Map conformational landscapes of intrinsically disordered proteins with polymer physics quantities

Disordered proteins are conformationally flexible proteins that are biologically important and have been implicated in devastating diseases such as Alzheimer's disease and cancer. Unlike stably folded structured proteins, disordered proteins sample a range of different conformations that needs to be accounted for. Here, we treat disordered proteins as polymer chains, and compute a dimensionless quantity called instantaneous shape ratio (R_s), as R_s = R_ee²/R_g², where R_ee is end-to-end distance and R_g is radius of gyration. Extended protein conformations tend to have high R_ee compared with R_g, and thus have high R_s values, whereas compact conformations have smaller R_s values. We use a scatter plot of R_s (representing shape) against R_g (representing size) as a simple map of conformational landscapes. We first examine the conformational landscape of simple polymer models such as Random Walk, Self-Avoiding Walk, and Gaussian Walk (GW), and we notice that all protein/polymer maps lie within the boundaries of the GW map. We thus use the GW map as a reference and, to assess conformational diversity, we compute the fraction of the GW conformations (f_C) covered by each protein/polymer. Disordered proteins all have high f_C scores, consistent with their disordered nature. Each disordered protein accesses a different region of the reference map, revealing differences in their conformational ensembles. We additionally examine the conformational maps of the nonviral gene delivery vector polyethyleneimine at various protonation states, and find that they resemble disordered proteins, with coverage of the reference map decreasing with increasing protonation state, indicating decreasing conformational diversity. We propose that our method of combining R_s and R_g in a scatter plot generates a simple, meaningful map of the conformational landscape of a disordered protein, which in turn can be used to assess conformational diversity of disordered proteins.