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This plug-in is a compilation of general co-localization indicators and more recently published methods. Depending on the ticked boxes, the plug-in will evaluate co-localization on two images according to the selected methods which are briefly outlined here after.
While the following text gives a short description of functionalities, you may have a look at our review to get a more detailled view of co-localization methods and their limits.
Pearson's coefficient (Manders et al. 1992):
The linear equation describing the relationship between the intensities in two images is calculated by linear regression. The slope of this linear approximation provides the rate of association of two fluorochromes. In contrast, the Pearson's coefficient provides an estimate of the goodness of this approximation. Its value can range from 1 to -1, with 1 standing for complete positive correlation and -1 for a negative correlation, with zero standing for no correlation.
Overlap coefficient, k1& k2 (Manders et al. 1992):
Introduced by Manders and colleagues, those three coefficients rely on Pearson's coefficient. The overlap coefficient is calculated as the Pearson's coefficient with the mean intensity value of both channels being taken out of the expression. k1 and k2 are defined as two components of the overlap coefficient, the former being related to the first channel total intensity, the later being related to the second channel total intensity.
M1 & M2 coefficients (Manders et al. 1992):
Manders' overlap coefficient is based on the Pearson's correlation coefficient with average intensity values being taken out of the mathematical expression (Manders 1992). This new coefficient will vary from 0 to 1, the former corresponding to non-overlapping images and the latter reflecting 100% co-localisation between both images. M1 is defined as the ratio of the 'summed intensities of pixels from the green image for which the intensity in the red channel is above zero' to the 'total intensity in the green channel' and M2 is defined conversely for red. Therefore, M1 (or M2) is a good indicator of the proportion of the green signal coincident with a signal in the red channel over its total intensity, which may even apply if the intensities in both channels are really different from one another.
Costes' automatic threshold(Costes et al. 2004):
To calculate this automatic threshold, limit values for each channel are initialised to the maximum intensity of the each channel and progressively decremented. The Pearson's coefficient is concomitantly calculated for each increment. The final thresholds are then set to values which minimize the contribution of noise (i.e. Pearson's coefficient under the threshold being null or negative).
Van Steensel's CCF (Van Steensel et al. 1996):
The authors apply a cross correlation analysis by shifting the green image in x-direction pixel per pixel relative to the red image and calculating the respective Pearson's coefficient. The Pearson's coefficient is then plotted as the function of dx (pixel shift) and the authors obtain by this a cross correlation function (CCF).
Cytofluorogram:
A simple way to measure the dependency of pixels in dual channel images is to plot the pixel grey values of two images against each other. Results are then displayed in a pixel distribution diagram called scatter plot or fluorogram. The intensity of a given pixel in the green image is used as the x-coordinate of the scatter plot and the intensity of the corresponding pixel in the red image as the y-coordinate.
Li's ICA (Li et al. 2004):
They first assume that the overall difference of pixels intensities to the mean intensity of a single channel is equal to zero: S_{n pixels} (A-a)=0 and S_{n pixels}(B-b)=0 with uppercase character being the current pixels intensity andlower case being the current channels mean intensity. As a consequence,the product of the two equalities should tend to zero. Now if weconsider a co-localizing pixel this product should be positive becauseof each difference to the mean being of the same sign. The differencesof intensities between both channels are scaled down by fitting thehistogram of both images to a 0 to 1 scale. The intensity correlationanalysis (ICA) results are then presented as a set of two graphs eachshowing the normalized intensities (from 0-1) as a function of theproduct (A_i-a)(B_i-b) for each channel. In this representation the x-axis reflects the covariance of the current channel and the y-axis reflects the intensity distribution of the current channel. As previously stated, in the case of co-localization the product (A_i-a)(B_i-b) is positive and therefore the dot cloud is mostly concentrated on the right side of x=0 line, while adopting a C shape. Its spread is dependent on the intensity distribution of the current channel as a function of the covariance of both channels intensities. Intensity correlation quotient (ICQ) is defined as the ratio of positive (Ai-a)(Bi-b) products divided by the overall products subtracted by 0.5. As a consequence, the ICQ varies from 0.5 (co-localisation) to -0.5 (exclusion) while random staining and images impeded by noise will give a value close to zero.
Costes' randomization (Costes et al. 2004):
Costes et al. (2004) introduce a new statistic analysis based on image randomization and evaluation of Pearson's coefficient. The authors point out that a single image reflects a particle distribution with sizes above optical resolution. These particles appear as a collection of adjacent pixels with intensities correlated to their neighbours. The intensity distribution depends on the PSF of the acquisition system and the approximate particle size may be calculated using the FWHM of the fluorescence intensity curve. The FWHM defines the area over which a signal belonging to a single particle is spread out, given the fact that the particle size is convolved by the PSF of the optical system. The authors create a randomized image by shuffling pixel blocks with the dimensions defined by the FWHM for the image of the green channel. This process is done 200 times for a single image and the Pearson's coefficient is calculated each time between the random images of the green channel against the original image of the red channel. The Pearson's coefficient for the original non-randomized images is then compared to the Pearson's coefficients of the randomized images and the significance (p-value) is calculated. The p-value, expressed as a percentage, is inversely correlated to the probability of getting the specified PC by hazard (i.e. on randomized image pairs). This value is calculated as the integrated area under the PC distribution curve, from the minimum PC value obtained from randomization to the PC obtained from original images.
Formulas:
In the following, channel A and channel B grey values of voxel i will be noted A_i & B_i respectively and the corresponding average intensities over the full image a & b.Pearson's coefficient:
r_P= (S_i ((A

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_i-a)x(B_i-b)))/…(S_i (A_i-a)²x S_i (B_i-b)²)
Overlap coefficient:
Same as previous except that mean value is not subtracted:
r= (S_i (A_ixB_i))/…(S_i (A_i-a)²x

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S_i (B_i-b)²)
k₁ and k₂ coefficients:
r²=k₁xk₂ with k₁= (S_i (A_ixB_i))/ (S_i (A_i)²) & k₂=

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(S_i (A_ixB_i))/ (S_i (B_i)²)
M₁ & M₂ coefficient:
k₁= (S_i (A_{i, coloc}))/ (S_i A_i) & k₂= (S_i (B_{i, coloc}))/ (S_i B_i)

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With A_{i, coloc} being A_i if B_i>0 and 0 if B_i=0; and B_{i, coloc} being B_i if A_i>0 and 0 if A_i=0.
References:

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Manders, E., Stap, J., Brakenhoff, G., van Driel, R., and Aten, J. (1992). Dynamics of three-dimensional replication patterns during the S-phase, analysed by double labelling of DNA and confocal microscopy. J Cell Sci 103, 857-862.Costes SV, Daelemans D., Cho EH, Dobbin Z, Pavlakis G, Lockett S. (2004). Automatic and quantitative measurement of protein-protein colocalization in live cells. Biophys J. 86, 3993-4003.
Li, Q., Lau, A., Morris, T. J., Guo, L., Fordyce, C. B., and Stanley, E. F. (2004). A Syntaxin 1, G_alphao, and N-Type Calcium Channel Complex at a Presynaptic Nerve Terminal: Analysis by Quantitative Immunocolocalization. J. Neurosci. 24, 4070-4081.
Van Steensel, B., van Binnendijk, E., Hornsby, C., van der Voort, H., Krozowski, Z., de Kloet, E., and van Driel, R. (1996). Partial colocalization of glucocorticoid and mineralocorticoid receptors in discrete compartments in nuclei of rat hippocampus neurons. J Cell Sci 109, 787-792.