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 \(\nu = 0.598\). 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

\[\frac{\langle R_g^2 \rangle}{\langle R_e^2 \rangle} = \frac{\gamma(\gamma + 1)}{2(\gamma + 2\nu)(\gamma + 2\nu + 1)}, \qquad \gamma = 1.1615,\ \nu = 0.598.\]

The exponent \(\nu\) here is the same one that sets \(R_{ee}\) above - both read the object’s nu attribute - so the \(R_g\) and \(R_e\) returned by the model always describe the same chain.

Parameters

Parameter	Default	Meaning and typical values
`prefactor`	5.5 Å	Sets the absolute per-monomer length scale in \(R_{ee} = \texttt{prefactor}\,N^{0.598}\). The normalisation constants \(a\) and \(b\) are chosen so that \(R_{ee}\) is exactly the root-mean-square end-to-end distance of the distribution. Values around 5-6 Å are reasonable, but the prefactor should be tuned to match explicit excluded-volume simulations for quantitative work. Must be > 0 (a `SAWException` is raised otherwise).

The scaling exponent is fixed at the good-solvent value (\(\nu = 0.598\), held on the object as nu). To vary \(\nu\) continuously, use the nu-dependent SAW instead - evaluated at \(\nu = 0.598\) with the same prefactor it reproduces this model.

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.

3D ensembles

sample_conformations() generates a Gaussian approximation to the SAW: bead coordinates are drawn exactly from the Gaussian chain whose mean-squared distances are the model’s, \(\langle r_{ij}^2 \rangle = (\texttt{prefactor}\,|i-j|^{0.598})^2\). The size and scaling of every inter-residue distance are therefore right, but the distance distributions are Gaussian rather than the des Cloizeaux form above, there is no excluded volume (beads can overlap), and the radius of gyration comes out about 5% below the universal-ratio value (whatever the chain length). Genuine self-avoiding conformations would need an explicit simulation. check_ensemble() and afrc-ensemble -m saw (see Command-line tools) check an ensemble against the exact mean-squared distances and show these differences as context.

Citations

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