(x) – g(x)|} x

17. Wq(X, Y) = [inf S || x – g(x) ||∞]1/q
|| (x1,…,xn) – (y1,…,yn) ||∞ = max{|x1 – y1|,…,|xn - yn|}
Given sets X, Y and bijection g: X  Y,
Wasserstein distance:
Wq(X, Y) = [inf S || x – g(x) ||∞]1/q q
g x in X

18. Wq(D(f), D(g)) ≤ C|| f − g ||∞
Stability theorem:
Let X be a triangulable space whose triangulations grow polynomially with constant exponent j.
Let f, g : X  R be tame Lipschitz functions .
Then there are constants C and k > j no smaller than 1 such that persistence diagrams satisfy
Wq(D(f), D(g)) ≤ C|| f − g ||∞
for every q ≥ k.
1 – k/q

19. http://www. plosone. org/article/info%3Adoi%2F10. 1371%2Fjournal. pone
2008
And section 9.1 in Computational Topology: An Introduction By Herbert Edelsbrunner, John Harer

20. Goal: To determine what genes are involved in a particular periodic pathway
Application: segmentation clock of mouse
1 somite develops about every 2 hours
What genes are involved in somite development?

21. Data from:

22. During the formation of each somite, Lfng is expressed in the PSM as a wave that sweeps across the tissue in a posterior-to-anterior direction (1). Therefore, by visually comparing the anteroposterior position of the Lfng expression stripes in the PSM in stained embryos, it is possible to define an approximate chronological order of the embryos along the segmentation clock oscillation cycle (3, 4). We collected PSM samples from 40 mouse embryos ranging from 19 to 23 somites and used their Lfng expression patterns as a proxy to select 17 samples covering an entire oscillation cycle

23. Fig. 2. Identification of cyclic genes based on the PSM microarray time series.
Identification of cyclic genes based on the PSM microarray time series. (A) Left side of the 17 mouse embryos, whose right posterior PSMs (below red hatched line) were dissected for microarray analysis. Embryos were ordered along one segmentation clock cycle according to the position of Lfng stripes in their left PSM as revealed by in situ hybridization (fig. S1). (B) Log2 ratios of the expression levels of the Hes1 (blue) and Axin2 (red) cyclic genes in each microarray of the time series. (C) Phaseogram of the cyclic genes identified by microarray and L-S analysis. Blue, decrease in gene expression; yellow, increase in gene expression; pink squares, genes validated by in situ hybridization; and orange circles, nonvalidated genes, that is, not evidently cyclical as detected by in situ hybridization.
M Dequéant et al. Science 2006;314:
Published by AAAS

24.
accession number E-TABM-163

26. Figure 1. Identification of benchmark cyclic genes in the top 300 probe set lists of the five methods.
Dequéant M-L, Ahnert S, Edelsbrunner H, Fink TMA, et al. (2008) Comparison of Pattern Detection Methods in Microarray Time Series of the Segmentation Clock. PLoS ONE 3(8): e2856. doi: /journal.pone

27. Figure 2. Comparison of the intersection of the top 300 ranked probe sets from the five methods.
Dequéant M-L, Ahnert S, Edelsbrunner H, Fink TMA, et al. (2008) Comparison of Pattern Detection Methods in Microarray Time Series of the Segmentation Clock. PLoS ONE 3(8): e2856. doi: /journal.pone

28. Persistence:
For each of 7549 genes, create
fk: S1  R, k = 1, …, 7549
fk (time point i) = amount of RNA at time point i for gene k

29. 17 time points  17 equally space time points
microarry expression of gene k at time i  ranked
order of microarry expression of gene k at time i
Ex: (0.41, 0.63, 0.11, 0.23, 0.59)  (3, 5, 1, 2, 4).

30. g(xi) =[ p(fk)(xi) – 1] / (17 – 1), for i = 1, …, 17.
fk (time point i) = RNA intensity at time point i for gene k.
p(fk) = replace RNA intensity with rank order.
g(xi) =[ p(fk)(xi) – 1] / (17 – 1), for i = 1, …, 17.
g(x) obtained by linear interpolation for x ≠ xi for some i.
Note: 0 ≤ g(x) ≤ 1

31. Figure 8. Function g(x) for the expression pattern of Axin2.
Dequéant M-L, Ahnert S, Edelsbrunner H, Fink TMA, et al. (2008) Comparison of Pattern Detection Methods in Microarray Time Series of the Segmentation Clock. PLoS ONE 3(8): e2856. doi: /journal.pone

37. Indeed, due to technical issues, the right PSM samples of the time series were dissected from mouse embryos belonging to five consecutive somite cycles, and they were ordered based on their phase of Lfng expression pattern (revealed by in situ hybridization on the left PSM of each dissected mouse embryo) to reconstitute a unique oscillation cycle [5].