dna topology
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/1.jpg)
DNA TOPOLOGY
De Witt Sumners
Department of Mathematics
Florida State University
Tallahassee, FL
![Page 2: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/2.jpg)
Pedagogical School: Knots & Links: From Theory to Application
![Page 3: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/3.jpg)
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
![Page 4: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/4.jpg)
DNA Site-Specific Recombination
• Topological Enzymology
• Rational tangles and 4-plats
• The Tangle Model
• Analysis of Tn3 Resolvase Experiments
• Open tangle problem
![Page 5: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/5.jpg)
Site-Specific RecombinationSite-Specific Recombination
RecombinaseRecombinase
![Page 6: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/6.jpg)
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
![Page 7: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/7.jpg)
Topological Enzymology
Mathematics: Deduce enzyme binding and mechanism from
observed products
![Page 8: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/8.jpg)
GEL ELECTROPHORESIS
![Page 9: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/9.jpg)
Rec A Coating Enhances EM
![Page 10: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/10.jpg)
RecA Coated DNA
![Page 11: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/11.jpg)
DNA Trefoil Knot
Dean et al. J. Biol. Chem. 260(1985), 4795Dean et al. J. Biol. Chem. 260(1985), 4795
![Page 12: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/12.jpg)
DNA (2,13) TORUS KNOT
Spengler et al. Cell 42(1985), 325Spengler et al. Cell 42(1985), 325
![Page 13: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/13.jpg)
T4 TWIST KNOTS
Wasserman & Cozzarelli, J. Biol. Chem. Wasserman & Cozzarelli, J. Biol. Chem. 3030(1991), 20567 (1991), 20567
![Page 14: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/14.jpg)
GIN KNOTS
Kanaar et al. CELL Kanaar et al. CELL 6262(1990), 553(1990), 553
![Page 15: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/15.jpg)
SITE-SPECIFIC RECOMBINATION
![Page 16: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/16.jpg)
Enzyme Bound to DNA
![Page 17: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/17.jpg)
DIRECT vs INVERTED REPEATS
![Page 18: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/18.jpg)
RESOLVASE SYNAPTIC COMPLEX
![Page 19: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/19.jpg)
DNA 2-STRING TANGLES
![Page 20: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/20.jpg)
2-STRING TANGLES
![Page 21: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/21.jpg)
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:
![Page 22: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/22.jpg)
RATIONAL TANGLES
![Page 23: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/23.jpg)
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
![Page 24: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/24.jpg)
TANGLE OPERATIONS
![Page 25: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/25.jpg)
RATIONAL TANGLES AND 4-PLATS
![Page 26: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/26.jpg)
4-PLATS (2-BRIDGE KNOTS AND LINKS)
![Page 27: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/27.jpg)
4-PLATS
![Page 28: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/28.jpg)
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)
![Page 29: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/29.jpg)
TANGLE EQUATIONS
![Page 30: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/30.jpg)
SOLVING TANGLE EQUATIONS
![Page 31: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/31.jpg)
SOLVING TANGLE EQUATIONS
![Page 32: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/32.jpg)
RECOMBINATION TANGLES
![Page 33: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/33.jpg)
SUBSTRATE EQUATION
![Page 34: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/34.jpg)
PRODUCT EQUATION
![Page 35: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/35.jpg)
TANGLE MODEL SCHEMATIC
![Page 36: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/36.jpg)
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
![Page 37: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/37.jpg)
DISTRIBUTIVE RECOMBINATION
![Page 38: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/38.jpg)
PROCESSIVE RECOMBINATION
![Page 39: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/39.jpg)
RESOLVASE PRODUCTS
![Page 40: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/40.jpg)
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
![Page 41: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/41.jpg)
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
![Page 42: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/42.jpg)
1st and 2nd ROUND PRODUC TS
![Page 43: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/43.jpg)
RESOLVASE SYNAPTIC COMPLEX
![Page 44: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/44.jpg)
Of = 0
![Page 45: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/45.jpg)
THEOREM 1
![Page 46: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/46.jpg)
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
![Page 47: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/47.jpg)
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
![Page 48: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/48.jpg)
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.
![Page 49: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/49.jpg)
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:
![Page 50: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/50.jpg)
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)
![Page 51: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/51.jpg)
3rd ROUND PRODUCT
![Page 52: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/52.jpg)
THEOREM 2
![Page 53: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/53.jpg)
4th ROUND PRODUCT
![Page 54: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/54.jpg)
THEOREM 3
![Page 55: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/55.jpg)
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
![Page 56: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/56.jpg)
REFERENCES
![Page 57: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/57.jpg)
JMB COVER
![Page 58: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/58.jpg)
XER RECOMBINATION
Tangle analysis produces 3 solutions
Vazquez et al, J. Mol. Bio. 346 (2005), 493-504
![Page 59: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/59.jpg)
TANGLES ARE PROJECTION DEPENDENT
P R
![Page 60: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/60.jpg)
3 XER SOLUTIONS ARE SAME TANGLE, PROJECTED DIFFERENTY
![Page 61: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/61.jpg)
UNSOLVED TANGLE PROBLEM
• Let A be a rational tangle; how many other rational tangles can be obtained from A by choosing another projection?
![Page 62: DNA TOPOLOGY](https://reader035.vdocuments.net/reader035/viewer/2022062315/56815b93550346895dc997a2/html5/thumbnails/62.jpg)
Thank You
•National Science Foundation
•Burroughs Wellcome Fund