  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.
