Worm-like polymer loops and Fourier knots

Abstract

Every smooth closed curve can be represented by a suitable Fourier sum as a function of an arbitrary parameter τ . We show that the ensemble of curves generated by randomly chosen Fourier coefficients with amplitudes inversely proportional to spatial frequency (with a smooth exponential cutoff) can be accurately mapped on the physical ensemble of inextensible worm-like polymer loops. The τ → s mapping of the curve parameter τ on the arc length s of the inextensible polymer is achieved at the expense of coupling all Fourier harmonics in a non-trivial fashion. We characterize the obtained ensemble of conformations by looking at tangent-tangent and position-position correlations. Measures of correlation on the scale of the entire loop yield a larger persistence length than that calculated from the tangent-tangent correlation function at small length scales. The topological properties of the ensemble, randomly generated worm-like loops, are shown to be similar to those of other polymer models. , use the analogy between conformation of a chain molecule and a Brownian random walk (BRW). In mathematically idealized form, such a random walk is thought of as a Wiener trajectory r(τ ) generated by the measure P {r(τ )} ∝ exp −const (∂r/∂τ ) 2 dτ , where τ is a parameter running along the trajectory. Despite its successes, the BRW model cannot reproduce the finite extensibility of polymer chains stretched by a strong force [2] and fails miserably in the study of polymers with knots. Indeed, simulations of discrete polymer models, or random polygons with N steps, show [3][4][5] that the probability of trivial knot configuration decays exponentially with N as P 0 (N) ∼ exp(−N/N * ), where N * defines the crossover from an unknotted to a knotted regime. It was argued that Wiener trajectory polymer models cannot be used to calculate

Differential geometry of polymer models: worm-like chains, ribbons and Fourier knots

“…Nevertheless, the observation of two persistence lengths reported in ref. [6] and the present finding that…”

“…Nevertheless, the observation of two persistence lengths reported in ref. [6] and the present finding that Fourier knots have finite torsion, suggest that the statistical properties of this mathematical ensemble (notice that persistence lengths can be measured directly from the ensemble of conformations of the space curves, just as is done in AFM experiments [15]) are quite similar to those of a physical ensemble of conformations of polymers with both bending and torsional rigidity. The detailed exploration of this analogy is the subject of future work.…”

“…In ref. [6] we suggested that the ensemble of configurations generated by the Fourier knot algorithm is equivalent to a physical ensemble of polymers which possess both bending and twist rigidity and, while the short range properties of the tangent-tangent correlation function are determined by the bending persistence length only, both bending and twist persistence length control its long distance behavior.…”

Models of random knots

“…In the search of a more numerically accessible model for worm-like loops, Rappaport et al (2006) and Rappaport and Rabin (2007) suggested the following mathematical model. One way to construct a Brownian bridge in R 3 is by the following Fourier series, with w k ¼ 1.…”

Buckling of stiff polymer rings in weak spherical confinement

“…This inspired to use the rather weak confinement of artificial giant vesicles as a versatile and well-controllable model system for the investigation of polymer and polymer bundle characteristics [10][11][12]. Especially but not only in these biomimetic systems, that investigate both biological processes and new nano-biomaterials, polymer rings become of larger and larger importance, stirring theoretical studies of semiflexible polymer rings [13][14][15][16][17][18][19][20]. DNA on the one hand naturally occurs in ring form [21,23] while actin and actin bundles self-assemble into rings under various conditions [11,12,22,24,25].…”