Analytical Flory Random Coil

The Analytical Flory Random Coil (AFRC), exposed through AnalyticalFRC, describes a polypeptide behaving as an ideal (Gaussian) chain: its apparent scaling exponent is \(\nu = 0.5\), as for a chain in a \(\theta\)-solvent, and it has no finite-size (“dangling-end”) effects. It has no free parameters - it is fully determined by the amino acid sequence. The model and its parameterization are described in Alston et al. (2023).

Origin: numerical Flory Random Coil simulations

The AFRC is a closed-form fit to numerical Flory Random Coil (FRC) ensembles. These are generated by all-atom Monte Carlo using Flory’s rotational isomeric state (RIS) approximation: at each step a residue is chosen at random and its backbone dihedrals (\(\phi, \psi\)) are reassigned to one of a set of residue-specific, sterically allowed states taken from Ramachandran maps. Every move is accepted, and there are no through-space interactions of any kind.

Two consequences make an analytical description possible:

  • The chain is ideal. Every monomer is blind to its surroundings, so global and internal dimensions both scale with \(\nu = 0.5\).

  • There are no end effects. Terminal residues sample the same conformations as internal ones, so internal scaling profiles for chains of every length superimpose, and one set of expressions describes the chain at all length scales.

From simulations to a closed-form model

Residue-specific prefactors are fit once, against homopolymer FRC simulations:

  • Inter-residue distances follow \(\sqrt{\langle r_{ij}^2 \rangle} = A_0\,|i-j|^{\nu}\). Fitting the internal scaling profiles gives the per-residue prefactor \(A_0\) (the \(R_0\) and \(R_0^{\mathrm{rms}}\) constants in this package).

  • The radius of gyration prefactor \(X_0\) is fit so that the Lhuillier distribution matches the simulated \(P(R_g)\).

Because each residue is independent in the RIS construction, a heteropolymer uses the composition-weighted average of the homopolymer prefactors. This was validated against FRC simulations of hundreds of sequences (10-500 residues), reproducing both end-to-end and \(R_g\) distributions to sub-angstrom accuracy.

Mathematical formalism

End-to-end distance. The end-to-end distribution is the Gaussian chain result

\[P(r) = 4\pi r^2 \left( \frac{3}{2\pi \langle R_e^2 \rangle} \right)^{3/2} \exp\!\left( -\frac{3 r^2}{2 \langle R_e^2 \rangle} \right), \qquad \sqrt{\langle R_e^2 \rangle} = R_0^{\mathrm{rms}}\, N^{1/2},\]

where \(R_0^{\mathrm{rms}}\) is the composition-weighted prefactor. The mean end-to-end distance uses a second prefactor, \(\langle R_e \rangle = R_0\, N^{1/2}\).

Radius of gyration. The \(R_g\) distribution uses the fractal-polymer form of Lhuillier (1988):

\[P(R_g) \propto N^{-\nu d}\, \frac{\rho}{N^{\nu}} \exp\!\left[ -\left(\frac{N^{\nu}}{\rho}\right)^{\alpha d} -\left(\frac{\rho}{N^{\nu}}\right)^{\delta} \right],\]

with \(\rho = X_0\, R_g\), \(d = 3\), \(\nu = 1/2\), \(\alpha = 1/(\nu d - 1) = 2\) and \(\delta = 1/(1-\nu) = 2\). The mean \(R_g\) is either the expectation of this distribution (calculation_mode='distribution') or the scaling law \(\langle R_g \rangle = R_0^{g}\, N^{1/2}\) (calculation_mode='scaling law'), where \(R_0^{g}\) is back-calculated from the same fits. The two agree to within 0.1%.

Note

The scaling law deliberately does not use \(R_g = \langle R_e \rangle/\sqrt{6}\). That relation holds between root-mean-square radii; applied to the mean end-to-end distance it under-estimates \(\langle R_g \rangle\) by around 5%.

Inter-residue distances. The distance between residues \(i\) and \(j\) is treated as the end-to-end distance of a chain of \(|i-j|\) residues, with prefactors taken from the composition of the sequence between them. This gives distance maps, the internal scaling profile, and the quantities below. Note that the whole-chain end-to-end distance uses all \(N\) residues, so it is slightly larger than the distance between the first and last residue (\(N-1\)).

Hydrodynamic radius. \(R_h\) comes from either the Kirkwood-Riseman equation or the empirical \(R_g\)-to-\(R_h\) conversion of Nygaard et al. (2017). The Kirkwood-Riseman estimate is

\[R_h = \left\langle \frac{1}{r_{ij}} \right\rangle_{i \neq j}^{-1}, \qquad \left\langle \frac{1}{r_{ij}} \right\rangle = \sqrt{\frac{6}{\pi \langle r_{ij}^2 \rangle}},\]

where the outer average runs over every pair of residues and the inner one is the exact mean inverse distance for the Gaussian inter-residue distribution. This is the same form used by Nygaard et al. (2017), Pesce et al. (2023) and SOURSOP. Note that it averages \(1/r_{ij}\), not \(r_{ij}\): for a Gaussian chain \(\langle 1/r \rangle = (4/\pi)/\langle r \rangle\), so inverting the mean distance would over-estimate \(R_h\) by 27%. The result gives \(R_g/R_h \approx 1.3\)-\(1.4\), as expected for a theta-state chain, which is larger than the empirical Nygaard ratio for sequences of typical IDR length.

Contact fractions. The contact fraction between two residues is the probability that their distance is below a threshold, i.e. the cumulative inter-residue distribution at that threshold.

PRE profiles. For a spin label at a given residue, distances to every other residue are sampled from the AFRC and converted, conformer by conformer, into transverse relaxation rates \(\Gamma_2 \propto r^{-6}\) and intensity ratios, which are then averaged. Averaging per conformer matters because relaxation depends very non-linearly on distance. The spin-label linker is not modelled.

3D ensembles. Because every inter-residue distance is Gaussian, the AFRC defines a Gaussian chain, and explicit conformations (one bead per residue) can be drawn from it exactly. The bead coordinates are jointly Gaussian; for each of \(x\), \(y\) and \(z\) the covariance is

\[C = -\frac{1}{6} J D J, \qquad D_{ij} = \langle r_{ij}^2 \rangle, \qquad J = I - \frac{1}{N}\mathbf{1}\mathbf{1}^T,\]

so factoring \(C = A A^T\) once and computing \(A z\) for standard normal vectors \(z\) gives centred, randomly oriented conformations. Every inter-residue distance in the resulting ensemble follows the AFRC’s distribution exactly, for all pairs at once. \(C\) must be positive semi-definite for this to work; it is for every sequence we have tested, including sequences that alternate or block the residues with the largest and smallest prefactors. A few things follow from the model rather than the method:

  • adjacent beads follow the AFRC’s neighbour distance distribution (mean ~5.8 Å), not a fixed 3.8 Å Cα spacing, and - as for any ideal chain - beads can overlap;

  • the first-to-last bead distance is the \((0, N-1)\) inter-residue distance, so it is slightly shorter than the whole-chain \(\langle R_e \rangle\);

  • \(R_g\) is that of the Gaussian chain, within ~2% of the AFRC’s separately calibrated \(\langle R_g \rangle\).

The afrc-ensemble command-line tool generates such an ensemble and checks it against these exact expectations (see Command-line tools).

Behaviour and relationship to other models

The AFRC behaves like a nu-dependent SAW at \(\nu = 0.5\) whose prefactor is set to the AFRC’s \(R_0^{\mathrm{rms}}\); the two distributions then sit essentially on top of one another. It is around 30% more expanded than the worm-like chain at a persistence length of 3 Å (matching it needs \(L_p \approx 5\) Å), and substantially more compact than the good-solvent self-avoiding walk, so it sits at the theta point between collapsed and fully solvated chains.

Intended use

Note

The AFRC is a reference (null) model, not a predictor of unfolded-protein dimensions. Real dimensions depend on sequence-encoded chain-chain and chain-solvent interactions that the AFRC deliberately omits. Deviations of a simulation or experiment from the AFRC therefore report directly on those interactions, and normalizing to the AFRC puts chains of different length and composition on a common footing.

Parameters

The AFRC has no model parameters: the per-residue constants (\(R_0\), \(R_0^{\mathrm{rms}}\), \(X_0\), \(R_0^{g}\)) are fixed, and the sequence is the only input.

Argument

Default

Meaning

adaptable_P_res

False

Numerical only. If True the distribution grid spacing is \(d_{max}/500\) (with \(d_{max} = 3.7N\)) rather than the fixed 0.05 Å. Changes the discretization, not the model.

What to expect for a protein. With \(R_0 \approx 6\) Å and \(R_0^{g} \approx 2.5\) Å, a disordered region of \(N\) residues has \(R_e \approx 6\sqrt{N}\) Å and \(R_g \approx 2.5\sqrt{N}\) Å. Real IDRs scatter around these values: in the original study, simulated or measured dimensions ranged from roughly 0.7 (more compact) to 1.4 (more expanded) times the AFRC, so treat it as a touchstone rather than a bound.

Citations

  1. Alston, J. J., Ginell, G. M., Soranno, A., & Holehouse, A. S. (2023). The Analytical Flory Random Coil is a simple-to-use reference model for unfolded and disordered proteins. The Journal of Physical Chemistry B, 127(21), 4746-4760. https://doi.org/10.1021/acs.jpcb.3c01619

  2. Flory, P. J. (1969). Statistical Mechanics of Chain Molecules. Wiley-Interscience.

  3. Mao, A. H., Lyle, N., & Pappu, R. V. (2013). Describing sequence-ensemble relationships for intrinsically disordered proteins. Biochemical Journal, 449(2), 307-318.

  4. Lhuillier, D. (1988). A simple model for polymeric fractals in a good solvent and an improved version of the Flory approximation. Journal de Physique, 49(5), 705-710.

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

  6. Nygaard, M., Kragelund, B. B., Papaleo, E., & Lindorff-Larsen, K. (2017). An efficient method for estimating the hydrodynamic radius of disordered protein conformations. Biophysical Journal, 113(3), 550-557.

  7. Kirkwood, J. G., & Riseman, J. (1948). The intrinsic viscosities and diffusion constants of flexible macromolecules in solution. The Journal of Chemical Physics, 16(6), 565-573.

  8. Pesce, F., Newcombe, E. A., Seiffert, P., Tranchant, E. E., Olsen, J. G., Grace, C. R., Kragelund, B. B., & Lindorff-Larsen, K. (2023). Assessment of models for calculating the hydrodynamic radius of intrinsically disordered proteins. Biophysical Journal, 122(2), 310-321.

  9. Meng, W., Lyle, N., Luan, B., Raleigh, D. P., & Pappu, R. V. (2013). Experiments and simulations show how long-range contacts can form in expanded unfolded proteins with negligible secondary structure. Proceedings of the National Academy of Sciences, 110(6), 2123-2128.