The entropic cost to tie a knot

The entropic cost to tie a knot M Baiesi1, E Orlandini2 and A L Stella3 1 Dipartimento di Fisica and sezione CNISM, Universit`a di Padova, Padova, Italy E-mail: baiesi@pd.infn.it 2 Dipartimento di Fisica and sezione CNISM, Universit`a di Padova, Padova, Italy E-mail: orlandini@pd.infn.it 3 Dipartimento di Fisica, and sezione INFN, Universit`a di Padova, Padova, Italy E-mail: stella@pd.infn.it Abstract. We estimate by Monte Carlo simulations the configurational entropy of N-steps polygons in the cubic lattice with fixed knot type. By collecting a rich statistics of configurations with very large values of N we are able to analyse the asymptotic behaviour of the partition function of the problem for different knot types. Our results confirm that, in the large N limit, each prime knot is localized in a small region of the polygon, regardless of the possible presence of other knots. Each prime knot component may slide along the unknotted region contributing to the overall configurational entropy with a term proportional to ln N. Furthermore, we discover that the mere existence of a knot requires a well defined entropic cost that scales exponentially with its minimal length. In the case of polygons with composite knots it turns out that the partition function can be simply factorized in terms that depend only on prime components with an additional combinatorial factor that takes into account the statistical property that by interchanging two identical prime knot components in the polygon the corresponding set of overall configuration remains unaltered. Finally, the above results allow to conjecture a sequence of inequalities for the connective constants of polygons whose topology varies within a given family of composite knot types. PACS numbers: 02.10.Kn, 36.20.Ey, 36.20.-r, 87.15.A- Submitted to: JSTAT Keywords: Knots in polymers, self-avoiding walks, Monte Carlo. arXiv:1003.5134v1 [cond-mat.stat-mech] 26 Mar 2010 The entropic cost to tie a knot 2 1. Introduction A long, flexible polymer chain in good solvent can be highly self-entangled [1, 2] and, if a ring closure reaction occurs, or if its extremities are hold tied by some device, the entanglement can be trapped as a knot [3, 4]. Moreover, because of the excluded volume interaction, a knotted molecule cannot change its topological status, without breaking and reconnecting back chemical bonds. This for example is the situation one encounters in biological systems where special enzymes, called topoisomerases, can pass one strand of the double stranded circular DNA through another and knot or unknot the molecule, to facilitate elementary cellular processes [5, 6]. In general, however, there is no spontaneous transition between different knotted statuses and in the most common experimental situations the topology of the ring does not change in time. Clearly, the presence of topological constraints limits the configurational space available to the ring, with a consequent reduction of the entropy of the system, compared to the topologically unconstrained case [7]. It is then interesting to precisely quantify this entropy loss and to determine how it depends on the particular topology (i.e. knot type) considered. Unfortunately, most of the theoretical studies performed so far refer to the ensemble in which the rings may assume all the topologies. The reason is that polymer rings in good solvent can be modelled as self-avoiding polygons (SAPs or simply polygons), which are in turn mapped to a magnetic system at its critical point and studied by renormalization group techniques [1, 8, 9]. This approach has led to the well established result that the number of Z(N) of N-steps SAPs grows, for large N, as Z(N) ≃AµNN α−2 (1) where the amplitude A and the connective constant µ are non-universal quantities that depend on the microscopic features of the chain while α is a universal exponent given by α = 2 −dν, where d is the dimensionality of the space and ν the metric exponent [9]. In d = 3 dimensions, numerical simulations [10] give for self avoiding loops the estimate ν ≃0.587597(7), and consequently α ≃0.237209(21), in agreement with field theoretical results [8]. Since for the subset of SAPs with a given knot type k the above mentioned mapping is not valid anymore, there is no field theory argument to establish a scaling similar to (1) for Zk(N). However, it is reasonable to expect that Zk(N) ≃AkµN k N αk−2 (2) where µk and αk are, respectively, the connective constant and the entropic exponent of the subset of SAPs with fixed knot type k. For a generic knot type k there is no rigorous relation between µk and µ but in the case of unknotted polygons (i.e. SAPs with trivial topology, k = ∅) it is possible to prove rigorously that µ∅< µ [7] whereas numerical estimates of α∅suggests the intriguing identity α ≃α∅[11], although results presented so far are not sharp enough to rule out completely a possible, although small, discrepancy between the two entropic exponents, i.e. α∅≃α. One

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