Self-avoiding walk

The self-avoiding walk (SAW), exposed through SAW, models a chain in a good solvent, where excluded-volume interactions swell the chain relative to an ideal coil. It uses the universal end-to-end distribution form of des Cloizeaux as implemented by O’Brien et al. (2009), at the fixed good-solvent scaling exponent \(\nu \approx 0.588\). It is composition-independent.

Mathematical formalism

The end-to-end distribution is the scaling form

\[P(r) = \frac{a}{R_{ee}} \left( \frac{r}{R_{ee}} \right)^{2+\theta} \exp\!\left[ -b \left( \frac{r}{R_{ee}} \right)^{\delta} \right],\]

with the small-\(r\) exponent \(\theta = 0.3\), the large-\(r\) exponent \(\delta = 2.5\), normalisation constants \(a = 3.679\), \(b = 1.232\), and a chain-length-dependent scale

\[R_{ee} = \texttt{prefactor} \cdot N^{0.598}.\]

The radius of gyration is obtained from the end-to-end size via the universal ratio

\[\frac{\langle R_g^2 \rangle}{\langle R_e^2 \rangle} = \frac{\gamma(\gamma + 1)}{2(\gamma + 2\nu)(\gamma + 2\nu + 1)}, \qquad \gamma = 1.1615,\ \nu = 0.589.\]

Parameters

Parameter

Default

Meaning and typical values

prefactor

5.5 Å

Sets the absolute per-monomer length scale in \(R_{ee} = \texttt{prefactor}\,N^{0.598}\). Values around 5-6 Å are reasonable, but the prefactor should be tuned to match explicit excluded-volume simulations for quantitative work.

The scaling exponent is fixed at the good-solvent value (\(\nu \approx 0.588\)). To vary \(\nu\) continuously, use the nu-dependent SAW instead.

What to expect for a protein. Because \(\nu \approx 0.6 > 0.5\), the SAW is more expanded than the AFRC and is an appropriate reference for a strongly solvated, expanded IDR. The prefactor sets where the absolute dimensions land; with a value near 5.5 Å the SAW gives end-to-end and \(R_g\) values noticeably larger than the theta-state AFRC.

Citations

  1. O’Brien, E. P., Morrison, G., Brooks, B. R., & Thirumalai, D. (2009). How accurate are polymer models in the analysis of Förster resonance energy transfer experiments on proteins? The Journal of Chemical Physics, 130(12), 124903.

  2. des Cloizeaux, J. (1974). Lagrangian theory for a self-avoiding random chain. Physical Review A, 10(5), 1665-1669.