# Internalization of Phospholipid-Coated Gold Nanoparticles
## 2 Materials and Methods
### Principles of Calculations of Image Correlation Functions  
The theoretical foundation for calculating correlation functions from images was established with the introduction of image correlation spectroscopy by [PERSON] and co-workers [57]. The auto-correlation function can be calculated directly as the sum of the products of all the pair-wise intensities \(\mathrm{i}(x,y)\) and \(\mathrm{i}(x+\xi,y+\eta)\) as indicated by the following equation:  
\[\mathrm{g}(\xi,\eta)=\frac{\big{\langle}(\mathrm{i}(x,y)-\big{\langle}\mathrm{ i}(x,y)\big{\rangle})\big{(}\mathrm{i}(x+\xi,y+\eta)-\big{\langle}\mathrm{i}(x,y) \big{\rangle}\big{\rangle}\big{\rangle}}{\big{\langle}\mathrm{i}(x,y)\big{ \rangle}^{2}} \tag{1}\]where the angular brackets indicate averaging over all spatial coordinates, x and y. For images of any size, this becomes computationally prohibitive, so the alternative is to recognize that the auto-correlation function is the reverse Fourier transform (FFT\({}^{-1}\)) of the power spectrum of the intensities in the image and that the power spectrum can be calculated as the Fourier transform [FFT] of the image multiplied by its complex conjugate [57]. Thus, the following applies:  
\[\overline{i}(\xi^{\prime},\eta^{\prime})=\mathrm{FFT}[i(x,y)] \tag{2}\]  
and  
\[g(\xi,\eta)=\frac{\mathrm{FFT}^{-1}\big{[}\overline{i}(\xi^{\prime},\eta^{ \prime})\overline{i}^{*}(\xi^{\prime},\eta^{\prime})\big{]}}{\mathrm{i}((x,y) )^{2}} \tag{3}\]  
where the asterisk denotes the complex conjugate of the Fourier transform. Correspondingly, the cross-correlation function between two images, say red, r, and green, g, is calculated as the reverse Fourier transform of the joint power spectrum [57], that is:  
\[\overline{i}_{\mathrm{r}}(\xi^{\prime},\eta^{\prime})=\mathrm{FFT}[i_{ \mathrm{r}}(x,y)] \tag{4}\]  
and  
\[g_{\mathrm{r},g}(\xi,\eta)=\frac{\mathrm{FFT}^{-1}\big{[}\overline{i}_{ \mathrm{r}}(\xi^{\prime},\eta^{\prime})\overline{i}_{\mathrm{g}}^{*}(\xi^{ \prime},\eta^{\prime})\big{]}}{\langle i_{\mathrm{r}}(x,y)\rangle\big{\langle} i_{\mathrm{g}}(x,y)\rangle} \tag{5}\]  
where the asterisk denotes the complex conjugate of the Fourier transform. Importantly, it does not matter which image is used for the complex conjugate. Once the auto- or cross-correlation functions have been calculated, they are fit to a two-dimensional Gaussian function to extract the amplitude of the correlation function, g(0,0). Thus,  
\[g(\xi,\eta)=g(0,0)\,\exp\left(\frac{-2(\xi^{2}+\eta^{2})}{w^{2}}\right)+g_{0} \tag{6}\]  
where g\({}_{0}\) is a factor that allows for incomplete decay of the correlation function and w is the width of the Gaussian function. Note that the use of a Gaussian function is determined by the use of a laser beam whose cross-sectional profile is Gaussian, thus w is the width of the laser beam for the auto-correlation function and the geometric mean of the two laser beams for the cross-correlation function. In this study, the image correlation toolkit in ImageJ software was utilized and the following pieces of information from images were obtained: the normalized auto- and cross-correlation function amplitudes, the average image intensity, and the laser beam radius in microns. Regions of interest were selected to \(256\times 256\) of the \(512\times 512\) image to avoid dark regions of the image where there is no cell. Cross correlation of images from two channels from two _different_ cells was used to determine the cross-correlation amplitude that can arise from uncorrelated fluctuations. This represents a lower limit for the significance of cross correlation amplitude arising from two channels from the _same_ cell. The first application of triple cross correlations to measure three-way interactions in solution was provided by [PERSON] and co-workers [58; 59], who studied ribosomal assembly using components labeled with three distinct chromophores. The extension of this to study three-way interactions using three images has been developed theoretically by [PERSON] and [PERSON] [unpublished].