Isabel K. Darcy Mathematics Department Applied Mathematical and Computational Sciences (AMCS) University of Iowa ©2008 I.K. Darcy. All rights reserved This work was partially supported by the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF ). Hyeyoung Moon, Michigan Rob Scharein, Hypnagogic Software Guanyu Wang, University of Iowa Danielle Washburn, University of Iowa Joint with:

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

=≠ Tangle = 3-dimensional ball containing strings where the endpoints of the strings are fixed on the boundary of the ball. Protein = 3-dimensional ball DNA = strings

=≠ Tangle = 3-dimensional ball containing strings where the endpoints of the strings are fixed on the boundary of the ball. Protein = 3-dimensional ball DNA = strings For geometry: see 12-12:30 -Mary Therese Padberg, Exploring the conformations of protein-bound DNA: adding geometry to known topology, Wednesday, March 14, 12-12:30pm Exploring the conformations of protein-bound DNA: adding geometry to known topology and poster.

Cellular roles of DNA topoisomerases: a molecular perspectiveCellular roles of DNA topoisomerases: a molecular perspective, James C. Wang, Nature Reviews Molecular Cell Biology 3, (June 2002) Topoisomerase II performing a crossing change on DNA:

Topoisomerases are involved in Replication Transcription Unknotting, unlinking, supercoiling. Targets of many anti-cancer drugs.

Topoisomerases are proteins which cut one segment of DNA allowing a second DNA segment to pass through before resealing the break.

Knot distance Unknotting number Crossing Change

Example Figure: courtesy of Hyeyoung Moon

There are undetermined values in the knot distance table. For example, Slide courtesy of Hyeyoung Moon

Knot distance tabulation The distances between two knots up to mirror images are tabulated. Slide courtesy of Hyeyoung Moon

Knot distance tabulation

7) [D, Moon] Jones polynomial

Tangle Equations

Determining upper bounds

Rational Tangles Rational tangles alternate between vertical crossings & horizontal crossings. k horizontal crossings are right-handed if k > 0 k horizontal crossings are left-handed if k < 0 k vertical crossings are left-handed if k > 0 k vertical crossings are right-handed if k < 0 Note that if k > 0, then the slope of the overcrossing strand is negative, while if k < 0, then the slope of the overcrossing strand is positive. By convention, the rational tangle notation always ends with the number of horizontal crossings.

Rational tangles can be classified with fractions.

A knot/link is rational if it can be formed from a rational tangle via numerator closure. N(2/7) = N(2/1) Note 7 – 1 = 6 = 2(3)

when B = c/d, E = f/g, and |cg – df| > 1

Cover: Visual presentation of knot distance metric created using the software TopoICE-X within KnotPlot. A pair of knots in this graph is connected by an edge if they can be converted into one another via a single intersegmental passage. This graph shows all mathematically possible topoisomerase reaction pathways involving small crossing knots. D, Scharein, Stasiak. (Nucleic Acids Res., 2008; 36: 3515– 3521).3515– 3521 TopoICE in Rob Scharein’s KnotPlot.com

Tangle table is joint work with Rob Scharein, Danielle Washburn, Guanyu Wang, Melanie DeVries, et. al.

A tangle which is not generalized Montisinos

Parity 0 Parity ∞ Parity 1 Table of 4-crossing parity zero 2-string tangles. D, Melanie DeVries, Danielle Washburn, Guanyu Wang, Rob Scharein, et al.

Parity ∞ Table of 4-crossing parity infinity 2-string tangles..

Parity 1 parity one 2-string tangles.

Table of parity zero tangles 4 crossings: 6 tangles 5 crossings: 44 tangles 6 crossings: 228 tangles 7 crossings: 1430 tangles 8 crossings: 8868 tangles 9 crossings: tangles Note the table currently contains many repeats

Right-handed Crossing +1 Left-handed Crossing Crossing Sign Determination Right-hand Rule

positive crossing negative crossing Signed crossing changes

Signed knot distances

Right-handed Crossing +1 Left-handed Crossing Crossing Sign Determination Right-hand Rule

TopoICE-R +1 tangle corresponds to a negative crossing since h + q + p(b+1) is odd

Recombination:

from the wall of the Pisa Cathedral. Photo courtesy of Rob Scharein

Montesinos knot/link

Solving tangle equations Theorem [Hyeyoung Moon, D]

Solving tangle equations

Theorem [Hyeyoung Moon, D]

Solving tangle equations Theorem 2.3 [Hyeyoung Moon, D]