Preparation of aqueous solutions of Fos-10 micelles doped with 10-bromodecan-1-ol

1μl of 10-bromodecan-1-ol was added to 60 μl of a 1.1 M Fos-10 stock solution in buffer A (5 mM sodium phosphate at pH 6.8, 10 mM NaCl, 90% H_{2}O/10% D_{2}O, 0.3 mM NaN_{3}). This mixture was repeatedly stirred and heated to 50 °C until a clear solution was obtained. To prepare samples with Fos-10 concentrations in the range between 0.1 and 200 mM, the stock solution was diluted with buffer A.

Reconstitution of OmpX in Fos-10 micelles

[^{2}H,^{15}N]-labeled OmpX was expressed as inclusion bodies in *Escherichia coli* and purified to yield a solution of the unfolded protein in 6 M urea. NMR samples were obtained using a previously established microscale protocol that yields 1 mM solutions of [^{2}H,^{15}N]-labeled OmpX reconstituted in Fos-10 micelles in buffer A.^{4}

NMR Spectroscopy

All NMR experiments were recorded at 25 °C on a Bruker DRX-700 spectrometer equipped with a 1.7 mm TXI microprobe (Bruker, Billerica, MA). One-dimensional (1D) ^{1}H NMR spectra were collected with the following parameters: data size = 16 K complex points, acquisition time = 1.38 s; number of scans = 128; sweep width = 11900 Hz. All NMR data were processed with the software TOPSPIN 1.3 (Bruker).

Translational diffusion coefficients for Fos-10 micelles at variable solution conditions were measured using the ^{1}H PFG-STE NMR experiment.^{5,6} For each dataset a series of 16 diffusion-weighted 1D ^{1}H PFG-STE spectra were recorded in a two-dimensional manner, using a pair of gradient pulses of duration δ = 4.5 ms that were separated by a delay of Δ = 50 ms, with gradient strengths, G_{D}, ranging from 3 to 55 Gcm^{–1}. Translational diffusion coefficients, *D*_{t}, for particles in solution were determined using the expression (1) for the signal attenuation, Ψ_{Q}, in PFG-STE NMR experiments,^{7}

with *Q* = (*γs*G_{D}*δ*)^{2}, where γ is the proton gyromagnetic ratio and *s* describes the shape of the diffusion gradient, and *T*_{diff} = Δ–δ / 3. All NMR spectra were recorded at 25 °C. G_{D} was calibrated with the residual ^{1}H signal in 99.9 % D_{2}O by use of a self-diffusion coefficient for HDO at 25 °C of (1.902 ± 0.002) × 10^{−9} m^{2}s^{−1}.^{8}

Analysis of the PFG-STE NMR experiments with Fos-10 solutions

Assuming fast exchange of detergent monomers between the pools of Fos-10 monomers (m) and Fos-10 micelles (em) present at detergent concentrations above about 2 × c.m.c, i.e.,

*τ*_{em}/

*T*_{diff} 1, where

*τ*_{em} is the mean lifetime of Fos-10 monomers in the micelles, we derived the expression (2) to describe the weighted average represented by the apparent diffusion constant,

, measured by observation of Fos-10 NMR signals.

^{9,10}where [Fos-10] is the total Fos-10 concentration in mM units with respect to monomers, c.m.c. the critical micelle concentration in mM, and Θ(x) the Heaviside step function with Θ(x) = 0 for x < 0 and Θ(x) = 1 for x ≥ 0.

Due to obstruction caused by the increasing density of solute particles in the solution,

*D*_{t,m} and

*D*_{t,em} are concentration-dependent, as given by

Eqs.(3) and

(4),

^{11-13}and

and

are the diffusion coefficients at infinite dilution of Fos-10 monomers and Fos-10 micelles, respectively,

is the volume fraction of the micelles, and

*k*_{em} is the slope of plots of

*D*_{t} versus the detergent concentration.

^{12,14} is given by

Eq. (5),

^{14} where *V*_{m} = 0.494 nm^{3} is the volume of a Fos-10 monomer^{15} and *N*_{A} is the Avogadro constant.

For spherical non-interacting particles, the Stokes–Einstein relationship describes the dependence of

on the solution viscosity,

*η,* and on the hydrodynamic radius of the particle of interest,

*R*_{h},

where *k*_{B} is the Boltzmann constant and *T* the absolute temperature. To eliminate the dependence on *η*, we determined the “relative diffusivity” for Fos-10 micelles, *d*_{em}, as^{3,16,17}

where

is a reference diffusion coefficient for a particle of known size at infinite dilution. The aggregation number,

*N*_{a,em}, for Fos-10 micelles can then be determined as

where *V*_{ref} is the volume of the reference compound.