Fluorescence heterogeneity across individual cells was quantified using a coefficient of variation (CV)–based metric computed from image-segmented single-cell intensity measurements.

Cell nuclei or GFP-positive regions were segmented using a hybrid approach combining pre-trained machine learning segmentation model (for cell boundary initialization) and intensity thresholding in the GFP channel for refinement.

The variability in expression within a population was expressed as the coefficient of variation (CV) of pixel intensity for each image field or individual cell population.

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Before you start

All conda .ymal were exported and tested on apple silicon OSX-ARM64 platform

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Installation

You can resolve environment using Conda.

cd [the location of src folder]

activate this for segmentation and calculation #1 and #3 script#

conda env create --file=co_var_image_analysis.yaml 

activate this for plotting #2 and #4 script#

conda env create --file=co_var_image_analysis_plot.yaml 

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Usage

conda activate co_var_image_analysis
python ~/src/1_analyze_images_EGFP.py

&

conda activate co_var_image_analysis_plot
python ~/src/2_plot_EGFP_CV_raincloud_vertical.py

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Definition

For each segmented nucleus $$\sigma_i$$, let $$\bar I_{i} = { I_{i1}, I_{i2}, \dots, I_{in_i} }$$ be the set of GFP pixel intensities within that nucleus.

The coefficient of variation (CV) is defined as:

$$ \text{CV}_i = \frac{\sigma_i}{\mu_i^{\text{corr}}} $$

where: $$\sigma_i$$: standard deviation of GFP intensities within nucleus $$\sigma_i$$

$$ \sigma_i = \sqrt{\frac{1}{n_i - 1} \sum_{j=1}^{n_i} (I_{ij} - \bar{I}_i)^2} $$

$$\bar{I}i$$ : mean GFP intensity inside the nucleus

$$ \bar I_{i} = \frac{1}{n_i} \sum{j=1}^{n_i} I_{ij} $$

$$\mu_i^{\text{corr}}$$ : background-corrected mean intensity

$$\mu_i^{\text{corr}} = \max(\bar{I}_i - \bar{B}_i,, 0)$$ where $$\bar{B}_i$$ is the mean intensity of the local background ring surrounding nucleus $$\sigma_i$$.

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Interpretation

The numerator $$\sigma_i$$ represents intranuclear variability of GFP intensity — higher values indicate stronger spatial heterogeneity (e.g., speckles or puncta).

The denominator $$\mu_i^{\text{corr}}$$ normalizes this variability to the corrected mean brightness, producing a dimensionless ratio independent of absolute intensity.

Thus, $$\text{CV}_i$$ reflects the relative dispersion of signal within each nucleus and serves as a quantitative descriptor of texture or punctate patterning.

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Final Expression

The full background-corrected coefficient of variation is:

$$ \boxed{ \text CV_{i} = \frac{ \sqrt{ \dfrac{1}{n_i - 1} \sum{j=1}^{n_i} (I_{ij} - \bar{I}_i)^2 } }{ \max(\bar{I}_i - \bar{B}_i, 0) } } $$

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Calculation for other channels

eg. Cy5/Cy3 • EGFP were used as a metric to include the cell for channal CV calculation. See 3_Cy5_CV_from_EGFP_Positive.py and input your desired channel.

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Output Example