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 only sets the number of residues.
Mathematical formalism
With contour length \(L_c = N b\), the end-to-end distribution is
where the Gaussian core is the flexible-limit result \(\langle R^2 \rangle = 2 L_p L_c\) and \(\zeta(r)\) is Zhou’s polynomial correction series (Zhou 2004, Eq. 5) in powers of \(r/L_c\) and \(L_p/L_c\). Mean and root-mean-square end-to-end distances are computed numerically from \(P(r)\).
Note
Because \(\zeta(r)\) is an expansion, it is only accurate when \(L_c \gg L_p\) - with the default parameters, chains of more than roughly 10-20 residues. In that regime the distribution reproduces the exact worm-like chain \(\langle R^2 \rangle = 2 L_p L_c - 2 L_p^2 (1 - e^{-L_c/L_p})\) to high precision. No probability is assigned beyond \(L_c\), and negative values of the series are set to zero. If \(L_c\) is comparable to or shorter than \(L_p\) the expansion has no valid region, and requesting the distribution raises a WLCException.
Parameters
Parameter |
Default |
Meaning and typical values |
|---|---|---|
|
3.0 Å |
Persistence length - the distance over which the chain “forgets” its direction. Larger values give a stiffer, more extended chain. Values of 3-5 Å are common for unfolded polypeptides (4 Å is also frequent in the literature). |
|
3.8 Å |
Contour length per residue, \(b\), so \(L_c = N b\). 3.8 Å is the Cα-Cα distance. |
What to expect for a protein. Because \(\langle R^2 \rangle \approx 2 L_p L_c\), the chain grows as \(\sqrt{L_p}\) and shows Gaussian-coil scaling when \(L_c \gg L_p\). With \(L_p\) in the 3-4 Å range the WLC is roughly 10-25% more compact than the AFRC; matching the AFRC’s dimensions needs \(L_p \approx 5\) Å.
3D ensembles
sample_conformations() generates the worm-like chain itself rather than the Zhou approximation: each residue is split into short straight sub-segments, each bending away from the previous one with a fixed mean cosine, and only the bead at the end of each residue is kept. That cosine is either the worm-like chain’s own tangent correlation, \(e^{-s/L_p}\) for sub-segments of length \(s\), or - for chains that are flexible on the scale of a residue - \((2L_p - s)/(2L_p + s)\), which reproduces the chain’s long-range size exactly and so needs far fewer sub-segments (221 rather than 807 per residue at \(L_p = 0.1\) Å); whichever needs fewer is used. The number of sub-segments is chosen automatically so the mean-squared distances match the exact worm-like chain \(\langle r_{ij}^2 \rangle = 2 L_p L - 2 L_p^2 (1 - e^{-L/L_p})\) (with \(L = |i-j|\, b\)) to within 0.02% at every separation. WormLikeChain and WormLikeChain2 generate identical ensembles. check_ensemble() and afrc-ensemble -m wlc (see Command-line tools) check an ensemble against those mean-squared distances and finite extensibility, and compare it with the Zhou \(P(r)\) as context.
Citations
Zhou, H.-X. (2004). Polymer models of protein stability, folding, and interactions. Biochemistry, 43(8), 2141-2154.
Rubinstein, M., & Colby, R. H. (2003). Polymer Physics. Oxford University Press.