We study the equilibrium shapes of prime and composite knots confined to two
dimensions. Using rigorous scaling arguments we show that, due to self-avoiding
effects, the topological details of prime knots are localised on a small
portion of the larger ring polymer. Within this region, the original knot
configuration can assume a hierarchy of contracted shapes, the dominating one
given by just one small loop. This hierarchy is investigated in detail for the
flat trefoil knot, and corroborated by Monte Carlo simulations.Comment: 4 pages, 3 figure

A. Dobay

A. Grosberg

A. L. Kholodenko

A. van Oudenaarden

A. Yu. Grosberg

Andreas Hanke

B. Alberts

B. Duplantier

B. Maier

E. Ben-Naim

E. Guitter

E. Orlandini

H. L. Frisch

J. Zinn-Justin

J.-P. Sauvage

K. Koniaris

K. Ohno

L. H. Kauffman

L. Schäfer

M. Delbrück

M. Doi

M. Otto

Mehran Kardar

O. Viro

P.-G. de Gennes

Paul G. Dommersnes

R. C. Ball

Ralf Metzler

S. Tabachnikov

V. I. Arnold

V. Katritch

W. E. Moerner

Y. Arai

Y. Kafri

Yacov Kantor

English

arXiv

Crossref

Physical Review Letters

Equilibrium Shapes of Flat Knots

The equilibrium shapes of prime and composite knots were studied. It was showed that the topological details of prime knots were localized on a small portion of the larger ring polymer due to self-avoiding effects. The original knot configuration could assume a hierarchy of contracted shape, the dominating one given by one small loop, within this region. For the flat trefoil knot, this hierarchy was studied in detail and corroborated by Monte Carlo simulations

Metzler, Ralf

Hanke, Andreas

Dommersnes, Paul G.

Kantor, Yacov

Kardar, Mehran

Scholarworks@UTRGV Univ. of Texas RioGrande Valley

University of Texas Rio Grande Valley ScholarWorks @ UTRGV Physics and Astronomy Faculty Publications and Presentations College of Sciences 5-6-2002 Equilibrium shapes of flat knots Ralf Metzler Andreas Hanke Paul G. Dommersnes Yacov Kantor Mehran Kardar Follow this and additional works at: https://scholarworks.utrgv.edu/pa_fac  Part of the Astrophysics and Astronomy Commons Recommended Citation Ralf Metzler, et. al., (2002) Equilibrium shapes of flat knots.Physical Review Letters88:181881011. DOI: http://doi.org/10.1103/PhysRevLett.88.188101 This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Physics and Astronomy Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact justin.white@utrgv.edu, william.flores01@utrgv.edu. arXiv:cond-mat/0110266v1  [cond-mat.stat-mech]  12 Oct 2001Equilibrium shapes of flat knotsRalf Metzler,1 Andreas Hanke,1 Paul G. Dommersnes,1 Yacov Kantor,2, 1 and Mehran Kardar1, 31Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA2School of Physics and Astronomy, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel3Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106(Dated: 29th October 2018)We study the equilibrium shapes of prime and composite knots confined to two dimensions. Usingrigorous scaling arguments we show that, due to self-avoiding effects, the topological details of primeknots are localised on a small portion of the larger ring polymer. Within this region, the originalknot configuration can assume a hierarchy of contracted shapes, the dominating one given by justone small loop. This hierarchy is investigated in detail for the flat trefoil knot, and corroborated byMonte Carlo simulations.PACS numbers: 05.20.-y, 02.10.Kn, 87.15.Aa, 87.15.Ya, 82.35.PqThe static and dynamic behaviour of single polymerchains, such as DNA, and multichain systems like gelsand rubbers, is strongly influenced by knots and perma-nent entanglements [1, 2, 3, 4]. Topological constraintsare created with probability one during the polymerisa-tion of long closed chains [5]; more generally, knots andentanglements are a ubiquitous element of higher molec-ular multi-chain melts and solutions. This has profoundconsequences, reaching far into biology and chemistry.For instance, knots in DNA impede the separation ofthe two strands of the double helix during transcription,and therefore the access to the genetic code [6]. Chemi-cally, even single closed polymers may exhibit quite dif-ferent properties if they have different topology [7]. In thenanosciences, recent experimental techniques allow sin-gle polymer molecules (with fixed topology) to be probedand manipulated; e.g., by single molecule spectroscopy[8], or by optomicroscopical imaging of small latex beadsattached to a molecule [9]. These tools provide impe-tus for the theoretical understanding of the behaviourof macromolecules under topological constraints. How-ever, analytical studies, such as the statistical mechan-ics of a knotted polymer, are difficult since topologicalconstraints require knowledge of the complete shape ofthe curve. Such global constraints are hard to imple-ment, and a complete statistical mechanical descriptionof knots remains unattained [10, 11, 12].The mathematical discipline of knot theory providestools for the classification of knots. In particular, differ-ent knots can be distinguished by their projections ontoa 2D plane, keeping track of crossings according to whichsegment passes on top of another [3, 4]. By a sequence ofso-called Reidemeister moves [3], which leave the topol-ogy unchanged, the number of crossings can be reducedto a minimum, which is a simple topological invariant[3, 4]. For instance, in Fig. 1 we depict the minimal pro-jection of the trefoil knot, classified as 31, with its 3 cross-ings. Such quasi 2D projections, which we call flat knots ,can be physically realized by compressing originally 3Dknots by forces normal to the projection plane. Examplesinclude polymers adsorbed on a surface or membrane byelectrostatic or other adhesive forces [13]; or confined be-tween parallel walls. In these cases the flat polymer knotcan still equilibrate in 2D. Another experimental realiza-tion comes from Ref. [14], in which macroscopic knottedchains are flattened by gravity onto a vibrating plane.The equilibrium shapes, and their scaling properties, ofsuch flat knots are studied in this paper. Flat knots havethe additional advantage of being easy to image by mi-croscopy. They are also more amenable to numeric stud-ies than their 3D counterparts, and have in fact beenalready studied in Ref. [15].There is growing numerical evidence that prime knotsare tight in the sense that the topologically entangled re-gion is statistically likely to be localised on a small por-tion of the longer chain [15, 16, 17]. Indirect numericalevidence of this was originally obtained by simulations in-dicating that the radius of gyration of a long polygon in3D is asymptotically independent of its knot type; whilethe presence of the knot increases the number of config-urations by a factor related to the number of positions ofthe tight region around the remaining loop [17]. Simula-tions of 2D polygons in Ref. [15] provide quite convincingvisual evidence of localised knot regions. In this paper,we quantify the tightness of flat knots , using scaling ar-guments to obtain the power law size distributions for ahierarchy of possible equilibrium shapes. For the trefoil,Fig. 1 shows this hierarchy of shapes and the correspond-ing exponents for the distribution of knot size.To demonstrate the entropic origin of tight shapes, weinitially consider a simple ring of length L with one cross-ing. The effect of the crossing consists in creating twoloops of lengths ℓ and L − ℓ, respectively, while the ori-entation of the crossing is irrelevant. In this sense, the231I II III IV Va Vb VI`L-`s1s2s3s5s1s3s2s4G  1  2d + 4 3d + 24 4d + 34 3d + 6 4d + 4+ 6 4d + 8m 0 2 4 1 3 24316= 2:68755416= 3:3756516= 4:06258816= 5:59916= 6:187514516= 9:0625Figure 1: Standard minimal projection of the trefoil knot 31, followed by its different possible contractions, arranged accordingto higher scaling orders. The uncontracted trefoil geometry is found at position III of the hierarchy. At I, the figure-eightstructure is drawn. The diagrams II-VI show the multiply connected knot region of total length ℓ =∑N−1i=1si where theprotruding legs indicate the outgoing large loop of length sN = L − ℓ. Below the individual contractions, we include thenetwork exponents γG , the number m of independent integrations, and the exponents c defined via the PDF, p(ℓ) ∼ ℓ−c.crossing can be replaced with a vertex with four outgoinglegs. We denote the resulting network by GI (see posi-tion I in Fig. 1). In fact, we can more generally considera sliding ring, or slip-link [18], to force two points of thechain to be close to each other, in d dimensions. With-out self-avoiding constraints (ideal chains), the numberof configurations ωI(ℓ, L) scales as [1, 19]ωI(ℓ, L) ∼ µLℓ−d/2(L− ℓ)−d/2 , (1)where, on a lattice, µ is the effective connectivity con-stant for Gaussian random walks. The average loopsize is given by 〈ℓ〉 =∫ L−aa dℓℓωI(ℓ, L)/∫ L−aa dℓωI(ℓ, L),where a is a short-distance cutoff set by the lattice con-stant. Note that 〈ℓ〉 = L/2 due to symmetry. However,the corresponding probability density function (PDF) isstrongly peaked at ℓ = 0 and ℓ = L, and a typical shapeconsists of one tight and one large loop. In d = 2, themean size of the smaller loop, 〈ℓ〉< ∼ L/| ln(a/L)|, isstill rather large. It is instructive to compare to higherdimensions: one has weak localisation, 〈ℓ〉< ∼ a1/2L1/2,in d = 3, and strong localisation, 〈ℓ〉< ∼ a, in d > 4.Thus, for ideal chains, tightness of the smaller loop ismore pronounced in higher dimensions.To include self-avoiding interactions, we use results forgeneral polymer networks obtained by Duplantier [20],and in Refs. [21, 22]: In a network G consisting of Nchain segments of lengths s1, . . . , sN and total length L =∑Ni=1 si, the number of configurations ωG scales as [23]ωG(s1, . . . , sN ) = µLsγG−1N YG(s1sN, . . . ,sN−1sN), (2)where YG is a scaling function, and µ is the effective con-nectivity constant for self-avoiding walks. The exponentγG is given by γG = 1 − dνL +∑N≥1nNσN , where ν isthe swelling exponent, L is the number of independentloops, nN is the number of vertices with N outgoing legs,and σN is an exponent associated with such a vertex. Ind = 2, σN = (2−N)(9N + 2)/64 [20].The network GI corresponds to the parameters N = 2,L = 2, n4 = 1, s1 = ℓ, and s2 = L − ℓ. By virtue ofEq. (2), the number of configurations of GI with fixed ℓfollows the scaling formωI(ℓ, L) = µL(L− ℓ)γI−1X(ℓL−ℓ), (3)where γI = 1 − 2dν + σ4. In the limit ℓ ≪ L, ωI(ℓ, L)should reduce to the number ωcrw(L) ∼ µLL−dν of closedrandom walks of length L which start and end at a givenpoint in space [19, 24]. This implies X (x) ∼ xγI−1+dν asx→ 0, such thatωI(ℓ, L) ∼ µL(L − ℓ)−dνℓ−c , ℓ≪ L , (4)where c = −(γI− 1+ dν) = dν − σ4. Using σ4 = −19/16and ν = 3/4 in d = 2, we find c = 43/16 = 2.6875.In d = 3, σ4 ≈ −0.48 and ν ≈ 0.588, so that c ≈ 2.24[21, 24]. In both cases the result c > 2 implies that theloop of length ℓ is strongly localised in the sense definedabove. This justifies the a priori assumption ℓ≪ L, andmakes the analysis self-consistent. Note that for self-avoiding chains, in d = 2 the localisation is even strongerthan in d = 3, in contrast to the corresponding trend forideal chains.We performed Monte Carlo (MC) simulations (see be-low for details) of a chain in d = 2 confined to the GI(figure-eight) structure by a slip-link. As shown in Fig.2, the size distribution for the small loop can be fittedto a power law with exponent c = 2.7± 0.1 [25], in goodagreement to the above prediction.For the configuration GI , the probability for the sizeof each loop is peaked at ℓ → 0 and ℓ → L. For morecomplicated projections, the joint probability to find theindividual segments with given lengths si is expected topeak at the edges of the higher-dimensional configuration310 10010−510−410−310−210−1 slope = −2.7slope = −3.4trefoil knotfigure−eight structurecontraction Icontraction II`p(`).Figure 2: Power law tails in PDFs for the size ℓ of tightsegments: As defined in the figure, we show results for thesmaller loop in a figure-eight structure, the overall size ofthe trefoil knot, as well as the two leading contractions of thelatter. The insets show typical configurations of the small loopfor a figure-eight (the arrow points to the slip-link consistingof 3 tethered beads), and the trefoil.hyperspace. Some analysis is necessary to find the opti-mal shapes; as presented here for the simplest non-trivialknot, the (flat) trefoil knot 31 (see Fig. 1). Each of thethree crossings is replaced with a vertex with four outgo-ing legs, and the resulting network is assumed to separateinto a large loop and a multiply connected region whichincludes the vertices. Let ℓ =∑5i=1 si be the total lengthof all segments contained in the multiply connected knotregion (see Fig. 1, position III). Accordingly, the lengthof the large loop is L− ℓ. In the limit ℓ≪ L, the numberof configurations of the network GIII can be derived in asimilar way as above, yieldingω′III ∼ µL(L− ℓ)−dνℓγIII−1+dνW(s1ℓ,s2ℓ,s3ℓ,s4ℓ), (5)where γIII = 1− 4dν + 3σ4 and W is a scaling function.The prime on ωIII indicates that each of the segmentlengths si is fixed. In order to obtain the number ofconfigurations ωIII(ℓ, L) for the case we are interested in,where only the total length ℓ is fixed, we should inte-grate ω′III over all distributions of lengths si under theconstraint∑5i=1 si = ℓ. This leads to the resultωIII(ℓ, L) ∼ µL(L− ℓ)−dνℓ−c, (6)with c = −(γIII− 1+ dν)−m, where m = 4 correspondsto the number of independent integrations over si. Thus,c = 3dν − 3σ4 − 4 =6516(see Fig. 1, position III).However, some care is necessary in performing theseintegrations, since the scaling function W in Eq. (5) mayexhibit non-integrable singularities if one or more of itsarguments tend to 0 or 1. The geometries correspondingto these limits (edges of the configuration hyperspace)represent contractions of the original trefoil network GIIIin the sense that the length of one or more of the seg-ments si is of the order of the short-distance cutoff a.If such a short segment connects different vertices, theycannot be resolved on larger length scales, but melt intoa single, new vertex, in the context of our scaling analy-sis [26]. Thus, each contraction corresponds to a differentnetwork G, which may contain a vertex with six or eveneight outgoing legs. For each of these networks, one cancalculate the corresponding exponent c in a similar way asabove, and using the Euler relations 2N =∑N≥1NnNand L =∑N≥112(N − 2)nN + 1, we obtaincG = 2 +∑N≥4 nN[N2(dν − 1) + (|σN | − dν)]. (7)Our scaling analysis relies on an expansion in a/ℓ≪ 1,and the values of c determine a sequence of contractionsaccording to higher orders in a/ℓ: The smallest value ofc corresponds to the most likely contraction, while theothers represent corrections to this leading scaling be-haviour, and are thus less and less probable (see Fig. 1).To lowest order, the trefoil behaves like a large ring poly-mer at whose fringe the point-like knot region is located.At the next level of resolution, it appears contracted tothe figure-eight shape (see Fig. 1, position I). If more ac-curate data are available, the higher order shapes II toVII may be found with decreasing probability. Inter-estingly, the original uncontracted trefoil configurationranks third in the hierarchy of shapes. Note that thecontractions shown in Fig. 1 may occur in different topo-logical variants. For instance, the smaller loop in contrac-tion I could be inside the larger loop. However, this doesnot make a difference in terms of the scaling analysis.To check these predictions, we performed MC simu-lations of a flat trefoil knot using a standard bead-and-tether chain model in 3D. The polymer was flattened bya “gravitational” field V = −kBTh/h∗ perpendicular toa hard wall, where h is the height and h∗ was set to 0.3times the bead diameter. With 512 beads, around 1011MC steps were performed to generate equilibrium states.Configurations corresponding to contraction I are thenselected by requiring that besides a large loop, they con-tain only one segment larger than a preset cutoff length(taken to be 5 monomers), and similarly for contractionII. The size distributions for such contractions, as wellas for all possible knot shapes are shown in Fig. 2. Thetails of the distributions are indeed consistent with thepredicted power laws, although the data (especially forcontraction II) is too noisy for a definitive statement.These scaling results pertain also to other prime knots.In particular, the dominating contribution for any primeknot corresponds to the figure-eight contraction GI. Thiscan be shown by noting that Eq. (7) predicts a larger43181931#31Figure 3: Typical equilibrium configurations for the trefoil 31, the prime knot 819, and the composite knot 31#31 consisting oftwo trefoils, in d = 2. The initial conditions were symmetric.value of the scaling exponent c for any network G otherthan GI. Figure 3 demonstrates the tightness of the primeknot 819. Composite knots, however, can maximise thenumber of configurations by splitting into their prime fac-tors as indicated in Fig. 3 for 31#31. Each prime factoris tight and located at the fringe of one large loop, andaccounts for an additional factor of L for the number ofconfigurations, as compared to a ring of length L withouta knot. Indeed, this gain in entropy leads to the tightnessof knots.In conclusion, we find that the trefoil knot, as well ashigher order prime and composite knots, are sharply lo-calized when forced to lie flat. In the most likely shapes,each prime factor is tightened into a loop (a figure-eightcontraction). It is natural to speculate that entropic fac-tors also confine knots in d = 3 by power law distributionsin size. Direct checks of such behaviour are hamperedby the difficulty of identifying the knotted region of acurve [16] in d = 3. One may instead search for indirectsignatures of localized knots in detailed dependencies ofgyration radius and other polymeric quantities on length.RM and AH acknowledge financial support from theDFG. PGD acknowledges financial support from theResearch Council of Norway. Support from the NSF(DMR-01-18213 and PHY99-07949) and the US-IsraelBSF (1999-007) is also acknowledged.[1] P.-G. de Gennes, Scaling concepts in polymer physics(Cornell University Press, Ithaca, New York, 1979).[2] M. Doi and S.F. Edwards The Theory of Polymer Dy-namics (Clarendon Press, Oxford, 1986).[3] L.H. Kauffman, Knots and physics (World Scientific, Sin-gapore, 1993).[4] A.Yu. Grosberg and A.R. Khokhlov, Statistical Mechan-ics of Macromolecules (AIP Press, New York, 1994).[5] H.L. Frisch and E. Wasserman, J. Am. Chem. Soc. 83,3789 (1961); M. Delbru¨ck, Proc. Symp. Appl. Math. 14,55 (1962).[6] B. Alberts et al., The molecular biology of the cell (Gar-land, New York, 1994).[7] J.-P. Sauvage and C. Dietrich-Buchecker, Molecular cate-nanes, rotaxanes, and knots: a journey through the worldof molecular topology (Wiley-VCH, Weinheim, 1999).[8] W.E. Moerner and M. Orrit, Science 283, 1670 (1999).W.E. Moerner and L. Kador, Phys. Rev. Lett. 62, 2535(1989).[9] A. van Oudenaarden and J.A. Theriot, Nat. Cell. Biol.1, 493 (1999); J.A. Theriot et al., Nature 37, 257 (1992).[10] T.A. Vilgis and M. Ott, Phys. Rev. Lett. 80, 881 (1998).[11] A.Yu. Grosberg, Phys. Rev. Lett. 85, 3858 (2000).[12] A.L. Kholodenko and T.A. Vilgis, Phys. Rep. 298, 251(1998).[13] B. Maier and J.O. Ra¨dler, Phys. Rev. Lett. 82, 1911(1999).[14] E. Ben-Naim et al., Phys. Rev. Lett. 86, 2001. (Whethersuch shaking results in equilibrium states is not clear.)[15] E. Guitter and E. Orlandini, J. Phys. A 32, 1359 (1999).[16] V. Katritch et al., Phys. Rev. E 61, 5545 (2000).[17] E. Orlandini, M.C. Tesi, E.J. Janse van Rensburg, andS.G. Whittington, J. Phys. A 31, 5953 (1998).[18] M. Doi and S.F. Edwards, J. Chem. Soc. Faraday Trans.II 74, 1802 (1978); R.C. Ball et al., Polymer 22, 1010(1981).[19] Following Refs. [15, 17], we consider two configurations asdistinct if they cannot be superimposed by translation.This eliminates the degrees of freedom due to transla-tional invariance of the whole structure.[20] B. Duplantier, Phys. Rev. Lett. 57, 941 (1986); J. Stat.Phys. 54, 581 (1989).[21] L. Scha¨fer et al., Nucl. Phys. B 374, 473 (1992).[22] K. Ohno and K. Binder, J. Phys. (Paris) 49, 1329 (1988).[23] This result is valid if the network has at least one vertexwith N 6= 2 outgoing legs. In contrast, for a simple ringpolymer of length L, one has ω(L) = µLL−dν−1 [19].[24] Y. Kafri et al., Phys. Rev. Lett. 85, 4988 (2000); preprintcond-mat/0108323 (2001).[25] The quoted errors reflect our subjective estimate of pos-sible systematic errors.[26] In field theory, this is an example of an operator productexpansion; see J. Zinn-Justin, Quantum Field Theory andCritical Phenomena (Clarendon Press, Oxford, 1989).

Equilibrium shapes of flat knots

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Equilibrium shapes of flat knots

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