Modified Pearson Correlation Coefficient for Two-color Imaging in Spherocylindrical Cells

The revolution in fluorescence microscopy enables sub-diffraction-limit (“superresolution”) localization of hundreds or thousands of copies of two differently labeled proteins in the same live cell. In typical experiments, fluorescence from the entire three-dimensional (3D) cell body is projected along the z -axis of the microscope to form a 2D image at the camera plane. For imaging of two different species, here denoted “red” and “green”, a significant biological question is the extent to which the red and green spatial distributions are positively correlated, anti-correlated, or uncorrelated. A commonly used statistic for assessing the degree of linear correlation between two image matrices R and G is the Pearson Correlation Coefficient (PCC). PCC should vary from –1 (perfect anti-correlation) to 0 (no linear correlation) to +1 (perfect positive correlation). However, in the special case of spherocylindrical bacterial cells such as E . coli or B . subtilis , we show that the PCC fails both qualitatively and quantitatively. PCC returns the same +1 value for 2D projections of distributions that are either perfectly correlated in 3D or completely uncorrelated in 3D. The PCC also systematically underestimates the degree of anti-correlation between the projections of two perfectly anti-correlated 3D distributions. The problem is that the projection of a random spatial distribution within the 3D spherocylinder is non-random in 2D, whereas PCC compares every matrix element of R or G with the constant mean value R or G . We propose a modified Pearson Correlation Coefficient (MPCC) that corrects this problem for spherocylindrical cell geometry by using the proper reference matrix for comparison with R and G . Correct behavior of MPCC is confirmed for a variety of numerical simulations and on experimental distributions of HU and RNA polymerase in live E . coli cells. The MPCC concept should be generalizable to other cell shapes.