Abstract
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 overline{R} or overline{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.
Highlights
In widefield and superresolution fluorescence microscopy of eukaryotic and prokaryotic cells, the fluorescent species occupy a three-dimensional (3D) volume
We have proposed a modified Pearson Correlation Coefficient (MPCC) that corrects this problem for spherocylindrical cell geometry by employing the proper reference matrices, Two dimension (2D) projection matrices derived from independent Three dimension (3D) random distributions in spherocylinders, for comparison with the images under analysis
MPCC can be employed for 2D superresolution as well as widefield images, conventionally acquired for wide variety of studies
Summary
In widefield and superresolution fluorescence microscopy of eukaryotic and prokaryotic cells, the fluorescent species occupy a three-dimensional (3D) volume. For two-color imaging of two different species, called the “red species” and the “green species”, an important biological question is the degree to which the red and green spatial distributions are positively correlated, anti-correlated, or uncorrelated with each other. Positive correlation may suggest binding to each other or to a common cytoplasmic element such as a membrane or the chromosomal DNA. It may suggest common sites of production, action, or degradation. Negative correlation may suggest a physical or biochemical mechanism that sequesters red and green species from each other [1, 2]. A number of different procedures for assessing co-localization between two images are described in a recent review [3]
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