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), 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 May 17;109(4):

vol. 110 no. 46, 18566–18571, 2013

Background

Recombination:

Homologous recombination

books.com/MoBio/Free/Ch8D2. htm

Homologous recombination

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 Cat Dog Rat Cow Where do we get distances from?

Perfectly “ tree-like ” distances CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow

Perfectly “ tree-like ” distances CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow

Perfectly “ tree-like ” distances CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow

Perfectly “ tree-like ” distances CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow

Perfectly “ tree-like ” distances CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow

Perfectly “ tree-like ” distances CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow

CatDogRat Dog3 Rat45 Cow676 Cat Dog Rat Cow RatDogCat Dog3 Cat45 Cow676 Rat Dog Cat Cow

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

Linking algebraic topology to evolution. Chan J M et al. PNAS 2013;110: ©2013 by National Academy of Sciences

Linking algebraic topology to evolution. Chan J M et al. PNAS 2013;110: ©2013 by National Academy of Sciences Reticulation

Multiple sequence alignment

Reassortment

Homologous recombination

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: ©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: ©2013 by National Academy of Sciences 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

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: ©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

/activities/Carlsson-Gunnar/lecture14.pdf

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

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

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

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: