Affinity of Skp to OmpC revealed by single-molecule detection

Outer membrane proteins (OMPs) are essential to gram-negative bacteria, and molecular chaperones prevent the OMPs from aggregation in the periplasm during the OMPs biogenesis. Skp is one of the molecular chaperones for this purpose. Here, we combined single-molecule fluorescence resonance energy transfer and fluorescence correlation spectroscopy to study the affinity and stoichiometric ratio of Skp in its binding with OmpC at the single-molecule level. The half concentration of the Skp self-trimerization (C1/2) was measured to be (2.5 ± 0.7) × 102 nM. Under an Skp concentration far below the C1/2, OmpC could recruit Skp monomers to form OmpC·Skp3. The affinity to form the OmpC·Skp3 complex was determined to be (5.5 ± 0.4) × 102 pM with a Hill coefficient of 1.6 ± 0.2. Under the micromolar concentrations of Skp, the formation of OmpC·(Skp3)2 was confirmed, and the dissociation constant of OmpC·(Skp3)2 was determined to be 1.2 ± 0.4 μM. The precise information will help us to quantitatively depict the role of Skp in the biogenesis of OMPs.


Supplementary note 1: Data processing in pscFCS
First, the high-resolution (0.96 μs-bintime) fluorescence traces of the samples were recorded.
The traces were binned to 1.008 ms bintime to generate the normal histogram of smFRET efficiency. Next, we added the original high-resolution fluorescence traces of the donor channel and the acceptor channel together, and selected the whole fluorescence bursts that belong to a specific subpopulation in the smFRET histogram. The rest in the original highresolution fluorescence traces was replaced by the Poisson noises. Then the FCS curves were calculated by using the synthesized high-resolution traces and fitted to yield the apparent diffusion time of the specific subpopulation.
The fluorescence fluctuation due to the singlet-triplet transition or other relaxations may have an impact on extracting the diffusion time. To evaluate the effect due to the FRET process, we simulated 20 sets of two-component Brownian motion with an accompanying FRET process of different relaxation times. The cross correlation was not used since the cross correlation between the donor and acceptor has obvious anti-correlation components from the FRET process, which strongly interfere with the diffusion time. The autocorrelations of acceptor traces only, donor traces only, and donor + acceptor traces together were compared. The FCS curves were fitted by either a model of 2-dimentional diffusion with 1 relaxation term (2D1R), 3 or a model of 3-dimentional diffusion with 1 relaxation term (3D1R), where  app is the apparent diffusion time, R is the relaxation time, 0 is the inverse of the number of fluorescent molecules in the laser focus, is the amplitude for the relaxation and is the ratio of the beam waist in the direction over the beam waist in the plane.
From the simulation we found that in the autocorrelation of acceptor traces or donor traces, the apparent diffusion time decreased when the relaxation exists, and the apparent diffusion time depended on specific relaxation dynamics of molecules. However, the autocorrelation of the donor + acceptor traces together was not influenced by the existence of the FRET relaxation ( Supplementary Fig. S1). We also found that the 2D1R fitting yielded systematically smaller diffusion time than the theoretical values, whereas the diffusion time of the 3D1R fitting matched the theoretical values well. However, since we used the ratio on the diffusion time to derive the stoichiometric ratio of complexes, the difference in that ratio between the 2D1R and 3D1R models was not significant ( Supplementary Fig. S2).
In the calculation of the smFRET histogram, a threshold on the fluorescence counts (named as the peak threshold) was taken to identify the qualified fluorescence bursts. Namely, the fluorescence bursts with a maximum photon counts below the peak threshold were where 0 is the laser intensity at the center, xy is the beam waist in the plane and z is the beam waist in the direction. Thus, an empirical polynomial equation was used to get the unbiased diffusion time, where is the peak threshold, is the unbiased diffusion time and is a fitting parameter.
Our simulation demonstrated that the first-order derivative of  app at the origin was zero.
Therefore, we did not include the linear term in the equation.
We exemplified Cy3B to examine which model is better to reduce the effect of the singlettriplet transition ( Supplementary Fig. S3). The transition usually occurs on the s timescale, so we compared the results of the 10 μs-started FCS curves and 1 μs-started FCS curves by using the 2-dimensional diffusion with (2D1R model, equation (1)) or without (2D model, equation (5)) a relaxation term.
For the 10 μs-started FCS curves, the singlet-triplet transition was mostly dropped out, so that the 2D model was suitable to fit ( Supplementary Fig. S4) and yielded the apparent diffusion 5 time of 151±2 μs as shown in Fig. 1d in the main text. For the 1 μs-started FCS curves, the singlet-triplet transition was obvious, and the 2D model could not fit the curves well ( Supplementary Fig. S5). The extrapolated unbiased diffusion time ( Supplementary Fig. S6) was 143±2 μs. For the 2D1R model, although the 1 μs-started FCS curves could be well fitted ( Supplementary Fig. S7), the apparent diffusion time was irregular due to large fitting errors ( Supplementary Fig. S8). The standard diffusion time of Cy3B measured by the conventional FCS was 162±8 μs ( Supplementary Fig. S9). Therefore, we chose to use the 2D model to fit the 10 μs-started pscFCS curves of the synthesized traces in our experiments.

Supplementary note 2: Derivation of dissociation constants of Skp self-trimerization
The Skp self-trimerization is described by where is the dissociation constant. When the dye-labelled Skp is mixed with far excessive wild-type Skp, the trimerization reaction with Skp * as a component becomes where Skp * is the dye-labelled Skp, Skp 3 * is the Skp trimer which contains one dye-labelled Skp monomer, and * is the respective dissociation constant. Because the wild-type Skp is 6 able to form trimer among themselves, reaction (8) and reaction (6) where nt=3 and nm=1 by definition. Substituting equation (9) into equation (10) we have where [Skp]0 is the total concentration of the Skp. Combining equations (7) and (12), we derive that which has a single real root as 7 where Δ is The as a function of [Skp]0 can be fitted by substituting equation (14) into equation (11) to obtain both the dissociation constants * and . The proportion of Skp in the trimer form Substituting equation (12) and equation (16) into equation (13), the relation between [Skp]0 and y is derived to be In the measurement of the equilibrium constant of the self-trimerization between Skp and Skp3, we labelled fluorescent dye Cy3B to mutant Skp D128C. Skp D128C-Cy3B of 0.4 nM was mixed with the wild-type Skp ranging from 0 to 10 3 nM in buffer C (50 mM PB, 100 mM NaCl, pH 7.0). The FCS curve was recorded on a home-built confocal microscope and the data were fitted by the 2D1R model ( Supplementary Fig. S10). The titration curve of n as a function of [Skp]0 was plotted and fitted by using above equations ( Supplementary Fig. S11).
This titration curve showed that the Skp concentration at which half Skp * was monomer was 37 nM, and the derived dissociation constant for Reaction (8) was * =(1.2±0.1)×10 3 nM 2 .
It is interesting to observe that different values of K and * were obtained. However, the deviation is merely superficial caused by whether or not identical molecules are involved, and it does not suggest that the physical properties of labeled and unlabeled molecules are different. To clearly demonstrate this point, let us analyze a dimerization reaction by a simple model.

Consider a dimerization reaction
We assert that the labeling does not alter any physical properties. So, the dissociation rate constants of reactions (18) and (19)  In the case of trimerization, it is hard to assess the relationship between Reactions (6) and (8) with a simple model, because the results will be dependent on specific reaction mechanisms.
However, the same logic will lead to the same conclusions that K and K * are different.

Skp complexes
For the equilibrium For the equilibrium The fitting of the smFRET peak areas positioned at 0.3 in Supplementary Fig. S13 was carried out to obtain f. Then, the normalized f as a function of [Skp]0 was plotted and fitted to obtain D ′ (Fig. 4d in the main text).

Supplementary Figures
Supplementary Figure S1 The 2D1R fit (with the smFRET efficiencies ranging from -0.1 to 0.1) at different peak thresholds.
The black lines and red lines represent the experimental data and fitted curves, respectively.
Supplementary Figure S8 The plot of the apparent diffusion time extracted from the 2D1R fit of the 1 μs-started FCS curves of Cy3B against the peak threshold. The data were irregular and could not be described by equation (4) (11), (14) and (15).  Supplementary Table S2.