Probing the Spatial Organization of Molecular Complexes Using Triple-Pair-Correlation

Super-resolution microscopy coupled with multiplexing techniques can resolve specific spatial arrangements of different components within molecular complexes. However, reliable quantification and analysis of such specific organization is extremely problematic because it is frequently obstructed by random co-localization incidents between crowded molecular species and the intrinsic heterogeneity of molecular complexes. To address this, we present a Triple-Pair-Correlation (TPC) analysis approach for unbiased interpretation of the spatial organization of molecular assemblies in crowded three-color super-resolution (SR) images. We validate this approach using simulated data, as well as SR images of DNA replication foci in human cells. This demonstrates the applicability of TPC in deciphering the specific spatial organization of molecular complexes hidden in dense multi-color super-resolution images.

them blinks stochastically CH1 times on average and registers one detection for each blinking event. The probability density function of having one blinking event detected at location is governed by the average localization uncertainty of the fluorophores in Channel 1 ( CH1 ) and written as: Accordingly, the average density of detections at location in Channel 1 is given by: where is the area of interested, and 〈 CH1 〉 = CH1 CH1 ⁄ .

Supplementary Note 2
Triple-Correlation function calculates the average probability for simultaneously observing three molecules, each in a different channel, as a function of their relative displacement, given by: where CH (R) is the surface density of detections at position R = | , ⟩ in channel . 〈 〉 denotes averaging operation over R. The Fourier Transform of the Triple-Correlation, known as its bispectrum is given by: ̂( k 1 ,k 2 ) =̂C H1 (k 1 +k 2 )̂C H2 (k 1 )̂C H3 (k 2 ) where ̂( k 1 ,k 2 ) and ̂C H (k) are the Fourier Transforms of (r 1 ,r 2 ) and CH (r), respectively, with the corresponding spatial frequency of . (r 1 ,r 2 ) is calculated via an Inverse Fourier Transform from ̂( k 1 ,k 2 ), which involves a 4D FFT operation with each dimension of ~ 800 elements, and goes far beyond the computing capability of most current cluster computers. To address the insufficient RAM problem and conduct the computation on a desktop computer we implemented the following indirect procedure: According to the definition of the Triple-Correlation per Equation ( 1 ), at each fixed , this can be written as: with CH12 (R|r 1 ) = CH1 (R) CH2 (R + r ) being a function of a single variable R. The Fourier Transform of Equation ( 2 ) can therefore be derived as: ̂( k 2 |r 1 ) =̂C H3 (k 2 )̂C H1 (k 2 |r 1 ) and a 2D Triple-Correlation map at a fixed r 1 (Equation ( 2 )) can be subsequently calculated by inverse Fourier Transform of Equation ( 3 ). We then carried a 2D scan of r 1 , yielding the 4D map of the Triple-Correlation ( 1 , 1 , 2 , 2 = 1 + ∆ ) (Main Text), which was then integrated along 1 to generate the final 3D ( 1 , 2 , ∆ ) cube.
We note that during scanning of r 1 , CH2 (R + r ) can co-localize with CH3 (R) at a specific r 1 , and consequentially result in a co-localization between the product CH12 (R|r 1 ) and CH3 (R). We also note that, at this r 1 , value, CH12 (R|r 1 ) = CH1 (R) CH2 (R + r ) stays non-zero, especially in a highly dense image, due to the random co-localization between CH1 (R) and CH2 (R + r ). Therefore, the specific co-localization between the non-zero CH12 (R|r 1 ) and CH3 (R) can result in a residual correlation signal at r 2 = 0 at this specific r 1 , such as the correlation at r M-Y = 10, and r M-C = 0 in Figure   2d(iii) in the main text. However, since in this case the CH12 (R|r 1 ) represents random co-localizations between CH1 (R) and CH2 (R + r ), this residual signal is insignificant compared to the true correlation (e.g. the correlation at r M-Y = 10, r M-C = 20 in Figure 2d(iii)), and can be easily excluded by the r 2 − ∆ correlation map (e.g. Figure 2d(ii)).

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To improve the accessibility of the Triple Correlation cube, we further integrate ( 1 , 2 , ∆ ) along each dimension to make it a combination of three 2D correlation maps (Fig. 1c). Specifically, we transformed 1 -∆ and 2 -∆ maps from Cartesian coordinates to Polar coordinates via 2D cubic interpolation algorithm (MATLAB, MathWorks).
All three maps were then further integrated along each dimension so that 1 , 2 , and ∆ can be fitted into a 1D modified Gaussian profile (Supplementary Note 3).

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Supplementary Note 3 The 1D correlation profile as a function of 1 (or 2 ) was obtained via integrating ( 1 , 2 , ∆ ) along 2 (or 1 ) and ∆ , with the latter performed radially through [− , ] The 1D correlation profile 1 ( 1 ) (or 2 ( 2 )) was approximated via a modified1D Gaussian distribution as a function of 1 (or 2 ) where 0 > 0 and > 0 denotes the center and the standard deviation of the Gaussian function, respectively. Note that in Polar coordinates, a Gaussian distribution cannot extend to the < 0 region. Instead, the ' < 0' portion of a normal Gaussian distribution centering at = 0 > 0 is considered as the ' > 0' portion of another normal Gaussian distribution centering at = − 0 < 0. Therefore, the radial integration along ∆ through [− , ] yields a modified 1D Gaussian profile (Equation ( 4 )) where the first and the second terms originate from the > 0 portion of a normal Gaussian distribution centered at = 0 > 0 and another Gaussian distribution centered at = − 0 < 0, respectively.
The correlation distance was then calculated as follows: The 1D correlation profile as a function of ∆ was obtained via integrating ( 1 , 2 , ∆ ) along 2 and 1 ∆ (∆ ) = ∬ ( 1 , 2 , ∆ )d 1 d 2 Since the molecular pattern can be imaged as a pair of reflectional symmetries, with ∆ (∆ ) being symmetric with respect to ∆ = 0, we used a modified Gaussian distribution similar to Equation ( 4 ) where ∆ 0 > 0 and > 0 denotes the center and the standard deviation of the Gaussian function, respectively.

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Supplementary Figure  does not converge to accurately represent the correlation distance (c). Therefore, we take the correlation distance obtained in (c), which integrates ∆ without 2D interpolation beforehand, as more accurate correlation distance. (d)

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Schematic illustration of the spatial organization of nascent DNA, RPA, and PCNA within a replication fork derived from the Triple Correlation analysis.

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Supplementary Figure