Difference topology experiments and skein relations Isabel K. Darcy Candice Price Mathematics Department Applied Mathematical and Computational Sciences (AMCS) University of Iowa http://www.math.uiowa.edu/~idarcy This work was partially supported by  the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285). ©2008 I.K. Darcy. All rights reserved

Mathematical Model Protein = = = = DNA =

protein = three dimensional ball protein-bound DNA = strings. 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. Protein-DNA complex Heichman and Johnson Slide (modified) from Soojeong Kim

Shailja Pathania, Makkuni Jayaram and Rasika Harshey Solving tangle equations Equation A is from: Path of DNA within the Mu Transpososome Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils, Cell 2002 Shailja Pathania, Makkuni Jayaram and Rasika Harshey

Recombination:

Recombination:

Recombination:

Recombination: Our focus today

A difference topology experiment:

Old Pathway: New Pathway (post tangle analysis): Path of DNA within the Mu Transpososome: Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils. Shailja Pathania, Makkuni Jayaram and Rasika M Harshey Cell 2002 New Pathway (post tangle analysis): Interactions of Phage Mu Enhancer and Termini that Specify the Assembly of a Topologically Unique Interwrapped Transpososome. Zhiqi Yin, Asaka Suzuki, Zheng Lou, Makkuni Jayaram and Rasika M. Harshey

A difference topology experiment:

Skein Triple: K- K+ K0

Skein Triple: K+ K- Topoisomerase action K0 Recombinase action

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)

a = c and b – d is a multiple of a or bd – 1 is a multiple of a.

Thm (Price, D): Suppose K+ and K- are rational knots Thm (Price, D): Suppose K+ and K- are rational knots. Then if K+ = N(a/b), then there exists relatively prime integers p and q, where p may be chosen to be positive, such that K- = and K0 = N a− 2p2b+2pqa b + 2q2a−2pqb N a− p2b+pqa b + q2a−pqb or K- = and K0 = # N −a− 2p2b+2pqa −b + 2q2a−2pqb N pb−qa da−jb N −p j where d and j are any integers such that pd - qj = 1 K0 K+ K-

Thm (Price, D): Suppose K+ and K- are rational knots Thm (Price, D): Suppose K+ and K- are rational knots. Then if K- = N(a/b), then there exists relatively prime integers p and q, where p may be chosen to be positive, such that K+ = and K0 = N a + 2p2b − 2pqa b − 2q2a+2pqb N a+ p2b−pqa b − q2a+pqb or K+ = and K0 = # N −a+ 2p2b−2pqa −b − 2q2a+2pqb N pb−qa da−jb N −p j where d and j are any integers such that pd - qj = 1 K0 K+ K-

Thm (Price, D): If K± = unknot = and K+ = right-handed trefoil knot = Then K0 = hopf link =

Thm (Price, D): If K± = unknot = and K+ = right-handed trefoil knot = Then K0 = hopf link = Assuming a flattened 2D model