*4.5. Image Analysis*

The confocal images were analyzed using an in-house algorithm, written as a Java plugin for ImageJ (https://github.com/cecilefradin/BidBax\_Simulation\_and\_Analysis, accessed on 30 July 2021). This algorithm detects single particles in one channel, determines their size, shape and fluorescence intensity, then calculate for each one a single particle cross-correlation coefficient, which allows establishing whether it is bound to another particle in the second channel. The intensity distribution for a population of particles inform about the distribution of stoichiometries in that population, while the surface concentrations of interacting and non-interacting particles allow calculating two-dimensional dissociation constants.

#### 4.5.1. Single Particle Detection

Particle detection and identification was done as detailed in [29]. Images were first searched for local maxima. Each maximum with intensity above the threshold value *IT* = *IB* + 0.2*B* (with *B* the value of the molecular brightness measured for the solubleprotein by FCS on the same confocal instrument, and *IB* initially chosen as the average intensity in a region of 5 × 5 pixels centred around the pixel with lowest intensity in the image) was considered in turn, starting with the brightest one, and fitted with a two-dimensional Gaussian function:

$$H(\mathbf{x}, y) = I\_p \cdot \exp\left\{-\frac{2(\mathbf{x} - \vec{x}\_p)^2}{w\_{\mathbf{x}, p}^2}\right\} \cdot \exp\left\{-\frac{2(y - \vec{y}\_p)^2}{w\_{\mathbf{y}, p}^2}\right\} + I\_{\mathbf{B}, p}.\tag{4}$$

This fit returned the particle position *<sup>x</sup>*¯*p*, *y*¯*p* with sub-pixel precision, the particle fluorescence intensity (*Ip*), the 1/*e*<sup>2</sup> radii of the image of the particle along the *x* and *y* direction ( *wx*,*<sup>p</sup>* and *wy*,*<sup>p</sup>*) an estimate of the local background intensity (*IB*,*<sup>p</sup>*), and a normalized *χ*-squared value ( *χ <sup>N</sup>*,*<sup>p</sup>*). Once the fit was over, a square region of 0.5 μm × 0.5 μm around the particle was erased from the image and the next most intense remaining local maximum was considered, using an updated threshold value (with *IB* now calculated as the average of all the local background intensities estimated from previous fits). Only particles for which the fit was judged acceptable ( *χ N*,*p* < 2 , *Ip* + *IB*,*<sup>p</sup>* > *IT* + √*IT*) were selected for further analysis.

#### 4.5.2. Single Particle Cross-Correlation Coefficient

To determine whether a particle detected in a given channel interacts with another particle in the other channel, a single particle cross-correlation coefficient was calculated for that particle. First, an area of size 7 × 7 pixel was delimited around the detected particle. Then the intensities *Ich*1 and *Ich*2 recorded at each pixel within that area for either channels were cross-correlated via:

$$\chi = \frac{\langle (I\_{ch1} - \langle I\_{ch1} \rangle) \cdot (I\_{ch2} - \langle I\_{ch2} \rangle) \rangle}{\sqrt{\sigma\_{ch1}^2 - \langle I\_{ch1} \rangle} \sqrt{\sigma\_{ch2}^2 - \langle I\_{ch2} \rangle}} \tag{5}$$

with

$$
\sigma\_{ch(k)}^2 = \left\langle (I\_{ch(k)} - \left\langle I\_{ch(k)} \right\rangle)^2 \right\rangle,\tag{6}
$$

and where averages are taken over the box drawn around the particle. Equation (5) includes a correction for the photon noise, via subtraction of the variance of the Poisson distributed photon noise ( *Ich*(*k*) ) from the total variance ( *<sup>σ</sup>*2*ch*(*k*)), leaving only the contribution due to the average signal of the particle, as discussed in Friaa and Fradin [33].

4.5.3. Dissociation Constant

A value of the two-dimensional dissociation constant, 2D-*KD*, was obtained by comparing surface concentrations of uncorrelated particles with the channel-averaged surface concentration of the correlated particles. It can be written as:

$$2D - K\_D = \frac{\mathcal{C}\_{\text{Bax}} \cdot \mathcal{C}\_{\text{tBid}}}{\mathcal{C}\_{\text{Bax} \cdot \text{tBid}}},\tag{7}$$

where *c*Bax and *c*tBid are the respective sums of particles per area with *χ* < 0.3 (no correlated signal in the other channel) and *c*tBid-Bax is the sum of particles per area with *χ* > 0.3 (correlation of the signals in both channels) corrected for the expected number of incidental correlation predicted by the simulations.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/1422-006 7/22/15/8240/s1.

**Author Contributions:** Conceptualization, C.F.; methodology, M.R., M.K., M.M. and C.F.; software, M.R., M.M., S.W. and J.M.M.-M.; formal analysis, M.R. and C.F.; investigation, M.R., M.K. and M.M.; writing—original draft preparation, M.R.; writing—review and editing, C.F.; visualization, M.R.; funding acquisition, C.F. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by grants FRN86657 from the Canadian Institutes of Health Research (CIHR) and RGPIN-2015-06362 from the Natural Sciences and Engineering Research Council of Canada (NSERC) to C.F.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** We thank D.W. Andrews and B. Leber for their help and guidance with this project.

**Conflicts of Interest:** The authors declare no conflict of interest.
