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 normalisation prefactors \(A_1\), \(A_2\) expressed through gamma functions of \(g\) and \(\delta\) (Soranno 2020, Eq. 9b). The size scale is

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

Note

The factor of \(\pi\) in \(R_{ee}\) is an empirical correction (noted in the source) that brings the model into quantitative agreement with the AFRC at \(\nu = 0.5\) and with the SAW at \(\nu \approx 0.598\).

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. 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). Around 5-6 Å is typical.

What to expect for a protein. At \(\nu = 0.5\) the model reproduces theta-state (AFRC-like) dimensions; 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 \(0.6\).

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.