TOPOLOGICAL METHODS IN PHYSICAL VIROLOGY

FSU-UF TOPOLOGY MEETINGFEB. 23, 2013

De Witt SumnersDepartment of Mathematics

Florida State University

Tallahassee, FL 32306

sumners@math.fsu.edu

DNA Replication

TOPOLOGICAL VIROLOGY

• Using DNA plasmids as an assay for site-specific recombination—deduce viral enzyme binding and mechanism

• Using DNA knots to elucidate packing geometry and ejection of DNA in viral capsids

A Little Entanglement Can Go a Long Way

DNA KNOTTING IS LETHAL IN BACTERIA

• Promotes replicon loss by blocking DNA replication

• Blocks gene transcription

• Causes mutation at a rate 3 to 4 orders of magnitude higher than an unknotted plasmid

Diebler et al, BMC Molecular Biology (2007) 8:44

Crossover Number

CHIRALITY

Knots and Catenanes

Prime and Composite Knots

http://www.pims.math.ca/knotplot/zoo/

A Knot Zoo By Robert G. Scharein

© 2005 Jennifer K. Mann

T ORUS KNOTS

TWIST KNOTS

Topological Enzymology

Mathematics: Deduce enzyme binding and mechanism from

observed products

Strand PassageStrand Passage

TopoisomeraseTopoisomerase

Strand ExchangeStrand Exchange

RecombinaseRecombinase

GEL ELECTROPHORESIS

RecA Coated DNA

DNA Trefoil Knot

Dean et al., J BIOL. CHEM. Dean et al., J BIOL. CHEM. 260260(1985), 4975(1985), 4975

DNA (2,13) TORUS KNOT

Spengler et al. CELL Spengler et al. CELL 4242 (1985), 325 (1985), 325

T4 TOPOISOMERASE TWIST KNOTS

Wassserman & Cozzarelli, J. Biol. Chem. 266 (1991), 20567Wassserman & Cozzarelli, J. Biol. Chem. 266 (1991), 20567

PHAGE GIN KNOTS

Kanaar et al. CELL Kanaar et al. CELL 6262(1990), 553(1990), 553

Topoisomerase Knots

DDeeaann eett aall..,, JJ BBIIOOLL.. CCHHEEMM.. 226600((11998855)),, 44997755

Topoisomerase Knots

Dean et al., J BIOL. CHEM. Dean et al., J BIOL. CHEM. 260260(1985), 4975(1985), 4975

GEL VELOCITY IDENTIFIES KNOT COMPLEXITY

Vologodskii et al, JMB Vologodskii et al, JMB 278278 (1988), 1 (1988), 1

SITE-SPECIFIC RECOMBINATION

Biology of Site-Specific Recombination

• Integration and excision of viral genome into and out of host genome

• DNA inversion--regulate gene expression & mediate phage host specificity

• Segregation of DNA progeny at cell division

• Plasmid copy number regulation

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

TANGLE OPERATIONS

RATIONAL TANGLES AND 4-PLATS

4-PLATS (2-BRIDGE KNOTS AND LINKS)

4-PLATS

TANGLE EQUATIONS

RECOMBINATION TANGLES

SUBSTRATE EQUATION

PRODUCT EQUATION

TANGLE MODEL SCHEMATIC

Ernst & Sumners, Math. Proc. Camb. Phil. Soc. 108 (1990), 489

Tn3 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

• 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

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

• Proof shows there is NO OTHER explanation of the data

REFERENCES

JMB COVER

Crisona et al, J. Mol. Biol. 289 (1999), 747

BACTERIOPHAGE STRUCTURE

T4 EM

HOW IS THE DNA PACKED?

SPOOLING MODEL

RANDOM PACKING

P4 DNA has cohesive ends that form closed circular molecules

GGCGAGGCGGGAAAGCAC

CCGCTCCGCCCTTTCGTG…...

….

GGCGAGGCGGGAAAGCAC CCGCTCCGCCCTTTCGTG

Liu et al P2 Knots (33kb)

VIRAL KNOTS REVEAL PACKING

• Compare observed DNA knot spectrum to simulation of knots in confined volumes

EFFECTS OF CONFINEMENT ON DNA KNOTTING

• No confinement--3% knots, mostly trefoils

• Viral knots--95% knots, very high complexity--average crossover number 27!

MATURE vs TAILLESS PHAGE

Mutants--48% of knots formed inside capsidMutants--48% of knots formed inside capsid

Arsuaga et al, PNAS Arsuaga et al, PNAS 99 99 (2002), 5373(2002), 5373

P4 KNOT SPECTRUM

97% of DNA knots had crossing number > 10!97% of DNA knots had crossing number > 10!Arsuaga et al, PNAS Arsuaga et al, PNAS 99 99 (2002), 5373(2002), 5373

2D GEL RESOLVES SMALL KNOTS

Arsuaga et al, PNAS Arsuaga et al, PNAS 102 (2005), 9165102 (2005), 9165

PIVOT ALGORITHM

• Ergodic—can include volume exclusion and bending rigidity

• Knot detector—knot polynomials (Alexander, Jones, KNOTSCAPE)

VOLUME EFFECTS ON KNOT SIMULATION

• On average, 75% of crossings are extraneous

Arsuaga et al, PNAS Arsuaga et al, PNAS 99 99 (2002), 5373(2002), 5373

SIMULATION vs EXPERIMENT

Arsuaga et al, PNAS Arsuaga et al, PNAS 102 (2005), 9165102 (2005), 9165

n=90, R=4n=90, R=4

EFFECT OF WRITHE-BIASED SAMPLING

Arsuaga et al, PNAS Arsuaga et al, PNAS 102 (2005), 9165102 (2005), 9165

n=90, R=4n=90, R=4

CONCLUSIONS

• Viral DNA not randomly embedded (41and 52 deficit, 51 and 71 excess in observed knot spectrum)

• Viral DNA has a chiral packing mechanism--writhe-biased simulation close to observed spectrum

• Torus knot excess favors toroidal or spool-like packing conformation of capsid DNA

• Next step--EM (AFM) of 3- and 5- crossing knots to see if they all have same chirality

NEW PACKING DATA—4.7 KB COSMID

• Trigeuros & Roca, BMC Biotechnology 7 (2007) 94

CRYO EM VIRUS STRUCTUREJiang et al NATURE 439 (2006) 612

DNA-DNA INTERACTIONS GENERATE KNOTTING AND SURFACE ORDER

• Contacting DNA strands (apolar cholosteric interaction) assume preferred twist angle

Marenduzzo et al PNAS 106 (2009) 22269

SIMULATED PACKING GEOMETRY

Marenduzzo et al PNAS 106 (2009) 22269

THE BEAD MODEL

• Semiflexible chain of 640 beads--hard core diameter 2.5 nm

• Spherical capsid 45 nm

• Kink-jump stochastic dynamic scheme for simulating packing

KNOTS DELOCALIZED

Marenduzzo et al PNAS 106 (2009) 22269

Black—unknot; 91—red; complex knot--green

SIMULATED KNOT SPECTRUM

Marenduzzo et al PNAS 106 (2009) 22269

DNA-DNA INTERACTION CONCLUSIONS

• Reproduce cryo-em observed surface order

• Reproduce observed knot spectrum—excess of torus knots over twist knots

• Handedness of torus knots—no excess of right over left at small twist angles—some excess at larger twist angles and polar interaction

REFERENCES

• Nucleic Acids Research 29(2001), 67-71.• Proc. National Academy of Sciences USA

99(2002), 5373-5377.• Biophysical Chemistry 101-102 (2002), 475-484.• Proc. National Academy of Sciences USA

102(2005), 9165-9169.• J. Chem. Phys 124 (2006), 064903• Biophys. J. 95 (2008), 3591-3599• Proc. National Academy of Sciences USA

106(2009), 2269-2274.

JAVIER ARSUAGA, MARIEL VAZQUEZ, CEDRIC, EITHNE

CHRISTIAN MICHELETTI, ENZO ORLANDINI, DAVIDE MARENDUZZO

ANDRZEJ STASIAK

COLLABORATORS

Mathematics: Claus Ernst, Mariel Vazquez, Javier Arsuaga, Steve Harvey, Yuanan Diao, Christian Laing, Nick Pippenger, Stu Whittington, Chris Soteros, Enzo Orlandini, Christian Micheletti, Davide Marenduzzo

Biology: Nick Cozzarelli, Nancy Crisona, Sean Colloms, Joaquim Roca, Sonja Trigeuros, Lynn Zechiedrich, Jennifer Mann, Andrzej Stasiak

Thank You

•National Science Foundation

•Burroughs Wellcome Fund

UNKNOWN P4 KNOT

UNKNOWN P4 KNOTS

AFM Images of Simple DNA Knots (Mg2+)

μmμm

μm

Ercolini, Ercolini, Dietler EPFL LausanneDietler EPFL Lausanne