We present a software plugin to analyze and quantify spatial patterns of objects in images using the free open-source image-processing platform ImageJ [1] or its distribution Fiji [2]. The spatial arrangement of objects relative to each other is a rich source of phenotypic information. This ranges from spatial patterns of sub-cellular structures or proteins, to the spatial patterns formed by cells in tissues, to spatial patterns of organisms in ecosystems. The mathematical framework of spatial statistics allows quantifying and analyzing such patterns, comparing them with each other, and performing statistical tests on them [3-5]. This for example allows testing whether the distribution of a set of objects is significantly different from random, or whether the objects in one set are distributed independently of the objects in another set. Significant deviations from spatial randomness are indicative of interactions (of some sort) between the objects, as formalized in the framework of spatial interaction analysis [6].

Biology has long relied on co-localization analysis in order to quantify the spatial distribution of one set of objects with respect to another one. This includes pixel-based and object-based co-localization analysis methods [7]. Pixel-based methods typically use a correlation measure between the pixel intensities in different images in order to quantify the degree of overlap or co-localization between the object distributions represented in the images. Object-based methods first detect and delineate the objects of interest in the images and then quantify their degree of co-localization using an overlap measure. While pixel-based measures are easy to compute, they are difficult to interpret. They are also sensitive to blurring and noise in the image. Moreover, in some observations, like those from Photo-Activation Localization Microscopy (PALM) and STochastic Optical Reconstruction Microscopy (STORM) [8-10], it is not obvious what constitutes an image [11], as these methods provide locations and localization uncertainties of individual molecules. Pixel-based methods are hence not directly applicable, unless one first renders a synthetic image from the observed point locations. This, however, leads to a loss of information as nearby molecule detections fuse into one blob in the rendered image. Object-based methods are more intuitive to interpret, as they directly work with locations of objects. However, they depend on prior object detection and segmentation. It is also not clear when two object should be considered “overlapping”. This requires defining a distance threshold, which typically is ≈200 nm for diffraction-limited data [12]. Single-molecule imaging, like PALM and STORM, directly provides point locations, rendering a separate object-detection step unnecessary.

While co-localization analysis captures the amount of overlap between objects, it is not sufficient to describe spatial patterns and interactions [6]. Also, co-localization measures do not account for the fact that accidental overlap occurs even in randomly distributed objects and becomes more frequent as the object density increases. This typically leads to a density bias in the final co-localization score.

Spatial interaction analysis

Helmuth et al. [6] have generalized co-localization analysis to interaction analysis, which corrects for accidental overlaps, is robust against imaging noise and image-processing errors, does not require defining a distance threshold, and is able to capture patterns also at larger length scales than just immediate overlap. They used a nearest-neighbor interaction model based on spatial Gibbs statistics, which characterizes an observed pattern by an interaction potential. This is the potential that is most likely to produce the observed distribution of nearest-neighbor (NN) distances between objects if the objects interact pair-wise according to this potential. Compared to object-based co-localization analysis, the interaction analysis model uses additional information that is present in the data, namely nearest-neighbor distance distributions also between non-overlapping objects. The model is hence able to better fit an observed distance distribution than a co-localization score would be (see Figure 1a). The method also provides standardized ways to (1) correct for the influence of the distribution of points within one set onto the distance distribution to another set; (2) infer parameters of the interaction potential, such as the strength and length scale of the interaction; (3) perform statistical hypothesis tests for the presence of an interaction.

The notion of “interaction” used here is purely geometric. We say that two sets of objects interact if the spatial distribution of one set is not independent of the distribution of the other set (see Figure 1b). The objects do not interact if the distribution of one set can be explained by a random distribution that does not depend on the other, the reference set (see Figure 1c). These interactions or spatial patterns need not imply causal interactions.

The plugin is written in Java. It uses the open-source Java library Weka [13] for efficiently computing NN distances using kd-trees and for kernel density estimation of probability distributions. The plugin further uses CMA-ES (the Covariance Matrix Adaptation Evolution Strategy) [14] for parameter optimization. This optimizer is less prone to get stuck in local optima than the simplex method used in the original publication [6]. The plugin has been tested with both ImageJ and Fiji on Windows, MacOS X, and Linux. It should run on any platform where Java and ImageJ are available. Running a complete analysis using the present plugin takes between a few seconds and a few minutes, depending on the number of objects present in the image.

The user interface of MosaicIA is shown in Figure 2. Its workflow is explained in the flowchart in Figure 3. The plugin reads either images (2D or 3D), or comma-separated text files containing the coordinates of objects. The user can create or load a mask to restrict the analysis to a certain region of interest, if necessary. If the analysis is based on images, bright points in the images are first detected using the feature-point detector by Sbalzarini and Koumoutsakos [15], which is also available in Java as an ImageJ plugin and processes both 2D and 3D images. Then, the NN distance distribution D between the two point sets is computed and the context q(d) estimated using grid sampling [6]. This means that q(d) is approximated by computing the NN distance from each point on a regular Cartesian lattice to the nearest neighbor in Y. The grid resolution has to be set by the user. Finer grids lead to more accurate results, but require more computer time. Smooth representations of the observed (empirical) NN distance distribution $\widehat{p}\left(d\right)$, and of the sampled context q(d) are obtained by kernel density estimation. The estimator kernel widths are set by the user, but the software provides an initial guess calculated with Silverman’s rule [16].

The plugin currently works with point objects only, even though the interaction analysis framework generalizes to extended objects too [6]. In order to work with non-point objects in the present plugin, they can be segmented using other software, e.g. the Region Competition plugin for ImageJ [17], and their centers of mass can be read into the present plugin. Representing extended objects by their centers of mass does not significantly change the result of the interaction analysis, as shown in the PALM example below.

The type of the potential function can be selected from a drop-down menu. Both parametric and non-parametric functions are provided. The plugin then estimates the potential or its parameters from the data by minimizing the ℓ² difference $\parallel \widehat{p}\left(d\right)-p\left(d\right){\parallel }_2$ between the observed NN distance distribution $\widehat{p}\left(d\right)$ and the one predicted from the interaction model p(d). The results, including the residual fitting error, are then shown along with a plot of $\widehat{p}\left(d\right),p\left(d\right)$, and q(d), as shown as in Figure 2b,c.

The plugin also provides hypothesis tests to check whether the estimated interaction is statistically significant. The test statistic is computed from K Monte Carlo samples of point distributions corresponding to the null hypothesis of “no interaction”. The test statistic from the actually observed distribution is ranked against these K random samples. If it ranks higher than ⌈(1 - α)K⌉-th, the null hypothesis of “no interaction” is rejected on the significance level α[6].

We validated and tested the plugin on synthetically generated point distributions in the presence and absence of interactions and confirmed that interactions were correctly detected (results not shown). We show here the application of the plugin to two real-world cases: interactions between viruses and endosomes in HER-911 cells as inferred from fluorescence confocal microscopy images, and clathrin-GPCR interactions as inferred from PALM super-resolution data in HeLa cells.

Application to virus-endosome interaction from confocal images

We apply the plugin to analyze the interactions between human adenoviruses of serotype 2 (Ad2), stained with ATTO-647, and Rab5-EGFP-stained endosomes in HER-911 cells (image data: Greber lab, University of Zurich). Similar data has also been used in Ref. [6].

The results are shown in Figure 4. Figure 4a shows the image X of the virus particles after object detection in the plugin. The image Y of the endosomes after object detection is shown in Figure 4b. Figure 4c shows the observed NN distance distribution (blue curve), the expected distribution if viruses were distributed at random and independently of the endosomes, i.e., the context (red curve), and the best fit (green curve) with the Plummer potential shown in Figure 4e. Figure 4d shows the results when using the step potential shown in Figure 4f, corresponding to a context-corrected object-based co-localization count. The residual fitting error when using the Plummer potential is about 4-fold lower than when using the step potential, even though the latter also corrects for the context. The improvement stems from using continuous distance information. The error when using classical co-localization analysis without correcting for the context would be even larger.

Application to GPCR-clathrin interaction from PALM data

Super-resolution microscopy techniques such as PALM and STORM do not provide images, but produce point clouds by measuring the coordinates of individual fluorophores and their localization uncertainties. Directly working with these position data provides more information than first rendering an artificial image from them and then working with that image [11].

G-protein-coupled receptors (GPCRs) are important signaling proteins that are transported in clathrin-coated vesicles. The sizes of these vesicles are typically below the resolution limit of classical microscopy, rendering them a good system to be studied with super-resolution techniques like PALM and STORM.

We analyze the prototypical GPCR β2-adrenergic receptor (β2-AR) and its internalization in clathrin-coated vesicles post stimulation with the agonist isoproterenol in HeLa cells [19]. β2-AR is labelled with PSCFP2, and clathrin light chains are labeled with PAMCherry1 [19]. Figure 5a shows an exemplary rendered probability map from dual-color PALM after setting a clustering threshold to remove localized molecules that are not within a cluster of at least the threshold size [19]. The estimated locations of these individual fluorescent molecules are shown as dots in Figure 5b, without the localization uncertainty distributions. Circles mark clusters of fluorophores with the cluster centers given by the crosses.

We can either analyze the interactions between the individual molecules in one color channel with those in the other channel, or we may exploit the biological knowledge that the actual interaction acts at the organelle level rather than the molecular level. For the latter, we hence analyze the interactions between the centers of the detected clusters across the two color channels. For the sake of simplicity, we do not explicitly model localization uncertainty, registration errors, and limited detection efficiency.

The results corresponding to the parametric potential that provided the best fit (in this case an L1 linear potential) are shown in Figure 5. Figures 5c,d show the results of the analysis based on cluster centers, Figures 5e,f those based on individual molecules. In both cases, the residual fitting error is 5-fold lower than when using a step potential. Figures 5g,h show controls obtained by randomly scrambling the cluster-center locations. We see that: (1) the plugin correctly infers an interaction (estimated ε=1.85) in the data with cluster centers, but detects no interaction in the randomized control (ε=0.6; the null hypothesis of no interaction has rank 512 of 1000 and cannot be rejected). (2) The results when applying the analysis to individual molecules or to cluster centers are similar, with the former estimating ε=2.0, indicating that the analysis is robust against clustering effects and correctly identifies the length scale of the interaction. A randomization control based on individual molecules yields ε=0.3 (data not shown), and the null hypothesis of no interaction cannot be rejected (rank 202 of 1000). This indicates that analyzing interactions between organelles by considering their cluster centers, rather than individual molecules, may be sufficient to distinguish an interacting pattern from a non-interacting one. Analysis with other potential shapes provided similar results.

We presented MosaicIA, an ImageJ/Fiji plugin for spatial point pattern and interaction analysis. The plugin takes as an input two 2D or 3D images showing the spatial distributions of two sets of bright, spot-like objects, or it reads the coordinates of two sets of objects from files. It then uses a nearest-neighbor Gibbs interaction model from spatial statistics in order to infer the pair-wise interaction potential that is most likely to create the observed distribution of objects. Compared to classical pixel-based or object-based co-localization analysis, this makes better use of the information present in the image and hence provides superior statistical detection power [6]. The analysis also accounts for the context created by the shape of the space within which the objects are distributed and by the object distribution within the reference set. Estimating interaction models is more robust against imaging noise and image-processing errors than classical co-localization analysis [20]. Statistical tests are provided by the plugin in order to check the significance of an interaction. The estimated interaction parameters provide a quantitative way of comparing spatial patterns across different samples, conditions, and perturbations.

The plugin has been tested on both synthetic and real-world data. We demonstrated its application to virus trafficking data obtained with confocal fluorescence microscopy and to PALM super-resolution data of the interaction between clathrin and β2-AR. In the latter case, we compared the results from applying the analysis to individual points with the results obtained from cluster centers. In all tested cases, the best interaction potential explained the data 4 to 5-fold better than a step potential, i.e., than context-corrected object-based co-localization analysis. Without context correction, the fits would be even worse.

Despite the fact that we have here only demonstrated the plugin for fluorescence microscopy and PALM data, it can be applied to distributions of any type of objects (organelles, cells, organisms), as long as their positions can be extracted from images or read from files. For example, in the case of cells in a tissue, it can be applied with the help of segmentation methods that can provide the spatial location that best fits a cell [21]. Comparing the model fits obtained with different parametric potentials can also be used to test and compare hypothetical interaction mechanisms directly on the data.

Future work includes extending the plugin to also handle extended objects that are not blob-like, and to explicitly account for localization uncertainties and registration (aberration) errors. Other useful extensions could be testing for spatial randomness within a single set of objects from the intra-set NN distance histogram, and automatic estimation of algorithm parameters from the data.

The present plugin is available as part of the MOSAICsuite for ImageJ and Fiji.

Project name: MosaicIA Project home page: http://mosaic.mpi-cbg.de/?q=downloads/imageJ Fiji update site:http://mosaic.mpi-cbg.de/Downloads/update/Fiji/MosaicToolsuiteOperating system(s): Linux, MacOS X, Windows Programming language: Java Other requirements: ImageJ or Fiji License: GNU LGPL v3. Please cite this paper as well as Ref. [6] in any publications that use this software. Any restrictions to use by non-academics: licenseneeded