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
RANDOM KNOTTING
• Proof of the Frisch-Wasserman-Delbruck
conjecture--the longer a random circle, the more
likely it is to be knotted
• Knotting of random arcs
• Writhe of random curves and arcs
• Random knots in confined volumes
TOPOLOGAL ENTANGLEMENT
IN POLYMERS
WHY STUDY RANDOM ENTANGLEMENT?
• Polymer chemistry and physics: microscopic
entanglement related to macroscopic chemical
and physical characteristics--flow of polymer
fluid, stress-strain curve, phase changes (gel
formation)
• Biopolymers: entanglement encodes information
about biological processes--random entanglement
is experimental noise and needs to be subtracted
out to get a signal
BIOCHEMICAL MOTIVATION
Predict the yield from a random cyclization experiment in a
dilute solution of linear polymers
SEMICRYSTALLINE POLYMERS
CHEMICAL SYNTHESIS OF
CIRCULAR MOLECULES
Frisch and Wasserman JACS 83(1961), 3789
MATHEMATICAL PROBLEM
• If L is the length of linear polymers in dilute
solution, what is the yield (the spectrum of
topological products) from a random cyclization
reaction?
• L is the # of repeating units in the chain--# of
monomers, or # of Kuhn lengths (equivalent
statistical lengths, persistence lengths)--for
polyethylene, Kuhn length is about 3.5
monomers. For duplex DNA, persistence length
is about 300-500 base pairs
MONTE CARLO RANDOM
KNOT SIMULATION
• Thin chains--random walk in Z3 beginning at 0,
no backtracking, perturb self-intersections away,
keep walking. Force closure--the further you are
along the chain, the more you want to go toward
the origin.
• Detect knot types: D(-1)
Vologodskii et al, Sov. Phys. JETP 39 (1974), 1059
IMPROVED MONTE CARLO
• Start with phantom closed circular polymer (equilateral polygon),
point mass vertices, semi-rigid edges (springs), random thermal
forces (random 3D vector field), molecular mechanics, edges pass
through each other, take snapshots of equilibrium distribution,
detect knots with D(-1).
Michels & Wiegel Phys Lett. 90A (1984), 381
THIN CHAIN RANDOM KNOTTING
FRISCH-WASSERMANDELBRUCK CONJECTURE
• L = # edges in random polygon
• P(L) = knot probability
lim P(L) = 1
L
Frisch & Wasserman, JACS 83(1961), 3789
Delbruck, Proc. Symp. Appl. Math. 14 (1962), 55
RANDOM KNOT MODELS
• Lattice models: self-avoiding walks (SAW) and
self-avoiding polygons (SAP) on Z3, BCC, FCC,
etc--curves have volume exclusion
• Off-lattice models: Piecewise linear arcs and
circles in R3--can include thickness
RANDOM KNOT METHODS
• Small L: Monte Carlo simulation
• Large L: rigorous asymptotic proofs
SIMPLE CUBIC LATTICE
PROOF OF FWD CONJECTURE
THEOREM:
P(L) ~
1 - exp(-lL)
l>0
Sumners & Whittington, J. Phys. A: Math. Gen. 23
(1988), 1689
Pippenger, Disc Appl. Math. 25 (1989), 273
KESTEN PATTERNS
Kesten, J. Math. Phys. 4(1963), 960
TIGHT KNOTS
3
Z
TIGHT KNOT ON
19 vertices, 18 edges
3
Z
TREFOIL PATTERN FORCES
KNOTTING OF SAP
• Any SAP which contains the trefoil Kesten
pattern is knotted--each occupied vertex is the
barycenter of a dual 3-cube (the Wigner-Seitz
cell)--the union of the dual 3-cubes is
homeomorphic to B3, and this B3 contains a red
knotted arc.
SMALLEST TREFOIL
L = 24
SLIGHTLY LARGER TREFOIL
L = 26
RANDOM KNOT QUESTIONS
• For fixed length n, what is the distribution of knot
types?
• How does this distribution change with n?
• What is the asymptotic behavior of knot
complexity--growth laws ~bna ?
• How to quantize entanglement of random arcs?
KNOTS IN BROWNIAN FLIGHT
• All knots at all scales
Kendall, J. Lon. Math. Soc. 19 (1979), 378
ALL KNOTS APPEAR
Every knot type has a tight Kesten pattern representative on
Z3
ALL KNOTS APPEAR: PROOF
Take a projection of your favorite knot on Z2. Bump up crossovers,
saturate holes to get a decorated pancake
LONG RANDOM KNOTS
MEASURING KNOT COMPLEXITY
LONG RANDOM KNOTS ARE
VERY COMPLEX
THEOREM: All good measures of knot complexity
diverge to +  at least linearly with the length--the longer
the random polygon, the more entangled it is.
Examples of good measures of knot complexity:
crossover number, unknotting number, genus, bridge
number, braid number, span of your favorite knot
polynomial, total curvature, etc.
WRITHE
KESTEN WRITHE
Van Rensburg et al., J. Phys. A: Math. Gen 26 (1993), 981
WRITHE GROWS LIKE n
WRITHE GROWTH: PROOF
WRITHE GROWTH: PROOF
GROWTH OF WRITHE FOR
FIGURE 8 KNOT
RANDOM KNOTS ARE CHIRAL
ENTANGLEMENT COMPLEXITY
OF GRAPHS AND ARCS
DETECTING KNOTTED ARCS
LONG RANDOM ARCS (GRAPHS)
ARE VERY COMPLEX
NEW SIMULATION: KNOTS IN
CONFINED VOLUMES
• Parallel tempering scheme
• Smooth configuration to remove extraneous
crossings
• Use KnotFind to identify the knot--ID’s prime
and composite knots of up to 16 crossings
• Problem--some knots cannot be ID’d--might be
complicated unknots!
SMOOTHING
UNCONSTRAINED KNOTTING
PROBABILITIES
CONSTRAINED UNKNOTTING
PROBABILITY
CONSTRAINED UNKOTTING
PROBABILITY
CONSTRAINED TREFOIL
KNOT PROBABILITIES
CONSTRAINED TREFOIL
PROBABILITY
ANALYTICAL PROOFS FOR
KNOTS IN CONFINED
VOLUMES
• CONJECTURE: THE KNOT PROBABILITY
GOES TO ONE AS LENGTH GOES TO
INFINITY FOR RADOM CONFINED KNOTS-AND THAT THE KNOTTING PROBABILITY
GROWS MUCH FASTER THAN RANDOM
KNOTTING IN FREE 3-SPACE
RANDOM KNOTTING IN
HIGHER DIMENSIONS
• A similar result should hold for d-spheres in Zd+2, d > 2.
• CONJECTURE: For d > 2, let Sd be a
“plaquette” d-sphere containing the origin in
Zd+2. If V(Sd) denotes the d-volume of Sd, and
P(Sd) denotes the probability that Sd is knotted,
then P(Sd) goes to 1 as V(Sd) goes to infinity.
COLLABORATORS
•
•
•
•
•
•
•
Stu Whittington
Buks van Rensburg
Carla Tesi
Enzo Orlandini
Chris Soteros
Yuanan Diao
Nick Pippenger
•
•
•
•
•
•
•
Javier Arsuaga
Mariel Vazquez
Joaquim Roca
P. McGuirk
Christian Micheletti
Davide Marenduzzo
Enzo Orlandini
REFERENCES
• J. Phys. A: Math. Gen. 21(1988), 1689
• J. Phys. A: Math. Gen. 25(1992), 6557
• Math. Proc. Camb. Phil. Soc. 111(1992), 75
• J. Phys. A: Math. Gen. 26(1993), L981
• J. Stat. Phys. 85 (1996), 103
• J. Knot Theory and Its Apps. 6 (1997), 31
• Proc. National Academy of Sciences USA 99(2002), 5373-5377.
• Proc. National Academy of Sciences USA 102(2005), 9165-9169.
• J. Chem. Phys. 124(2006), 064903
• Biophys. J. 95 (2008), 3591-3599
DNA KNOTS IN VIRAL
CAPSIDS
De Witt Sumners
Department of Mathematics
Florida State University
Tallahassee, FL 32306
sumners@math.fsu.edu
THE PROBLEM
• Ds DNA of bacteriophage is packed to nearcrystalline density in capsids (~800mg/ml), and
the pressure inside a packed capsid is ~50
atmospheres.
• PROBLEM: what is the packing geometry?
T4 EM
HOW IS THE DNA PACKED?
THE METHOD: VIRAL KNOTS
REVEAL PACKING
• Compare observed DNA knot spectrum to simulation of knots in
confined volumes
Crossover Number
CHIRALITY
Knots and Catenanes
+ TORUS KNOTS
DNA (2,13) + TORUS KNOT
Spengler et al. CELL 42 (1985), 325
TWIST KNOTS
T4 TWIST KNOTS
Wassserman & Cozzarelli, J. Biol. Chem. 266 (1991), 20567
GEL VELOCITY IDENTIFIES
KNOT COMPLEXITY
Vologodskii et al, JMB 278 (1988), 1
P4 DNA has cohesive ends that form
closed circular molecules
GGCGAGGCGGGAAAGCAC
…... CCGCTCCGCCCTTTCGTG
GGCGAGGCGGGAAAGCAC
CCGCTCCGCCCTTTCGTG
….
EFFECTS OF CONFINEMENT
ON P4 DNA KNOTTING (10.5 kb)
• No confinement--3% knots, mostly trefoils
• Viral knots--95% knots, very high complexity-average crossover number 27!
PIVOT ALGORITHM
•
•
•
•
Ergodic--no volume exclusion in our simulation
D(-1), D(-2), D(-3) as knot detector
Reject swollen conformations—a problem!!
Space filling polymers in confined volumes--very
difficult to simulate
VOLUME EFFECTS ON KNOT
SIMULATION
• On average, 75% of crossings are extraneous
Arsuaga et al, PNAS 99 (2002), 5373
2D GEL RESOLVES SMALL
KNOTS
Arsuaga et al, PNAS 102 (2005), 9165
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 torus knots
to see if they all have same chirality
NEW SIMULATION SCHEME
• Multiple chain Monte Carlo (parallel tempering)
• Several Monte Carlo evolutions run in parallel,
each evolution labeled by temperature (pressure)
• Periodically, conformations are swapped
(Metropolis energy probability) between adjacent
runs
• Scheme greatly enhances exploration of phase
space
SMOOTH CONFORMATIONS
• Need to reduce complexity—70% of crossings are
extraneous
• Smoothing of conformations to reduce crossing
number
IDENTIFYING KNOTS
• Use KNOTFIND to identify knots of 16
crossings or less
• Problem--some knots cannot be ID’d--might
be complicated unknots!
CONCLUSIONS FROM MCMC
• Get precise determination of knot type (if < 16
crossings after smoothing!), volume exclusion,
model tuned to P4 DNA bending rigidity
• Get confining volume down to 2.5 times P4
capsid volume
• Get excess of chiral knots
• Do not get excess torus over twist knots
NEW PACKING DATA—4.7
KB COSMID
• Trigeuros & Roca, BMC Biotechnology 7 (2007)
94
CRYO EM VIRUS STRUCTURE
J
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N
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E
4
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
Thank You
•National Science Foundation
•Burroughs Wellcome Fund
UNCONSTRAINED KNOTTING
PROBABILITIES
CONSTRAINED UNKNOTTING
PROBABILITY
CONSTRAINED UNKOTTING
PROBABILITY
CONSTRAINED TREFOIL
KNOT PROBABILITIES
CONSTRAINED TREFOIL
PROBABILITY
4 vs 5 CROSSING PHASE
DIAGRAM
CONFINED WRITHE
GROWTH OF CONFINED
WRITHE
Thank You
•National Science Foundation
•Burroughs Wellcome Fund