Genoa, Italy               StatPhys 23            July 12, 2007The Elusiveness of Polymer Knots

Paul Dommersnes, Ralf Metzler, Andreas HankeYacov Kantor, Oded Farago, Roya Zandi

Peter Virnau, Grigory Kolesov, Leonid Mirny

OutlineI.       Classification of knots                                         II.      The tightness of charged knots                                     III.     Entropic tightening of slip-links and ‘flat’ knots          IV.     Open polymers: pulling on knots; model polyetheleneV.      Rarity of knotted proteins

Classification of Knots

Knots are usually classified according to the minimal number of crossings in planar projection. [P.G. Tait, Trans. Roy. Soc. Edinburgh 28, 145 (1876-7)]

Examples of `prime knots'

Example of  `composite knot'            31#31

Identification of a knot is difficult, because it is a global property that depends on the entire shape of the curve (need knot invariants):

• J.W. Alexander (1923) polynomialFirst algorithm which can distinguish between (some) knots

• Jones, HOMFLY, Kauffman polynomials, ...

Charged Knots

• Can the knot type be determined from the ‘ideal shape’ of the curve that minimizes a particular (two-body) potential?

• Jonathan Simon in Ideal Knots (1998): “Suppose you have a knotted loop on a string, and you spread an electric charge along the string and then let go; what will happen? …This question has been a common ‘cocktail party’ topic among knot theorists for many years, …”

• The Coulomb interaction is not useful for this purpose, since …

• Charged prime knots are tight:

• Charged composite knots are factored:

Screened Interactions• Do tight knots disappear if the Coulomb interaction is screened?

• Bjerrum length•

• The answer depends on the ‘electorstatic (Odijk) persistence length:’The tight knot is the global energy minimum as long as lc

is comparable to size of the open chain.The tight knot is a local energy minimum as long as lc

is larger than the bare persistence length.

Entropic Tightening• Even without interactions, knots could be ‘tight:’

A. Yu Grosberg, et al, Phys. Rev. E 54, 6618 (96). [Flory theory]

E. Orlandini, et al, J. Phys. A 31, 5953 (98). [simulations]

V. Katrich, et al, Phys. Rev. E 61, 5545 (00). [phantom walks]• This can be motivated by examining a Figure-8 (slip-link):

• The tendency for tight loops is characterized by an exponent c :c=d/2 for an ideal polymer (random walk) in d-dimensionsc=2.69… for a self-avoiding polymer in d=2c=2.26… for a self-avoiding polymer in d=3[B. Duplantier, Phys. Rev. Lett. 57, 941 (1986)]

‘Flat’ Knots• Topologically constrained polymers in two dimensions, with a fixed number of crossings, e.g. polymers absorbed to a surface.

A.Yu Grosberg and S.K. Nachaev, J. Phys. A25, 4659 (92).

E. Guitter and E. Orlandini, J. Phys. A 32, 1359 (99).• The Flat Figure-8- Theory predicts:• This is confirmed by simulations:

• Also observed in simulations on vibrated chains:

M.B. Hastings, Z.A. Daya, E. Ben-Naim, and R.E. Ecke,

Phys. Rev. Lett. E 66, 025102(R) (2002).

• The Flat Figure-8- Theory predicts:

• The Flat knot- Theory predicts that all flat knots are tight in the swollen phase,

• The Flat knot- Theory predicts that the trefoil is loose in the compact phase,

E. Orlandini, A.L. Stella, and C.Vanderzande, PRE 68, 031804 (2003)

Pulling knotted polymers• An indirect probe of the size of the 3-d knot:

O. Farago, Y. Kantor and M. Kardar, Europhys. Lett. 60, 53 (2002).

• According to scaling, for an unknotted polymer of length N,

• Comparison of simulation results for sizes N=225 (diamonds), 335 (triangles), 500 (squares), 750 (circles), with (solid) and without (open) a knot in the polymer:

• We interpreted the results as a reduction in the length of the polymer, by the extentof the knot, as , with t~0.5

Model Polyethylene (CH2)n• Monte Carlo (MC) simulations of a coarse-grained model for polyethylene

Bead-spring chain (LJ+FENE): 1 bead ≅ 3 CH2Equilibrium configurations generated with standard MC techniques

(pivot, reptation, local moves) Qualitative results in   coil (swollen),   globule (compact),  and   confined  sates.

• Knots are rare in the swollen phase (1% for 3000 CH2) but common in a dense phases (80% for 3000 CH2) 

• Knots are tight in the swollen phase but loose in a dense phases

P. Virnau, Y. Kantor and M. Kardar, J. Am. Chem. Soc. 127, 15102 (2005)

Biopolymers• Knots are rare for DNA in good solvent 

(0.5%-4% for 10000 base pairs) • Knots are also rare in globular proteins  

(~1% - 273/32,853 in PDB structures, 1/3/2007)

Intricate Knots in Proteins: Function and Evolution

P. Virnau, L. Mirny, and M. Kardar, PLOS Comp. Biol. 2, e122 (2006)]

In contrast to globular polymers, knots are extremely rare in globular proteins, and their occurrence is likely connected to protein function in as yet unexplored fashion.

We analyzed all experimentally known protein structures and discovered several unknown knots, including the most complicated knot found to date (Fig.1 right). In this particular case, we believe that the occurrence of the knot might be related to the role of the enzyme in protein degradation.

While protein knots are typically preserved across species and sometimes even across kingdoms, we also identified an example of a knot which is not present in a closely related structure (Fig.2). The emergence of this knot is accompanied by a shift in the enzymatic function of the protein. It is also easy to imagine how this alteration happened: a simple insertion extends the loop and modifies the folding pathway of the protein.

Examples of the three different types of knots found in proteins.

Structures of Transcarbamylase from X. campestris with knot (left); and from Human without knot (right).

Genoa, Italy               StatPhys 23            July 12, 2007The Elusiveness of Polymer Knots

Paul Dommersnes, Ralf Metzler, Andreas HankeYacov Kantor, Oded Farago, Roya Zandi

Peter Virnau, Grigory Kolesov, Leonid Mirny

SummaryI.       Knots are source of fascination of mystery in arts, nature, and scienceII.      Charged knots are tight, stable or metastable depending on rigidityIII.     ‘Flat’ knots are (weakly) tightened by entropic effects     IV.     Knots are rare and tight in swollen polymers; abundant and loose when compact