[Paper Review] Under-knotted and Over-knotted Polymers: Unrestricted Loops

Nathan Moore, Rhonald C. Lua+1 more|arXiv (Cornell University)|Mar 18, 2004

Geometric and Algebraic Topology5 references3 citations

TL;DR

This study uses Monte Carlo simulations to investigate the gyration radius distributions of ideal, closed polymers with fixed knot topology, revealing that topologically constrained loops exhibit narrower radius distributions than phantom chains, indicating entropic rigidity. The key finding is that for N > N₀ ≈ 2500, all simple knots (e.g., trefoil, figure-eight) approach a universal gyration radius distribution, with trivial knots scaling like self-avoiding walks due to topological constraints mimicking excluded volume.

ABSTRACT

We present computer simulations to examine probability distributions of gyration radius for the no-thickness closed polymers of N straight segments of equal length. We are particularly interested in the conditional distributions when the topology of the loop is quenched to be a certain knot, K. The dependence of probability distribution on length, N, as well as topological state K are the primary parameters of interest. Our results confirm that the mean square average gyration radius for trivial knots scales with N in the same way as for self-avoiding walks, where the cross-over length to this "under-knotted" regime is the same as the characteristic length of random knotting, N_0. Probability distributions of gyration radii are somewhat more narrow for topologically restricted under-knotted loops compared to phantom loops, meaning knots are entropically more rigid than phantom polymers. We also found evidence that probability distributions approach a universal shape at N>N_0 for all simple knots.

Motivation & Objective

• To understand how fixed knot topology influences the size and conformational entropy of closed polymers.
• To determine whether topological constraints lead to scaling behavior similar to self-avoiding walks.
• To clarify the distinction between phantom (annealed) and non-phantom (quenched) polymers in the context of knotting and size statistics.
• To investigate the transition from under-knotted to over-knotted regimes as a function of chain length N.
• To test whether probability distributions of gyration radius become universal for long, topologically constrained loops.

Proposed method

• Computer simulations of ideal, closed polymers composed of N equal-length segments in 3D space.
• Use of the pivot algorithm to generate equilibrium ensembles of closed polygons with fixed knot type.
• Calculation of the gyration radius Rg for each configuration to build probability distributions P(Rg | K) for a given knot K.
• Application of the δ-function constraint in the path integral formalism to enforce closure and knot type.
• Analytical derivation of characteristic functions K(s) for chains and loops via Fourier transform of the distribution.
• Numerical inversion of the characteristic function to obtain P(ρ), where ρ = Rg² / (Nℓ²), enabling comparison with analytical asymptotics.

Experimental results

Research questions

• RQ1 How does the mean square gyration radius ⟨Rg²⟩ scale with N for a polymer with a fixed non-trivial knot?
• RQ2 Do probability distributions of gyration radius for topologically constrained loops approach a universal shape at large N?
• RQ3 What is the difference in conformational entropy between phantom loops and topologically quenched loops of the same knot type?
• RQ4 How does the crossover length N₀, associated with random knotting, relate to the onset of under-knotted behavior?
• RQ5 To what extent do topological constraints mimic excluded volume effects in determining polymer size?

Key findings

• The mean square gyration radius for trivial knots scales as ⟨Rg²⟩ ∼ N²ν with ν ≈ 0.588, matching the scaling of self-avoiding walks.
• For N > N₀ ≈ 2500, the probability distribution of gyration radius becomes universal across all simple knots, indicating a collapse to a common scaling form.
• Topologically constrained loops exhibit narrower gyration radius distributions than phantom loops, indicating entropic rigidity due to knot constraints.
• The asymptotic form of P(Rg) at small ρ (collapsed states) is identical in form for chains and loops, with entropy scaling as −ln P ∼ 9Nℓ²/(24Rg²).
• At large ρ (extended states), loop entropy is four times larger than that of a chain of the same size, due to the two-fold symmetry of the closed loop.
• The analytical form of the characteristic function for loops, K_loop(s) = (2 sin(z/2)/z)^{-3} with z² = 8i s, enables exact derivation of the full distribution and its asymptotics.