Conformational Heterogeneity and FRET Data Interpretation for Dimensions of Unfolded Proteins

A mathematico-physically valid formulation is required to infer properties of disordered protein conformations from single-molecule F\"orster resonance energy transfer (smFRET). Conformational dimensions inferred by conventional approaches that presume a homogeneous conformational ensemble can be unphysical. When all possible---heterogeneous as well as homogeneous---conformational distributions are taken into account without prejudgement, a single value of average transfer efficiency $\langle E\rangle$ between dyes at two chain ends is generally consistent with highly diverse, multiple values of the average radius of gyration $\langle R_{\rm g}\rangle$. Here we utilize unbiased conformational statistics from a coarse-grained explicit-chain model to establish a general logical framework to quantify this fundamental ambiguity in smFRET inference. As an application, we address the long-standing controversy regarding the denaturant dependence of $\langle R_{\rm g}\rangle$ of unfolded proteins, focusing on Protein L as an example. Conventional smFRET inference concluded that $\langle R_{\rm g}\rangle$ of unfolded Protein L is highly sensitive to [GuHCl], but data from small-angle X-ray scattering (SAXS) suggested a near-constant $\langle R_{\rm g}\rangle$ irrespective of [GuHCl]. Strikingly, the present analysis indicates that although the reported $\langle E\rangle$ values for Protein L at [GuHCl] = 1 M and 7 M are very different at 0.75 and 0.45, respectively, the Bayesian $R^2_{\rm g}$ distributions consistent with these two $\langle E\rangle$ values overlap by as much as $75\%$. Our findings suggest, in general, that the smFRET-SAXS discrepancy regarding unfolded protein dimensions likely arise from highly heterogeneous conformational ensembles at low or zero denaturant, and that additional experimental probes are needed to ascertain the nature of this heterogeneity.