measuring the size and shape of polymers · eric rawdon measuring the size and shape of polymers....

Measuring the size and shape of polymers

Eric Rawdon

University of St. ThomasSt. Paul, MN

[email protected]

http://george.math.stthomas.edu/rawdon/

April 16, 2008

Eric Rawdon Measuring the size and shape of polymers

Quick Summary

Joint work with Akos Dobay, John Kern, Ken Millett,Michael Piatek∗, Patrick Plunkett∗, and Andrzej Stasiak

Goals

Grand goal – understand what polymers “look like”

Question – How does topology affect the size and shape?

Compare shapes of

All knots (phantom polygons)Those knots with a fixed topology (e.g. just the trefoils)

Polymer model

Freely jointed model (larger length scales)

Equilateral closed polygons

No repulsion or attraction between edges


Transitions in Behavior

Effect of length on size and knotting

With few edges, closure condition is strong, so polygons arequite constrained (or not possible)

With increasing edges, polygons are “more free” and morelikely to be knotted

Goal: determine transition from compressed to swollen

Depends on the knot typeDepends on the knot model (lattice knots, Gaussian knots)Depends on what you measure

Equilibrium Length

Equilibrium length is where phantom = knot type

Original question: is the equilibrium length universal?


Example

box length – max distance between vertices

4

6

8

10

12

14

16

18

20

22

50 100 150 200 250 300 350 400 450 500

Box

Len

gth

Number of Edges

3.1phantom

compressed

swollen

Equilibrium length

Intersection point, where phantom = 31

Releasing topological constraint yields no net change inaverage box length


Quantities

Measuring size and shape

Smallest enclosing boxes

Standard boxSkinny box

Miniball (smallest enclosing sphere)

Convex hull (smallest enclosing polyhedron)

Radius of gyration

Average crossing number


Standard Box


Skinny Box (more economical)


Convex Hull (Eric’s recommendation as “Best Value”)


Radius of Gyration


Miniball


Another Example – 16 edge Trefoil


And Radius of Gyration


Other Quantities

Other measures of polymers

Average crossing number

Radius of gyration

Total curvature (total bending)

Total torsion (total twisting)

Thickness (self-avoiding)


Data Generation

How

Hedgehog method

From 50 edges to 500 edges by 10

400,000 knots for each number of edges

Knot types “determined” using Ewing/Millett HOMFLY code

Computations took several weeks on 40 node cluster


Who

Thanks to Rob Scharein and KnotPlotwww.knotplot.com

01 31 41 51

52 61 62 63


Scaling

Fits

Fitting function: xdν(A + B/√

x + C/x)Orlandini, Tesi, Janse van Rensburg, WhittingtonJ Phys A Math Gen 31:5935–5967, 2005.

Lengths (d = 1)

phantom dν = 0.5knots dν = 0.588

Surface areas (d = 2)


Volumes (d = 3)


Use Monte Carlo Markov Chains to find eq. lengths andestimate errors


Probability Data: Unknot, Trefoil, and Figure-8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

50 100 150 200 250 300 350 400 450 500

Pro

babi

lity

Number of Edges

013141


Probability Data: 5-Crossing Knots

0

0.005

0.01

0.015

0.02

0.025

0.03

50 100 150 200 250 300 350 400 450 500

Pro

babi

lity

Number of Edges

5152


Probability Data: 6-Crossing Knots

0

0.002

0.004

0.006

0.008

0.01

50 100 150 200 250 300 350 400 450 500

Pro

babi

lity

Number of Edges

616263


Scaling of Miniball Radius

2

3

4

5

6

7

8

9

10

11

12

100 200 300 400 500 600

Min

iba

ll R

ad

ius

Number of Edges

7.53

8.03277 282


Results

Equilibrium Length ± Error31 41 51 52 61 62 63

ACN 169 ± 1 250 ± 1 337 ± 2 336 ± 2 431 ± 5 441 ± 5 447 ± 7

RGN 187 ± 1 278 ± 2 368 ± 5 369 ± 4 475 ± 12 486 ± 12 487 ± 16SBL 190 ± 1 279 ± 2 363 ± 5 367 ± 4 461 ± 10 470 ± 10 472 ± 17BXL 191 ± 1 279 ± 2 363 ± 5 367 ± 3 461 ± 10 469 ± 10 474 ± 12MBR 191 ± 1 280 ± 2 363 ± 4 367 ± 3 460 ± 9 469 ± 10 472 ± 13SBW 195 ± 1 282 ± 2 366 ± 5 366 ± 3 460 ± 11 468 ± 9 474 ± 13BXW 195 ± 1 281 ± 2 366 ± 4 366 ± 3 458 ± 10 465 ± 10 477 ± 13BXH 198 ± 1 280 ± 3 359 ± 5 361 ± 4 445 ± 12 458 ± 12 460 ± 16SBH 198 ± 1 280 ± 2 359 ± 5 360 ± 4 436 ± 11 458 ± 11 451 ± 14

CHA 198 ± 1 287 ± 2 370 ± 3 373 ± 3 465 ± 7 477 ± 7 478 ± 9SBA 198 ± 1 286 ± 2 371 ± 3 373 ± 3 464 ± 7 476 ± 7 477 ± 10BXA 198 ± 1 286 ± 2 370 ± 3 373 ± 2 465 ± 8 474 ± 7 480 ± 12

SBV 203 ± 1 292 ± 1 377 ± 4 380 ± 3 469 ± 7 484 ± 8 484 ± 11CHV 203 ± 1 292 ± 1 376 ± 3 379 ± 3 471 ± 7 483 ± 7 483 ± 10BXV 203 ± 1 292 ± 2 377 ± 4 380 ± 3 471 ± 8 483 ± 8 490 ± 12

THI 232 ± 1 305 ± 4 358 ± 8 361 ± 6 432 ± 19 440 ± 18 430 ± 17

TCU 226 ± 4 241 ± 6 265 ± 10 275 ± 8 287 ± 17 301 ± 14 289 ± 18TTO 229 ± 2 245 ± 5 279 ± 8 276 ± 5 292 ± 10 303 ± 9 307 ± 10


Equilibrium Lengths

0

100

200

300

400

500

THITTO

TCUBXV

CHVSBV

BXASBA

SBHCHA

BXHBXW

SBWM

BRBXL

SBLRGN

ACN

Equ

ilibr

ium

Len

gth

31415152616263


Observations

Groupings

ACN

Linear dimensions, possibly excluding RGN

Quadratic dimensions

Cubic dimensions

Thickness

Total curvature and total torsion

Other observations

Standard ordering: 31 → 41 → 51 → 52 → 61 → 62 → 63

Interrelationships?

Relationships to other measurements of complexity?


Linear relationship?

150

200

250

300

350

400

450

150 200 250 300 350 400 450 500

Ave

rage

Cro

ssin

g N

umbe

r

Miniball Radius

200

250

300

350

400

450

150 200 250 300 350 400 450 500

Thi

ckne

ss R

adiu

s

Box Length

200

250

300

350

400

450

500

150 200 250 300 350 400 450 500

Con

vex

Hul

l Vol

ume

Radius of Gyration

200

250

300

350

400

450

500

220 230 240 250 260 270 280 290 300 310

Thi

ckne

ss R

adiu

s

Total Curvature


Inertial Ellipsoid


Are You Suddenly Hungry For Fish?

What fish is this?


Are You Suddenly Hungry For Fish?

What fish is this?

The Northern Pike


Mini Ellipsoid


Comparison to Electrophoretic Separation

200

220

240

260

280

300

0 1 2 3 4 5 6

Ave

rag

e T

C/T

T E

qu

ilib

riu

m L

en

gth

Gel Separation

Direction of

electrophoresis

19.82x – 173.93


Comparison to minimum ropelength

30

35

40

45

50

55

60

150 200 250 300 350 400 450

Min

imum

Rop

elen

gth

Average Crossing Number

30

35

40

45

50

55

60

150 200 250 300 350 400 450 500

Min

imum

Rop

elen

gth

Miniball Radius

30

35

40

45

50

55

60

200 250 300 350 400 450 500

Min

imum

Rop

elen

gth

Convex Hull Volume

30

35

40

45

50

55

60

200 250 300 350 400 450

Min

imum

Rop

elen

gth

Thickness Radius


Ropelength Problem and Ideal Knots

Ropelength problem

Question: Is it possible to tie a nontrivial knot with 2 feet of1-inch radius rope?

Goal: Find the least amount of rope needed to tie aconformation of a given knot type

Idealized rope:

Tube is made of circular disks perpendicular to the knotThickness radius – largest non-self intersecting radius


Constraints Due to Thick Tube

Determining ropelength

Curvature constraint – knot can bend back on itself

Distance constraint – two portions of the knot cannot becloser than twice the radius of the tube

Curvature Constraint Distance Constraint


Understanding Ropelength

Definition of R(K )

K a smooth knot, C 2 or C 1,1

MinRad(K ) – minimum radius of curvature = 1max κ

dcsd(K ) – doubly critical self-distance

R(K ) – called thickness radius

Rope(K ) = Length(K )/R(K ) – called ropelength

Ideal or Tight – conformation minimizing ropelength within aknot/link type


Characterization of Thickness/Ropelength

Theorem

The thickest non self-intersecting tube about K has radiusR(K ) = min{MinRad(K ), dcsd(K )/2} .

31 41


Applications: DNA Topology

31

61


Gel Electrophoresis

Topology determines speed through gel

Typically, length determines speed through the gel

DNA separates into bands by knot types


Topological Effects in Gel Electrophoresis

DNA

DNA is all the same length, knot type determines the speed

y -axis is ACN of tight knots

Stasiak, et al., Ideal knots and their relation to the physics of real knots, Ideal Knots, 1998.


Applications: Breaking Point

Pieranski, et al., Localization of breakage points in knotted strings, New J. Phys., 2001.


Applications: Glueballs

Kephart and Buniy

Glueballs are hadrons containing no valance quarks

Hypothesis: glueballs are tight knotted and linked QCD fluxtubesObserved a linear relationship between energy and length ofmost simple knots and links

Collaboration: compute table of shortest knots and links

Thousands of knots and linksPredict existence of new glueballs

Kephart, Buniy, A Model of Glueballs, Phys. Lett., 2003.


What Is Known?

Not much!

Unknot minimized by a circle, Rope = 2π

Links with planar unknotted components

Conjectures for tight clasp and Borromean rings

For nontrivial knots, there are no exact results

Cantarella, et al., Criticality for the Gehring Link Problem, Geom. Topol., 2006.


Polygonal Ropelength

MinRad(P) = minRad(vi ) = min length(edge)2 tan(θi/2)

θi

Rad(vi )

vi

dcsd(P) = minimum distance over pairs like this


More Polygonal Ropelength

R(P) = min{MinRad(P), dcsd(P)/2} (thickness radius)

Rope(P) = Length(P)/R(P) (ropelength)

31 41


Simulations

Optimization

Vertex perturbations

Descent: shake and checkSimulated annealing: temperature and energy differencedetermine the extent to which you take “bad” steps

Ridgerunner

Problem – need coordinated movement, especially with links

ridgerunner: Ashton, Cantarella, Piatek, Rawdon


Simulated Annealing

300.85 127.80 93.08

76.25 59.73 48.56

44.39 35.71 33.60


ridgerunner

Tightening algorithm based on constrained lengthminimization

Construct a length minimizing gradient

Place a strut between close pairs (tensegrity theory)

balanced movement

Turns into a big linear algebra problem (tsnnls)

Resolves forces due to balancing

Ashton, et al., Knot tightening by constrained gradient descent, in preparation, 2008.


Movie


Theorems

Theorem: If polygons → smooth thenropelength(polygons) → ropelength(smooth).

Theorem: There is a smooth curve inscribed in a polygon sothat ropelength(smooth) ≈ ropelength(polygon).

This allows us to find upper bounds for the minimumropelength

Rawdon, Can computers discover ideal knots, Experiment. Math., 2003.

Rawdon, Approximating smooth thickness, J. Knot Theory Ramifications, 2000.


Ropelength Upper Bounds for Trefoil

Upper bounds for minimum ropelength

34.18 (Pieranski 2001)

32.77 (Rawdon 2003)

32.74446 (Maddocks 2005)

32.74391 (Baranska, Pieranski, Rawdon 2005)

32.74339 (Baranska, Pieranski, Przybyl, Rawdon 2005)


Ropelength Lower Bounds

Lower bounds for minimum ropelength

4π ≈ 12.57 (Fenchel, Milnor, 1950-1970)

5π ≈ 15.71 (Litherland, Simon, Durumeric, Rawdon, 1999)

4π + 2π√

2 ≈ 21.45 (Cantarella, Kusner, Sullivan, 2001)

> 24 (Diao, 2002)

32.68 (Denne, Diao, Sullivan, 2004)

Current bounds

32.68 < Rope(trefoil) < 32.74339


Measuring the Effectiveness of ridgerunner

Ropelength upper bounds are the best, except for trefoil


The ridgerunner Trefoil

http://george.math.stthomas.edu/rawdon/data.php


Offshoot Projects

Ropelength with stiff rope

s = 10 s = 3 s = 2 s = 1

Hydrodynamic properties of tight knots

Geometric analysis of the shape of tight knots

Holding and breaking strength of suture knots


Stiff Rope

Rs(K ) = min

{

1

s· MinRadK , dcsdK/2

}

, s ≥ 1

Ropes(K ) = Length(K )/Rs(K )

0

100

200

300

400

500

0 0.2 0.4 0.6 0.8 1

Min

imum

Rop

elen

gth

Flexibility

314151

818819821939949


Knots with Stiff Rope

31

41

819

s = 10 s = 3 s = 2 s = 1

Buck, Rawdon, Role of flexibility in entanglement, Phys. Rev. E, 2004.


Flexibility and Tensile Strength

There is little difference between the minimum ropelength ofknots made of the materials that can bend like

Flexible = poor tensile strength

Flexibility of 1/2 should maximize strength for a knottedmaterial


Arbitrarily Stiff Links


Torus knots

Conjecture: (n, n + 1)-torus knots exhibit the same behavior

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Rop

elen

gth

Flexibility

(2,3) torus knot(3,4) torus knot(4,5) torus knot(5,6) torus knot(6,7) torus knot


Thanks

Collaborators:

Roman Buniy (University of Indiana)Jason Cantarella (University of Georgia)Akos Dobay (Ludwig-Maximillians-Universitat, Munich)Tom Kephart (Vanderbilt University)John Kern (Duquesne University, Pittsburgh)Ken Millett (University of California, Santa Barbara)Andrzej Stasiak (University of Lausanne, Switzerland)

Students:

Ted Ashton (University of Georgia)Pat Plunkett (University of California, Santa Barbara)Michael Piatek (University of Washington)

National Science Foundation

IMA


measuring the size and shape of polymers · eric rawdon measuring the size and shape of polymers....

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