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Topological entanglement of long polymer chains at equilibrium

Lecture IV:Topological entanglement in ropes

The rope in the picture is spatially organized in a quiteintricated manner. Suppose that the two ends are tied together.If the rope is sufficiently flexible, we can try to simplify itsspatial arrangment by smooth deformations. This would remove most of the geometrical entanglement.

after some simplification the rope may look like or, by simplifying a little more, we can get something like

2

31

or something like

915

or

or

more frequent situation…..

KNOTS

only TOPOLOGICAL ENTANGLEMENT is left. This is described in terms of

…once all the geometrical entanglement has been removed

To remove topological entanglementwe have to cut with scissors the rope and let the strands of the rope to cross each others!!

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A knot is any simple closed (diff) curve embedded in R3

Definition (KNOT)

A knot is any polygonal curve embedded in R3

or

unknot

31 31 31

Knot as equivalent class (under ambient isotopy)

Ambient isotopy

Ambient isotopy

Ambient isotopy

Ambient isotopy

unknot unknot

Knot projections

knot diagrams

regular

singular

Minimal knot diagram

II

III

I I

II

Reideimester moves

Moves on knot diagrams that corresponds to smooth deformations in 3D

4

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Minimal knot diagrams

Non minimal knot diagrams

Minimal knot diagrams

Minimal ? Non minimal ?

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By playing with Reideimester moves….

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Main problem in knot theory: given an embedding of a closed loop (or its diagram) determine to which knot class it belongs to.

Design topological invariants, i.e. quantities that can be build algebraically, combinatorially or from the theory of manifolds whose properties do not depend on the embedding but only on the knot (topological) class to which this embedding belong to.

(some) Topological invariants

Minimal crossing number; unknotting number …

Genus of the Seifert surface associated to a knot

Knot polynomials

Knot polynomials

Alexander polynomial, Jones polynomialHOMFLY polynomial

∆(t)=1 - t + t2 J(q)= - q-4 + q-3 + q-1

∆(t) = 1 - 3t + t2 J(q)= q-2 - q-1 + 1 - q + q2

∆(t)= (1 - t + t2 )2

J(q)= - q-5 + q-4 - q-3 + 2q-2 - q-1 + 2 - q820

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The perfect knot invariant has not been found yet. What all this has to do with polymers ?

Physics and Chemistry

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Connectivity

Flexible

Polymers

On largelength scales

Entangledin space

Cyclicizationprocess

31

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unknot

Polymers are characterized by excludedvolume interactions that does not allow strandpassages.

BiologyEscherichia coli

4Mbp of circular DNA

?

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Topoisomerases: enzymes that simplify topology

Enzyme

Topo II cuttinga ds DNA

ds Strand passage operation

Dean et al., J BIOL. CHEM. Dean et al., J BIOL. CHEM. (1985)(1985) SpenglerSpengler et al. CELL et al. CELL (1985).(1985).

(2,13) (2,13) TorusTorus knotknot

Knotted Circular Bacterial DNA

The amount of Topoisomerase II in the eukaryotic cellis cyclic, with peaks (phase M) and minima (phase G1)

By stabilizing the cleavable complex topoII-DNA it ispossible to increase the action of topo II i.e. the numberof strand breaks. This cause the death of the cell if thesebreaks are not promptly repaired.

This is the way in which some drugs based on antraciclineor epidofillotoxins etc. acts to kill cells.

Acting on topo II functions in cancer therapy

Back to Physics

Which are the essential features thatwe should take into account in order to modeland study properly the topologicalentanglement of long polymers ?

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Entropy (Free energy) is important

Statistical mechanics approach to thetopological properties is natural

Unlike ropes polymers are mesoscopic chainsthat live in solution (thermal fluctuations)

Coarse grained models

Coarse grained models

Lattice polygons

Small number of possibleorientations.

Equilateralrandompolygons

Gaussianrandompolygons

Similar to Equilateral random polygonsbut each edge is a Gaussian randomvector in R3

Excluded volume

Statistical approach to the problem

What is the probability that a polymer ring (in equilibrium) with N units is knotted ?

What is the probability that a N unit linear polymer (in equilibrium) becomes knotted under cyclization ?

How knotting probability and knot complexitydepends on:

Degree N of polymerization

Quality of the solvent

Confinement

What is the probability that a ring (in equilibrium) with N unit is knotted ?

Difficult to give a definitive answer for any finite N.In general one must rely on numerical approachesbased on MC or MD simulations on coarse grained models

In the (asymptotic) limit N >> 1, however, it ispossible to be more precise….

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FRISCH-WASSERMAN-DELBRUCK CONJECTURE

P(N) = knotting probability for N units polymer

lim P(N) = 1N → ∞

Frisch & Wasserman, JACS 83 (1961), 3789Delbruck, Proc. Symp. Appl. Math. 14 (1962), 55

PROOF OF FWD CONJECTURE

THEOREM:

P(N) ~ 1 - exp(-α0 N +o(N)) α0 > 0

Sumners & Whittington (1988)Pippenger (1989)

Dia et al. (1995)

Estimates of α0 and finite N behaviour by Monte Carlo simulations for example on SAP’s

A skecth of the proof for polygons on the cubic lattice

First notice that….

0

PN= 1 - PN = 1 -0 pN

pN

therefore the FWD Conjecture can be reformulated as

lim P0n = 0

n → +∞

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1st ingredient: antiknots do not exist

For any knot type κ there does not exist a knot type κ’such that κ # κ’ = unknot

There are no antiknots that can untie a given knot.

The proof relies on the additivity of the genus under connected sum

κ κ’

2nd ingredient: tight knots

Knot untied by the red part

Local knot

We need local tight knot (no space available)

The rest of the polygon(because of self-avoidance) cannotreenter this region and untie the knot.

Knotted ball pair correspondingto the trefoil

3rd ingredient: Pattern theoremKesten, J. Math. Phys (1963)

Kesten pattern T: a subwalkthat can occur at least three times on a sufficientlylong polygon

Τ Τ Τ

not a Kesten pattern

Kesten’s Pattern Theorem (Kesten 1963) :Let T be a Kesten pattern. Then there exists ε > 0 such that T appears at least εn times on all except exponentially few sufficiently long SAPs.

lim sup n-1 log pn(T) < σn→+∞

Mixing all the ingredients…

Τ = Τ Τ Τ

σ0 < σ0

PN =0 pN

pN= e-(σ− σ0)N + o(N)

0α0 = σ−σ0 > 0

pN(T) = # polygons that do not contain the pattern T (tight trefoil )

pN(31) = # polygons that do not contain any trefoil

p0n pn(31) pn(T) ===>

N → +∞

n → +∞

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Long polymers are badly knotted too !!

THEOREM: All good measures of knot complexity diverge to + ∞ at least linearly with N

Examples of good measures of knot complexity:unknotting number, genus, braid number, span of your favorite knot polynomial, etc.

The longer the random polygon,

the more entangled it is.

More complicated or open questions

-We know that α0 > 0 but little else is known rigorously (upper and lower bounds ? ). Is α0 lattice dependent ?

- knotting probability for fixed knot type. We don’t even know if σ(τ) exist for fixed knot type τ. Reasonably, we expect σ0<= σ(τ) < σ.

Open problem: prove that actually σ0=σ(τ).

- knotting probability for finite small N

- knot spectrum (distribution on knot types for fixed N)

- dependence of the knot spectrum on N

Can be tackled by numerical approaches

Monte Carlo study of knotting probability

based on two-point pivot moves

inversion

Reflection (x=-y)

Ergodic on the whole class of polygons

Very efficient in sampling unweighted polygons

Good for computing knotting probabilities, but a knot detector is needed

Alexander polynomial: less perfect but less costly numerically

P0N = e-α0N + o(N)

60 10−≈α

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Deguchi & Tsurusaki (1997) seeabove

Koniaris & Muthukumar (1991), J. Chem. Phys. 95, 2873-2881

Rod bead model3.7 x 10-3 (r=0.05)

1.25 x 10-6 (r=0.499)

Deguchi & Tsurusaki (1997), Phys. Rev E. 55, 6245-6248

Gaussian2.9 x 10-3

Michels and Wiegel (1986) Proc. Roy. Soc. A, 403, 269-284

Gaussian (Langevin. N up to 320)

(3.6 +/- 0.02) x 10-3

Janse van Rensburg (2002) seeabove

SAPs BCC (5.82 +/- 0.37) x 10-6

Janse van Rensburg (2002)Contemporary Mathematics Vol. 304 (American Mathematical Society, Providence, Rhode Island), pp. 125-135.

SAPs FCC (5.91 +/- 0.32) x 10-6

Yao et al (2001), J. Phys. A: Math. Gen. 34, 7563-7577

SAPs SC (up to 3000)(4.0 +/- 0.5) x 10-6

Janse van Rensburg & Whittington(1990), J. Phys. A: Math. Gen. 23, 3573-3590

SAPs CC (up to 1600)(7.6 +/- 0.9) x 10-6

refsmodelαααα0000 = 1/Ν0

α0 estimates

…and what about knotsin adsorbing polymers ??

Adsorbing positive 3D polygons

pN+ (Ns)= # positive N-edges

Polygons with Ns visits

( ) ∑=

+=N

N

NsNN

s

seNpZ0

)( αα ( ) ( )αα FZN N

N=

∞→log

1lim

F( )α

α c α

κ

α= αc the free energy isnot analytic (phase transition)

desorbed

adsorbed

Knotting probability of adsorbing polygons

( ))(

)(1)(1

00

αααα

N

NNN Z

ZPP −=−=

s

s

NN

NsNN eNpZ αα ∑

=

=0

00 )()( Partition function of adsorbing UNKNOTTED polygons

( ) ( )αα 00log1

lim FZN N

N=

∞→It is possible to show

Which is the relation between

( ) ( ) ? and αα FF 0

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Theorem: (C. Vanderzande, J. Phys. A 38, 3681 (1995))

∞<∀α ( ) ( )αα FF <0

( )( )

( )( ) ( )( )NFF

NF

NF

N

NNN e

e

e

Z

ZPP αα

α

α

αααα

0

0

11)(

)(1)(1

00 −−−=−=−=−=

∞<∀α ( ) 1 → ∞→NNP α

Sufficiently long adsorbing polygons are almost surely knotted for any finite adsorbing energy.

In the limit ∞→α and fixed N, polygons are 2D objects and no knots are possible.

It is plausible to expect 0cc αα ≤

Numerical results

( ) ( ) ( )( )NFFN eP ααα

0

1 −−−=

1/T

F(α)−F0(α) x 10 6

Janse van Rensburg, Contemporary Mathematics, 304, (2002)

N=500

N=1000-1500

Given a knotted ring, is the hosted knot

ortight loose ?

The size of knots in polymers

Where is the knot?How can we measure its length?Practical problem:

Knot localizationSuppose we are able to measure the length, λ, of a knot

hosted by a N-size loop

tAN≈λ

t = 1 (τ = 1) ⇒ delocalization0 < t < 1 (1 < τ < 2) ⇒ weak localizationt = 0 (τ > 2) ⇒ strong localization

τλλ −≈)(P

Average knot size

Knot size distribution

The problem of measuring λ is an hard one

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Why is the problem hard ?•Knots are well defined only for closed curves•To localize the knot one has to extract an open substring and see if it is knotted

•But …. open strings are always unknotted!!!!!

And what about un unknotted closed string “hosting” a knot???

The closed curve is globally unknotted, but an arc extracted from it looks likea knotted arc.

A knot is in general a NON-local object: a crossing that in a knot diagram is well localized could be generated by two strands that are very far apart in 3D space.

This is not the case for knotted polymers confined in quasi two-dimensional spaces (knots in very thin slabs, strongly adsorbed rings ..)

If the confinement is severe one would expect knots that are essentially knot diagrams with the minimal number crossings compatible with the topology nd small excursion for overpassings

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the knotted part of the flat knot is composed by theL-1 smaller loops.

Flat Knots: a model of quasi-2D knots

crossings vertices

λ := length of the L-1 smaller loops

Let L = #loops ; V= # vertices;

Results for flat knots

λ total lengthof theL - 1 smaller loops

τλλ −≈)(P

Dominant contribution fron the 8-graph

τ ≈ 2,69

⇒ strong localization for flat knots.

τ predicted using polymer network theory.

B. Duplantier, J. Stat. Phys. 54, 581 (1988); R. Metzler et al, Phys. Rev. Lett. 88, 188101 (2002)

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3D knots: Closure schemes

Deterministic Random

c

3D: Our method for direct measure

Look for the minimal arc which,once reclosed in a ``topologically neutral” way,

still contains the knot.

J. Janse van Rensburg et al, J. Phys. A 25, 6557 (1992); K. Katritch er al, Phys. Rev. E 51, 5545 (2000) B. Marcone et al, J. Phys. A 38, L15 (2005)

Example Model and simulation

• Polymer model ⇒ Self-Avoiding Polygon in cubic lattice.

Simulation technique:

• Equilibrium Monte Carlo at fixed temperature• Two Monte Carlo algorithms

BFACF(N varying, topology preserved)Pivot (N fixed, topology varying)

...andMMC for better statistics

B. Berg, D. Foerster, Phys. Lett. B 106, 323 (1983); C. Aragao de Carvalho, S. Caracciolo, J. Phys. 44, 323 (1983)

M. Lal, Molec. Phys. 17, 57 (1969); Geyer C. J. Computing science and statistics: Proc 23rd Symp. Interface, 156 (1991)

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Results for direct measure

knot t

31 00..7722±±00..0033

41 00..7777±±00..0044

51 00..7799±±00..0033

52 00..7766±±00..0044

• Equilibrium simulations at T = ∞ show weak localizaion.• exponent apparently independent of knot type.

B. Marcone et al, J. Phys. A 38, L15 (2005)

Filling the gap

Different degrees of localization!• Adsorbed knots have excursions of arbitrary length for every

T > 0

• They also have

Must rely on simulation.

( )knotNN crossingscrossings min>• Direct treatment required to understand realistic adsorbed

knots!

• No theoretical tool available!

2D: Flat knots

stong localization

3D: “true” knots

weak localization

Polymer adsorption• Model: SAP confined on upper half plane z ≥ 0 and rooted• Plane z = 0 cannot be crossed

• Energy -1 to each contact:sNE −=

• Adsorption 2nd order transition at T = Tc ≈ 3.5:

φNN s =

• φ very close to 1/2.• Unknots exponentially rare like in bulk!

<>

>=

∞→cs

cs

N TTTf

TT

N

N

,0)(

,0lim

C. Vanderzande, J. Phys. A 38, 3681 (1995); T. W. Burkhardt et al, Nucl. Phys. B 316, 559 (1989)

• Special regime at the transition:

Desorbed phase

Adsorbed phase

Knotted polymer adsorption • Tc unaffected by knottedness.

• Crossover exponent φ unchanged, too:

φNN s = • φ = (0.53±0.02)

• The knot behaves like the whole chain • φknot = (0.49±0.03)

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Results at T=∞ and T=Tc

• At T = ∞ the exponent t,tAN≈λ

Thus, simply lowering T has no effect on knot localization up to the transition point.

is the same found

for bulk polymers.

•At T = Tc≈ 3.5, the exponent appears to be the same, too.

Results below Tc

• Below Tc, t decreases.

• At T ≈ 2.0: localization still weak; τ ≈ 1.9. • At lower T, strong localization regime appears.

• At T=1.25: τ exponent appears to have the flat-knots value (within errors!)

Conclusions

• HIGH T : weak localization• LOW T: strong localization

Adsorption =

Knot localization transition!

• At the transition: bulk behaviour

• Adsorption Tc and exponent φ unaffected by knottedness• Flat knots exponents found for 0 < T < Tc.

Ercolini et al. PRL 98 (2007)

AMF images of localized knots in adsorbed polymers !!!!

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The end