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

2. Mathematical Model
Protein = = = =
DNA =

3. 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),
protein = three dimensional ball
protein-bound DNA = strings.
Protein-DNA complex
Heichman and Johnson
Slide (modified) from Soojeong Kim

4. 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

5. Recombination:

6. Recombination:

7. Recombination:

8. Recombination:
Our focus today

9. A difference topology experiment:

10. 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

11. A difference topology experiment:

15. Skein Triple: K- K+ K0

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

17. 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.

18. Rational tangles can be classified with fractions.

19. 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)

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

21. 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-

22. 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-

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

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