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DNA TOPOLOGY De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL [email protected] Pedagogical School: Knots & Links: From Theory to Application 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 = a2k + 1/(a2k-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(a,b) where b/a = 1/(c1+1/(c2+1/…)…) b(a,b) = b(a’,b’) as unoriented knots and links) iff a = a’ and b+1 b’ (mod a ) Schubert Math. Z. (1956) TANGLE EQUATIONS SOLVING 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 Of = 0 THEOREM 1 PROOF OF THEOREM 1 • Analyze 2-fold branched cyclic cover T* of tangle T--T is rational iff T* = S1 x D2 • 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 1st round product (Hopf link) contain no local knots, so Ob, 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(Ob+P) = [1] so N(Ob+P)* = [1]* = S3 If Ob is prime, the P is rational, and Ob* is a knot complement in S3. One can similarly argue that R and (R+R) are rational; then looking at the 2fold branched cyclic covers of the 1st 2 product equations, we have: Proof that Tangles are Rational N(Ob+R) = [2] so N(Ob+R)* = [2]* = L(2,1) N(Ob+R+R) = [2,1,1] so N(Ob+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 Ob* is a Seifert Fiber Space. Dehn surgery on a SFS cannot produce L(2,1) unless Ob* is a solid torus, hence Ob is a rational tangle. 3rd ROUND PRODUCT THEOREM 2 4th ROUND PRODUCT THEOREM 3 UTILITY OF TANGLE MODEL • Precise mathematical language for recombinationallows 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), 493-504 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