Decoupling of size and shape fluctuations in heteropolymeric sequences reconciles discrepancies in SAXS vs. FRET measurements

Significance

Conformational properties of unfolded and intrinsically disordered proteins (IDPs) under native conditions are important for understanding the details of protein folding and the functions of IDPs. The average dimensions of these systems are quantified using the mean radius of gyration and mean end-to-end distance, measured by small-angle X-ray scattering (SAXS) and single-molecule Förster resonance energy transfer (smFRET), respectively, although systematic discrepancies emerge from these measurements. Through holistic sets of studies, we find that the disagreements arise from chemical heterogeneity that is inherent to heteropolymeric systems. This engenders a decoupling between different measures of overall sizes and shapes, thus leading to discrepant inferences based on SAXS vs. smFRET. Our findings point the way forward to obtaining comprehensive descriptions of ensembles of heterogeneous systems.

Abstract

Unfolded states of proteins and native states of intrinsically disordered proteins (IDPs) populate heterogeneous conformational ensembles in solution. The average sizes of these heterogeneous systems, quantified by the radius of gyration (R_G), can be measured by small-angle X-ray scattering (SAXS). Another parameter, the mean dye-to-dye distance (R_E) for proteins with fluorescently labeled termini, can be estimated using single-molecule Förster resonance energy transfer (smFRET). A number of studies have reported inconsistencies in inferences drawn from the two sets of measurements for the dimensions of unfolded proteins and IDPs in the absence of chemical denaturants. These differences are typically attributed to the influence of fluorescent labels used in smFRET and to the impact of high concentrations and averaging features of SAXS. By measuring the dimensions of a collection of labeled and unlabeled polypeptides using smFRET and SAXS, we directly assessed the contributions of dyes to the experimental values R_G and R_E. For chemically denatured proteins we obtain mutual consistency in our inferences based on R_G and R_E, whereas for IDPs under native conditions, we find substantial deviations. Using computations, we show that discrepant inferences are neither due to methodological shortcomings of specific measurements nor due to artifacts of dyes. Instead, our analysis suggests that chemical heterogeneity in heteropolymeric systems leads to a decoupling between R_E and R_G that is amplified in the absence of denaturants. Therefore, joint assessments of R_G and R_E combined with measurements of polymer shapes should provide a consistent and complete picture of the underlying ensembles.

Quantitative characterizations of the sizes, shapes, and amplitudes of conformational fluctuations of unfolded proteins under denaturing and native conditions are directly relevant to advancing our understanding of the collapse transition during protein folding. These types of studies are also relevant to furthering our understanding of the functions and interactions of intrinsically disordered proteins (IDPs) in physiologically relevant conditions (1). Polymer physics theories provide the conceptual foundations for analyzing conformationally heterogeneous systems such as IDPs and unfolded ensembles of autonomously foldable proteins (2–4). Specifically, order parameters in theories of coil-to-globule transitions and analytical descriptions of conformational ensembles (5, 6) are based on ensemble-averaged values of radii of gyration (R_G) and amplitudes of fluctuations measured by end-to-end distances (R_E).

Estimates of R_G are accessible through small-angle X-ray scattering (SAXS) measurements because scattering intensities are directly related to the global protein size (Fig. 1) (7, 8). At finite concentrations, assuming the absence of intermolecular interactions, R_G is proportional to the square root of the mean square of interatomic distances within individual molecules averaged over the conformations of all molecules in solution (see SI Appendix, Table S1 for details). Estimates of R_E can be made from single-molecule Förster resonance energy transfer (smFRET) experiments. Here, donor and acceptor fluorophores are covalently attached to N- and C-terminal ends of the protein of interest and the measured mean FRET efficiencies ($\langle E_{FRET}\rangle$) are used to infer the mean distances between dyes (R_E,L) (Fig. 1D). This serves as a useful proxy for estimating R_E although it requires the assumption of an a priori functional form for the distribution of interdye distances, which is often based on the Gaussian chain model (9–13). Because dyes are attached to the protein sidechain via flexible linkers, R_E,L is different from the actual end-to-end distance R_E, which we denote as R_E,U (Fig. 1B). Similarly, the R_G of an unlabeled protein, R_G,U, should be numerically different from the R_G of a labeled protein, R_G,L (compare Fig. 1 A and C).

Fig. 1.

The combined SAXS/smFRET approach. Proteins are depicted as a chain of beads (blue), where each bead represents an amino acid residue. Donor dye (Alexa488), acceptor dye (Alexa594), and their linkers are shown in green, red, and black traces, respectively. (A) The radius of gyration of an unlabeled protein, R_G,U, can be estimated from a SAXS profile were the intensity of scattered X-rays is recorded as a function of the scattering vector q. (B) The end-to-end distance of the polymer, R_E,U, is not directly accessible by smFRET. (C) The radius of gyration of a labeled protein, R_G,L, can also be measured by SAXS. (D) The donor-to-acceptor distance, R_E,L, can be estimated via the FRET efficiencies (E_FRET) measured by smFRET upon assumption of a model. R_E,L and R_G,L can be related to each other via the G ratio (R_E²/R_G²). (E) Mean charge of the 10 proteins used in this study plotted against their mean hydrophobicity. Dashed lines show the theoretical prediction separating IDPs (Left) from folded proteins (Right).

Proteins that fold autonomously under physiological conditions can be denatured in high concentrations of urea or guanidinium hydrochloride (14). Upon dilution of denaturants, proteins collapse and fold to form compact structures. An unresolved issue is the nature of the collapse transition (2, 4, 13, 15, 16). Inferences from smFRET measurements suggest that proteins, including IDPs, undergo continuous contraction as the denaturant concentration is decreased (4, 16, 17). The implication for protein folding is that the acquisition of persistent local and nonlocal contacts might follow barrierless collapse that leads to the formation of globules. Inferences from SAXS measurements provide a discrepant view of the collapse transition for protein folding and for IDPs (15, 18, 19). In these experiments, the measured R_G values are shown to change minimally over a wide range of denaturant concentrations. Therefore, one might conclude that the collapse transition is virtually nonexistent for IDPs and abrupt and concomitant with the rate-limiting folding transition for autonomously folding proteins. The discrepancies in interpretations regarding the collapse transition for protein folding and for IDPs have led to numerous debates (4, 9, 15, 20–23).

Why do SAXS and smFRET lead to apparently conflicting inferences regarding the collapse transition and the nature of heterogeneous ensembles, especially under physiologically relevant conditions and away from high concentrations of denaturants? Both techniques have distinct strengths and weaknesses (15, 20–23). Strengths of smFRET measurements include the ultralow protein concentrations, at which experiments can be conducted, the ability to resolve distinct conformational populations, and the advantage of following motions across timescales that range from the nanosecond to the millisecond regimes. Weaknesses of smFRET experiments derive from the possibility that fluorescent dyes, tethered via flexible linkers to protein sidechains, could engender nontrivial alterations to the dimensions of unfolded proteins and IDPs. Also, with typical dye pairs, smFRET affords accurate estimates of distances that are limited to the range of ∼2 nm to ∼10 nm. In contrast, SAXS measurements do not require the attachment of labels and the measured scattering intensities are weighted averages over all of the protein molecules in solution, thus enabling direct investigations of chain dimensions. However, SAXS experiments require higher protein concentrations and the averaging over the conformations of all molecules in solutions makes it difficult to obtain assessments of conformational populations and insights regarding fluctuations that are smaller than the global dimensions of the protein. Here, we ask whether the discrepancies between inferences drawn from SAXS vs. smFRET measurements are due to the perceived weaknesses of the methods themselves or because the two methods provide complementary insights that have to be analyzed jointly to obtain a robust quantitative assessment of conformational features of heterogeneous systems.

We performed SAXS measurements on labeled and unlabeled IDPs as well as chemically denatured proteins. Inferences from these measurements were compared with those from smFRET measurements of labeled molecules. Atomistic Monte Carlo simulations based on the ABSINTH (self-assembly of biomolecules studied by an implicit, novel, and tunable Hamiltonian) implicit solvation model (24) were used to generate quantitative insights and to aid in the joint analysis of SAXS and smFRET data. We made rigorous comparisons between R_E,L, calculated from smFRET measurements and atomistic simulations of dye-labeled proteins, and the values of R_G,U and R_G,L obtained from SAXS measurements. We find that the dyes do not significantly influence the SAXS measurements, under either native conditions or denatured conditions. Instead, estimates of R_G and R_E yield different inferences because these quantities interrogate distinct length scales and are influenced by very different types of averaging. For finite-sized heteropolymeric sequences, we show that large changes in R_E are compatible with negligible changes in R_G (22, 25). We discuss that such differences are minimized in long homopolymers and long block copolymers that are characterized by the chemical similarity of the interacting units (25). Accordingly, the estimates of R_G and R_E lead to mutually consistent inferences regarding conformational preferences and the physics of coil-to-globule transitions for long homopolymers (26). A similar robustness prevails for proteins in highly denaturing environments where preferential interactions between denaturants and chain units appear to have a homogenizing effect on the pattern of intrachain interactions (3, 23, 27–29), in line with the observations we report from the different methods.

Therefore, at a minimum, it becomes important to measure both R_G and R_E if we are to obtain a reliable description of global chain density through R_G, amplitudes of fluctuations through R_E, and deviations from uniform expansion/contraction by assessments of the overall shape that can be estimated by quantifying the ratio G = (R_E²/R_G²). Alternatively, we show that a more rigorous assessment of overall shapes and the decoupling between shape and size fluctuations can be derived from analysis of the entire SAXS profile. This provides a more complete description compared with extracting estimates of R_G alone. However, if intermolecular interactions at high protein concentrations required for SAXS are an issue, then global analysis of data from multiple, independent smFRET measurements performed using constructs distinguished by different linear separations between dye pairs would be a promising route to pursue (3).

Results

The Protein Set, Labeling Scheme, and Experimental Design.

We selected a set of 10 protein sequences with lengths between 38 residues and 178 residues, covering different amino acid compositions and physicochemical properties (Fig. 1E and SI Appendix, Table S2 and Note S1). Three of the 10 proteins fold to form stable structures under native conditions whereas the other 7 are IDPs that remain disordered in the absence of denaturant. To avoid potential uncertainties that can (30), but must not (31) arise from random labeling of proteins, we exploited the advantages of site-specific, unambiguous dual labeling. Specifically, the donor dye Alexa488 (SI Appendix, Fig. S1A) was attached via oxime ligation to the unnatural amino acid p-acetylphenylalanine, engineered at the penultimate position of the polypeptide chain using amber suppression technology (32). The acceptor fluorophore Alexa594 (SI Appendix, Fig. S1B) was reacted with a cysteine residue located at the second position via maleimide chemistry. Single-molecule measurements were made using the doubly labeled proteins under strongly denaturing conditions (6 M urea) and in (near)-native conditions with urea virtually absent (see buffer details in SI Appendix, Note S2 and experimental smFRET details in SI Appendix, Note S3).

SAXS measurements were performed using unlabeled and labeled samples (see experimental SAXS details in SI Appendix, Note S4). As an example of experimental results, we show the SAXS profiles (Fig. 2 A and B) and Guinier fits (Fig. 2 C and D) for the IDP NUS, under denaturing (Fig. 2 A and C) and native conditions (Fig. 2 B and D). The R_G is typically calculated from a plot of the SAXS intensity I(q) vs. the momentum transfer q, using the Guinier approximation (SI Appendix, Note S4):

$$\ln \left[I\left(q\right)\right]=\text{ln}\left[I\left(0\right)\right]-q^2{R}_G^2/3.$$ [1]

Fig. 2.

Representative experimental results and estimating G. The data correspond to the IDP NUS. (A) SAXS profile of unlabeled (black line) and labeled NUS (red line) measured in denaturing buffer. (B) SAXS profile of unlabeled (black line) and labeled NUS (red line) measured in native buffer. In A and B the dashed green lines are fits to a mass fractal dimension (for labeled proteins only) to obtain the parameter ν, related to the scaling of internal distances (Eq. 4). (C) Guinier fits of the SAXS profiles shown in A. R_G values are directly proportional to the slope of such plots (Eq. 1). (D) Guinier fits to the SAXS profiles shown in B. In A and C the dashed green lines are fits to a mass fractal dimension (for labeled proteins only) to obtain the parameter ν, related to the scaling of internal distances (Eq. 4). In C and D dashed cyan lines and dashed yellow lines represent the Guinier fits of unlabeled and labeled proteins, respectively. (E) E_FRET histogram of NUS measured in denaturing buffer. (F) E_FRET histogram of NUS measured in native buffer. In E and F the blue lines are fits using a double Gaussian function to get mean FRET values (<E_FRET>). (G) Probability distribution functions used to infer R_E,L from the mean E_FRET shown in E, using Eq. 2. (H) Probability distribution functions used to infer R_E,L from the mean E_FRET shown in F. In G and H the models are as follows: Gaussian chain (solid line) and CAMPARI simulations reweighted to match <E_FRET> and R_G,U² (dashed line). (I) <E_FRET> as a function of R_G,L under denaturing (dark violet circles) and native conditions (light violet circles). Each circle corresponds to exactly the same protein (i.e., double labeled) measured by smFRET and SAXS. Fits to a distribution of distances according to a Gaussian chain model (with SI Appendix, Eq. S15 and Eq. 2) are shown as dark violet lines (G = 7.1 ± 0.5, proteins denatured in urea) and light violet lines (G = 4.3 ± 0.4, IDPs in native buffer).

Alternatively, R_G can be estimated from the pair–distance distribution function (SI Appendix, Note S4). R_G,U and R_G,L calculated from either the Guinier approximation or the pair–distance distribution function were found to be similar to one another (values in SI Appendix, Table S3). Fig. 2 also shows the smFRET histograms (Fig. 2 E and F) and the most common distance distribution functions used to infer R_E,L from $\langle E_{FRET}\rangle$ (Fig. 2 G and H) corresponding to the same protein (NUS) under denaturing (Fig. 2 E and G) and native (Fig. 2 F and H) conditions. The peak at E_FRET near zero in the smFRET histograms arises from donor-only species (33), whereas the second population, originating from molecules containing an active donor–acceptor pair, appears at E_FRET ∼ 0.55 for native NUS. The parameter R_E,L quantifies the ensemble-averaged root mean-squared distance between the donor and acceptor dyes and it is related to $\langle E_{FRET}\rangle$ via

$$\langle E_{FRET}\rangle ={\displaystyle {\int }_0^{\infty }\frac{1}{1+{\left(r_{D,A}/R_0\right)}^6}P\left(r_{D,A};R_{E,L}\right)d{r}_{D,A}}.$$ [2]

Here, R₀ or the Förster distance (the distance at which FRET efficiency is 50%) depends on the specific dye pair and it is usually around 5 nm (our measured values are in SI Appendix, Table S4); P(r_D,A; R_E,L) is a probability distribution function that quantifies the likelihood of realizing values of interdye distances, within an interval r_D,A and r_D,A + dr_D,A given a mean donor-to-acceptor distance of R_E,L. The form for P(r_D,A; R_E,L) is unknown a priori and is usually chosen from a list of polymer models that includes the Gaussian chain model, the self-avoiding random walk (SARW) model, or a distribution of points inside a sphere of fixed diameter (34) (SI Appendix, Notes S3 and S8). These models are parameterized in terms of R_E,L, which reflects the contribution of the first (mean) and second (variance) moments of the distribution P(r_D,A; R_E,L) (35).

Fig. 2 shows illustrative datasets from smFRET and SAXS measurements. The complete sets of data from smFRET measurements for all proteins and conditions are shown in SI Appendix, Table S4 (FRET parameters); SI Appendix, Table S5 (anisotropies); SI Appendix, Fig. S2 (gamma and quantum yields); and SI Appendix, Fig. S3 (FRET efficiencies). Similarly, the complete SAXS data are shown in SI Appendix, Fig. S4 A–D (SAXS profiles, Guinier plots, Kratky plots, and pair distance distribution function, respectively). Importantly, to deal with the fact that smFRET and SAXS measurements were performed at very different concentrations, we carried out additional experiments to ensure that the large differences in concentration are not the source of discrepancies in inferences drawn from these measurements (SI Appendix, Note S5 and Fig. S5). Analyses of the datasets, which include information regarding $\langle E_{FRET}\rangle$ (originating from smFRET), R_G,L (measured by SAXS), and R_G,U (also from SAXS), are presented in the following sections, first for denatured proteins and then for IDPs under native conditions.

Measurements of R_G and Estimates of R_E from Measurements of $\langle E_{\mathit{FRET}}\rangle$ Yield Mutually Consistent Inferences for Denatured Proteins.

We performed SAXS experiments using labeled and unlabeled molecules to quantify the impact of fluorescent dyes on the global dimensions of flexible polymers. SI Appendix, Fig. S6A shows R_G,U,D (yellow points) and R_G,L,D (red points) calculated from the Guinier approximation as a function of the number of residues (N_RES) for eight proteins denatured in 6 M urea. Here, the letters L and U in the subscripts refer to labeled vs. unlabeled molecules and D refers to denaturing conditions (and N refers to native). Our dataset includes five IDPs and three proteins that fold autonomously. The differences between R_G,L,D and R_G,U,D were generally small, with a root mean-squared deviation (rmsd) of ∼0.3 nm between both datasets. For flexible polymers, a scaling law governs the value of R_G whereby

$$R_G\propto {\left(N_{RES}\right)}^{\nu }.$$ [3]

Here, N_RES is the number of residues in the chain. The exponent ν quantifies the correlation length and is governed by the solvent quality. In good, theta (indifferent), and poor solvents the values of ν for long homopolymers are 0.59, 0.5, and 0.33, respectively (26). Scattering data for a given protein can be analyzed within an intermediate q range to quantify ν (SI Appendix, Fig. S7A) because

$$I\left(q\right)\propto q^{-1/\nu }.$$ [4]

For reference, the full form factor is shown in SI Appendix, Eqs. S19 and S20 (36). An example of the fitting of SI Appendix, Eq. 4 to the experimental SAXS profile is shown in Fig. 2A for denatured NUS (all proteins can be found in SI Appendix, Fig. S4A). In 6 M urea, we find that ν = 0.55 ± 0.04 for unlabeled proteins. Within error, this value is similar to the value for labeled samples, ν = 0.58 ± 0.03 (SI Appendix, Table S6). These findings suggest that the dyes do not fundamentally alter the balance of chain–chain and chain–solvent interactions (SI Appendix, Fig. S7B), thus leaving the solvent quality unchanged. For the analysis that follows, we used an average value of ν_D = 0.57 ± 0.03 for proteins in 6 M urea. This value for ν is in line with the expected value for the SARW model and the analysis of larger datasets from previous measurements (37, 38), which suggest that high concentrations of denaturants are good solvents for generic protein sequences (3, 23). To test whether smFRET measurements yield similar inferences regarding solvent quality, we calculated the values of G = R_E²/R_G². For chains in a good solvent G ∼ 7 (26), and obtaining such a value would require accurate estimates of R_E from the smFRET data. In Fig. 2I we plot $\langle E_{FRET}\rangle$ against R_G,L,D, which is extracted from SAXS using exactly the same labeled proteins. The data were analyzed using a Gaussian chain model for the distribution of interdye distances (9–12), with G as the fitting parameter (SI Appendix, Eq. S15 and Note S3). For denatured proteins we obtained G_D = 7.1 ± 0.5. This value is in line with theoretical expectations for a swollen chain in good solvent (39) and is larger than the value of 6 expected for random coils (40) in theta solvents (R_E,L values in SI Appendix, Table S7 and G values in SI Appendix, Table S8). Taken together, our analyses of SAXS and smFRET data yield mutually consistent inferences regarding solvent quality for denatured proteins in 6 M urea. Importantly, our data establish that the dyes do not materially impact the analysis of chain dimensions of denatured proteins.

SAXS and smFRET Yield Discrepant Inferences Regarding IDP Dimensions in Native Conditions.

We applied the analyses described above to the set of seven IDPs under native conditions to calculate ν_N and G_N. Analysis of SAXS profiles for each of the labeled and unlabeled IDPs yielded similar values for ν_N (SI Appendix, Fig. S4A), suggesting that dyes do not have a major impact on the dimensions of IDPs under native conditions. The mean value of ν_N = 0.50 ± 0.04 (SI Appendix, Table S6) is in line with values reported for IDPs with similar compositional biases (3, 7, 41). This suggests that for a class of IDP sequences, the effects of chain–chain and chain–solvent interactions are, on average, mutually compensatory, thus unmasking statistics that are similar to those of chains in theta solvents (29, 41)—a result that has previously been described for unfolded protein ensembles under folding conditions (3). For G, we obtained a mean value of G_N = 4.3 ± 0.4, and this is different from the value of 6 that is expected for chains in theta solvents (35, 39, 40). To test whether the anomalous value of G reflects differences in the changes of R_G vs. R_E, we quantified the swelling ratios that compare the dimensions in 6 M urea vs. native conditions. The swelling ratios are defined as

$$\alpha \left(R_{E,L}\right)=R_{E,L,D}^2/R_{E,L,N}^2\hspace{1em}\text{and}\hspace{1em}\alpha \left(R_{G,L}\right)=R_{G,L,D}^2/R_{G,L,N}^2.$$ [5]

The inferred values of R_E,L of denatured IDPs (R_E,L,D) are considerably larger than those of native IDPs (R_E,L,N). However, the values of R_G,L for denatured IDPs (R_G,L,D) are only moderately yet systematically different from those of native IDPs (R_G,L,N). This is evidenced by larger values of α(R_E,L) vs. smaller values of α(R_G,L) (on average 2.02 ± 0.18 vs. 1.27 ± 0.12, respectively; individual values given in SI Appendix, Table S9). These findings are concordant with previous results, which point to disagreements between inferences from SAXS/small-angle neutron scattering (SAXS/SANS) and smFRET measurements at low denaturant concentrations (15, 21). SAXS measurements of labeled vs. unlabeled molecules rule out the dyes as the source of the discrepancy. Once we rule out specific errors with the smFRET measurements, which are presented in Discussion, we are left with three other possible sources for the observed discrepancies: (i) the nature of the averaging that goes into the calculation of R_G is likely to make this quantity relatively insensitive to small changes in solvent quality (20), especially for heteropolymers that transition between coil-like ensembles corresponding to ν ∼ 0.59 and ν ∼ 0.5 (42); (ii) because R_E quantifies the average distance between a pair of residues, as opposed to an average over all interresidue distances, it is possible that this quantity is more sensitive to fluctuations due the dangling ends of chains (43); and (iii) it is also possible that the inferred values of R_E are subject to errors due to assumptions of a Gaussian chain model for the distribution of interdye distances. Each of these factors contributes to the discrepancies between inferences drawn from analysis of SAXS vs. smFRET data. We demonstrate this by analyzing conformational distributions extracted from atomistic simulations that accounted for the presence of fluorescent dyes.

Source of the Discrepant Inferences Regarding the Extent of Collapse Observed Using SAXS vs. smFRET.

We performed all-atom Metropolis Monte Carlo thermal replica exchange simulations for five of the IDPs, using the ABSINTH implicit solvation model and force-field paradigm (24). This combination has proved to be useful for the analysis of conformationally heterogeneous IDPs (42, 44). Details of the simulations are described in SI Appendix, Note S6. For each sequence, we used the measured values of $\langle E_{FRET}\rangle$ in native conditions to generate reweighted ensembles that match the experimental data. Then, we selected the ensemble corresponding to the lowest simulation temperature (SI Appendix, Table S10) that best matched the experimental observable of interest (more details in SI Appendix, Note S6). To calculate $\langle E_{FRET}\rangle ,$ we incorporated atomistic descriptions of rotamers of fluorescent dyes into the simulated ensembles. For each conformation of a specific sequence, we placed roughly 10³ distinct dye rotamers in different mutual orientations and distances and calculated FRET efficiencies for each conformation. This process was repeated across the entire ensemble to calculate $\langle E_{FRET}\rangle$ across the ensemble. Conformations were reweighted based on the agreement between the measured and calculated values of $\langle E_{FRET}\rangle .$ The reweighting of ensembles based on experimental data was performed using COPER (45), which is a maximum-entropy reweighting method that attempts to give conformations similar weights while simultaneously attempting to match an experimental observable or a set of experimental observables.

Fig. 3A shows the values of R_E and R_G that were extracted from the unbiased ensembles (denoted as R_E,S and R_G,S) and the ensembles reweighted to match $\langle E_{FRET}\rangle$ (R_E,SW and R_G,SW) corresponding to native conditions. The subscript S refers to values obtained from simulations and W refers to cases where the simulation values were weighted to match an experimental observable. Here, R_E was calculated as the distance between the C_α atoms of the first and last residues and R_G was calculated only over the protein atoms. The reweighting procedure reveals an interesting decoupling between the values of R_G and R_E. Ensembles that were reweighted to match $\langle E_{FRET}\rangle$ showed minimal changes between R_G,S and R_G,SW and large changes between R_E,S and R_E,SW (Fig. 3B). This is consistent with the idea that large changes to $\langle E_{FRET}\rangle$ and hence R_E are compatible with minimal changes to R_G. If true, then the discrepant inferences between SAXS and smFRET measurements must originate in the ability to decouple measures of specific pairwise distances such as R_E from the averaging over the square of all pairwise distances, which is the case with R_G².

Fig. 3.

Simulated ensembles reweighted to match <E_FRET> suggest decoupling between R_G and R_E. (A) R_G and R_E values extracted from unbiased (R_G,S and R_E,S) and reweighted (R_G,SW and R_E,SW) ensembles for N49, NLS, NUS, IBB, and NUL. Here, reweighted ensembles refer to the ensembles generated by reweighting to <E_FRET> values under native conditions. Error bars indicate the SEM over three independent simulations. The experimental R_G,U values determined under native conditions are plotted for reference. R_G,U is used as a reference given that for the simulated ensembles R_G is calculated only over the protein. (B) The relative change in R_G and R_E between unbiased and reweighted ensembles calculated as |R_G,SW − R_G,S|/R_G,S and |R_E,SW − R_E,S|/R_E,S, respectively. (C) Internal scaling plots for NUS simulated ensembles reweighted to match R_G,U² and one of the following <E_FRET> values: [0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8]. For every pair of residues at a given sequence separation (|j–i|) the average through-space distance for that given sequence separation (<<R(|j–i|)>>) is plotted. Here, i and j are the residue positions. FRC denotes the internal scaling profile of the Flory random coil (Gaussian chain) reference and EV denotes the internal scaling profile of the excluded volume coil reference.

To put the proposed decoupling between R_G and R_E on a quantitative footing, we reweighted the NUS ensembles at 360 K to match the experimentally derived $R_{G,U}^2$ and one of the following target values for mean FRET efficiencies: $\langle E_{FRET}\rangle$ = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]. Here, $R_{G,U}^2$ in the simulations is the weighted mean square of the R_G values calculated over the protein atoms alone. If R_G and R_E can be decoupled, then ensembles should be generated that satisfy a single value of $R_{G,U}^2$ and a range of values of $\langle E_{FRET}\rangle .$ Indeed, we find that with the exception of the most extreme $\langle E_{FRET}\rangle$ values (0.1 and 0.9), NUS ensembles can be generated that match $R_{G,U}^2$ and a given $\langle E_{FRET}\rangle$ value with minimal changes to the force field (SI Appendix, Fig. S8 A and B and Note S6). This suggests that, under certain conditions, an entire spectrum of $\langle E_{FRET}\rangle$ and therefore multiple R_E values are consistent with a given R_G value (22). This result is consistent with the finding that large differences in G are virtually indistinguishable by SAXS (SI Appendix, Fig. S7C). Such a result emerges from the combination of two effects: (i) at low to intermediate values of R_G small changes in R_E (∼1 nm) can lead to large changes in G (SI Appendix, Fig. S9A) and (ii) large, potentially informative fluctuations at the ends of chains have little effect on the global conformational properties measured by SAXS (SI Appendix, Fig. S9 C and D).

The preceding findings do not imply that the ensembles generated to match different $\langle E_{FRET}\rangle$ values have the same conformational properties. To make this point, we characterized the overall shapes of polymers and scaling of internal distances for ensembles of NUS that match the experimentally derived $R_{G,U}^2$ and one of the following target values for mean FRET efficiencies: $\langle E_{FRET}\rangle$ = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]. We quantified overall shape preferences by calculating conformation-specific and ensemble averaged values of asphericity, δ*, that is given in terms of the eigenvalues, λ₁, λ₂, and λ₃ of conformation-specific gyration tensors (46, 47). Here,

$${\delta }^{\ast }=1-3\frac{({\lambda }_1{\lambda }_2+{\lambda }_2{\lambda }_3+{\lambda }_1{\lambda }_3)}{{({\lambda }_1+{\lambda }_2+{\lambda }_3)}^2}.$$ [6]

For rod-like conformations δ* ∼ 1 and for a perfect sphere δ* ∼ 0 (26, 47). Distributions of δ*_SW\ (SI Appendix, Fig. S8D) show that δ*_SW decreases as $\langle E_{FRET}\rangle$ increases, whereas distributions of R_G,SW are similar for all $\langle E_{FRET}\rangle$ values (SI Appendix, Fig. S8C). The decrease in δ*_SW observed with decreasing R_E suggests that ensembles become more spherical to account for the same R_G albeit with smaller R_E values. SI Appendix, Fig. S10 shows a comparison of shape characterization in terms of G and δ*. These parameters are weakly coupled although, on average, an increase in〈G〉implies an increase in〈δ*〉. The weak coupling results from the fact that G is highly sensitive to large fluctuations at the ends of chains, whereas δ* is only mildly sensitive to such fluctuations and changes in δ* depend on the sequence separation at which the fluctuations emerge (SI Appendix, Fig. S9 B–D). To extract further insights regarding the distributions of internal distances, we calculated internal scaling profiles that serve as formal order parameters in more nuanced theories of coil-to-globule transitions (48).

Internal scaling profiles quantify the mean spatial separation between all residues i and j that are |j–i| apart along the linear sequence. Fig. 3C shows that all ensembles, irrespective of the target $\langle E_{FRET}\rangle$ value used for reweighting, show similar scaling in spatial separation for |j–i| < 40. However, the spatial separations start to diverge from one another at larger sequence separations. These internal scaling profiles highlight an important point: based on Lagrange’s theorem (39) we know that the mean-squared R_G can be written as the mean-squared sum over all internal distances (definition in SI Appendix, Table S1). Thus, if a majority of internal distances change negligibly, then the value of R_G will change minimally. In contrast, the overall shape shows intermediate changes and distances corresponding to larger sequence separations will show large fluctuations (SI Appendix, Fig. S9 C and D).

Because we measured R_G and $\langle E_{FRET}\rangle$ for each IDP under native and denatured conditions, we can analyze the ensembles that were reweighted to match both experimental observables. Fig. 4 shows the two-dimensional histograms of R_G,SW vs. δ*_SW for ensembles reweighted to match both $R_{G,U}^2$ and $\langle E_{FRET}\rangle$ for each IDP under native (Fig. 4 F–J) and denatured (Fig. 4 A–E) conditions. For all IDPs, δ*_SW increases under denaturing conditions, indicating that the ensembles become less spherical. This is consistent with the larger G values extracted from denatured compared with native conditions. Internal scaling plots of the simulated ensembles (Fig. 4 K–O) show that the denatured ensembles diverge from native ensembles to prefer larger spatial separations for larger sequence separations. The sequence separation at which this divergence occurs is specific to each IDP sequence, thus highlighting the contribution of sequence-specific interactions to chain deformations under denaturing conditions. To visualize the change in shape between native and denatured ensembles, we extracted 100 representative conformations with the highest weights for NUS when reweighted to match the experimental observables under either native (Fig. 5A) or denatured (Fig. 5B) conditions. The results show that NUS adopts more elongated and less spherical conformations under denaturing conditions compared with native conditions.

Fig. 4.

Quantification of the shape (δ*_SW), size (R_G,SW), and scaling of simulated ensembles reweighted to match both <E_FRET> and R_G,U² for native and denatured conditions. (A–E) Two-dimensional histograms of R_G,SW vs. δ*_SW extracted from simulated ensembles reweighted to match both <E_FRET> and R_G,U² for denatured conditions. (F–J) Two-dimensional histograms of R_G,SW vs. δ*_SW extracted from simulated ensembles reweighted to match both <E_FRET> and R_G,U² for native conditions. The percentages indicate the percentage of δ*_SW that falls between different δ*_SW limits (dashed horizontal lines). For all IDPs studied, δ*_SW increases under denatured conditions, indicating that the ensembles become less spherical and more elongated. (K–O) Internal scaling plots comparing the native (N) and denatured (D) profiles from simulated ensembles reweighted to match both <E_FRET> and R_G,U². FRC and EV denote the internal scaling profiles generated from the Flory random coil and excluded volume coil references, respectively. Error bars denote the SEM over three independent simulations.

Fig. 5.

Representative ensembles of NUS under native and denatured conditions. (A and B) The 100 conformations with the highest weights from the simulated ensembles reweighted to match both <E_FRET> and R_G,U² for native (A) and denatured (B) conditions.

We also note that simulations can be used to estimate the error associated with inferences of R_E,L from smFRET that are based on the use of the Gaussian chain or other generic polymer models for P(r_D,A; R_E,L) (49). Fig. 2 G and H shows the distance distributions corresponding to the Gaussian chain model together with the distance distributions obtained from the simulations by restraining the ensembles to match $\langle E_{FRET}\rangle$ and R_G,U². The results suggest that the Gaussian chain model tends to overestimate R_E,L for denatured proteins and underestimate R_E,L for IDPs under native conditions (SI Appendix, Table S7). These results are consistent with the findings of O’Brien et al. (49) and Borgia et al. (23). Accordingly, the final α(R_E,L) values (SI Appendix, Table S9) are overestimated.

Analysis of the Full SAXS Profiles Beyond R_G and ν.

If ensembles of chemically denatured proteins display larger asphericities compared with the native IDPs, then this should be discernible in the SAXS data as well. We tested this by performing a model-independent comparison of the experimental data. Indeed, if one computes a size-independent version of scattering profiles by plotting log[I(q)/I(0)] vs. qR_G (Fig. 6A), then the curves corresponding to bodies with changing asphericity display a rather systematic trend, from the right (aspherical polymers) to the left (spherical polymers) of the plot. We plotted the experimental data for unlabeled native and chemically unfolded proteins (Fig. 6 B–F). For the two smallest proteins N49 and NLS, the differences are within the level of statistical noise, whereas the three larger proteins display a systematic shift of the size-independent scattering patterns from the right (higher asphericity for chemically denatured proteins) to the left (more spherical shapes for IDPs under native conditions). The results of this analysis are important because they were obtained solely from the experimental data.

Fig. 6.

The full SAXS profile (but not R_G alone) is sensitive to the changes in the ensemble shape that occur upon IDP collapse. To remove the contribution of size to the SAXS profiles and visualize exclusively the influence of shape, size-independent SAXS curves are constructed by plotting the normalized scattering intensities (log[I(q)/I(0)]) as a function of q times R_G. (A) Theoretical SAXS profiles predicted for different values of asphericities (δ*): 0.1 (solid line), 0.3 (dashed line), 0.5 (dashed-dotted line), and 0.9 (dashed dotted-dotted line) (see SI Appendix, Fig. S7 for more details on color scale). Note that asphericity increases from left (more spherical) to right (more anisometric) and that a given R_G is compatible with many asphericities; i.e., R_G is not sensitive to shape. The experimental SAXS profiles of unlabeled native (red lines) and unlabeled denatured (blue lines) IDPs are shown in B (N49), C (NLS), D (NUS), E (IBB), and F (NUL).

We further tested the proposed change in asphericity, using size-independent maps of scattering profiles that were generated using the reweighted ABSINTH ensembles. CRYSOL (50) was used to convert each conformation to a SAXS profile and these were combined to generate the final weighted SAXS profile. The profiles generated from the reweighed ABSINTH ensembles consistently show an increase in asphericity for denatured IDPs compared with native IDPs (SI Appendix, Fig. S11D). This recapitulates the direct calculations of asphericities from reweighted ASBSINTH ensembles, without consideration of the full scattering curves. However, it is noteworthy that ensembles showing divergence in the internal scaling profiles between native and denatured conditions at larger sequence separations show less pronounced differences in the scattering patterns (Fig. 4 and SI Appendix, Fig. S11). This is consistent with the observation that asphericity is mainly sensitive to changes in spatial separation in the intermediate sequence separation regime (SI Appendix, Fig. S9 C and D). Fluctuations at the ends of the chain will have minimal impact on the overall asphericity. It is also noteworthy that the reweighted ABSINTH ensembles, which were reweighted to match R_G² and <E_FRET> values, also resemble the experimentally derived SAXS profiles (SI Appendix, Fig. S11A).

In another independent and unbiased approach, the ensemble optimization method (EOM) (51) was used to analyze the SAXS data. The EOM analysis used the unweighted pool of ABSINTH conformations to select subensembles of conformers such that their mixture accurately matches the experimental SAXS data (SI Appendix, Fig. S11A). The EOM-selected ensembles unveiled substantial conformational heterogeneity (displayed as essentially broader, not necessarily monomodal size distributions) compared with the reweighted ABSINTH ensembles. Further, EOM ensembles showed an increase in both conformational heterogeneity (SI Appendix, Fig. S11B) and asphericity (SI Appendix, Fig. S11D) under denaturing conditions compared with native conditions. The increase in asphericity was more pronounced for the longer constructs, in agreement with the results shown in Fig. 6. The EOM ensembles did not always reproduce the experimentally measured <E_FRET> values (SI Appendix, Fig. S11C). This highlights the distinctive nature of the information that is gleaned from SAXS vs. smFRET measurements. Specifically, information about the end-to-end distance may be diluted or lost in SAXS profiles. This is not unexpected given that SAXS measurements yield integral information on averaged distance distributions over conformational ensembles as opposed to “differential” averaging of a single end-to-end distance in smFRET. Hence, ensembles generated to match SAXS data are likely to be incompatible with inferences that are based on smFRET measurements, especially away from denaturing conditions. Overall, these results emphasize the importance of gathering SAXS and smFRET data and the joint use of both methodologies for generating mutually compatible ensembles that provide a more complete picture of the overall shapes, sizes, and conformational fluctuations.

Estimating N_RES from SAXS and smFRET Data.

Although our work raises caution regarding the use of generic polymer models when analyzing smFRET data for heteropolymers, these models afford the practical convenience required to obtain quick estimates of R_E,L from measured $\langle E_{FRET}\rangle$ values for IDPs as well as denatured states. It is useful to quantify the contribution that dyes (N_DYES) make in terms of equivalent residues to the polypeptide chain (N_RES). Previous estimates of N_DYES have varied from 0- to 20-residue equivalents (3, 12, 52, 53). Given direct access to $R_{G,L},$$R_{G,U},$ and estimates of $R_{E,L}$ we can quantify N_DYES using these data.

Because the scaling behavior of R_G,L depends on the actual number of amino acids in both the polypeptide chain (N_RES) and N_DYES, we rewrite Eq. 3 as follows for R_G,L (a similar reasoning can be used for R_E,L):

$$R_{G,L}=\sqrt{1/G}\hspace{0.17em}{\rho }_E{\left(N_{RES}+N_{DYES}\right)}^{\nu }$$ [7]

and

$$R_{E,L}=\sqrt{G}\hspace{0.17em}{\rho }_G{\left(N_{RES}+N_{DYES}\right)}^{\nu }.$$ [8]

Here, the preexponential factors ρ_E and ρ_G are related to the size of the repeating unit. Whereas dye labeling does not substantially affect R_G as detected by SAXS, we can perform a global fit of the six experimental datasets to extract the contributions that dyes make to R_E,L for both denatured proteins and native IDPs. This allowed us to obtain estimates of $N_{DYES}=5\pm 3$ (SI Appendix, Fig. S6 and Table S11).

Discussion

SAXS and smFRET are two powerful experimental tools that provide useful insights regarding disordered systems such as IDPs and unfolded ensembles of autonomously foldable proteins (7, 54). However, the two measurements yield discrepant inferences when going from denatured to native conditions, with SAXS detecting minimal changes and smFRET suggesting discernible reduction in R_E,L as measured by an increase in <E_FRET>. We obtained good agreement between inferred R_E,L and R_G,L values at high denaturant concentrations in terms of the scaling behavior and inferred solvent quality. However, we find a clear “mismatch” in inferences regarding chain sizes in the absence of denaturant: either the inferred values of R_E,L appear to be too small or the measured values of R_G,L are too large.

Our insights were derived by combining experimentally derived R_G values and mean FRET efficiencies with simulations that also include the effects of dyes. A major conclusion from the simulations is that many disordered ensembles with substantially different R_E can have similar values of R_G (Figs. 3–6). This result was also demonstrated by Song et al. (22) for heteropolymeric systems and it implies that the discrepant expansion factors inferred from SAXS and smFRET measurements are not a consequence of any intrinsic weaknesses of these methods. Instead, they represent a fundamental decoupling between R_G, a globally averaged quantity, and R_E as well as other distances between dangling ends that are not averaged across the entire sequence. This decoupling is amplified in finite-sized heteropolymeric sequences in the absence of denaturant.

Advanced theories that account for the effects of chain connectivity to describe excluded volume effects demonstrate that chains can undergo nonuniform expansion/compaction (55). The dangling ends of chains (43) experience fewer restrictions on fluctuations. Hence, inferences regarding chain dimensions can be different when quantified in terms of R_G vs. R_E or distances near the ends of chains. The use of R_G and R_E as equivalent measures of chain dimensions dates back to Flory-style mean-field theories that reduce polymers to collections of uncorrelated monomers or Kuhn segments (5). This is a powerfully simplifying approach that affords convenient analytical descriptions. In contrast, Lifshitz-style theories recognize the decoupling between R_G and R_E and rely on the radial density profile (equivalent to the internal scaling profile) as an order parameter for coil-to-globule transitions (56, 57).

Effects of Dyes in smFRET Measurements.

For native and denatured conditions we showed that the behavior of labeled proteins is not different from that of their unlabeled counterparts, at least in terms of the scaling of internal distances manifested by similar values of ν for unlabeled and labeled samples (SI Appendix, Table S6). The parallel axes theorem is a useful theoretical construct to describe the relationship between R_G,U, R_G,L, and R_E,L. A full theoretical treatment of this can be found in SI Appendix, Note S7. The main conclusion from this analysis is that for many values of R_E,L, dyes do not cause a measurable change of R_G,L relative to R_G,U (SI Appendix, Note S7 and Fig. S12A). However, as G increases, the difference between R_G,L and R_G,U is predicted to increase, with larger changes observed for shorter chain lengths (SI Appendix, Fig. S12B). This prediction is consistent with the experimental trends we observe (SI Appendix, Fig. S12C). Combining the results from SAXS and smFRET with simulations, we estimated the contribution of dyes to R_E,L expressed in terms of extra residues as N_DYES = 5 ± 3 (SI Appendix, Fig. S6). Such a value is likely to be generally useful for smFRET analysis, irrespective of the particular fluorescent dye pair used, because the actual size of each fluorophore has a limited influence on the inferred distances (SI Appendix, Eq. S29 and Fig. S1C). To further rule out the possibility of artifacts due to the dyes themselves, we discuss potential sources of errors in our experimental design and broader implications.

Case A.

Dyes might experience hindered rotations such that the orientation parameter κ², and hence the Förster distance R₀, deviates from the isotropic averaging condition (58). We tested this via anisotropy measurements. The low values we observe for anisotropies (<0.1, SI Appendix, Table S5) support free dye rotation under all assayed conditions. Therefore, it appears to be reasonable to assume that rotational averaging is allowed, and thus the assumption of κ² = 2/3 in the FRET equation is valid (SI Appendix, Table S4).

Case B.

The dyes might be drawn toward one another through cohesive forces. The analysis of scaling exponents should make such an effect easy to detect. We do not observe such a trend under either denaturing conditions or native conditions (SI Appendix, Fig. S6).

Case C.

It is known that the dynamics of dyes can affect E_FRET measurements (9, 31, 33, 59–63). For unfolded proteins of similar size and in similar solvents to the ones studied here (including NUS), chain reconfiguration times have been shown to be in the range of ∼100 ns (3, 64), which is well above the donor lifetimes of ∼4 ns and well below the transit times through the confocal volume, ∼1 ms. As a result, a major role of dynamics in the measured intensity-based E_FRET values seems unlikely. Taken together, we conclude that the dyes alone cannot explain the large changes to R_E,L that we observe upon protein denaturation in contrast to the modest changes of R_G,L (SI Appendix, Table S9).

Choice of Polymer Models for Analyzing smFRET Data.

Our findings highlight the need for caution in coopting models for distributions of R_E or R_G that have been designed for infinitely long flexible homopolymers—a point that has been made in previous studies as well (22, 23, 49). Flory’s mean-field theory (5) yields a value of G = 6 for ν = 0.5 in theta solvents and SARWs yield G ∼ 7 for ν ∼ 0.6 (SI Appendix, Note S9 and Table S8). The values of ν (0.57) and the inferred values of G (6.6) for denatured proteins are in accord with the values for SARWs. For the native dataset we obtained G_N = 5.2 and ν_N = 0.5, respectively, when we used smFRET, SAXS, and simulations. This result suggests that according to the inferred value of G, IDPs under native conditions deviate from the Gaussian chain model, whereas the inferred scaling exponent suggests congruence with the statistics of the Gaussian chain model. In SI Appendix, Note S8 and Fig. S13 we show that the same issue persists when using other polymer models, thus highlighting the role of simulations in inferring self-consistent sets of distances and the need for caution in using generic polymer models for estimating R_E from measured FRET efficiencies, especially in the absence of denaturants. To overcome difficulties associated with the choice of generic polymer models, O’Brien et al. (49) proposed a self-consistency test that requires the measurement of FRET efficiencies by attaching dyes along different internal positions within a sequence. They showed that the use of multiple, independent measurements provides a rigorous test of the polymer model that is used to extract distance estimates from measured FRET efficiencies.

Connections to Recent Studies.

The discrepant inferences drawn from SAXS and smFRET measurements have stimulated numerous debates and independent investigations. Discrepancies were recently reported for nonbiological homopolymers like polyethylene glycol (PEG) (21). This study compared R_G values from SANS experiments to R_E values derived from smFRET. Unlike our study, the impact of dyes was not directly investigated as this would have required SANS measurements on PEG molecules with and without dyes. Additionally, the concentrations for SANS measurements correspond to the semidilute regime for PEG in water. In this regime, there are significant nonidealities such as the scaling of osmotic pressure as c^9/4, where c is the PEG concentration. The impact of these nonidealities on using PEG as a negative control remains unclear.

Aznauryan et al. (65) performed SAXS and smFRET measurements and combined these with distances extracted from structural ensembles based on data from NMR experiments. Their results point to consistent inferences for average distances and distributions of distances for ubiquitin in high concentrations of denaturant (65). A similar consistency regarding denaturant-mediated expansion was reported by Borgia et al. (23), who used a combination of smFRET, SAXS, dynamic light scattering, and two-focus fluorescence correlation spectroscopy to assess how conformational ensembles change as a function of denaturant concentration. They focused their measurements on the denatured state of the spectrin domain R17 and the IDP ACTR. All of their data support an expansion with increasing denaturant concentration. Borgia et al. (23) also showed that the inferred R_E and R_G values can be overestimated when using polymer-based models for proteins in denaturant. They argued that the inferred R_E and R_G appear to have different sensitivities to denaturant. In a third study, Zheng et al. (66) reported results from unbiased simulations to demonstrate consistency between inferences drawn from smFRET and SAXS measurements for proteins in increasing concentrations of denaturant. They noted that the dyes do not materially affect the degree of increase in R_G with increases in denaturant concentration. The work of Schuler and colleagues (23, 65, 66) highlights the mutual consistency of inferences from SAXS and smFRET measurements for denatured proteins, the insensitivity of estimates of the changes of R_G with denaturant to the presence or absence of dyes, and the possible overestimation of R_E and R_G values based on the polymer models that are used. Our results for denatured proteins and for IDPs in high concentrations of denaturants are consistent with those of Schuler and coworkers (23, 65, 66).

Working Hypothesis for the Decoupling Between R_G and R_E.

Flexible polymers can be described using the thermal blob model. R_G and R_E for a thermal blob will scale as g^½, where g is the number of residues per blob (67). By definition, the blob is a length scale where the intrablob interactions and blob-solvent interactions are counterbalanced. The blob size is approximately five to seven residues for most IDPs (41). In mean-field theories for polymers in dilute solutions, there are two interrelated parameters to consider: the surface tension per blob (γ_B) and the effective pairwise interactions between blobs (67). Depending on solvent quality, γ_B will be positive (poor solvent), zero (theta solvent), or negative (good solvent) and the pairwise interblob interactions will respectively be, negative, zero, or positive. All blobs are identical in homopolymers, and hence all interactions are uniform and a single parameter suffices to describe the overall chain statistics. Accordingly, in theta and good solvents, R_G and R_E will provide equivalent descriptions of chain behavior.

For heteropolymers, blobs can be quite different from one another and this depends on the amino acid composition and sequence patterning (68, 69). The chain could have blobs that encode negative, zero, or positive values of γ_B and these will in turn modulate the pattern of interblob interactions. Attractions can screen repulsions and this can give rise to relatively uniform density profiles that make R_G inert to changes in solution conditions but they will be manifest as differences in distances across specific length scales (Fig. 4). The effects of heteropolymericity can be captured as an interaction matrix as opposed to a single interaction parameter, and the key question is whether the variance across the values within the interaction matrix is smaller than, equivalent to, or larger than thermal energy. This variance will encode the extent of convergence or divergence between measures of chain dimensions averaged across the entire sequence (R_G) and measures that probe specific length scales, such as R_E. The blob-based analysis explains why despite water being a poor solvent for polypeptide backbones (29, 70), we now know that the apparent solvent quality for real IDPs deviates from that for backbones and is actually governed by charge and proline contents as well as the patterning of charged and proline residues (3, 17, 41, 42, 68, 69, 71).

Conclusion and Perspective

Given the high cost required to perform complete SAXS experiments with dye-labeled samples and the small contribution of the commonly used dyes to the total protein size, it is both impractical and unnecessary to measure SAXS profiles for labeled molecules on a routine basis. We have shown that, for many IDPs, R_G,U will be a reasonable approximation to R_G,L. Given the diversity of IDP sequences (68), it should be stressed that our measured values of G_N and δ_N* are unlikely to be universal. Therefore, R_E and R_G should be determined for each combination of solution condition and IDP through independent quantification of R_E,L by smFRET and R_G,U by SAXS or the measurement of multiple internal distances for different sequence separations by smFRET (3, 34) or through the joint use of intramolecular three-color FRET measurements (58). For SAXS measurements, this includes estimates of R_G (7) combined with analysis of protein shape preferences from the entire SAXS profile. These measurements can be augmented using methods such as anomalous SAXS (59) that introduce gold labels along the chain for extracting intramolecular distances. Measurements when complemented with computer simulations as performed here and in other efforts (66) can help in converting experimental observables into self-consistent molecular models of the conformational ensembles. The relevance of our work goes beyond IDPs under native conditions. In the protein-folding field there is lingering controversy over the earliest folding events arising from dissimilar FRET and SAXS experiments (15, 34); suggestions have been put forward for chain collapse preceding the folding transition—a view largely supported by FRET measurements—whereas the alternative position is that collapse is intimately coupled with the folding transition—a view supported by SAXS measurements. Based on our data, we propose that the earliest events are likely to be changes in shape (26, 46, 72) within the unfolded ensembles upon dilution from denaturant before folding and the formation of stable local as well as nonlocal contacts; decreased asphericity may be what smFRET measurements pick up as a “collapse” transition. This would be difficult to detect by SAXS using only R_G, but the full SAXS profile might be more useful for detecting changes in asphericity and directly estimating the correlation length via the scaling exponent ν. Therefore, we propose that the joint use of smFRET and SAXS, together with other structural biology methods, and the support of computational tools and advanced theories will improve our understanding of heterogeneous conformational ensembles.

Materials and Methods

In total, 10 proteins (abbreviated as N49, BBL, NLS, CSP, NUS, IBB, TRX, NUL, N98, and NSP) bearing a cysteine residue at the second position and the noncanonical amino acid p-acetylphenylalanine at the penultimate position were expressed recombinantly in Escherichia coli BL21 AI cells, purified, and double labeled with Alexa488 hydroxylamine and Alexa594 maleimide. Proteins were measured in two PBS buffer conditions: “denaturing” (in presence of 6 M urea) and “native” (with urea absent). SmFRET was done on a custom-built multiparameter spectrometer, using picomolar concentrations of labeled proteins. FRET efficiencies were analyzed burst-wise. SAXS profiles of labeled and unlabeled proteins at different concentrations (micromolar and beyond) were measured at the BioSAXS P12 beamline of Petra III (DESY). The scattering profiles were analyzed in full to obtain size (mean radius of gyration and its distribution) and shape (asphericity, correlation length) information. Molecular simulations of labeled proteins were performed using the CAMPARI package with the ABSINTH implicit solvation model and force-field paradigm. Experimental observables were used to restrain the conformational space sampled by the simulated ensembles. Comprehensive descriptions of the protein expression, purification, labeling, smFRET and SAXS measurements, atomistic simulations, and theoretical considerations are described in detail in SI Appendix, Notes S1–S9, Tables S1–S11, and Fig. S1–S13.