Freely rotating chain

The freely rotating chain, exposed through FreelyRotatingChain, models the chain as \(N\) bonds of length \(b\) joined at a fixed bond angle but with unrestricted (free) torsion angles. It is an ideal chain - Gaussian end-to-end statistics with scaling exponent \(\nu = 0.5\) - whose absolute size is set by a single stiffness parameter, the characteristic ratio \(C_\infty\). (The module is called frc, but note this is not the Flory Random Coil that the AFRC is built on.)

Mathematical formalism

The mean-squared end-to-end distance uses the exact finite-\(N\) freely-rotating-chain result

\[\langle R^2 \rangle = C_\infty N b^2 - 2 b^2\, \frac{\alpha\,(1 - \alpha^{N})}{(1 - \alpha)^2}, \qquad \alpha = \frac{C_\infty - 1}{C_\infty + 1},\]

where \(\alpha\) is the cosine of the angle between successive bonds and \(C_\infty = (1+\alpha)/(1-\alpha)\). The first term is the long-chain limit and the second is the finite-size correction (which vanishes when \(\alpha = 0\)). The end-to-end distribution is then the standard Gaussian chain form

\[P(r) = 4\pi r^2 \left( \frac{3}{2\pi \langle R^2 \rangle} \right)^{3/2} \exp\!\left( -\frac{3 r^2}{2 \langle R^2 \rangle} \right),\]

and the radius of gyration uses the ideal-chain relation \(\sqrt{\langle R_g^2 \rangle} = \sqrt{\langle R^2 \rangle}/\sqrt{6}\).

Parameters

Parameter	Default	Meaning and typical values
`b`	3.8 Å	Bond (segment) length. The default is the Cα-Cα distance, i.e. one virtual bond per residue, so that the contour length is \(N b\).
`c_inf`	2.0	Characteristic ratio \(C_\infty\) - a dimensionless stiffness. `c_inf = 1` (\(\alpha = 0\)) recovers the freely jointed chain; `c_inf = 2` corresponds to a tetrahedral bond angle. Larger values give a stiffer, more extended ideal chain. Must be > 0.

Note

A genuine freely rotating chain (free torsion) cannot reach the large characteristic ratio of a real polypeptide (\(C_\infty \approx 9\)); that value arises from hindered rotation between backbone dihedrals. The Analytical Flory Random Coil captures those local restrictions directly, so for a sequence-specific theta-state reference use the AFRC. The freely rotating chain is best thought of as a tunable, composition-independent ideal-chain reference.

What to expect for a protein. With one virtual bond per residue (\(b = 3.8\) Å), \(R_e \approx \sqrt{C_\infty}\, b\sqrt{N}\) and \(R_g \approx R_e/\sqrt{6}\). The default \(C_\infty = 2\) gives dimensions in the ballpark of the theta-state AFRC; raising \(C_\infty\) swells the chain while preserving ideal (\(\nu = 0.5\)) scaling.

3D ensembles

sample_conformations() builds freely rotating chains directly: bonds of exactly \(b\), every bond angle fixed (at \(180^\circ - \arccos\alpha\), e.g. 109.47° for c_inf = 2) and every torsion uniformly random. Its mean-squared distances are exactly the finite-\(N\) result above for every pair of residues, but for short separations its distance distributions are not Gaussian, unlike the analytical \(P(r)\). Note that c_inf = 1 gives a fixed 90° bond angle - the same mean-squared size as the freely jointed chain, but a different chain. check_ensemble() and afrc-ensemble -m frc (see Command-line tools) check an ensemble against the exact mean-squared distances, the fixed bond length and angle, and finite extensibility.

Citations

Flory, P. J. (1969). Statistical Mechanics of Chain Molecules. Wiley-Interscience.
Rubinstein, M., & Colby, R. H. (2003). Polymer Physics. Oxford University Press.