Recombination:. Different recombinases have different topological mechanisms: Xer recombinase on psi. Unique product Uses topological filter to only perform.

Recombination:

Different recombinases have different topological mechanisms: Xer recombinase on psi. Unique product Uses topological filter to only perform deletions, not inversions Ex: Cre recombinase can act on both directly and inversely repeated sites.

PNAS 2013

Tangle Analysis of Protein-DNA complexes

Mathematical Model Protein = DNA = = ==

Protein-DNA complex Heichman and Johnson C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489-515. protein = three dimensional ball protein-bound DNA = strings. Slide (modified) from Soojeong Kim

Solving tangle equations Tangle equation from: Path of DNA within the Mu transpososome. Transposase interactions bridging two Mu ends and the enhancer trap five DNA supercoils. Pathania S, Jayaram M, Harshey RM. Cell. 2002 May 17;109(4):425-36.

http://www.pnas.org/content/110/46/18566.full vol. 110 no. 46, 18566–18571, 2013

Background

http://ghr.nlm.nih.gov/handbook/mutationsanddisorders/possiblemutations

Recombination:

Homologous recombination http://en.wikipedia.org/wiki/File:HR_in_meiosis.svg

http://www.web- books.com/MoBio/Free/Ch8D2. htm

Distances can be derived from Multiple Sequence Alignments (MSAs). The most basic distance is just a count of the number of sites which differ between two sequences divided by the sequence length. These are sometimes known as p-distances. Cat ATTTGCGGTA Dog ATCTGCGATA Rat ATTGCCGTTT Cow TTCGCTGTTT CatDogRatCow Cat00.20.40.7 Dog0.200.50.6 Rat0.40.500.3 Cow0.70.60.30 Where do we get distances from? http://www.allanwilsoncentre.ac.nz/massey/fms/AWC/download/SK_DistanceBasedMethods.ppt

Perfectly “ tree-like ” distances CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow 1 1 2 24 http://www.allanwilsoncentre.ac.nz/massey/fms/AWC/download/SK_DistanceBasedMethods.ppt

CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow 1 1 2 24 RatDogCat Dog3 Cat45 Cow676 Rat Dog Cat Cow 1 1 2 24

CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow 1 1 2 24 RatDogCat Dog3 Cat45 Cow676 Rat Dog Cat Cow 1 1 2 24 CatDogRat Dog4 Rat44 Cow676

http://upload.wikimedia.org/wikipedia/commons/7/79/RPLP0_90_ClustalW_aln.gif Multiple sequence alignment

http://www.virology.ws/2009/06/29/reassortment-of-the-influenza-virus-genome/ Reassortment

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)

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/2 009/06/29/reassor tment-of-the- influenza-virus- genome/ For multiple segments, non-trivial H k k = 1, 2. Thus horizontal transfer via reassortment but not homologous recombination

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/

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.

TOP = Topological obstruction = maximum barcode length in non-zero dimensions TOP ≠ 0  no additive distance tree TOP is stable

ICR = irreducible cycle rate = average number of the one-dimensional irreducible cycles per unit of time Simulations show that ICR is proportional to and provides a lower bound for recombination/reassortment rate

Persistent homology Viral evolution Filtration value  Genetic distance (evolutionary scale)  0 at filtration value  Number of clusters at scale  Generators of H 0 A representative element of the cluster Hierarchical Hierarchical clustering relationship among H 0 generators  1 Number of reticulate events (recombination and reassortment)

Persistent homology Viral evolution Generators of H 1 Reticulate events Generators of H 2 Complex horizontal genomic exchange H k ≠ 0 for some k > 0 No phylogenetic tree representation Number of Lower bound on rate of higher-dimensional reticulate events generators over time (irreducible cycle rate)

MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Oct 16, 2013: Zigzag Persistence and installing Dionysus part I. 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

http://www.ima.umn.edu/2008-2009/ND6.15- 26.09/activities/Carlsson-Gunnar/lecture14.pdf http://www.ima.umn.edu/videos/?id=863

http://geometrica.saclay.inria.fr/workshops/TGDA_07_2009/ workshop_files/slides/deSilva_TGDA.pdf

Lee-Mumford-Pedersen [LMP] study only high contrast patches. Collection: 4.5 x 10 6 high contrast patches from a collection of images obtained by van Hateren and van der Schaaf Recall from Sept 20 lecture

M(100, 10) U Q where |Q| = 30 On the Local Behavior of Spaces of Natural Images, Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian, International Journal of Computer Vision 2008, pp 1-12.

The Theory of Multidimensional Persistence, Gunnar Carlsson, Afra Zomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbner bases, Gunnar Carlsson http://www.ima.umn.edu/videos/?id=862

Computing Multidimensional Persistence, Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian

http://www.mrzv.org/software/dionysus/

Time varying data X[t 0, t 1 ] = data points existing at time t for t in [t 0, t 1 ] X[t 1, t 2 ] X[t 2, t 3 ] X[t 0, t 2 ] X[t 1, t 3 ] X[t 2, t 4 ]

Time varying data X[t 0, t 1 ] = data points existing at time t for t in [t 0, t 1 ] X[t 1, t 2 ] X[t 2, t 3 ] X[t 0, t 2 ] X[t 1, t 3 ] X[t 2, t 4 ] VR(X[t 1, t 2 ], ε) VR(X[t 2, t 3 ], ε) VR(X[t 0, t 2 ], ε) VR(X[t 1, t 3 ], ε) VR(X[t 2, t 4 ], ε)

Time varying data X[t 0, t 1 ] = data points existing at time t for t in [t 0, t 1 ] X[t 1, t 2 ] X[t 2, t 3 ] X[t 0, t 2 ] X[t 1, t 3 ] X[t 2, t 4 ] VR(X[t 1, t 2 ], ε) VR(X[t 2, t 3 ], ε) VR(X[t 0, t 2 ], ε) VR(X[t 1, t 3 ], ε) VR(X[t 2, t 4 ], ε) C 0  C 1  C 2  C 3  C 4

C 1 C 3 C 0 C 2 C 4 H 1 H 3 H 0 H 2 H 4

C 0  C 1  C 2  C 3  C 4 H 0  H 1  H 2  H 3  H 4 H k i, p = Z k i /(B k i+p Z k i ) = L(i, i+p)( H k i ) U Persistent Homology: C 0  C 1  C 2  C 3  C 4 H 0  H 1  H 2  H 3  H 4 Zigzag Homology:

Lee-Mumford-Pedersen [LMP] study only high contrast patches. Collection: 4.5 x 10 6 high contrast patches from a collection of images obtained by van Hateren and van der Schaaf http://www.kyb.mpg.de/de/forschung/fg/bethgegroup/downloads/van-hateren-dataset.html

M(100, 10) U Q where |Q| = 30 On the Local Behavior of Spaces of Natural Images, Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian, International Journal of Computer Vision 2008, pp 1-12.

is a point in S 7 Data set M has over 8 × 10 6 points in S 7. Randomly choose 5000 points. Take the T% densest points. Choose a subset of 50 Landmark points.

comptop.stanford.edu/preprints/witness.pdf

Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. Normally L is a small subset, but in this example, L is a large red subset. U

Let D = set of point cloud data points. Choose L D, L = set of landmark points = vertices. U W ∞ (D) = Witness complex v 0,v 1,...,v k span a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v 0,v 1,...,v k and all the faces of {v 0,v 1,...,v k } belong to the witness complex. w is called a “weak” witness.

W ∞ (D) = Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points = vertices. U v 0,v 1,...,v k span a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v 0,v 1,...,v k and all the faces of {v 0,v 1,...,v k } belong to the witness complex. w is called a “weak” witness.

W 1 (D) = Lazy witness complex Let L = set of landmark points. 1-skeletion of W 1 (D) = 1-skeletion of W ∞ (D). Create the flag (or clique) complex: Add all possible simplices of dimensional > 1.

Choosing Landmark points: A.) Random B.) Maxmin 1.) choose point l 1 randomly 2.) If { l 1, …, l k-1 } have been chosen, choose l k such that { l 1, …, l k-1 } is in D - { l 1, …, l k-1 } and min {d( l k, l 1 ), …, d( l k, l k-1 )} ≥ min {d(v, l 1 ), …, d(v, l k-1 )}

Choosing Landmark points

Tamal K. Dey http://www.cse.ohio-state.edu/~tamaldey/ Graph Induced Complex: A Data Sparsifier for Homology Inferencehttp://www.cse.ohio-state.edu/~tamaldey/ Video: http://www.ima.umn.edu/videos/?id=2497http://www.ima.umn.edu/videos/?id=2497 Slides: http://web.cse.ohio-state.edu/~tamaldey/talk/GIC/GIC.pdfhttp://web.cse.ohio-state.edu/~tamaldey/talk/GIC/GIC.pdf Paper: http://web.cse.ohio-state.edu/~tamaldey/paper/GIC/GIC.pdfhttp://web.cse.ohio-state.edu/~tamaldey/paper/GIC/GIC.pdf Graph Induced Complex on Point Data T. K. Dey, F. Fan, and Y. Wang, (SoCG 2013) Proc. 29th Annu. Sympos. Comput. Geom. 2013, 107-116. Website: http://web.cse.ohio-state.edu/~tamaldey/GIC/gic.htmlhttp://web.cse.ohio-state.edu/~tamaldey/GIC/gic.html The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the Vietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size is huge--particularly in high dimensions. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structures. We investigate a complex called the {em graph induced complex} that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the power of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably. We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a very lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample. We provide experimental evidences in support of our theory.

HanTunHanTun software available at http://web.cse.ohio-state.edu/~tamaldey/handle/hantun.html

Figures from http://web.cse.ohio-state.edu/~tamaldey/shortloop-pictures.html Shortloop software (more general) available at http://web.cse.ohio-state.edu/~tamaldey/shortloop.html

http://web.cse.ohio-state.edu/~tamaldey/homology.html

Recombination:. Different recombinases have different topological mechanisms: Xer recombinase on psi. Unique product Uses topological filter to only perform.

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Recombination:. Different recombinases have different topological mechanisms: Xer recombinase on psi. Unique product Uses topological filter to only perform.

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