DNA TOPOLOGY: EXPERIMENTS AND ANALYSIS De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306 sumners@math.fsu.edu

Isomers

Using Topology in Science

SYNTHETIC KNOT Dietrich-Buchecker & Sauvage, Ang. Chemie 28 (1989), 189

KNOT IN A PROTEIN J. Am. Chem. Soc. 118(1996), 8945

What is Knot Theory?

Knots and Catenanes

Crossover Number

A Knot Zoo By Robert G. Scharein http://www.pims.math.ca/knotplot/zoo/ © 2005 Jennifer K. Mann

Prime and Composite Knots

CHIRALITY

CROSSING SIGN CONVENTION

LINKING NUMBERS

TWIST

WRITHE & AVERAGE CROSSING NUMBER Writhe --average the sum of signed crossings over all projections (average number of crossings over all projections)

LK = TW + WR

LK = TW + WR

LK = TW + WR

DNA Replication

DNA is Crowded in the Cell

Radial Loop Chromosome

Replication Obstruction

Strand Passage Topoisomerase

Strand Exchange Recombinase

* 07/16/96 Enzyme Bound to DNA *

Topological Enzymology * 07/16/96 Topological Enzymology Mathematics: Deduce enzyme binding and mechanism from observed products *

Information We Seek

TOPOLOGICAL ENZYMOLOGY React circular DNA plasmids in vitro (in vivo) with purified enzyme Gel electrophoresis to separate products (DNA knots & links) Electron microscopy of RecA coated products Use topology and geometry to build predictive models

GEL ELECTROPHORESIS

Rec A Coating Enhances EM

RecA Coated DNA

* 07/16/96 DNA Trefoil Knot *

DNA (2,13) TORUS KNOT

TOPOISOMERSE AND LINKING

TOPO I vs TOPO II

DNA PLASMID REPLICATION

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

Topoisomerase Knots

Topoisomerase Knots

Right and Left Hand Trefoils

Torus and Square Knots

Gel Mobility of DNA Knots

Conclusions

Crystal Structure of Topoisomerase

GEL VELOCITY IDENTIFIES KNOTS

Toposides--Chemotherapy Topoisomerase Replication Fork

SITE-SPECIFIC RECOMBINATION

* 07/16/96 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’

TANGLE OPERATIONS

RATIONAL TANGLES AND 4-PLATS

4-PLATS

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’) iff a = a’ and b+1 = b’ (mod a )

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 for processive recombination

1st and 2nd ROUND PRODUC TS

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 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

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