DNA TOPOLOGY De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL

Pedagogical School: Knots & Links: From Theory to Application

De Witt Sumners: Florida State University Lectures on DNA Topology: Schedule Introduction to DNA Topology Monday 09/05/11 10:40-12:40 The Tangle Model for DNA Site-Specific Recombination Thursday 12/05/11 10:40-12:40 Random Knotting and Macromolecular Structure Friday 13/05/11 8:30-10:30

DNA Site-Specific Recombination Topological Enzymology Rational tangles and 4-plats The Tangle Model Analysis of Tn3 Resolvase Experiments Open tangle problem

Site-Specific Recombination Recombinase

Biology of Recombination Integration and excision of viral genome into and out of host genome DNA inversion--regulate gene expression Segregation of DNA progeny at cell division Plasmid copy number regulation

Topological Enzymology Mathematics: Deduce enzyme binding and mechanism from observed products

GEL ELECTROPHORESIS

Rec A Coating Enhances EM

RecA Coated DNA

DNA Trefoil Knot Dean et al. J. Biol. Chem. 260(1985), 4795

DNA (2,13) TORUS KNOT Spengler et al. Cell 42(1985), 325

T4 TWIST KNOTS Wasserman & Cozzarelli, J. Biol. Chem. 30(1991), 20567

GIN KNOTS Kanaar et al. CELL 62(1990), 553

SITE-SPECIFIC RECOMBINATION

Enzyme Bound to DNA

DIRECT vs INVERTED REPEATS

RESOLVASE SYNAPTIC COMPLEX

DNA 2-STRING TANGLES

2-STRING TANGLES

3 KINDS OF TANGLES A tangle is a configuration of a pair of strands in a 3-ball. We consider all tangles to have the SAME boundary. There are 3 kinds of tangles:

RATIONAL TANGLES

RATIONAL TANGLE CLASSIFICATION q/p = a 2k + 1/(a 2k-1 + 1(a 2k-2 +1/…)…) Two tangles are equivalent iff q/p = q’/p’ J. Conway, Proc. Conf. Oxford 1967, Pergamon (1970), 329

TANGLE OPERATIONS

RATIONAL TANGLES AND 4-PLATS

4-PLATS (2-BRIDGE KNOTS AND LINKS)

4-PLATS

4-PLAT CLASSIFICATION 4-plat is b(  ) where  = 1/(c 1 +1/(c 2 +1/…)…) b(  b(  ’  ’  as unoriented knots and links) iff  ’  and     ’ (mod  ) Schubert Math. Z. (1956)

TANGLE EQUATIONS

SOLVING TANGLE EQUATIONS

RECOMBINATION TANGLES

SUBSTRATE EQUATION

PRODUCT EQUATION

TANGLE MODEL SCHEMATIC

ITERATED RECOMBINATION DISTRIBUTIVE: multiple recombination events in multiple binding encounters between DNA circle and enzyme PROCESSIVE: multiple recombination events in a single binding encounter between DNA circle and enzyme

DISTRIBUTIVE RECOMBINATION

PROCESSIVE RECOMBINATION

RESOLVASE PRODUCTS

RESOLVASE MAJOR PRODUCT MAJOR PRODUCT is Hopf link [2], which does not react with Tn3 Therefore, ANY iterated recombination must begin with 2 rounds of processive recombination

RESOLVASE MINOR PRODUCTS Figure 8 knot [1,1,2] (2 rounds of processive recombination) Whitehead link [1,1,1,1,1] (either 1 or 3 rounds of recombination) Composite link ( [2] # [1,1,2]--not the result of processive recombination, because assumption of tangle addition for iterated recombination implies prime products (Montesinos knots and links) for processive recombination

1st and 2nd ROUND PRODUC TS

RESOLVASE SYNAPTIC COMPLEX

O f = 0

THEOREM 1

PROOF OF THEOREM 1 Analyze 2-fold branched cyclic cover T* of tangle T--T is rational iff T* = S 1 x D 2 Use Cyclic Surgery Theorem to show T* is a Seifert Fiber Space (SFS) Use results of Dehn surgery on SFS to show T* is a solid torus--hence T is a rational tangle Use rational tangle calculus to solve tangle equations posed by resolvase experiments

Proof that Tangles are Rational 2 biological arguments DNA tangles are small, and have few crossings— so are rational by default DNA is on the outside of protein 3-ball, and any tangle on the surface of a 3-ball is rational

Proof that Tangles are Rational THE MATHEMATICAL ARGUMENT The substrate (unknot) and the 1 st round product (Hopf link) contain no local knots, so O b, P and R are either prime or rational. If tangle A is prime, then ∂ A* (a torus) is incompressible in A. If both A and B are prime tangles, then (AUB)* contains an incompressible torus, and cannot be a lens space.

Proof that Tangles are Rational THE MATHEMATICAL ARGUMENT N(O b +P) = [1] so N(O b +P)* = [1]* = S 3 If O b is prime, the P is rational, and O b * is a knot complement in S 3. One can similarly argue that R and (R+R) are rational; then looking at the 2- fold branched cyclic covers of the 1 st 2 product equations, we have:

Proof that Tangles are Rational N(O b +R) = [2] so N(O b +R)* = [2]* = L(2,1) N(O b +R+R) = [2,1,1] so N(O b +R+R)* = [2,1,1]* = L(5,3) Cyclic surgery theorem says that since Dehn surgery on a knot complement produces two lens spaces whose fundamental group orders differ by more than one, then O b * is a Seifert Fiber Space. Dehn surgery on a SFS cannot produce L(2,1) unless O b * is a solid torus, hence O b is a rational tangle. N(O b +R+R) = [] so N(O b +R)* = [2]* = L(5,3)

3rd ROUND PRODUCT

THEOREM 2

4th ROUND PRODUCT

THEOREM 3

UTILITY OF TANGLE MODEL Precise mathematical language for recombination- allows hypothesis testing Calculates ALL alternative mechanisms for processive recombination Model can be used with incomplete experimental evidence (NO EM)--crossing # of products, questionable relationship between product and round of recombination

REFERENCES

JMB COVER

XER RECOMBINATION Tangle analysis produces 3 solutions Vazquez et al, J. Mol. Bio. 346 (2005),

TANGLES ARE PROJECTION DEPENDENT P R

3 XER SOLUTIONS ARE SAME TANGLE, PROJECTED DIFFERENTY

UNSOLVED TANGLE PROBLEM Let A be a rational tangle; how many other rational tangles can be obtained from A by choosing another projection?

Thank You National Science Foundation Burroughs Wellcome Fund