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 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