Effects from the nonideality of the solution due to the high protein concentrations required were investigated systematically by measurements at different protein concentrations see SI Text and Figure S9a,b. Again, we observe an increase in R h with increasing denaturant concentration for all samples, very similar to that calculated from 2f-FCS Figure 4.
We note that, although both DLS and 2f-FCS can measure R h with high precision corresponding to reliable relative changes in chain expansion , systematic errors e. Accordingly, both statistical and systematic errors are reported in Figure 4. Note also that the values from 2f-FCS are generally slightly larger than from DLS, consistent with an increase in size due to the additional fluorophore attached to the protein for 2f-FCS.
This result further suggests that the labelling with our fluorophores only increase the protein size slightly, while dye-protein interactions do not exert a detectable effect on the change in unfolded state expansion. An additional benefit of using DLS is that it allowed us to quantify even small fractions of protein aggregates present in our samples, in a range that would be difficult to detect by SAXS. In summary, all four experimental techniques we have employed expose a clear change in unfolded state expansion with increasing denaturant concentration, for both proteins and both denaturants investigated i.
However, the relative changes in the quantities accessible from the different methods are significantly different Table S2 , raising the question whether these results can be accounted for consistently. We have analyzed all experiments by using standard techniques with the simplest possible models. However, each experiment carries its own uncertainties due to the way the data are interpreted. For example, FRET experiments must employ a specific model to obtain the distribution of donor-acceptor distances P r over which the transfer efficiency is averaged.
With SAXS, the extraction of the radius of gyration from raw data is relatively model-free. Ideally, one would use all of the available scattering data to estimate the molecular size. However, model-free analysis of the wider angle data is more challenging: P r distance distribution function and associated R g are commonly calculated via a regularized Fourier transform, which creates a regularizer bias towards distributions characteristic of globular folded particles and may thus be ill-suited for the analysis of structurally diverse IDPs.
Accurate extraction of R g via P r methods is further complicated by the inevitable underestimation of the maximum dimensions for an unfolded protein, motivating the development of ensemble refinement methods Lastly, we would ideally like to compare the results for R , R g and R h more directly, and there is no generally applicable analytical relation between them.
Is there a way to obtain all the desired parameters by employing the different experimental data at our disposal, and concomitantly overcoming the inherent uncertainties and limitations of each individual technique? One way to achieve all of these goals is to use an explicit molecular model that accounts for the expected conformational heterogeneity of unfolded proteins. We then apply a reweighting procedure to achieve agreement with the experimental data using the EROS method 62 , A key feature of the analysis is a regularization procedure to prevent overfitting of the data to the very large number of structures in the initial ensemble relative to the number of experimental data 76 , 81 , 87 — 90 described in SI Text and Figure S To test whether we are able to recover a representative ensemble, we first applied the procedure to synthetic FRET efficiencies and SAXS intensities calculated from all-atom, explicit solvent simulations of ACTR in urea 72 , in which case the true properties of the molecular ensemble are known.
We find that the distributions of R g , R , and R h recovered from the reweighting of the implicit-solvent model to match the synthetic FRET and SAXS data, agree very well with those estimated directly from the all-atom simulations 72 much better than the unweighted implicit solvent models Figure S Note that the differences between SAXS curves calculated from simulations with increasing denaturant concentrations are quite subtle, as in experiment, yet associated with a clear increase in R g.
We thus applied the same Bayesian ensemble reweighting approach to a joint analysis of the FRET and SAXS data, whose quantitative relation to the structure ensemble is more straightforward than for hydrodynamic data. We find that we are able to fit both data sets very well, showing that they are mutually compatible. The distributions of r g from the resulting ensembles reweighted using the experimental data, shown in Figure 5c , reveal a systematic expansion with increasing denaturant concentration.
As shown in Figure 4 , these back-calculated hydrodynamic radii and especially their changes with denaturant concentration are in reasonable agreement with those estimated from 2f-FCS and DLS measurements, further cross-validating the simulated ensembles and indicating the consistency of all four experimental techniques used. Accompanying the increase of R g and R h is a modest increase in asphericity Figure S14 with increasing denaturant concentration, consistent with theoretical expectations 66 , We have also tested whether the ensemble analysis is sensitive to the consistency of the experimental data sets with each other.
In the absence of denaturant, a pronounced increase in FRET efficiency was observed for ACTR labeled with these dyes, indicating further collapse relative to the commonly employed hydrophilic dyes containing charged groups e. Alexa Fluor and , Table S3.
While it is possible to select a sub-ensemble which fits both SAXS and FRET data using the hydrophobic dyes, there are two indications that the fit is poor Table S4 : First, a much stronger reweighting of the original ensemble is required, as measured by the lower fit entropy, compared to when the data based on the hydrophilic dyes are used. Secondly, if only the SAXS data are used to reweight the simulations, FRET for the hydrophilic chromophore pairs is in reasonable agreement with experiment, but the agreement for the hydrophobic chromophore pairs is poor.
Similarly, if only the FRET data are used for reweighting, reasonable agreement with the SAXS data is obtained for the hydrophilic dye pair, but not for the hydrophobic dyes. In summary, the analysis used here not only provides molecular ensembles compatible with all experimental data used, but it also enables inconsistent results to be identified. Overall, the results from all methods we employed indicate an expansion of the polypeptide chain with increasing denaturant concentration and are mutually compatible.
In all cases, we find an expansion with denaturant concentration, with the most pronounced changes occurring at the lower denaturant concentrations. The steeper increase in R g at lower denaturant concentrations is consistent with the expectations of a binding model of denaturant interactions where saturation must occur at some point and with previous FRET studies on unfolded and intrinsically disordered proteins 6 , 35 , 58 , Similarly, the polymer scaling exponents increase with denaturant concentration, whether estimated from FRET, SAXS fractal dimension, or the dependence of the intramolecular distances on the sequence separation in the reweighted ensembles Figure 2 — 3 ; Figure S A corresponding analysis of conformational ensembles from unbiased molecular simulations of ACTR yields similar trends Table S2.
These different measures of chain size thus result in different relative amplitudes upon expansion or collapse. Since FRET measurements are most directly related to R , the transfer efficiency is intrinsically most responsive to chain expansion. These differences in relative amplitudes are expected both from polymer theory, simulations, and previous experiments.
R g is also expected to exhibit a larger change upon expansion than R h from theory 68 , 94 , simulations 95 , and experiments on homopolymers The combination of these effects helps to explain why unfolded state expansion has invariably been detected in single-molecule FRET experiments. However, why do we observe an increase in R g using SAXS, while such an expansion was not resolved in some earlier studies?
The corresponding radii of gyration are shown in panels e-h. As our data illustrate, detecting changes of R g from SAXS data is challenging due to the subtle variations in the shape of I q with denaturant concentration and the large associated errors for each data set Figure S5 , as previously suggested The trend becomes clear only with repeated independent data collections for each combination of protein and denaturant concentration, sampling a sufficiently large number of denaturant concentrations, and careful control of the effects of inter-particle interference and protein self-association.
The latter necessitates the use of low protein concentrations, requiring high flux of the incident beam. Both a reduction of the applicable range of the Guinier approximation with the expansion of the protein at higher denaturant concentration, and the lack of reliable lowest- q data, reduce the apparent fitted R g progressively as the denaturant concentration increases.
A systematic analysis of the validity of the Guinier fit to the primary SAXS data is challenging because of the experimental noise, but we can use the smooth I q calculated from the structure ensembles to illustrate this point: in Figure 5e we show the dependence of the Guinier-fitted R g on q max R g. Finally, for equilibrium SAXS measurements, reliable radii of gyration can only be extracted well above the denaturation midpoint, due to the difficulty of accounting for native state scattering at lower denaturant concentration 18 — Our results are in fact consistent with the earlier findings of little variation in R g above typical midpoint denaturation concentrations To illustrate the difficulty of observing an R g change at high denaturant concentration, we fit the dependence of R g from the SAXS Guinier region on denaturant concentration Figure 6 with two linear models: one with both slope and intercept as free parameters and the other with only the intercept, and zero slope.
Since the model with two parameters always fits better, we introduce the Bayesian information criterion BIC 96 to evaluate whether the fit is significantly better if the slope is not fixed to zero. In Table S5 , we show that if we fit R g over all denaturant concentrations, the BIC score indicates with high significance that the two-parameter model with nonzero denaturant dependence is better.
However, when restricting the fit to data from denaturant concentrations above 3 M urea or GdmCl , in three of the four cases, the SAXS data fail to indicate a statistically meaningful change of R g with denaturant concentration, and in the fourth ACTR in GdmCl , the improvement when including denaturant dependence of R g is of marginal significance. These results stress the importance of making as many measurements of R g over as wide a range of denaturant conditions as possible in order to have the best chance of resolving any variation.
In principle, for stably folded proteins, time-resolved SAXS measurements could provide access to the low-denaturant region, where we find R g expansion to be most prominent. However, in most cases time-resolved SAXS measurements also suggest no collapse after denaturant dilution, even when the final denaturant concentration is very low 21 — We cannot comment directly on these results except to note that these measurements, performed with very short exposures, would have even larger errors than static scattering data.
The potential presence of small amounts of aggregates or other larger particles in the sample may distort the R g extracted from the SAXS measurements. To mitigate this problem, we firstly selected highly soluble proteins and secondly used sample aliquots coming from the exact same batch i. We note, however, that higher concentrations of aggregates could lead to an overestimation of R g , especially at the lowest denaturant concentrations, where aggregation is most likely to occur.
A potential complication in FRET experiments is whether the extrinsic fluorophores themselves influence the results, perhaps inducing collapse, although molecular simulations suggest this to be a small effect 72 , To probe for this contribution, we have tested some of the most hydrophobic chromophores currently available, which lead to a pronounced collapse of ACTR in the absence of denaturant.
However, the resulting transfer efficiencies are incompatible with our SAXS data, in the sense that the structural ensembles produced using both the SAXS data and the hydrophobic dyes require rather extreme reweighting, or ensembles produced with SAXS or FRET alone do not reproduce the respective other data set Table S3.
In contrast, the SAXS data are consistent with the FRET data collected from the protein labeled with the hydrophilic dyes used here and in many other experiments. Lastly, in many experiments using identical dye pairs, large differences in FRET-based intramolecular distances have been observed for different polypeptide sequences, demonstrating that changes in the charge composition and hydrophobicity of the polypeptide chain itself are dominant over any effects from the fluorophores 40 , 63 , For a quantitative determination of average distance, R , and radius of gyration, R g , from single-molecule FRET, and comparison with SAXS without using ensemble refinement, important considerations are the uncertainty in the transfer efficiency and the need to assume a specific polymer model.
Therefore, the greater challenge for the quantitative interpretation of single-molecule FRET experiments on unfolded proteins is the model-dependence of the conversion of E to R and R g. Our results indicate that using the P r of simple polymer models may overestimate the degree of expansion.
On the other hand, P r of a SAW tends to underestimate chain dimensions at low denaturant concentrations but provides a better approximation at high denaturant concentrations Figure 6. Using the distance distribution of a SAW, we find that we are able to more accurately recover the distance R and radius of gyration R g from the transfer efficiency, when applied to synthetic data from simulations Figure S4.
The conversion of R to R g involves additional assumptions regarding the ratio of the two quantities, which depends on solvent quality 68 , 99 , and thus introduces additional uncertainty. Because of the controversy we aim to address, our analysis has been focused mainly on the radius of gyration and related quantities that probe large-scale features and overall dimensions of the sampled molecular conformations.
Obtaining a consistent value of this most basic property of an unfolded or disordered protein states, when measured by different techniques, is clearly a prerequisite for developing structural models for these states. Nonetheless, the ensemble of states populated by an IDP or unfolded protein cannot always be reduced to a description in terms of simple polymer theories, and specific local interactions and structure may be important in many cases , Resolving the apparent disagreement between SAXS and FRET experiments opens the way to the integration of both types of data in detailed structural models of disordered proteins.
Previously, qualitative discrepancies regarding the effect of chemical denaturants on the dimensions of unfolded and disordered proteins have been reported, when comparing the results from SAXS and other experimental methods, especially FRET. However, the two methods had previously only been applied to one protein in common, protein L. In the present work, by comparing two different proteins in two different denaturants, and using four different experimental methods, we find that all results are self-consistent, and show an increase of the average distance between FRET labels, radius of gyration, polymer scaling exponent and hydrodynamic radius of the chains with increasing denaturant concentration.
These findings are consistent with expectations based on the improved solvent quality in concentrated denaturant solutions 33 , 42 , 43 , 49 , 50 , , We stress that, while the proteins considered here do collapse as the denaturant concentration is reduced, they do not form a fully collapsed globule in water.
A careful analysis of our results helps to explain the apparent discrepancies in earlier work. First, the FRET efficiency is inherently more sensitive to changes in protein expansion, due to the greater relative change of R with denaturant than R g or R h , and due to the nonlinear distance dependence of FRET.
In addition, the use of polymer-based distance distributions for obtaining average distance and radius of gyration from FRET can lead to an overestimation of the degree of chain expansion with denaturant. On the other hand, probing expansion by SAXS is complicated by several factors which may lead to an underestimation, including most prominently i the sensitivity of R g to the fitting range used in the Guinier analysis, and ii the difficulty of determining R g at the lowest denaturant concentrations, where the largest changes in protein dimensions occur, in equilibrium ensemble-averaged techniques such as SAXS that are restricted to measurements sufficiently far above the unfolding midpoint.
The integrated experimental approach presented here, combined with Bayesian ensemble refinement, suggests a plausible resolution to a long-standing controversy. We thank Dr. Xiaobing Zuo ANL for their expert support. The Advanced Photon Source, a U. Description of experiment and simulation methods, data analysis and additional figures are given in the Supporting Information.
J Am Chem Soc. Author manuscript; available in PMC Sep Best , 2 and Benjamin Schuler 1. Madeleine B. Robert B. Author information Copyright and License information Disclaimer. To whom correspondence should be addressed: Alessandro Borgia, hc. Copyright notice. The publisher's final edited version of this article is available at J Am Chem Soc.
See other articles in PMC that cite the published article. Associated Data Supplementary Materials Supporting text, figures and tables. Abstract There has been a long-standing controversy regarding the effect of chemical denaturants on the dimensions of unfolded and intrinsically disordered proteins: A wide range of experimental techniques suggest that polypeptide chains expand with increasing denaturant concentration, but several studies using small-angle X-ray scattering SAXS reported no such increase of the radius of gyration R g.
TOC image. Introduction Understanding the properties of unfolded and disordered proteins is an important goal in biophysics. Open in a separate window. Figure 1. Figure 2. SAXS X-ray scattering from dilute, monodisperse protein solutions provides rich information on the distributions of interatomic distances within each molecule.
Figure 3. Figure 5. Figure 4. Bayesian Reweighting of Structure Ensembles One way to achieve all of these goals is to use an explicit molecular model that accounts for the expected conformational heterogeneity of unfolded proteins. Discussion Overall, the results from all methods we employed indicate an expansion of the polypeptide chain with increasing denaturant concentration and are mutually compatible. Figure 6. Conclusions Previously, qualitative discrepancies regarding the effect of chemical denaturants on the dimensions of unfolded and disordered proteins have been reported, when comparing the results from SAXS and other experimental methods, especially FRET.
Supplementary Material Supporting text, figures and tables Click here to view. Supporting Information Description of experiment and simulation methods, data analysis and additional figures are given in the Supporting Information.
References 1. Nature Structural Biology. Schuler B, Eaton WA. Curr Opin Struct Biol. Chem Rev. Schuler B, Hofmann H. J Chem Phys. FEBS Lett. Chem Phys Chem. Protein Sci. J Mol Biol. Nat Struct Biol. Sherman E, Haran G. Series ISSN : Edition Number : 1. Number of Pages : XIV, Skip to main content.
Search SpringerLink Search. Editors: view affiliations R. Comprehensive overview on the fast developing field of single molecule detection Written by experts in the field Includes supplementary material: sn. Buying options eBook EUR Softcover Book EUR Hardcover Book EUR Learn about institutional subscriptions. Table of contents 11 chapters Search within book Search. Front Matter Pages i-xiii. Nanophotonics and Single Molecules W. Moerner, P.
James Schuck, David P. Fromm, Anika Kinkhabwala, Samuel J. Lord, Stefanie Y. Nishimura et al. Pages Ewa Snaar-Jagalska, Herman P.
To the composition of the program's approval committee. The Israel Academy of Sciences and Humanities established a prestigious program in to bring outstanding post-docs from around the world to Israel's research universities. The Israel Young Academy led by Prof.
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