dna topology de witt sumners department of mathematics florida state university tallahassee, fl...
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DNA TOPOLOGY
De Witt Sumners
Department of Mathematics
Florida State University
Tallahassee, FL
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: (-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 (-1).
Michels & Wiegel Phys Lett. 90A (1984), 381
THIN CHAIN RANDOM KNOTTING
FRISCH-WASSERMAN-DELBRUCK CONJECTURE
• L = # edges in random polygon• P(L) = knot probability
lim P(L) = 1 L
Frisch & Wasserman, JACS 83(1961), 3789Delbruck, 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(-L)
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), 960Kesten, J. Math. Phys. 4(1963), 960
TIGHT KNOTS
Z3
TIGHT KNOT ON Z3
19 vertices, 18 edges19 vertices, 18 edges
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
LL = 24 = 24
SLIGHTLY LARGER TREFOIL
LL = 26 = 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 ~n ?
• 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
THE PROBLEM
• Ds DNA of bacteriophage is packed to near-crystalline 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 Spengler et al. CELL 4242 (1985), 325 (1985), 325
TWIST KNOTS
T4 TWIST KNOTS
Wassserman & Cozzarelli, J. Biol. Chem. 266 (1991), 20567
GEL VELOCITY IDENTIFIES KNOT COMPLEXITY
Vologodskii et al, JMB Vologodskii et al, JMB 278278 (1988), 1 (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
• 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 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
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 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 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
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