Worm-like chain (Zhou)

The worm-like chain (WLC) describes a semiflexible polymer whose stiffness is set by a persistence length \(L_p\). This implementation, WormLikeChain, uses the closed-form approximation of Zhou (2004). It is composition-independent: the sequence enters only through the number of residues.

Mathematical formalism

With contour length \(L_c = N b\), the end-to-end distribution is

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

where the Gaussian core corresponds to the flexible-limit WLC result \(\langle R^2 \rangle = 2 L_p L_c\), and \(\zeta(r)\) is the polynomial correction series of Zhou (2004, Eq. 5) in powers of \(r/L_c\), \(L_p/L_c\). The series is accurate for \(r\) up to about the contour length; spurious negative values in the far tail are clamped to zero so the result remains a valid probability distribution. Mean and root-mean-square end-to-end distances are obtained by numerical integration over \(P(r)\).

Parameters

Parameter	Default	Meaning and typical values
lp	3.0 Å	Persistence length - the length scale over which the chain “forgets” its direction. Larger \(L_p\) means a stiffer, more extended chain. For unfolded/disordered polypeptides values of roughly 3-5 Å are commonly used (4 Å is also frequent in the literature).
aa_size	3.8 Å	Segment length \(b\) (the per-residue contribution to the contour length, \(L_c = N b\)). 3.8 Å is the Cα-Cα distance.

What to expect for a protein. With \(L_p\) in the 3-4 Å range the WLC produces dimensions comparable to a slightly expanded coil. Because \(\langle R^2 \rangle = 2 L_p L_c\), the chain becomes more extended as \(L_p\) increases, and recovers Gaussian-coil scaling when \(L_c \gg L_p\).

Citations

1. Zhou, H.-X. (2004). Polymer models of protein stability, folding, and interactions. Biochemistry, 43(8), 2141-2154.

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