Conformational Heterogeneity and FRET Data Interpretation for Dimensions of Unfolded Proteins

Abstract

A mathematico-physically valid formulation is required to infer properties of disordered protein conformations from single-molecule Förster 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 prejudgment, a single value of average transfer efficiency 〈E〉 between dyes at two chain ends is generally consistent with highly diverse, multiple values of the average radius of gyration 〈R_g〉. 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 〈R_g〉 of unfolded proteins, focusing on Protein L as an example. Conventional smFRET inference concluded that 〈R_g〉 of unfolded Protein L is highly sensitive to [GuHCl], but data from SAXS suggested a near-constant 〈R_g〉 irrespective of [GuHCl]. Strikingly, our analysis indicates that although the reported 〈E〉 values for Protein L at [GuHCl] = 1 and 7 M are very different at 0.75 and 0.45, respectively, the Bayesian R_g² distributions consistent with these two 〈E〉 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.

Figure 1

Unfolded-state dimensions of Protein L obtained from SAXS and various interpretations of smFRET experiments. Open and solid squares are results from previous time-resolved and equilibrium SAXS experiments by Plaxco et al. (46) at 2.7 ± 0.5° and 5 ± 1°C, respectively. The associated error bars represent 1 SD fitting uncertainties (kinetic data) or confidence intervals from two to three independent measurements (thermodynamic data). Subsequent equilibrium SAXS measurement at 22°C by Yoo et al. (43) produced essentially identical results. Open and solid diamonds are results from smFRET experiments, respectively, by Merchant et al. (17) (Eaton group, temperature not provided) and by Sherman and Haran conducted at “room temperature” (16). These prior experimental data were compared in a similar manner in (43). Here, the open and solid circles are from our analysis corresponding, respectively, to the most-probable R_g⁰ (21) and the root-mean-square $\sqrt{\langle R_g^2\rangle }$ value based on the experimental transfer efficiency 〈E〉 = 0.74 for [GuHCl] = 0 given by Merchant et al. (17), the 〈E〉 values for Protein L (corrected from the measured FRET efficiency 〈E_m〉) in SI Table 2 for the same reference, and the 〈E〉 values for [GuHCl] = 1 and 7 M in Sherman and Haran (16). A Förster radius of R₀ = 55 Å was used in our calculations. The error bars for the solid circles span ranges delimited by $\sqrt{\langle R_g^2\rangle \pm \sigma \left(R_g^2\right)}$, where σ(R_g²) is the SD of the distribution of R_g² at the given E value. The horizontal dashed line marks the R_g = 25.3 Å value we obtained from applying the scaling relation of Kohn et al. (71) to N = 74, where n = N + 1 = 75 is taken to be the equivalent number of amino acid residues for Protein L plus dye linkers. To see this figure in color, go online.

Figure 2

Large variations in dimensions among conformations with a given end-to-end distance R_EE. (a) Root-mean-square $\sqrt{\langle R_g^2\rangle }$ and (b) the square root of the SD of R_g² as functions of R_EE. The gray profile in (a) shows the theoretical transfer efficiency Eq. 1 for n = 75 and R₀ = 55 Å in a vertical scale ranging from zero to unity. (c–f) Example conformations with the darker shaded (red and blue online) beads marking the termini of n = 75 chains. They serve to illustrate the possible concomitant occurrences of (c) small R_EE = 19.7 Å and large R_g = 26.3 Å; (d) large R_EE = 80.1 Å and large R_g = 26.2 Å; (e) small R_EE = 19.7 Å and small R_g = 14.2 Å; and (f) large R_EE = 80.4 Å and small R_g =19.8 Å. These examples underscore that there is no general one-to-one mapping from 〈R_EE〉 to 〈R_g〉. To see this figure in color, go online.

Figure 3

Perimeters of inference on conformational dimensions from Förster transfer efficiency. (a) Distribution P(R_g, E) of conformational population as a function of R_g and E for n = 75 and R₀ = 55 Å. The distribution was computed using R_EE × R_g bins of 1.0 Å × 0.5 Å. White area indicate bins with no sampled population. (b) Shown here is the most-probable radius of gyration R_g⁰(〈E〉) from our previous subensemble SAW analysis (21) (black solid curve) compared against root-mean-square radius of gyration $\sqrt{\langle R_g^2\rangle \left(E\right)}$ (red solid curve) computed by considering 30 subensembles with narrow ranges of R_EE. The latter overlaps almost completely with 〈R_g〉(E) computed using the same set of subensembles (blue solid curve). Another set of R_g⁰ (〈E〉) values (black dotted curve) and another set of 〈R_g〉(E) values (blue dashed curve) were obtained from the distribution in (a), respectively, by averaging over E at given R_g values and by averaging over R_g at given E values. Variation of radius of gyration is illustrated by the red dashed curves for $\sqrt{\langle R_g^2\rangle \pm \sigma \left(R_g^2\right)}$ as functions of E. The essential coincidence between the black solid and dotted curves and between the blue solid and dashed curves indicate that these results are robust with respect to the choices of bin size we have made. Note that the black solid curve for R_g⁰(〈E〉) does not cover 〈E〉 values close to zero or close to unity because larger R_g bin sizes (∼1.1–3.6 Å) than the current R_g bin size of 0.5 Å were used (Table S5 of (21)), thus precluding extreme values of 〈E〉 to be considered in that previous n = 75 subensemble SAW analysis (21). This limitation is now rectified for n = 75 (black dotted curve).

Figure 4

Substantially overlapping distributions of conformational dimensions can be consistent with very different Förster transfer efficiencies. (a) Given here are hypothetical distributions P(R_EE) of end-to-end distance R_EE. Two hypothetical sharp distributions at two R_EE values (vertical bars) and two hypothetical broad Gaussian distributions (bell curves) are centered at these two R_EE values, with SD of the Gaussian distributions chosen to be 20.3 Å. (b) Given here is the corresponding distribution P(E) of Förster transfer efficiency E. The left and right sharp distributions of P(R_EE) in (a) lead, respectively, to E ≈ 0.745 (right) and E ≈ 0.447 (left) in (b). The corresponding P(E) for the hypothetical Gaussian distributions in (a) entail broad distributions in E in (b) with mean values at 〈E〉 = 0.735 (right) and 〈E〉 = 0.453 (left), respectively. (c) The left and right curves are the conditional distributions P(R_g²|E), respectively, for the sharply defined E ≈ 0.745 and E ≈ 0.447 in (b). (d) Similar to (c) except the distributions of R_g² are now for the two broad P(E) distributions in (b). We denote these distributions as P(R_g²|〈E〉). The R_g² bin size in (c and d) is 1.0 Å². The OVL of the two normalized distribution curves in (c and d) are, respectively, 0.747 and 0.754. The percentages of population with R_g² ≥ 625 Å in the distributions in (c and d) are, respectively, 9.2 and 10.1% for E ≈ 0.745 and 〈E〉 = 0.735, and 25.2 and 26.3% for E ≈ 0.447 and 〈E〉 = 0.453. To see this figure in color, go online.

Figure 5

Ambiguities in FRET inference of conformational dimensions. The heat map provides for n = 75 and R₀ = 55 Å the overlapping coefficient $\text{OVL}{\left(R_g^2\right)}_{E_1,E_2}$ of pairs of R_g² distributions conditioned upon FRET efficiencies E₁ and E₂. Contours on the heat map are for $\text{OVL}{\left(R_g^2\right)}_{E_1,E_2}$ = 0.8, 0.6, 0.4, and 0.2, as indicated by the scale on the right. To see this figure in color, go online.

Figure 6

Most probable and root-mean-square radius of gyration. Shown here is generalization of the R_g⁰(〈E〉) (solid black curve traversing across the shaded region in each panel at a steeper incline), and $\sqrt{\langle R_g^2\rangle \left(E\right)}$ (curve at a milder incline in the middle of the shaded region in each panel) for R₀ = 55 Å and n = 75 in Fig. 3 to other Förster radii R₀ and chain lengths n. The shaded areas are bound by $\sqrt{\langle R_g^2\rangle \left(E\right)\pm \sigma \left(R_g^2\right)\left(E\right)}$, which were represented by red dashed curves in Fig. 3. As discussed in the text, the $\sqrt{\langle R_g^2\rangle \left(E\right)}$ curves computed here for sharply defined E values are expected to apply also to $\sqrt{\langle R_g^2\rangle \left(\langle E\rangle \right)}$ for essentially symmetric distributions of E where 〈E〉 denotes the mean value of E in such distributions. As pointed out above for Fig. 3, the black R_g⁰(〈E〉) curves shown here do not cover 〈E〉 values close to zero or unity because of the relatively large R_g bin sizes used previously (21). To see this figure in color, go online.

Figure 7

A hypothetical resolution of the Protein L smFRET-SAXS puzzle. The two distributions depicted by the black and red curves are from Fig. 4d, for 〈E〉 = 0.74 and 〈E〉 = 0.45, respectively. For R_g² ≥ 625 Å, the area shaded in pink is under the 〈E〉 = 0.45 (red) distribution but above the $\langle E\rangle =0.74$ (black) distribution, whereas the area shaded in gray is under the 〈E〉 = 0.74 (black) distribution. The R_g² < 625 Å areas that are in lighter shades are mirror reflections of the corresponding R_g² ≥ 625 Å areas with respect to R_g² = 625 Å. The sum total of the pink-plus-gray area (∼50% of P(R_g²|〈E〉 = 0.45)) represents a hypothetical ensemble with 〈E〉 ≈ 0.45 and $\sqrt{R_g^2}\approx 25$ Å, whereas the gray area (∼20% of P(R_g²|〈E〉 = 0.74)) represents a hypothetical ensemble with 〈E〉 ≈ 0.74 but nonetheless the same $\sqrt{R_g^2}\approx 25$ Å. Shown on the right are example conformations in these restricted ensembles, as marked by the arrows. Both conformations have R_g² = 700 Å² (R_g = 26.5 Å), but their different R_EE values entail different E values of $\approx 0.45$ (top) and $\approx 0.74$ (bottom). See text and Fig. 4 for further details.