5.4. ENSEMBLE

ENSEMBLE 2.1 [31] was used to determine a subset of conformations from an initial pool of conformers created by the statistical coil generator TraDES[55, 56]. All modules were given equal rank, and all other ENSEMBLE parameters were left at their default values.

To achieve a balance between the concerns of over-fitting (under-restraining) and under-fitting (over-restraining) we performed multiple independent ENSEMBLE calculations with 100 conformers, N_conf = 100, as suggested by Ref [58], and averaged the results from independent ensemble calculation or combined them to form ensembles with larger numbers of conformers (e.g., N_conf = 500). To address the possibility that changing the ensemble size could affect the structural properties of the ensemble, or its agreement with experimental observables, we re-performed the Sic1 SAXS+PRE ensemble calculations, but varied the ensemble size, N_conf (details in the Supporting Information). The determination of polymer properties and the agreement with experimental observables is robust in a range of N_conf from ca. 50–100. Below N_conf ≈ 50, agreement with restraining data (SAXS and PRE) is worsened, and the ensembles do not agree with validating data (smFRET and CSs). Above N_conf ≈ 150, ensembles are in agreement with the experimental observables, though increased ensemble-to-ensemble variation suggests that 5 replicates (independently calculated ensembles with same set of restraints) is insufficient to ensure convergence. Larger ensembles are calculated quicker (> 72 hours for N_conf = 20 vs ca. 1 hour for N_conf = 100). Ensembles with 100 conformers were chosen to minimize the computational cost per ensemble calculation, and ensemble-to-ensemble variation.

NMR data were obtained from BMRB accession numbers 16657 (Sic1) and 16659 (pSic1)[29]. A total of 413 PRE restraints were used with a typical conservative upper- and lower-bound on PRE distance restraints of ±5 Å[57, 82]. This tolerance was used in computing the χ² metric for the PRE data. CSs were back-calculated using the SHIFTX calculator[51] and a total of 90 C_α CSs and 85 C_β CSs were used. The CS χ² metric was computed using the experimental uncertainty σ_exp and the uncertainty in the SHIFTX calculator (σ_SHIFTX = 0.98 ppm for C_α CSs and σ_SHIFTX = 1.10 ppm for C_β CSs[51]). CRYSOL[33] with default solvation parameters was used to predict the solution scattering from individual structures from their atomic coordinates. A total of 235 data points from q = 0.02 to q = 0.254 Å⁻¹ were used in SAXS-restrained ensembles. The SAXS χ² metric was computed using the experimental uncertainty in each data point.

Accessible volume (AV) simulations[34, 35] were used to predict the sterically accessible space of the dye attached to each conformation via its flexible linker (Figure 1D). These calculations were performed using the AvTraj[34] v0.0.9 and MDTraj[83] v1.9.3 packages in Python 3.7.6. In the quasi-static approximation, the inter-dye distance dynamics within the AVs for a particular conformation are quasi-static on the timescale of the donor excited state (τ_DA ≤ τ_D0 = 3.7 ns). The per-conformer mean FRET efficiency is therefore e=∫E(rDA)P(rDA)drDA, where P(r_DA) is the distribution of inter-dye distances resulting from the AV simulation for a particular conformation, and E(rDA)=(1+(rDA/R0)6)−1. End-to-end distance reconfiguration times for IDPs and unfolded proteins are typically in the range 50–150 ns [84], and so the end-to-end distance is also quasi-static on the timescale of τ_DA. The back-calculated ensemble-averaged ⟨E⟩_ens is calculated as the linear average of the per-conformer FRET efficiencies ⟨E⟩_ens = ⟨e⟩. The quasi-static approximation gives the same ⟨E⟩_ens within error as a more computationally demanding method which considers Monte-Carlo simulations of the photon emission process and Brownian motion simulations of dye translation diffusion within the accessible volume (detail in the Supporting Information). Further support for the quasi-static averaging approach used, comes from multiparameter E vs τ_DA histograms (Figure S2) which provide complementary information of inter-dye distances and dynamics, but with different experimental integration times.

The uncertainty in ⟨E⟩_ens, σ_E,ens, is ca. 0.01, which is a combination of SEM and uncertainty in R₀. Differences |〈E〉exp−〈E〉ens|≤σE,exp2+σE,ens2≈0.02 indicate no disagreement between back-calculated and experimental mean transfer efficiencies. A comprehensive description of the ENSEMBLE calculations, restraints and back-calculations can be found in the Supporting Information.