# Co-Density Distribution Maps for Advanced Molecule Colocalization and Co-Distribution Analysis

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sample Preparation and Image Acquisition

_{EX}= 488 nm, λ

_{EM}= 525 nm) and vesicular glutamate transporter 1 (VGLUT1, λ

_{EX}= 561 nm, λ

_{EM}= 595 nm), as described in [27], were sequentially acquired with a Nikon Ti-E A1R laser confocal fluorescence microscope (Nikon, Tokyo, Japan), equipped with a Plan Apo 60x/1.4 objective at a resolution of 512 × 512 × 9 pixels with a pixel size (XYZ) of 0.1 × 0.1 × 0.25 µm

^{3}(Pinhole size = 39.59 µm). (2) The 12 bit range images of human osteosarcoma MG-63 cells exposed to Keratin-based nanoparticles ([email protected]

_{ag}, λ

_{EX}= 649 nm, λ

_{EM}= 700 nm) were sequentially acquired with a confocal fluorescence laser scanning microscope Ti-E A1R (Nikon, Amsterdam, Netherlands), equipped with a 60×/NA 1.4 oil Plan-Fluo at a resolution of 1024 × 1024 × 19 pixels with a pixel size (XYZ) of 0.2 × 0.2 × 0.25 µm

^{3}(Pinhole size = 24.27 µm). MG-63 cells were indirectly immunostained against the Lysosomal-associated membrane protein 1 (Lamp-1, λ

_{EX}= 563 nm, λ

_{EM}= 595 nm) as described in [28]. 3) The 8 bit range images from rat spinal cord immunostained for neurofilaments (NFs, primary antibody: mouse anti-NF200, 1:800, Sigma Aldrich Saint Louis, MO; secondary antibody: Rhodamine Red™-X, 1:100, Jackson Immuno Research, Cambridgeshire, UK, λ

_{EX}= 570 nm, λ

_{EM}= 590 nm) and stained for myelin with FITC-Fluoromyelin™ (FM, Thermo Fisher, λ

_{EX}= 479 nm, λ

_{EM}= 598 nm) were acquired with a Nikon Eclipse E600 (Q Imaging, Surrey, BC, Canada), equipped with a Plan Apo 10x/0.4 objective and Q Imaging RETIGA-2000RV camera. For each sample, 10 images were acquired and stitched into a single mosaic (resolution: 3532 × 2384 pixels, pixel size: 0.74 × 0.74 µm

^{2}) with Photoshop (Adobe Suite, release 22.4.2).

#### 2.2. Image Segmentation

^{®}(R2019a v.9.7.0, The MathWorks, Natick, MA, USA). SYP and VGLUT1 signals are segmented by Isodata thresholding. Lamp-1 and Ce6 signals and NF200 and FM signals are segmented by Otsu method.

#### 2.3. Local Distribution and Co-Distribution Analysis, DDM and cDDM

^{2}− 1) to +(WS

^{2}− 1). Different LDI couples can result in the same cLDI (Figure 1b, red and green arrows). Negative cLDI values indicate pixels where the first marker signal is locally denser than the second one, the opposite holds for positive values. A cLDI equal to zero indicates pixels where the two markers are equally dense, hence defining the equi-density region, where the signals are in a 1:1 ratio. However, non-zero cLDIs cannot be considered indicators of a specific ratio, but rather, of a specific difference in the markers’ abundance that is, by definition, a more correct indication of the degree of colocalization than of pixel intensities correlation. Finally, mapping cLDIs back to the image domain in pseudo-colors also allows us to gain information about the markers’ spatial co-distribution.

#### 2.4. Pixel Density as a Measure of Colocalization

#### 2.5. Colocalization Analysis

_{1}and M

_{2}, and signals correlation by Pearson’s (ρ) and Spearman’s (ρ

_{s}) [29] coefficients. Of note, MOC’s informativeness as a co-occurrence estimator is actually an ongoing topic of discussion [4,30,31,32,33] and the MOC values reported hereafter should be carefully interpreted accordingly. The formulae and description for the mentioned coefficients can be found in Appendix A. In addition, we also evaluate:

- The markers overlap region through our co-occurrence maps (cOMs) built on top of segmented signals, highlighting in four different pseudo-colors the pixels where: (1) both markers are absent, (2) only the first marker is present, (3) only the second marker is present and (4) both markers are present (co-occurrence region).
- The local density and co-density of marked structures, by DDMs and cDDMs computation and analysis.

#### 2.6. Assessment of Results

_{s}) and overlap (by MOC, M

_{1}and M

_{2}) are calculated for both the signals’ intensities (i.e., between the pixel values in the two markers’ images) and the signals’ local density (i.e., between the pixel values in the two markers’ DDMs) to assess to what extent density and intensity are comparable descriptors of colocalization. The signals’ intensity correlation (and MOC) is calculated in three increasingly narrowed domains: the entire image, the co-occurrence region and the co-density region. As expected, the first narrowing, from the entire image to the co-occurrence region, always decreases the correlation coefficients value, excluding the random colocalization of the background (data not shown). M

_{1}and M

_{2}coefficients are calculated for signals’ intensity with respect to both the co-occurrence and the co-density regions, according to equations (3) and (4) of Appendix A, where the “colocalizing” pixels at the numerators are the co-occurring and the co-dense pixels, respectively. The signals’ density correlation and co-occurrence are calculated only for the co-occurrence region. Indeed, density computation is theoretically impossible before the co-occurrence region definition, whilst inside the co-density region, the coefficients values would be biased by the density-based nature of the refinement itself (i.e., all coefficients value would be set to 1).

_{s}) value, before and after pixel removing by erosion and pixel selection by cDDMs.

## 3. Results and Discussion

#### 3.1. Functional Implication of cDDMs

_{m1}− LDI

_{m2}= n, where n is a specific cLDI value (Figure 2, ③). If we now compute the correlation coefficients (ρ and ρ

_{s}) within each cLDI-defined subregion (Figure 2, central scattergram), we can see that correlation between signals intensities increases as cLDI moves from the highest (in absolute terms, i.e., |cLDI| = 8) to the equi-density condition (i.e., cLDI = 0). This proportionality confirms that cLDIs can serve as indicators of colocalization, just as ρ and ρ

_{s}, at least when they hold. Then, cDDMs can be applied for a density-based refinement of colocalization quantification by correlation coefficients, namely, by restricting their computation from the co-occurrence region to the equi-density one (Figure 2, ④). Apparently, the same restriction of the computational domain could be obtained by a simply binary erosion. However, even under the additional assumption of negligible colocalization at the edge of the co-occurrence region, a refinement by erosion would remove the outer pixels independently of their connection or the presence of colocalization. If this could produce a somewhat lightly divergent set of results when the co-occurrence region is dense (i.e., the edge pixels are a clear minority), the erosion would yield an increasingly invalid outcome as the border indentation of the co-occurrence region increases, or in the presence of small objects. Table 1 reports all of the results, from the initial whole co-occurrence mask to the final masks, achieved by erosions and cDDM, used to assess colocalization. Accordingly, the numbers of edge pixels are complementary (e.g., for NF200-FM the percentage of edge pixels is 34.65).

#### 3.2. cDDMs Disclose Information about the Degree of Colocalization

_{ag}) into late endosomes (marked by Lamp-1 staining) [29] (Figure 3a, top).

_{1}and M

_{2}’s value) hints at the capability of our method to selectively retain the colocalization between signals, more so than with false positives.

_{s}values (Table 2, third column), suggesting the existence of a real, although spatially limited, colocalization. Its detection by correlation coefficients is initially weakened by the scarcity of marked structures within the co-occurrence region, but subsequently strengthened by coDDM-driven increase in analysis specificity. Moreover, co-density analysis reveals that Ce6 signal tends to be locally denser than Lamp-1’s, as attested by the prevalence of negative values in the cDDM (Figure 3b). This last finding, in agreement with expectedly denser NPs due to their nanoformulation [28], also suggests that NPs’ internalization into late endosomes could occur at a ratio higher than 1:1, with many NPs entering the same endosomes at once. On one hand, this is positive for the pharmacokinetic-improving function of the developed system, but on the other, it opens up to the possibility that a different nanoformulation, producing less dense NPs, could result in better colocalization values and NPs internalization.

#### 3.3. cDDMs Open to the Formulation of New Biological Considerations

_{1}and M

_{2}coefficients. Most probably, these results Table 3, first and second columns) can be interpreted as an artifact of image resolution, which is not able to fully capture the concentric nature of the myelin signal, surrounding the axon, without overlapping. In any case, these results confirm the outcome of cDDM already seen in Lamp1-Ce6, where a reduction of the signals’ co-occurrence is coupled with a marked increase in correlation values (ρ and especially ρ

_{s}value, Table 3, third column). In fact, the resolution problem seems to be alleviated by our approach, indeed reducing the signals’ overlap, quantified by M

_{1}and M

_{2}of about 40%. The increase in correlation coefficients also indicates that markers intensities should not be assumed a priori to linearly correlate, according to the functional heterogeneity of axons’ and myelin’s distribution in the tissue. Even though most of the co-occurring pixels are also equally dense (58%, cLDI = 0, Figure 4b), a remarkable prevalence of positive values in cDDM indicates axons’ tendency to be denser than myelin, agreeing with the reduced myelin sheaths thickness observable for some pathways. Indeed, a lower myelin thickness reasonably reflects a lower local density of FM, but not of NF200 signal, therefore bringing higher cLDI values and decorrelating the two markers’ density (Table 3, second column). Moreover, by locally analyzing cDDM, we can see that the local density pattern depends on the nature of the anatomical pathway (Figure 4a, cDDM left magnification), specifically being enriched in low values (hence, in myelin) in the proximity of the dorsal median sulcus (DMS) and in high values (hence, in less myelinated axons) away from it (Figure 4c, line plot of the pixel values underlying the red line in cDDM motor pathway magnification). In conclusion, in addition to also exemplifying its applicability at the tissue level, here, the cDDM provides new biological information, revealing and mapping the spatial heterogeneity of the myelination pattern, which could not be derived from the original image. This makes the local co-density an effective indicator of the local degree of myelination and the cDDM a possible discriminator of neuronal pathways.

#### 3.4. GUI for cDDMs Creation

^{®}App Designer [26]. coDDMaker was conceived for the guided analysis of the distributions and co-distribution of marker pairs. Starting from RGB, greyscale or directly binary images and based on customed search window size, the software builds the markers’ DDMs, cDDM and cOM and tabulates their numerical content. With coDDMaker, we also introduce a module for the background correction of non-binary input images [35] and a module for their local segmentation to also be used as tools for image denoising. A detailed description of coDDMaker functionalities is provided in Appendix B. The software completes the colocalization analysis of a couple of images under standard setting (i.e., global image segmentation and WS = 3) in less than 30 s on entry-level computers, although the total elapsed time strongly depend on different factors (e.g., the size of the objects to be segmented), as exemplified in Supplementary Table S2. As much as DDMaker, or even more so, coDDMaker could serve as a checkpoint for long-lasting experiments, follow-up and large-scale studies, that can be monitored on-line and adjusted on the basis of the software feedbacks, therefore, optimizing time and costs. coDDMaker is available as a public open-source software written in MATLAB

^{®}and as a 64-bit stand-alone application (https://sourceforge.net/projects/coddmaker/ accessed on 10 September 2021).

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Correlation and Co-Occurrence Coefficients

#### Appendix A.1.1. Pearson’s Correlation Coefficient

#### Appendix A.1.2. Spearman’s Correlation Coefficient

_{s}, r

_{s}or SRCC) is defined as the Pearson’s correlation coefficient between the rank variables [29] and it is then computed simply by replacing x and y intensity values with intensity ranks of values in Equation (A1). By working on ranks, ρ

_{s}assesses how well the relationship between two variables can be described using a monotonic function, disregarding any assumption of linearity [37].

#### Appendix A.1.3. Mander’s Coefficients

_{1}and M

_{2}coefficients:

_{1}and M

_{2}separate the fluorophores contribution to colocalization, by calculating for each fluorophore the fraction of the total intensity that co-occurs.

_{1}and M

_{2}coefficients have since then been widely used to quantify signals co-occurrence in intensity images. However, doubts on the suitability of the MOC coefficient to the quantification of unbiased co-occurrence have been casted [30] and its use along with M

_{1}and M

_{2}coefficients is currently a topic under heavy discussion [4,31,32,33].

## Appendix B

#### coDDMaker: GUI Description

^{®}App Designer.

**Figure A1.**Main GUI of coDDMaker. The main window is divided into five sections: (1) Input: to select the input images’ folders; (2) Segmentation: to select the thresholding method, its locality of application and to eventually perform background correction before threshold calculation and image binarization; (3) DDM: to select the size of the search window for local density analysis and, to allow user creating and binarizing DDMs after setting the colorbar for pseudo-color DDMs visualization and the percentile for DDMs thresholding; (4) cDDM: to allow user creating and binarizing cDDMs after setting, the colorbar for pseudo-color cDDMs visualization and the tolerance for equi-density region segmentation; (5) Output: to visualize and save intermediates and outputs. From left to right: markers binary masks, cOM, pseudo-color DDMs, pseudo-color cDDM.

## References

- Landmann, L.; Marbet, P. Colocalization analysis yields superior results after image restoration. Microsc. Res. Tech.
**2004**, 64, 103–112. [Google Scholar] [CrossRef] - Zhou, L.; Cai, M.; Tong, T.; Wang, H. Progress in the correlative atomic force microscopy and optical microscopy. Sensors
**2017**, 17, 938. [Google Scholar] [CrossRef][Green Version] - Wells, K.S.; Sandison, D.R.; Strickler, J.; Webb, W.W. Quantitative fluorescence imaging with laser scanning confocal microscopy. In Handbook of Biological Confocal Microscopy; Springer: Boston, MA, USA, 1990; pp. 27–39. [Google Scholar]
- Aaron, J.S.; Taylor, A.B.; Chew, T.L. Image co-localization–co-occurrence versus correlation. J. Cell Sci.
**2018**, 131, jcs211847. [Google Scholar] [CrossRef][Green Version] - Akner, G.; Mossberg, K.; Wikström, A.C.; Sundqvist, K.G.; Gustafsson, J.Å. Evidence for colocalization of glucocorticoid receptor with cytoplasmic microtubules in human gingival fibroblasts, using two different monoclonal anti-GR antibodies, confocal laser scanning microscopy and image analysis. J. Steroid Biochem. Mol. Biol.
**1991**, 39, 419–432. [Google Scholar] [CrossRef] - Manders, E.M.M.; Verbeek, F.J.; Aten, J.A. Measurement of co-localization of objects in dual-colour confocal images. J. Microsc.
**1993**, 169, 375–382. [Google Scholar] [CrossRef] - Pike, J.A.; Styles, I.B.; Rappoport, J.Z.; Heath, J.K. Quantifying receptor trafficking and colocalization with confocal microscopy. Methods
**2017**, 115, 42–54. [Google Scholar] [CrossRef] - Dunn, K.W.; Kamocka, M.M.; McDonald, J.H. A practical guide to evaluating colocalization in biological microscopy. Am. J. Physiol. Cell Physiol.
**2011**, 300, C723–C742. [Google Scholar] [CrossRef][Green Version] - Samacoits, A.; Chouaib, R.; Safieddine, A.; Traboulsi, A.M.; Ouyang, W.; Zimmer, C.; Peter, M.; Bertrand, E.; Walter, T.; Mueller, F. A computational framework to study sub-cellular RNA localization. Nat. Commun.
**2018**, 2018. 9, 4584. [Google Scholar] [CrossRef][Green Version] - Silver, M.A.; Stryker, M.P. A method for measuring colocalization of presynaptic markers with anatomically labeled axons using double label immunofluorescence and confocal microscopy. J. Neurosci. Meth.
**2000**, 94, 205–215. [Google Scholar] [CrossRef] - Oheim, M.; Li, D. Quantitative colocalisation imaging: Concepts, measurements, and pitfalls. In Imaging Cellular and Molecular Biological Functions; Springer: Berlin/Heidelberg, Germany, 2007; pp. 117–155. [Google Scholar]
- Lachmanovich, E.; Shvartsman, D.E.; Malka, Y.; Botvin, C.; Henis, Y.I.; Weiss, A.M. Co-localization analysis of complex formation among membrane proteins by computerized fluorescence microscopy: Application to immunofluorescence co-patching studies. J. Microsc.
**2003**, 212, 122–131. [Google Scholar] [CrossRef] - Lagache, T.; Sauvonnet, N.; Danglot, L.; Olivo-Marin, J.C. Statistical analysis of molecule colocalization in bioimaging. Cytom. Part A
**2015**, 87, 568–579. [Google Scholar] [CrossRef] [PubMed] - Adler, J.; Pagakis, S.N.; Parmryd, I. Replicate-based noise corrected correlation for accurate measurements of colocalization. J. Microsc.
**2008**, 230, 121–133. [Google Scholar] [CrossRef] - Taylor, R. Interpretation of the correlation coefficient: A basic review. J. Diagn. Med. Sonog.
**1990**, 6, 35–39. [Google Scholar] [CrossRef] - Singan, V.R.; Jones, T.R.; Curran, K.M.; Simpson, J.C. Dual channel rank-based intensity weighting for quantitative co-localization of microscopy images. BMC Bioinform.
**2011**, 12, 407. [Google Scholar] [CrossRef][Green Version] - Herce, H.D.; Casas-Delucchi, C.S.; Cardoso, M.C. New image colocalization coefficient for fluorescence microscopy to quantify (bio-) molecular interactions. J. Microsc.
**2013**, 249, 184–194. [Google Scholar] [CrossRef][Green Version] - Sheng, H.; Stauffer, W.; Lim, H.N. Systematic and general method for quantifying localization in microscopy images. Biol. Open
**2015**, 5, 1882–1893. [Google Scholar] [CrossRef][Green Version] - Li, Q.; Lau, A.; Morris, T.J.; Guo, L.; Fordyce, T.B.; Stanley, E. A syntaxin 1, Galpha(o), and Ntype calcium channel complex at a presynaptic nerve terminal: Analysis by quantitative immunocolocalization. J. Neurosci.
**2004**, 24, 4070–4081. [Google Scholar] [CrossRef][Green Version] - Wang, S.; Arena, E.T.; Becker, J.T.; Bement, W.M.; Sherer, N.M.; Eliceiri, K.W.; Yuan, M. Spatially adaptive colocalization analysis in dual-color fluorescence microscopy. IEEE Trans. Image Process.
**2019**, 28, 4471–4485. [Google Scholar] [CrossRef] - Pearson, K. Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia. Philos. Trans. Roy. Soc. Lond. A
**1896**, 187, 253–318. [Google Scholar] [CrossRef][Green Version] - Gilles, J.F.; Dos Santos, M.; Boudier, T.; Bolte, S.; Heck, N. DiAna, an ImageJ tool for object-based 3D co-localization and distance analysis. Methods
**2017**, 115, 55–64. [Google Scholar] [CrossRef][Green Version] - Costes, S.V.; Daelemans, D.; Cho, E.H.; Dobbin, Z.; Pavlakis, G.; Lockett, S. Automatic and quantitative measurement of protein-protein colocalization in live cells. Biophys. J.
**2004**, 86, 3993–4003. [Google Scholar] [CrossRef][Green Version] - Bolte, S.; Cordelieres, F.P. A guided tour into subcellular colocalization analysis in light microscopy. J. Microsc.
**2006**, 224, 213–232. [Google Scholar] [CrossRef] - Cordelieres, F.P.; Bolte, S. Experimenters’ guide to colocalization studies: Finding a way through indicators and quantifiers, in practice. Methods Cell Biol.
**2014**, 123, 395–408. [Google Scholar] [CrossRef] - De Santis, I.; Zanoni, M.; Arienti, C.; Bevilacqua, A.; Tesei, A. Density Distribution Maps: A Novel Tool for Subcellular Distribution Analysis and Quantitative Biomedical Imaging. Sensors
**2021**, 21, 1009. [Google Scholar] [CrossRef] [PubMed] - Giuliani, A.; Sivilia, S.; Baldassarro, V.A.; Gusciglio, M.; Lorenzini, L.; Sannia, M.; Calzà, L.; Giardino, L. Age-related changes of the neurovascular unit in the cerebral cortex of alzheimer disease mouse models: A neuroanatomical and molecular study. J. Neuropat. Exp. Neurol.
**2019**, 78, 101–112. [Google Scholar] [CrossRef][Green Version] - Martella, E.; Ferroni, C.; Guerrini, A.; Ballestri, M.; Columbaro, M.; Santi, S.; Sotgiu, G.; Serra, M.; Donati, D.M.; Lucarelli, E.; et al. Functionalized keratin as nanotechnology-based drug delivery system for the pharmacological treatment of osteosarcoma. Int. J. Mol. Sci.
**2018**, 19, 3670. [Google Scholar] [CrossRef] [PubMed][Green Version] - Spearman, C. The proof and measurement of association between two things. Am. J. Psychol.
**1904**, 15, 72–101. [Google Scholar] [CrossRef] - Adler, J.; Parmryd, I. Quantifying colocalization by correlation: The Pearson correlation coefficient is superior to the Mander’s overlap coefficient. Cytom. Part A
**2010**, 77, 733–742. [Google Scholar] [CrossRef] [PubMed] - Adler, J.; Parmryd, I. Quantifying colocalization: The MOC is a hybrid coefficient–an uninformative mix of co-occurrence and correlation. J. Cell Sci.
**2019**, 132, jcs222455. [Google Scholar] [CrossRef][Green Version] - Aaron, J.S.; Taylor, A.B.; Chew, T.L. The Pearson’s correlation coefficient is not a universally superior colocalization metric. Response to ‘Quantifying colocalization: The MOC is a hybrid coefficient–an uninformative mix of co-occurrence and correlation’. J. Cell Sci.
**2019**, 132, jcs227074. [Google Scholar] [CrossRef][Green Version] - Adler, J.; Parmryd, I. Quantifying colocalization: The case for discarding the Manders overlap coefficient. Cytom. Part A
**2021**, 99, 910–920. [Google Scholar] [CrossRef] [PubMed] - Saliani, A.; Perraud, B.; Duval, T.; Stikov, N.; Rossignol, S.; Cohen-Adad, J. Axon and myelin morphology in animal and human spinal cord. Front. Neuroanat.
**2017**, 11, 129. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gherardi, A.; Bevilacqua, A.; Piccinini, F. Illumination field estimation through background detection in optical microscopy. In Proceedings of the 2011 IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB), Paris, France, 11–15 April 2011. [Google Scholar] [CrossRef]
- Lee Rodgers, J.; Nicewander, W.A. Thirteen ways to look at the correlation coefficient. Am. Stat.
**1988**, 42, 59–66. [Google Scholar] [CrossRef] - Artusi, R.; Verderio, P.; Marubini, E. Bravais-Pearson and Spearman correlation coefficients: Meaning, test of hypothesis and confidence interval. Int. J. Biol. Markers
**2002**, 17, 148–151. [Google Scholar] [CrossRef] [PubMed] - Sandberg, K. Introduction to image processing in Matlab. Dept. Appl. Math. Colo. BIODATA
**2007**, 1, 1–18. [Google Scholar]

**Figure 1.**Flowchart of cDDM creation pipeline for a couple of markers. (

**a**) The acquired images are segmented in binary masks and their pixel connectivity separately explored by local density analysis for the two pseudo-color DDMs building. Then, the cDDM is built through local co-density analysis, by comparing the single markers DDMs pixelwise. (

**b**) Details of local density (blue boxes) and co-density (orange boxes) analyses: after setting the search (moving) windows size, each foreground (FG) pixel of each binary mask is assigned a number representing the amount of FG pixels in its locality, this constituting the input to build the pseudo-color DDM (here shown with no “saturated” densities). Then, the local co-density analysis is performed by pixelwise subtraction of the two DDMs.

**Figure 2.**Functional implications of coDDM. Starting from a couple of marked images (m1 and m2), colocalization is usually quantified as a combination of markers overlap (by co-occurrence mask and Manders’ MOC, M

_{1}and M

_{2}coefficients computation, ①) and intensity correlation (primarily by ρ and ρ

_{s}correlation coefficients, ②). By cLDIs computation, co-occurrence pixels can be further partitioned by their local co-density and resulting groups visualized in a pseudo-color scattergram (③). When quantifying colocalization through markers’ intensity correlation, the analysis specificity can be increased by narrowing the computational domain from the co-occurrence to the equi-density region (i.e., made of pixels with cLDI = 0, ④). In addition, being based on density instead of intensity, cLDIs are more appropriate for estimating markers’ relative abundance (⑤). Finally, cDDM permits to preserve the spatiality of original images, additionally coding it with colors for the regional investigation of cLDI distribution (⑥). Details of presented scatterplot data in Supplementary Figure S1A–C.

**Figure 3.**cDDM discloses information about the degree of colocalization. (

**a**) Top: Exemplificative immunofluorescence (IF) images of MG-63 cells exposed to [email protected]

_{ag}nanoparticles, marked against late endosomes (Lamp-1), with Ce6 (NPs), or both (fusion). Middle: Lamp-1 and Ce6 signals’ binary masks (BW), whose combination produce the co-occurrence map (cOM). Bottom: Lamp-1 and Ce6 DDMs and cDDM. (

**b**) Bar graph of co-occurrence region partitioning by co-density, showing a prevalence of negative cLDI values that indicate NPs as generally denser than late endosomes.

**Figure 4.**cDDM opens to the formulation of new biological considerations. (

**a**) Top: Exemplificative immunofluorescence (IF) images of rats’ spinal cord, marked against the axonal (NF200), the myelin (FM) components of the cord, or both (fusion). Middle: NF200 and FM signal binary masks (BW), whose combination produce the co-occurrence map (cOM). Bottom: NF200 and FM DDMs and cDDM. (

**b**) Scattergram of NF200 and FM signals intensity color-coded by cLDI, showing a clear prevalence of equi-density pixels (grey, cLDI = 0). (

**c**) The line plot reports the cLDI values underlying the horizontal red arrow (x) inside the “motor pathway” magnification. The cLDI medio-lateral distribution is shown in function of the pixel distance (d, yellow line) from the dorsal median sulcus (DMS, white line), highlighting a progressive myelin thinning from spinal cord center to periphery.

**Table 1.**Comparison between binary erosion and co-density analysis in refining the correlation computation domain.

MASKS | NF200-FM | SYP-VGLUT1 | Lamp1-Ce6 | ||||
---|---|---|---|---|---|---|---|

Co-occurrence(before refinement) | Pixel nr (% ^{1}) | 1465036 | (100) | 9343 | (100) | 737 | (100) |

Object nr (%) | 19068 | (100) | 968 | (100) | 199 | (100) | |

ρ (ρ_{s}) | 0.5535 | (0.3760) | 0.2406 | (0.1286) | 0.1666 | (0.1656) | |

Binary erosion refinement (4-conn) ^{2} | Pixel nr (%) | 957332 | (65.35) | 3011 | (32.23) | 88 | (11.94) |

Object nr (%)ρ (ρ_{s}) | 11244 | (58.97) | 244 | (25.21) | 24 | (12.06) | |

0.6170 | (0.4456) | 0.3353 | (0.2112) | 0.1479 | (0.1459) | ||

Binary erosion refinement (8-conn) ^{2} | Pixel nr (%) | 810579 | (55.33) | 1865 | (19.96) | 31 | (4.21) |

Object nr (%) | 10162 | (53.29) | 158 | (16.32) | 9 | (4.52) | |

ρ (ρ_{s}) | 0.6416 | (0.4736) | 0.3707 | (0.2536) | 0.3454 | (0.3288) | |

cDDM refinement ^{3} | Pixel nr (%) | 851042 | (58.09) | 2394 | (25.62) | 99 | (13.43) |

Object nr (%) | 16300 | (85.48) | 378 | (39.05) | 46 | (23.12) | |

ρ (ρ_{s}) | 0.6508 | (0.5031) | 0.4824 | (0.4635) | 0.5156 | (0.4353) |

^{1}Percentages refer to the co-occurrence region (pixel or object number) before its refinement.

^{2}3 × 3 structuring element.

^{3}3 × 3 search window.

**Table 2.**Comparison between Lamp-1 and [email protected]

_{ag}intensity and local density colocalization analysis, before and after refinement for local co-density.

Lamp1-Ce6 | |||
---|---|---|---|

Co-Occurrence Region (n * = 737) | Co-Density Region (n * = 99) | ||

Intensity | Density | Intensity | |

ρ | 0.1666 | 0.1278 | 0.5156 |

ρ_{s} | 0.1656 | 0.1270 | 0.4353 |

MOC | 0.1564 | 0.1669 | 0.9059 |

M_{1} | 0.1852 | 0.1662 | 0.0246 |

M_{2} | 0.1712 | 0.1958 | 0.0275 |

**Table 3.**Comparison between NF200 and FM intensity and local density colocalization analysis, before and after refinement for local co-density.

NF200-FM | |||
---|---|---|---|

Co-Occurrence Region (n * = 1,465,036) | Co-Density Region (n * = 851,042) | ||

Intensity | Density | Intensity | |

ρ | 0.5535 | 0.2064 | 0.6508 |

ρ_{s} | 0.3760 | 0.2520 | 0.5031 |

MOC | 0.5741 | 0.7221 | 0.9782 |

M_{1} | 0.4909 | 0.5060 | 0.2983 |

M_{2} | 0.6772 | 0.6601 | 0.4212 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

De Santis, I.; Lorenzini, L.; Moretti, M.; Martella, E.; Lucarelli, E.; Calzà, L.; Bevilacqua, A. Co-Density Distribution Maps for Advanced Molecule Colocalization and Co-Distribution Analysis. *Sensors* **2021**, *21*, 6385.
https://doi.org/10.3390/s21196385

**AMA Style**

De Santis I, Lorenzini L, Moretti M, Martella E, Lucarelli E, Calzà L, Bevilacqua A. Co-Density Distribution Maps for Advanced Molecule Colocalization and Co-Distribution Analysis. *Sensors*. 2021; 21(19):6385.
https://doi.org/10.3390/s21196385

**Chicago/Turabian Style**

De Santis, Ilaria, Luca Lorenzini, Marzia Moretti, Elisa Martella, Enrico Lucarelli, Laura Calzà, and Alessandro Bevilacqua. 2021. "Co-Density Distribution Maps for Advanced Molecule Colocalization and Co-Distribution Analysis" *Sensors* 21, no. 19: 6385.
https://doi.org/10.3390/s21196385