TOPOLOGICAL METHODS IN
PHYSICAL VIROLOGY
FSU-UF TOPOLOGY MEETING
FEB. 23, 2013
De Witt Sumners
Department 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
A Knot Zoo
By Robert G. Scharein
http://www.pims.math.ca/knotplot/zoo/
© 2005 Jennifer K. Mann
T ORUS KNOTS
TWIST KNOTS
Topological Enzymology
Mathematics: Deduce enzyme
binding and mechanism from
observed products
Strand Passage
Topoisomerase
Strand Exchange
Recombinase
GEL ELECTROPHORESIS
RecA Coated DNA
DNA Trefoil Knot
Dean et al., J BIOL. CHEM. 260(1985), 4975
l DNA (2,13) TORUS KNOT
Spengler et al. CELL 42 (1985), 325
T4 TOPOISOMERASE TWIST
KNOTS
Wassserman & Cozzarelli, J. Biol. Chem. 266 (1991), 20567
PHAGE m GIN KNOTS
Kanaar et al. CELL 62(1990), 553
Topoisomerase Knots
D
Topoisomerase Knots
Dean et al., J BIOL. CHEM. 260(1985), 4975
GEL VELOCITY IDENTIFIES
KNOT COMPLEXITY
Vologodskii et al, JMB 278 (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 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
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 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
• 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 capsid
Arsuaga et al, PNAS 99 (2002), 5373
P4 KNOT SPECTRUM
97% of DNA knots had crossing number > 10!
Arsuaga et al, PNAS 99 (2002), 5373
2D GEL RESOLVES SMALL
KNOTS
Arsuaga et al, PNAS 102 (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 99 (2002), 5373
SIMULATION vs EXPERIMENT
n=90, R=4
Arsuaga et al, PNAS 102 (2005), 9165
EFFECT OF WRITHE-BIASED
SAMPLING
n=90, R=4
Arsuaga et al, PNAS 102 (2005), 9165
CONCLUSIONS
• Viral DNA not randomly embedded (41and 52 deficit, 51
and 71 excess in observed knot spectrum)
• Viral DNA has a chiral packing mechanism--writhebiased 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 STRUCTURE
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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
Black—unknot; 91—red; complex knot--green
Marenduzzo et al PNAS 106 (2009) 22269
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+)
μm
Ercolini,
Dietler EPFL Lausanne
μm
μm