Freely jointed chain

The freely jointed chain (FJC), exposed through FreelyJointedChain, models the chain as \(N\) rigid segments of length \(b\) connected by perfectly flexible joints. Unlike the Gaussian AFRC it uses the non-Gaussian Kuhn-Grün distribution, which respects the chain’s finite extensibility: the end-to-end distance can never exceed the contour length \(L = N b\).

Mathematical formalism

The radial end-to-end probability is

\[P(r) \propto 4\pi r^2 \exp\!\left[ -N \left( x\,\beta + \ln\frac{\beta}{\sinh\beta} \right) \right], \qquad x = \frac{r}{N b},\]

where \(x \in [0, 1)\) is the fractional extension and \(\beta = \mathcal{L}^{-1}(x)\) is the inverse Langevin function. The inverse Langevin is evaluated with the Cohen Padé approximant

\[\beta = \mathcal{L}^{-1}(x) \approx \frac{x\,(3 - x^2)}{1 - x^2},\]

which diverges as \(x \to 1\), correctly suppressing all probability beyond the contour length. At small extension the exponent reduces to \(\tfrac{3}{2} N x^2\), recovering the Gaussian chain with \(\langle R_e^2 \rangle = N b^2\); the root-mean-square size therefore approaches \(b\sqrt{N}\) from below. The radius of gyration uses the ideal-chain relation \(\sqrt{\langle R_g^2 \rangle} = \sqrt{\langle R_e^2 \rangle}/\sqrt{6}\).

Parameters

Parameter

Default

Meaning and typical values

b

3.8 Å

Segment (Kuhn) length. The default corresponds to the Cα-Cα distance, i.e. one residue per segment. To represent a stiffer effective chain one can use a larger Kuhn length (a polypeptide Kuhn length is often quoted around 7-10 Å); \(N\) and \(b\) together fix the contour length \(L = Nb\).

What to expect for a protein. With one segment per residue (\(b = 3.8\) Å) the FJC is an ideal chain: \(R_e \approx b\sqrt{N}\) and \(R_g \approx R_e/\sqrt{6}\), essentially matching the AFRC through the bulk of the distribution. The differences appear in the far tail (the FJC has a hard cutoff at \(L = Nb\)) and at short chain lengths, where finite extensibility pulls the mean and RMS slightly below the Gaussian values.

3D ensembles

sample_conformations() builds freely jointed chains directly: bonds of exactly \(b\) pointing in independent, uniformly random directions. The ensemble is the freely jointed chain, so its mean-squared distances are exactly \(\langle r_{ij}^2 \rangle = |i-j|\, b^2\); note that the analytical Kuhn-Grün distribution above is itself an approximation that becomes exact for long chains. check_ensemble() and afrc-ensemble -m fjc (see Command-line tools) check an ensemble against the exact mean-squared distances, the fixed bond length and finite extensibility.

Citations

  1. Kuhn, W., & Grün, F. (1942). Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe. Kolloid-Zeitschrift, 101(3), 248-271.

  2. Cohen, A. (1991). A Padé approximant to the inverse Langevin function. Rheologica Acta, 30(3), 270-273.

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