Worm-like chain (O’Brien)

A second worm-like chain, WormLikeChain2, using the closed form of O’Brien et al. (2009). It describes the same physics as the Zhou model, but enforces finite extensibility exactly, stays accurate for long and stiff chains, and also 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

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

with normalization constant

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

The \(\left(1 - (r/L_c)^2\right)\) factors enforce \(r < L_c\). Because \(C_1 \propto e^{\alpha}\) overflows for long chains, the code evaluates \(\log P(r)\) and normalizes numerically; \(C_1\) never enters the calculation.

The radius of gyration is given in closed form (with \(C_2 = 1/(2 L_p)\)):

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

This is the Benoit-Doty worm-like chain result, \(L_c L_p/3 - L_p^2 + 2 L_p^3/L_c - 2 L_p^4/L_c^2 \left(1 - e^{-L_c/L_p}\right)\), rewritten in terms of \(C_2\). It reduces to \(L_c L_p/3\) when \(L_c \gg L_p\) and to the rigid-rod value \(L_c^2/12\) when \(L_c \ll L_p\). 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

Contour length per residue, \(b\) (the Cα-Cα distance); sets \(L_c = N b\).

The contour length must be at least one persistence length (\(N b \ge L_p\)), otherwise the constructor raises a WLC2Exception.

What to expect for a protein. Results closely track the Zhou model for typical disordered-protein parameters, and match the exact worm-like chain \(\langle R^2 \rangle\) to within a fraction of a percent across chain lengths and stiffnesses.

3D ensembles

sample_conformations() generates the worm-like chain itself, exactly as described for the Zhou model (the two classes generate identical ensembles). check_ensemble() and afrc-ensemble -m wlc2 (see Command-line tools) check an ensemble against the exact worm-like chain mean-squared distances and finite extensibility, and compare it with the O’Brien \(P(r)\) and the Benoit-Doty \(R_g\) as context.

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. Benoit, H., & Doty, P. (1953). Light scattering from non-Gaussian chains. The Journal of Physical Chemistry, 57(9), 958-963.

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