dna topology: experiments and analysis de witt sumners department of mathematics florida state...
TRANSCRIPT
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DNA TOPOLOGY: EXPERIMENTS AND ANALYSIS
De Witt Sumners
Department of Mathematics
Florida State University
Tallahassee, FL 32306
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Isomers
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Using Topology in Science
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SYNTHETIC KNOT
• Dietrich-Buchecker & Sauvage, Ang. Chemie 28 (1989), 189
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KNOT IN A PROTEIN
J. Am. Chem. Soc. J. Am. Chem. Soc. 118118(1996), 8945(1996), 8945
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What is Knot Theory?
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Knots and Catenanes
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Crossover Number
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http://www.pims.math.ca/knotplot/zoo/
A Knot Zoo By Robert G. Scharein
© 2005 Jennifer K. Mann
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Prime and Composite Knots
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CHIRALITY
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CROSSING SIGN CONVENTION
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LINKING NUMBERS
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TWIST
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WRITHE & AVERAGE CROSSING NUMBER
Writhe --average the sum of signed crossings over all projections (average number of crossings over all projections)
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LK = TW + WR
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LK = TW + WR
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LK = TW + WR
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DNA Replication
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DNA is Crowded in the Cell
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Radial Loop Chromosome
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Replication Obstruction
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Strand PassageStrand Passage
TopoisomeraseTopoisomerase
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Strand ExchangeStrand Exchange
RecombinaseRecombinase
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Enzyme Bound to DNA
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Topological Enzymology
Mathematics: Deduce enzyme binding and mechanism from
observed products
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Information We Seek
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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
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GEL ELECTROPHORESIS
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Rec A Coating Enhances EM
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RecA Coated DNA
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DNA Trefoil Knot
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DNA (2,13) TORUS KNOT
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TOPOISOMERSE AND LINKING
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TOPO I vs TOPO II
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DNA PLASMID REPLICATION
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Topoisomerase I ExperimentDean et al. J. Biol. Chem. 260 (1985), 4795
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Topoisomerase Knots
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Topoisomerase Knots
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Right and Left Hand Trefoils
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Torus and Square Knots
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Gel Mobility of DNA Knots
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Conclusions
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Crystal Structure of Topoisomerase
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GEL VELOCITY IDENTIFIES KNOTS
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Toposides--ChemotherapyToposides--Chemotherapy
Replication ForkReplication Fork
TopoisomeraseTopoisomerase
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SITE-SPECIFIC RECOMBINATION
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Enzyme Bound to DNA
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DIRECT vs INVERTED REPEATS
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RESOLVASE SYNAPTIC COMPLEX
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DNA 2-STRING TANGLES
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2-STRING TANGLES
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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:
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RATIONAL TANGLES
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RATIONAL TANGLE CLASSIFICATION
q/p = a2k + 1/(a2k-1 + 1(a 2k-2 +1/…)…)
Two tangles are equivalent iff q/p = q’/p’
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TANGLE OPERATIONS
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RATIONAL TANGLES AND 4-PLATS
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4-PLATS
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4-PLATS
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4-PLAT CLASSIFICATION
4-plat is b() where = 1/(c1+1/(c2+1/…)…)
b(b(’’iff
’and ’ (mod )
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TANGLE EQUATIONS
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SOLVING TANGLE EQUATIONS
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SOLVING TANGLE EQUATIONS
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RECOMBINATION TANGLES
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SUBSTRATE EQUATION
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PRODUCT EQUATION
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TANGLE MODEL SCHEMATIC
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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
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DISTRIBUTIVE RECOMBINATION
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PROCESSIVE RECOMBINATION
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RESOLVASE PRODUCTS
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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
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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
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1st and 2nd ROUND PRODUC TS
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Of = 0
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THEOREM 1
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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
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3rd ROUND PRODUCT
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THEOREM 2
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4th ROUND PRODUCT
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THEOREM 3
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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
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REFERENCES
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JMB COVER