End-to-end distribution for a wormlike chain in arbitrary dimensions

We construct an efficient methodology for calculating wormlike chain statistics in arbitrary $\mathit{D}$ dimensions over all chain rigidities, from fully rigid to completely flexible. The structure of our exact analytical solution for the end-to-end distribution function for a wormlike chain in arbitrary $\mathit{D}$ dimensions in Fourier-Laplace space (i.e., Fourier-transformed end position and Laplace-transformed chain length) adopts the form of an infinite continued fraction, which is advantageous for its compact structure and stability for numerical implementation. We then proceed to present a step-by-step methodology for performing the Fourier-Laplace inversion in order to make full use of our results in general applications. Asymptotic methods for evaluating the Laplace inversion (power-law expansion and Rayleigh-Schrödinger perturbation theory) are employed in order to improve the accuracy of the numerical inversions of the end-to-end distribution function in real space. We adapt our results to the evaluation of the single-chain structure factor, rendering simple, closed-form expressions that facilitate comparison with scattering experiments. Using our techniques, the accuracy of the end-to-end distribution function is enhanced up to the limit of the machine precision. We demonstrate the utility of our methodology with realizations of the chain statistics, giving a general methodology that can be applied to a wide range of biophysical problems.

Figure 1

Plots of the eigenvalues ${\mathcal{E}}_l$ for $l=0,1,\dots ,5$ vs the Fourier variable $\mathit{K}$ for three dimensions $\left(D=3\right)$ and ${\mu }_1=0$. Panels A and B show the real and imaginary parts of ${\mathcal{E}}_l$, respectively. Three methods are employed to evaluate ${\mathcal{E}}_l$: small-$K$ asymptotic expansion (dashed curve), intermediate-$\mathit{K}$ matrix method (solid curve), and large-$\mathit{K}$ asymptotic expansion (dotted curve).

Figure 2

(Color online) The end distribution function or Green function $G\left(R;L\right)$ for the wormlike chain model in three dimensions vs the end extension $R/L$ and the number of Kuhn segments $N=L/\left(2l_p\right)$. This surface plot shows high probability in white and low probability in black (see the text for details).

Figure 3

(Color online) Individual Green function curves for a wormlike chain in three dimensions, extracted from Fig. 2. The chosen $\mathit{N}$ values $\left(N=0.3,1.1,1.9,2.7,3.5\right)$ span the transition from rigid to flexible behavior. The top panel gives $L^{3}G\left(R;L\right)$ versus $R/L$ on a linear scale; the same data is given on a logarithmic plot in the bottom panel.

Figure 4

(Color online) The end distribution function (Green function) for the wormlike chain model in three dimensions vs $1/\left(1-R/L\right)$, thus focusing on the behavior near full extension. The asymptotic form presented in Eq. (42) is plotted as dashed curves along with their corresponding Green-function curves (solid curves).

Figure 5

The rigid-flexible phase diagram for the wormlike chain model showing the chain behavior versus the chain length $N=L/\left(2l_p\right)$ and dimensionality $\mathit{D}$. The transition from rigid to flexible states occurs at $N_{RF}$ (solid curve). The spinodal curves for the rigid state $\left(N_{RS}\right)$ and the flexible state $\left(N_{FS}\right)$ are the dotted and dashed curves, respectively. These spinodal curves mark the limit of stability of these two states. The inset shows the same phase diagram versus the dimensional chain length $\tilde{N}=\left(D-1\right)N/2$ (see the text).