DNA TOPOLOGY

De Witt Sumners

Department of Mathematics

Florida State University

Tallahassee, FL

sumners@math.fsu.edu

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 RecombinationSite-Specific Recombination

RecombinaseRecombinase

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

DNA (2,13) TORUS KNOT

Spengler et al. Cell 42(1985), 325Spengler et al. Cell 42(1985), 325

T4 TWIST KNOTS

Wasserman & Cozzarelli, J. Biol. Chem. Wasserman & Cozzarelli, J. Biol. Chem. 3030(1991), 20567 (1991), 20567

GIN KNOTS

Kanaar et al. CELL Kanaar et al. CELL 6262(1990), 553(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 A tangle tangle is a configuration of a pair of strands in a 3-ball. We consider all is a configuration of a pair of strands in a 3-ball. We consider alltangles to have the SAME boundary. There are 3 kinds of tangles: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() where = 1/(c1+1/(c2+1/…)…)

b(b(’’as unoriented knots and links) iff ’and ’ (mod )

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 2-fold branched cyclic covers of the 1st 2 product equations, we have:

Proof that Tangles are RationalN(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.

N(Ob+R+R) = [] so N(Ob+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), 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