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RANDOM KNOTTING AND VIRAL DNA PACKING: THEORY AND EXPERIMENTS. De Witt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306 [email protected] RANDOM KNOTTING.

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Random knotting and viral dna packing theory and experiments l.jpg

RANDOM KNOTTING AND VIRAL DNA PACKING: THEORY AND EXPERIMENTS

De Witt Sumners

Department of Mathematics

Florida State University

Tallahassee, FL 32306

[email protected]


Random knotting l.jpg
RANDOM KNOTTING EXPERIMENTS

  • Proof of the Frisch-Wasserman-Delbruck conjecture--the longer a random circle, the more likely it is to be knotted

  • DNA knotting in viral capsids


Slide3 l.jpg

A Knot Zoo EXPERIMENTS

By Robert G. Scharein

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

© 2005 Jennifer K. Mann



Why study random entanglement l.jpg
WHY STUDY RANDOM ENTANGLEMENT? EXPERIMENTS

  • 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


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BIOCHEMICAL MOTIVATION EXPERIMENTS

Predict the yield from a random cyclization experiment in a dilute solution of linear polymers


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MATHEMATICAL PROBLEM EXPERIMENTS

  • 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)--for polyethylene, Kuhn length is about 3.5 monomers. For duplex DNA, Kuhn length is about 300-500 base pairs


Frisch wasserman delbruck conjecture l.jpg
FRISCH-WASSERMAN-DELBRUCK CONJECTURE EXPERIMENTS

  • 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 l.jpg
RANDOM KNOT MODELS EXPERIMENTS

  • 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 l.jpg
RANDOM KNOT METHODS EXPERIMENTS

  • Small L: Monte Carlo simulation

  • Large L: rigorous asymptotic proofs



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PROOF OF FWD CONJECTURE EXPERIMENTS

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 l.jpg
KESTEN PATTERNS EXPERIMENTS

Kesten, J. Math. Phys. 4(1963), 960


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TIGHT KNOTS EXPERIMENTS


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Z EXPERIMENTS3


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TIGHT KNOT ON Z EXPERIMENTS3

19 vertices, 18 edges



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RANDOM KNOT QUESTIONS EXPERIMENTS

  • 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 l.jpg
KNOTS IN BROWNIAN FLIGHT EXPERIMENTS

  • All knots at all scales

    Kendall, J. Lon. Math. Soc. 19 (1979), 378


All knots appear l.jpg
ALL KNOTS APPEAR EXPERIMENTS

Every knot type has a tight Kesten pattern representative on Z3


Long random knots l.jpg
LONG RANDOM KNOTS EXPERIMENTS



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LONG RANDOM KNOTS ARE VERY COMPLEX EXPERIMENTS

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.




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DNA Replication EXPERIMENTS


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Topological Enzymology EXPERIMENTS

Mathematics: Deduce enzyme binding and mechanism from observed products


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TOPOLOGICAL ENZYMOLOGY EXPERIMENTS

  • React circular DNA plasmids in vitro (in vivo) with purified enzyme

  • Gel electrophoresis to separate products (DNA knots & links)

  • Electron microscopy of RecA coated products

  • Use topology and geometry to build predictive models


Gel electrophoresis l.jpg
GEL ELECTROPHORESIS EXPERIMENTS


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RecA Coated DNA EXPERIMENTS


Dna trefoil knot l.jpg
DNA Trefoil Knot EXPERIMENTS

Dean et al., J BIOL. CHEM. 260(1985), 4975


Dna 2 13 torus knot l.jpg
DNA (2,13) TORUS KNOT EXPERIMENTS

Spengler et al. CELL 42 (1985), 325


Gel velocity identifies knot complexity l.jpg
GEL VELOCITY IDENTIFIES KNOT COMPLEXITY EXPERIMENTS

Vologodskii et al, JMB 278 (1988), 1


Chirality l.jpg
CHIRALITY EXPERIMENTS



Writhe average crossing number l.jpg
WRITHE & AVERAGE CROSSING NUMBER EXPERIMENTS

Writhe --average the sum of signed crossings over all projections (average number of crossings over all projections)


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VIRUS LIFE CYCLE EXPERIMENTS


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VIRUS ATTACKS! EXPERIMENTS


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VIRUS ATTACKS! EXPERIMENTS


T4 em l.jpg
T4 EM EXPERIMENTS


Virus attack l.jpg
VIRUS ATTACK EXPERIMENTS


T4 attack l.jpg
T4 ATTACK EXPERIMENTS



Spooling model l.jpg
SPOOLING MODEL EXPERIMENTS


Fold model l.jpg
FOLD MODEL EXPERIMENTS


Random packing l.jpg
RANDOM PACKING EXPERIMENTS


P4 dna has cohesive ends that form closed circular molecules l.jpg
P4 DNA has cohesive ends that form closed circular molecules EXPERIMENTS

….

GGCGAGGCGGGAAAGCAC

…...

CCGCTCCGCCCTTTCGTG

GGCGAGGCGGGAAAGCAC

CCGCTCCGCCCTTTCGTG


Viral knots reveal packing l.jpg
VIRAL KNOTS REVEAL PACKING EXPERIMENTS

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



Effects of confinement on dna knotting l.jpg
EFFECTS OF CONFINEMENT ON DNA KNOTTING Probability

  • No confinement--3% knots, mostly trefoils

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


Mature vs tailless phage l.jpg
MATURE vs TAILLESS PHAGE Probability

Mutants--48% of knots formed inside capsid

Arsuaga et al, PNAS 99 (2002), 5373


P4 knot spectrum l.jpg
P4 KNOT SPECTRUM Probability

97% of DNA knots had crossing number > 10!

Arsuaga et al, PNAS 99 (2002), 5373


2d gel l.jpg
2D GEL Probability


2d gel resolves small knots l.jpg
2D GEL RESOLVES SMALL KNOTS Probability

Arsuaga et al, PNAS 102 (2005), 9165


2d gel resolves small knots55 l.jpg
2D GEL RESOLVES SMALL KNOTS Probability

Arsuaga et al, PNAS 102 (2005), 9165


Pivot algorithm l.jpg
PIVOT ALGORITHM Probability

  • Ergodic--no volume exclusion in our simulation

    • D(-1), D(-2), D(-3) as knot detector

    • Space filling polymers in confined volumes--very difficult to simulate


Volume effects on knot simulation l.jpg
VOLUME EFFECTS ON KNOT SIMULATION Probability

  • On average, 75% of crossings are extraneous

Arsuaga et al, PNAS 99 (2002), 5373


Simulation vs experiment l.jpg
SIMULATION vs EXPERIMENT Probability

n=90, R=4

Arsuaga et al, PNAS 102 (2005), 9165


Effect of writhe biased sampling l.jpg
EFFECT OF WRITHE-BIASED SAMPLING Probability

n=90, R=4

Arsuaga et al, PNAS 102 (2005), 9165


Slide60 l.jpg

3 Probability1 Wr = 3.41

41 Wr = 0.00

51 Wr = 6.26

52 Wr = 4.54

61 Wr = 1.23

62 Wr = 2.70

63 Wr = 0.16

71 Wr = 9.15

31 # 31 Wr = 6.81

31 # -31 Wr = 0.01

31 # 41 Wr ~ 3. 41

primes

composites

Consistent with a high writhe of the packed DNA:

- Low amount of knot 41 and of composite 31# 41

- Predominance of torus knots (elevated writhe)


Conclusions l.jpg
CONCLUSIONS Probability

  • 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


Unknown p4 knot l.jpg
UNKNOWN P4 KNOT Probability


Unknown p4 knots l.jpg
UNKNOWN P4 KNOTS Probability


Afm images of simple dna knots mg 2 l.jpg
AFM Images of Simple DNA Knots Probability(Mg2+)

μm

Ercolini, Dietler EPFL Lausanne

μm

μm

Ercolini, Dietler, EPFL Lausanne


New simulation l.jpg
NEW SIMULATION Probability

  • 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!








Confined writhe l.jpg
CONFINED WRITHE Probability



Collaborators l.jpg

Stu Whittington Probability

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

COLLABORATORS


References l.jpg
REFERENCES Probability

• Nucleic Acids Research29(2001), 67-71.

• Proc. National Academy of Sciences USA99(2002), 5373-5377.

  • Biophysical Chemistry101-102 (2002), 475-484.

    • Proc. National Academy of Sciences USA102(2005), 9165-9169.

    • J. Chem. Phys 124 (2006), 064903


Thank you l.jpg

Thank You Probability

National Science Foundation

Burroughs Wellcome Fund


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