Worm-like chain (O’Brien)

A second worm-like chain implementation, WormLikeChain2, using the closed form of O’Brien et al. (2009). It is conceptually equivalent to the Zhou model but is numerically better behaved at large contour lengths and additionally provides a closed-form radius of gyration.

Mathematical formalism

With contour length \(L_c = N b\) and \(\alpha = 3 L_c / (4 L_p)\), the end-to-end distribution is

\[P(r) = \frac{4\pi C_1\, r^2}{L_c \left(1 - (r/L_c)^2\right)^{9/2}} \exp\!\left( -\frac{3 L_c}{4 L_p \left(1 - (r/L_c)^2\right)} \right),\]

where the normalisation constant is

\[C_1 = \left[ \pi^{3/2} e^{-\alpha} \alpha^{-3/2} \left( 1 + 3\alpha^{-1} + \tfrac{15}{4}\alpha^{-2} \right) \right]^{-1}.\]

The \(\left(1 - (r/L_c)^2\right)\) factors enforce finite extensibility (\(r < L_c\)). The radius of gyration is given in closed form (with \(C_2 = 1/(2 L_p)\)):

\[\langle R_g^2 \rangle = \frac{L_c}{6 C_2} + \frac{1}{4 C_2^2} + \frac{1}{4 C_2^3 L_c} - \frac{1 - e^{-L_c/L_p}}{8 C_2^4 L_c^2},\]

and get_mean_radius_of_gyration() returns \(\sqrt{\langle R_g^2 \rangle}\).

Parameters

Parameter

Default

Meaning and typical values

lp

3.0 Å

Persistence length (chain stiffness). As for the Zhou model, 3-5 Å is typical for unfolded polypeptides; larger values give a stiffer, more extended chain.

aa_size

3.8 Å

Segment length \(b\) (Cα-Cα distance); sets the contour length \(L_c = N b\).

The constructor additionally requires the sequence to be at least as long as the persistence length (\(N \ge L_p\)); otherwise a WLC2Exception is raised.

What to expect for a protein. Results closely track the Zhou model for typical disordered-protein parameters, with the practical advantages of numerical stability for long chains and a directly available \(R_g\).

Citations

  1. O’Brien, E. P., Morrison, G., Brooks, B. R., & Thirumalai, D. (2009). How accurate are polymer models in the analysis of Förster resonance energy transfer experiments on proteins? The Journal of Chemical Physics, 130(12), 124903.

  2. Rubinstein, M., & Colby, R. H. (2003). Polymer Physics. Oxford University Press.