DNA TOPOLOGY De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL...

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: (-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

sumners@math.fsu.edu

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