I am an undergrad working in biophysics research, and I was hoping you guys may have input on a simple but strange problem I'm facing. I don't know if this is appropriate for this forum, but it is not an especially specialized or complex problem and I am desperate.(adsbygoogle = window.adsbygoogle || []).push({});

I am currently conducting analysis on fluorescent images using image cross-correlation spectroscopy. I am trying to quantify the colocalization of two fluorescent tags in images of cells. Simply put, I have 2 pictures per image of the cell, one for each wavelength of light that each the fluorescent tag emits. They are simple black and white images 500x400, with pixels having certain intensities. In order to quantify how much the two fluorescent tags overlap, I am using this process which compares the two images and comes up with coefficients that quantify this overlap. The process uses a spatial correlation function:

r(ε,η) =

<(I_{1}(x,y) - <I_{1}>)>*<(I_{2}(x+ε,y+η) - <I_{2}>)>

<I_{1}>*<I_{2}>

I_{1}(x,y) is the intensity at each pixel (x,y) in image 1

<I_{1}> is the average intensity of all pixels in image 1

The brackets <> denote an average over all pixels, or ensemble average

(ε,η) are the spatial lag variables

This may seem complicated but its simple: When cross-correlating, you place image 2 directly on top of image 1 so there is no spatial lag and (ε,η) = (0,0), and run this function which gives you a negative or positive number indicating the correlation (positive if the two images correlate strongly negative if they don't). Image 2 is then shifted over by one pixel on top of image 1 and this function is run again, and again for every position (ε,η) image 2 can take on top of image 1.

When you use the correlation function on two images which are the same, this is auto-correlation and it gives a 2-D function which is ideally Gaussian, with a peak at the center point (0,0) (because the images are perfectly correlated at the center).

If you think about it and plug values into the function, an autocorrelation function can only take on positive values and a crosscorrelation function can take on either negative or positive values. If none of the fluorescent tag in image 1 colocalized with the tag in image 2, the crosscorrelation amplitude would be negative.

However, the theory of ICCS ultimately leads to this equation as well:

<N>_{1/2}= r_{1/2}(0,0) / [r_{1}(0,0) * r_{2}(0,0)]

where <N>_{1/2}is the average number of co-localized fluorescent particles per area

r_{1/2}(0,0) is the amplitude of the cross-correlation function between images 1 and 2

[r_{1}(0,0) is the amplitude of the auto-correlation function of image 1

[r_{2}(0,0) is the amplitude of the auto-correlation function of image 2

So this begs the question: if the cross-correlation amplitude can be positive or negative and the autocorrelation amplitudes have to be positive, how can this equation hold? <N>_{1/2}must be positive.

This means that ICCS theory states that the cross-correlation peak must positive. But, in my analysis, I get negative cross-correlation peaks, which can be proven to be possible from the correlation function. The only conclusion I can reach is that the assumptions ICCS makes were violated in my analysis. I looked long and hard through papers but found that the number of pixels I was using was ideal for ICCS, and I can't find any assumptions of ICCS theory that leads to the seconds equation that I've broken.

I'm just wondering if there is something I haven't realized mathematically about this problem, and the positive and negative values, or if anyone here knows anything about this type of analysis. You can also see a succinct and exact explanation of what I'm doing in this paper: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1779921/

If you go to the Theory section, then the ICCS subsection.

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# Negative amplitude of cross-correlation function in ICCS?

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