MATH:7450 (22M:305) Topics in Topology: Scientific and Engi - PowerPoint Presentation

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MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology

Nov

18,

2013:

Viral Evolution II and

Crickets via cubical homology.

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.htmlSlide2

http://

www.pnas.org

/content/110/46/18566.full

vol. 110 no. 46, 18566–18571, 2013

Monday December 09, 2013

10:15am-11:05am The Topology of Evolution Raul

Rabadan

(Columbia University) Slide3

http://

upload.wikimedia.org

/

wikipedia/commons/7/79/RPLP0_90_ClustalW_aln.gifMultiple sequence alignmentSlide4

Linking algebraic topology to evolution.

Chan J M et al. PNAS 2013;110:18566-18571

©2013 by National Academy of Sciences

ReticulationSlide5

http://

www.web-books.com

/MoBio

/Free/Ch8D2.htmHomologous recombinationSlide6

http://

www.virology.ws

/2009/06/29/reassortment

-of-the-influenza-virus-genome/ReassortmentSlide7

Reconstructing phylogeny from persistent homology of avian influenza HA. (A) Barcode plot in dimension 0 of all avian HA subtypes.

Chan J M et al. PNAS 2013;110:18566-18571

©2013 by National Academy of Sciences

Influenza:

For a single segment,

no

H

k

for k > 0

no horizontal transfer

(i.e., no homologous recombination)Slide8

Persistent homology of

reassortment

in avian influenza.

Chan J M et al. PNAS 2013;110:18566-18571

©2013 by National Academy of Sciences

www.virology.ws

/2009/06/29/

reassortment

-of-the-influenza-virus-genome/

For multiple segments,

n

on-trivial

H

k

k = 1, 2.

T

hus

horizontal transfer via

reassortment

but not homologous recombinationSlide9

Barcoding plots of HIV-1 reveal evidence of recombination in (A) env, (B), gag, (C) pol, and (D) the concatenated sequences of all three genes.

Chan J M et al. PNAS 2013;110:18566-18571

©2013 by National Academy of Sciences

HIV –

single segment

(so no

reassortment

)

Non-trivial

H

k

k = 1, 2.

Thus horizontal transfer via homologous recombination.Slide10

TOP = Topological obstruction

= maximum barcode length in non-zero dimensions

TOP 0

 no additive distance treeTOP is stableSlide11

ICR

=

irreducible cycle rate = average number of the one-dimensionalirreducible cycles per unit of

timeSimulations show that ICR is proportional to and provides a lower bound for recombination/

reassortment

rateSlide12

Persistent

homology

Viral evolution

Filtration value e Genetic distance (evolutionary scale)

b

0

at filtration value

e

Number of clusters at scale eGenerators of H0 A representative element of the clusterHierarchical Hierarchical clusteringrelationship among H0 generatorsb1 Number of reticulate events (recombination and reassortment)Slide13

Persistent

homology

Viral evolution

Generators of H1 Reticulate eventsGenerators

of

H

2

Complex horizontal genomic exchangeHk ≠ 0 for some k > 0 No phylogenetic treerepresentationNo. of Lower bound on rate of higher-dimensional reticulate events generators over time (irreducible cycle rate)Slide14

http://

upload.wikimedia.org

/

wikipedia/commons/8/8d/Cricket900ppx_crop.jpgCricketSlide15

2012

Thursday December 12, 2013

9:00am-9:50am Structure of the Afferent Terminals in Terminal Ganglion of a Cricket and Persistent Homology Tomas

Gedeon

(Montana State University) Slide16

2012

dense nerve center

Nerve endingsSlide17
Slide18

Terminal GanglionSlide19
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distal

vs

proximal

closest 15%

proximal data:

long, medium, short 42428, 27442, 29297

distal data: long

sparse since harder to obtain (6194).

furthest 30%Slide21

Proximal data was filtered:

1.) remove outliers (noise)

experimental error

using data obtained from many different crickets2.) remove (redundant) points in densest regions to improve computational speedPerformed comparison using data sampled from a

Gaussian Mixture Model.Slide22

cubical homology: for reducing memory requirements (bitmap).

To calculate persistent homology:

Mrozek

, Batko and Wanner’scubPersistenceMD

.

To find cycles:

Homcubes

in

CHomP

.Slide23

cubical homology

To calculate persistent homology:

Mrozek, Batko and

Wanner’s cubPersistenceMD.

www.ii.uj.edu.pl/~mrozek

/software/

homology.html

Kaczynski T,

Mischaikow

K, Mrozek M (2004) Computational Homology.Applied Mathematical Sciences. Springer.http://chomp.rutgers.eduhttp://www.sagemath.org/doc/reference/homology/sage/interfaces/chomp.htmlTo find cycles: Homcubes in CHomP.Slide24
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medium+small

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http://

www.pnas.org

/content/110/46/18566.

full

http://

www.sciencemag.org

/content/312/5772/380.

full

http

://www.virology.ws/2009/04/30/structure-of-influenza-virus/Slide49
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cubical homologySlide53

Amino-acid sequence alignment of

stomatin-orthologs

Archaea

PH1511

P. horikoshii

PAB1933

P. abyssi

Bacteria

PA0452 P. aeruginosa SMB20989 S.melilotiEukaryotes MEC-2 C. elegans Stomatin human