nu-dependent self-avoiding walk

The \(\nu\)-dependent SAW, exposed through NuDepSAW, generalises the self-avoiding walk so that the Flory scaling exponent \(\nu\) becomes a free parameter. This lets a single model span the full range from a collapsed globule to a fully solvated coil. It uses the form derived by Zheng et al. (2018) as written by Soranno (2020).

Mathematical formalism

The end-to-end distribution is

\[P(r) = \frac{4\pi A_1}{R_{ee}} \left( \frac{r}{R_{ee}} \right)^{2+g} \exp\!\left[ -A_2 \left( \frac{r}{R_{ee}} \right)^{\delta} \right],\]

with the exponents

\[g = \frac{\gamma - 1}{\nu}, \qquad \delta = \frac{1}{1 - \nu}, \qquad \gamma = 1.1615,\]

and the normalisation prefactors (Soranno 2020, Eq. 9b)

\[A_1 = \frac{\delta}{4\pi}\, \frac{\Gamma\!\left[\frac{5+g}{\delta}\right]^{(3+g)/2}} {\Gamma\!\left[\frac{3+g}{\delta}\right]^{(5+g)/2}}, \qquad A_2 = \left( \frac{\Gamma\!\left[\frac{5+g}{\delta}\right]} {\Gamma\!\left[\frac{3+g}{\delta}\right]} \right)^{\delta/2}.\]

\(A_1\) normalises \(P(r)\) to unity and \(A_2\) is fixed by requiring that the root-mean-square end-to-end distance of the distribution is exactly the size scale

\[R_{ee} = \texttt{prefactor} \cdot N^{\nu}.\]

Note

Because \(A_2\) is defined by that requirement, \(\sqrt{\langle R_e^2 \rangle} = \texttt{prefactor} \cdot N^{\nu}\) holds exactly, for any \(\nu\). Setting \(\nu = 0.598\) recovers the fixed-exponent SAW to within the small differences between the two models’ rounded exponents.

The radius of gyration uses the same universal ratio as the SAW, evaluated at the chosen \(\nu\).

Parameters

Parameter

Default

Meaning and typical values

nu

0.5

Flory scaling exponent. Must lie strictly between 0 and 1 (a NuDepSAWException is raised otherwise). Physically meaningful values run from \(\approx 1/3\) (collapsed globule, poor solvent), through \(0.5\) (ideal / theta solvent), to \(\approx 0.588\) (good solvent, fully expanded). Sweeping \(\nu\) lets you place a measured chain on the collapse-to-expansion axis.

prefactor

5.5 Å

Sets the absolute per-monomer length scale (as for the SAW): it is the root-mean-square end-to-end distance of a one-residue chain, and \(\sqrt{\langle R_e^2 \rangle} = \texttt{prefactor}\,N^{\nu}\). Around 5-6 Å is typical. Must be > 0 (a NuDepSAWException is raised otherwise).

What to expect for a protein. At \(\nu = 0.5\) the model gives theta-state (ideal-chain) scaling; increasing \(\nu\) toward 0.588 swells the chain to good-solvent dimensions, while decreasing toward 1/3 collapses it. Most aqueous IDRs sit somewhere between \(\nu \approx 0.5\) and \(0.6\).

Note that the absolute dimensions are set by prefactor, not by \(\nu\) alone. With the default 5.5 Å a \(\nu = 0.5\) chain comes out roughly 12% more compact than the AFRC, because the AFRC’s own composition-weighted prefactor (\(R_0^{\mathrm{rms}} \approx 6.3\) Å for a typical sequence) is larger. To use this model as a \(\nu\)-tunable version of the AFRC, set prefactor to the AFRC’s \(R_0^{\mathrm{rms}}\) for your sequence; the two then agree closely at \(\nu = 0.5\).

3D ensembles

As for the fixed-exponent SAW, sample_conformations() generates a Gaussian approximation: bead coordinates are drawn exactly from the Gaussian chain with \(\langle r_{ij}^2 \rangle = (\texttt{prefactor}\,|i-j|^{\nu})^2\) (fractional Brownian motion, valid for any \(0 < \nu < 1\)), so the size and scaling are right at every separation but the distributions are Gaussian and there is no excluded volume. check_ensemble() and afrc-ensemble -m saw-nu (see Command-line tools) check an ensemble against the exact mean-squared distances and show the differences from the SAW’s own \(P(r)\) and \(R_g\) as context.

Citations

  1. Zheng, W., Zerze, G. H., Borgia, A., Mittal, J., Schuler, B., & Best, R. B. (2018). Inferring properties of disordered chains from FRET transfer efficiencies. The Journal of Chemical Physics, 148(12), 123329.

  2. Soranno, A. (2020). Physical basis of the disorder-order transition. Archives of Biochemistry and Biophysics, 685, 108305.

  3. Le Guillou, J. C., & Zinn-Justin, J. (1977). Critical exponents for the n-vector model in three dimensions from field theory. Physical Review Letters, 39(2), 95-98.