The issue of separating contributions from global bias and local interactions is best illustrated with the alignment of two sets of variables that carry orientational information. Examples of co-orientation include the alignment of two filament networks (Drew et al., 2015; Gan et al., 2016; Nieuwenhuizen et al., 2015), or the alignment of cell velocity and traction stress, a phenomenon referred to as plithotaxis (Das et al., 2015; Tambe et al., 2011; Trepat and Fredberg, 2011). In these systems, global bias imposes a preferred axis of orientation on the two variables, which is independent of the local interactions between the two variables (Figure 1A).

Figure 1

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Similar observed alignments may arise from different levels of global bias and local interactions. This is demonstrated by simulation of two independent random variables X and Y, representing orientations (Figure 1B, left), from which pairs of samples x_i and y_i are drawn to form an alignment angle θ_i (Figure 1B, middle). Then, a local interaction between the two variables is modeled by co-aligning θ_i by ζ_i degrees, resulting in two variables x_i’ and y_i’ with an observed alignment θ_i - ζ_i (Figure 1B, right).

We show the joint distribution of X, Y for four simulations (Figure 1C) where X and Y are normally distributed with identical means but different standard deviations (σ), truncated to ${\displaystyle \left[-{90}^{\circ },{90}^{\circ }\right]}$, and different magnitudes of local interactions (ζ). The latter is defined as ζ = αθ (Figure 1B, α = 1 for perfect alignment). Throughout the simulations both σ and α are gradually increased (Figure 1C, left-to-right), implying that the global bias in the orientational variables is reduced while their local interactions increase. As a result, all simulations display similar observed alignments (mean values, 18.9°−19.5°). Figure 1D visualizes 100 samples from each of the two most distinct scenarios: low σ and no local interaction (σ = 17°, α = 0) leads to tendency of X and Y to align independently to one direction (left); higher variance together with increased interaction (σ = 40°, α = 0.5) leads to more diverse orientations of X and Y (right), while maintaining similar mean alignment. This simple example highlights the possibility of observing similar alignments arising from different mechanisms. While the described properties are well known and many others have used statistical post-processing to eliminate confounding factors for accurate assessment of local interactions (Drew et al., 2015; Helmuth et al., 2010; Karlon et al., 1999; Krishnaswamy et al., 2014; Reshef et al., 2011), we aim at directly quantifying the global bias, with the goal of extracting encapsulated information that is fundamental to the biological question.

DeBias models the observed marginal distributions X’ and Y’ as the sum of contributions by a common effector, i.e., the global bias, and by local interactions that effect the co-alignment of the two variables in every data point (Figure 2A).

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In a scenario without any global bias or local interaction between X’ and Y’, the observed alignment would be uniformly distributed (denoted uniform). Hence any deviation from the uniform distribution would reflect contributions from both the global bias and the local interactions. To extract the contribution of the global bias we constructed a resampled alignment distribution (denoted resampled) from independent samples of the marginal distributions X’ and Y’, which decouples matched pairs (x_i’, y_i’), and thus excludes their local interactions. The global bias is defined as the dissimilarity between the uniform and resampled distributions and accordingly describes to what extent elements of X’ and Y’ are aligned without local interaction (Figure 2B). If a local interaction exists then the distribution of the observed alignment angles will differ from the independently resampled alignment distribution. Hence, the uniform distribution will be less similar to the experimentally observed alignment distribution (denoted observed) than to the resampled distribution. Accordingly, the local interaction is defined by the difference of dissimilarity between the observed and uniform distributions and dissimilarity between the resampled and uniform distributions (Figure 2B).

The Earth Mover's Distance (EMD) (Peleg et al., 1989; Rubner et al., 2000) was used to calculate dissimilarities between distributions. The EMD of 1-dimensional distributions is defined as the minimal 'cost' to transform one distribution into the other (Kantorovich and Rubinstein, 1958). This cost is proportional to the minimal accumulated number of moving observations to adjacent histogram bins needed to complete the transformation. Formally, we calculate ${\displaystyle EMD\left(A,B\right)={\sum }_{i=1,\dots ,K}|{\sum }_{j=1,\dots ,i}{a}_j-{\sum }_{j=1,\dots ,i}{b}_j|}$, with K - number of histogram bins, a_j and b_j - fraction of observations in bin j for distributions A and B correspondingly. Introducing the EMD defines scalar values for the dissimilarities and allows us to define the EMD between resampled and uniform alignment distributions as the global index (GI) and the local index (LI) as the difference of EMD between observed and uniform and the GI (Figure 2B).Figure 2C, demonstrates how the GI and LI recognize the global bias and local interactions between the matched variable pairs (x_i’, y_i’) established in Figure 1C. For a scenario with no local interaction (α = 0) DeBias correctly reports LI~0 and GI~3. For a scenario with gradually wider distributions X,Y, i.e., less global bias, and gradually stronger local interactions (α > 0), the LI increases while the GI decreases. Additional simulations showed that similar properties apply for negative local interactions ζ = αθ (Figure 1B) were α < 0 (Figure 2—figure supplement 1).

In the previous illustrations, changes in spread of the distributions X and Y were compensated by changes in the local interactions. When leaving the interaction parameter α constant while changing the spread of X and Y, a weak, but intrinsically negative correlation between LI and GI becomes apparent (Figure 2D–E). Thus, while DeBias can correctly distinguish scenarios with substantial shifts from global bias to local interactions, the precise numerical values estimating the contribution of LI varies between scenarios with a low versus high global bias. To address this issue we propose to exploit the variation between experiments for modeling the relation between LI and GI. This is demonstrated by comparing two distinct values of the interaction parameter, α, emulating different experimental settings (Figure 2F). Within experiments variation was obtained by drawing multiple values of σ from a normal distribution. Due to the negative correlation between LI and GI the experimental patterns can only be discriminated by combining LI and GI into a two-dimensional descriptor (Figure 2F). This point will be further demonstrated in one of the following case studies and in the Discussion.

To characterize the properties of DeBias we used theoretical statistical reasoning. The first limiting case is set by the scenario in which observations from X and Y are independent. The expected values of the observed and resampled alignments are identical; accordingly, LI converges to 0 for large N (Appendix 1, Theorem 1). The second limiting case is set by the scenario in which X and Y are both uniform. The corresponding resampled alignment is also uniform; accordingly, GI converges to 0 for a large N (Appendix 1, Theorem 2). The third limiting case occurs with perfect alignment, i.e., x_i = y_i for all i. In this case the observed alignment distribution is concentrated in the bin containing θ = 0. We examine two scenarios of perfect alignment: (1) When all the locally matched measurements are identical (x_i = y_j for all i, j), the resampled distribution is also concentrated in the bin θ = 0 implying that LI = 0 and GI assumes the maximal possible value: $GI=\frac{1}{K}{\displaystyle \sum }_{i=1,\mathrm{..},K}\left(i-1\right)=\frac{K-1}{2}$, where K is the number of quantization bins (Appendix 1, Theorem 3.I). (2) When X, Y are uniform (and x_i = y_i for all i), the resampled distribution is uniform, thus GI = 0 and LI reaches its maximum value: $LI=\frac{1}{K}{\displaystyle \sum }_{i=1,\mathrm{..},K}\left(i-1\right)=\frac{K-1}{2}$, (Appendix 1, Theorem 3.II). Generalizing this case, we prove that LI is a lower bound for the actual contribution of the local interaction to the observed alignment (Appendix 1, Theorem 4). Complementarily, GI is an upper bound for the contribution of the global bias to the observed alignment.

Last, we show that when X and Y are truncated normal distributions, or when the alignment distribution is truncated normal, GI reduces to a limit of 0 as σ → ∞, when σ is the standard deviation of the normal distribution before truncation (Appendix 1, Theorem 5). Simulations complement this result demonstrating that σ and GI are negatively associated, i.e., GI decreases with increasing σ (Figure 2E). This final property is intuitive, because resampling from more biased distributions (smaller σ) tends to generate high agreement between (x_i, y_i) leading to reduced alignment angles and increased GI.

The modeling of the observed alignment as the sum of GI and LI allowed us to assess the performance of DeBias from synthetic data. By using a constant local interaction parameter ζ (ζ = c), we were able to retrieve the portion of the observed alignment that is attributed to the local interaction and to compare it with the true predefined ζ (Appendix 2, Appendix 2—figure 1). These simulations demonstrated again the need for a GI-dependent interpretation of LI (first shown in Figure 2E–F). Simulations were also performed to assess how the choice of the quantization parameter K (i.e., number of histogram bins) and number of observations N affect GI and LI (Appendix 2, Appendix 2—figures 2–3), and this was also verified in our experimental data (Figure 3—figure supplement 1). In summary, by combining theoretical considerations and simulations we demonstrated the properties and limiting cases of DeBias in decoupling paired matching variables from orientation data.

We applied DeBias to investigate the degree of alignment between vimentin intermediate filaments and microtubules in polarized cells. Recent work using genome-edited Retinal Pigment Epithelial (RPE) cells with endogenous levels of fluorescently tagged vimentin and α-tubulin showed that vimentin provides a structural template for microtubule growth, and through this maintains cell polarity (Gan et al., 2016). The effect was strongest in cells at the wound front where both vimentin and microtubule networks collaboratively align with the direction of migration (Figure 3A–C). An open question remains as to how much of this alignment is caused by the extrinsic directional bias associated with the collective migration of cells into the wound as opposed to a local interaction between the two cytoskeleton systems.

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Analysis of the GI and LI revealed that most of the discrepancy in vimentin-microtubule alignment originated from a shift in the global bias (Figure 3D), suggesting that the local interaction between the two cytoskeletons is unaffected by the cell position or knock-down of vimentin. Instead, the reduced alignment between the two cytoskeletons is caused by a loss of cell polarity in cells away from the wound edge, probably associated with the reduced geometric constraints imposed by the wound edge. In a similar fashion, reduction of vimentin expression relaxes global cell polarity cues that tend to impose alignment.

To corroborate our conclusion that the global state of cell polarity is encoded by the GI, we performed a live cell imaging experiment, in which single cells at the edge of a freshly inflicted wound in a RPE monolayer were monitored for 80 min after scratching. DeBias was applied to calculate a time sequence of LI and GI. Cells at the wound edge tended to gradually increase their polarity and started migrating during the imaging time frame (Figure 3E, Video 1). Accordingly, the GI increased over time (Figure 3F). We also used this data set to verify that the reported shifts in GI are independent of the number of data points in and the binning of the distribution (Figure 3—figure supplement 1). This demonstrates the capacity of DeBias to distinguish fundamentally different effectors of cytoskeleton alignment.

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Collective cell migration requires intercellular coordination, achieved by mechanical and chemical information transfer between cells. One mechanism for cell-cell communication is plithotaxis, the tendency of individual cells to align their velocity with the maximum principal stress orientation (He et al., 2015; Tambe et al., 2011; Zaritsky et al., 2015). As in the previous example of vimentin and microtubule interaction, much of this alignment is associated with a general directionality of velocity and stress field parallel to the axis of collective migration (Zaritsky et al., 2015).

Using a wound healing assay, Das et al. (Das et al., 2015) screened 11 tight-junction proteins to identify pathways that promote motion-stress alignment (Figure 4A). Knockdown of Merlin, Claudin1, Patj and Angiomotin (Amot) reduced the alignment of velocity direction and stress orientation (Das et al., 2015). Further inspection of these hits showed that the stress orientation remained stable upon depletion of these proteins, but the velocity direction distribution was much less biased towards the wound edge (Zaritsky et al., 2015). Here, we further analyze this data to demonstrate the capacity of DeBias to pinpoint tight-junction proteins that alter specifically the global or local components that induce velocity-stress alignment.

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By distinguishing GI and LI we generated a refined annotation of the functional alteration that depletion of these tight-junction components caused in mechanical coordination of collectively migrating cells (Figure 4B–C). First, we confirmed that the four hits reported by (Das et al., 2015) massively reduced the GI, consistent with the notion that absence of these proteins diminished the general alignment of velocity to the direction induced by the migrating sheet (Figure 4B, red dashed rectangle). Merlin, Patj and Angiomotin reduced the LI to values close to 0, suggesting that the local dependency between stress orientation and velocity direction was lost. Depletion of Claudin1, or of its paralog Claudin2, which was not reported as a hit in the Das et al. screen, reduced the LI to a lesser extent, similarly for both proteins, but had very different effects on the GI (Figure 4B, purple dashed rectangle). This suggested that the analysis by (Das et al., 2015) missed effects that do not alter the general alignment of stress or motion, and implied the existence of a local velocity-stress alignment mechanism that does not immediately change the collective aspect of cell migration.

When assessing the marginal distributions of stress orientation and velocity direction we observed that depletion of Claudin1 reduced the organization of stress orientations and of velocity direction, while Claudin2 reduced only the latter. The LI values of depletion conditions were similar and lower than control (Figure 4D). Merlin depletion is characterized by an even lower LI and marginal distributions with aligned stress orientation and almost uniform alignment distribution (Figure 4D). Since we think that aligned stress is transformed to aligned motion (He et al., 2015; Zaritsky et al., 2015), we propose that in this data the LI quantifies the effect of local mechanical communication on parallelizing the velocity among neighboring cells. Accordingly, stress-motion transmission mechanism is impaired to a similar extent by reduction of Claudin1 and Claudin2, albeit less than by reduction of Merlin.

Using LI as a discriminative measure also allowed us to identify a group of new hits (Figure 4C). ZO1, ZO2, ZO3, Occludin and ZONAB are all characterized by small reductions in GI but a substantial increase in LI relative to control (Figure 4B, orange dashed rectangle). A quantitative comparison of control and ZO1 depleted cells provides a good example for the type of information DeBias can extract: both conditions yield similar observed alignment distributions with nearly identical means, yet ZO1 depletion has an 83% increase in LI and 8% reduction in GI, i.e., the mild loss in the marginal alignment of velocity or stress is compensated by enhanced local alignment in ZO1 depleted cells (Figure 4D). This might point to a mechanism, in which stress orientation is reduced by tight-junction depletion, but enhanced by transmission of stress orientation into motion orientation, leading to comparable alignment. Notably, all paralogs, ZO1, ZO2 and ZO3 fall into the same cluster of elevated LI and slightly reduced GI relative to control experiments. This phenotype is in agreement with the outcomes of a screen that found ZO1 depletion to increase both motility and cell-junctional forces (Bazellières et al., 2015).

Protein-protein co-localization is another ubiquitous example of correlating spatially matched variables in cell biology. To quantify GI and LI for protein-protein co-localization, we normalized each channel to intensity values between 0 and 1. The 'alignment' θ_i of matched observations (x_i,y_i) was replaced by the difference in normalized fluorescent intensities x_i - y_i (Materials and methods). Simulations demonstrated that stronger interactions in co-localization are translated to larger LI values and validated that the choice of K (number of histogram bins) and N (number of observations) marginally affect GI and LI (Appendix 3, Appendix 3—figure 1). While LI could serve as a measure to assess co-localization, the interpretation of GI is less intuitive. In the following, we present two examples of applying DeBias for protein-protein co-localization, and demonstrate the type of information that can be extracted from the combined GI and LI analysis.

To test the potential of DeBias in quantification of pixel-based co-localization, we analyzed the effect of fluorescence resonance energy transfer (FRET) in the C kinase activity reporter (CKAR), which reversibly responds to PKC activation and deactivation (Violin et al., 2003). Reduced PKC activity leads to energy transfer from CFP to YFP_CFP, resulting in reduced FRET ratio ($\frac{CFP}{YF{P}_{CFP}}$) (Figure 5A). Assuming that the CFP signal is dominant (CFP > YFP_CFP), this alteration should reduce the difference between the CFP and YFP_CFP channels, which would in DeBias yield an increased LI (Figure 5A, Materials and methods).

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To test this we labeled hTERT-RPE-1 cells with CKAR and imaged CFP and YFP_CFP channels before and after specific inhibition of PKC with HA-100 dihydrochloride (Figure 5B, Materials and methods), leading to reduced pixel differences in their normalized fluorescent intensities (Figure 5C). As expected, the $\frac{CFP}{YF{P}_{CFP}}$ ratio decreased (Figure 5D), LI values increased (Figure 5E) and seemed more sensitive to the FRET. Surprisingly, DeBias indicated a shift in the GI values (Figure 5F), reflected in a more homogeneous marginal distribution of both channels before inhibition (Figure 5G). Control experiments with cytoplasmic GFP and mCherry expression did not show the shifts observed in LI or GI (Figure 5H). Thus, we conclude that PKC inhibition changes the localization of PKC towards a more random spatial distribution. One possible mechanism for this behavior is that deactivation releases the kinase from the substrate. This example illustrates DeBias’ capabilities to simultaneously quantify changes in local interaction and global bias in pixel-based co-localization.

Clathrin-mediated endocytosis (CME) is the major pathway for entry of cargo receptors into eukaryotic cells. Cargo receptor composition plays an important role in regulating clathrin-coated pit (CCP) initiation and maturation (Liu et al., 2010; Loerke et al., 2009). The clustering of transferrin receptors (TfnR), the classic cargo receptor used to study CME, promotes CCP initiation, in concert with clathrin and adaptor proteins (Liu et al., 2010). Recent evidence suggests a diversity of mechanisms regulating endocytic trafficking, including cross-talks between signaling receptors and components of the endocytic machinery (Di Fiore and von Zastrow, 2014). For example, the oncogenic protein kinase Akt has been shown to play an important role in mediating CME in cancer cells (Liberali et al., 2014; Reis et al., 2015), but not in normal epithelial cells (Reis et al., 2015). Here we tested how the decoupling by DeBias of global and local contributions to the overall intensity alignment of clathrin and TfnR, can be used to simultaneously investigate co-localization and predict CCP dynamics, using fixed cell fluorescence imaging.

We used fluorescence images of fixed non‐small lung cancer cells (H1299) or untransformed human retinal pigment epithelial cells (ARPE-19) expressing clathrin light chain A fused to eGFP (eGFP-CLCa) as a CCP marker (Figure 6A–B). Cells were either treated with DMSO or with an AKT inhibitor (Akt inhibitor X, ‘ten’), and imaged by Total Internal Reflection Fluorescence Microscopy (TIRFM). CCPs were reported in the eGFP-CLCa channel and TfnR was visualized by immunofluorescence in a second channel (Materials and methods). For single cells, the location of fluorescent signals of CLCa and TfnR were recorded and the data were pooled and processed by DeBias (Materials and methods).

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LI values, indicative of the co-localization between TfnR and CLCa, were significantly lower in Akt-inhibited H1299 compared to control cells (Figure 6C). In contrast, Akt inhibition increased the LI values in ARPE-19 cells but this effect was less prominent (Figure 6D). Akt inhibition resulted in increased GI values for both cell lines, to a much greater degree in H1299 cells (Figure 6C–D). To test whether GI enhances the ability to distinguish between control and Akt-inhibited cells, we applied Linear Discriminative Analysis (LDA) classification to calculate the true positive rate versus the false positive rate for LI alone (black lines, Figure 6E–F) or the pair (GI, LI), (orange lines, Figure 6E–F). The area under these curves (AUC) provided a direct measure of the ability of each method to accurately classify the experimental condition of single cells. AUC for the (GI, LI) representation was superior to using LI alone for both cell lines (H1299: 0.96 versus 0.88, Figure 6E; ARPE-19: 0.83 versus 0.72, Figure 6F). Similar benefit in discrimination was achieved when using the GI to complement Pearson’s correlation as an alternative to LI for measuring local interaction (Figure 6—figure supplement 1A–F). Such improved discrimination is indicative of distinct molecular processes that were altered upon Akt inhibition. We also used this data set to experimentally validate the independence of GI and LI of the number of observations (N, Figure 6—figure supplement 1G,H) and the choice of the number of histogram bins (K, Figure 6—figure supplement 1I,J).

To interpret the increased GI values for Akt-inhibited cells, we examined the joint and marginal distributions of CLCa and TfnR. Upon Akt-inhibition, the joint distributions were more biased toward regions of low TfnR intensities (Figure 6G–H). This was clearly observed in the marginal distributions (Figure 6I–J). Hence, although the CLCa distribution appeared not to change upon AKT inhibition, the frequency of CCPs with fewer TfnRs increased. Given the positive relation between TfnR cargo quantities and CCPs maturation (Loerke et al., 2009), we wondered whether the increased frequencies of CCPs containing less TfnR might alter CCPs dynamics. Indeed, live-imaging of H1299 cells showed a higher frequency of CCPs with shorter lifetimes upon Akt-inhibition, which was not seen in normal ARPE-19 cells (Figure 6K–L, Materials and methods). It has previously been shown that Akt inhibition reduces the rate of TfnR CME uptake in H1299 cells, but not in ARPE-19 cells ([Reis et al., 2015], see also Figure 6M–N); therefore, these findings indicate that the reduced levels of TfnR in CCPs upon Akt inhibition results in an increase in short-lived, most likely abortive events, and hence a decrease in CME efficiency.

Altogether, DeBias could distinguish alterations in the regulation of CME between two cell types. The decoupling to GI and LI indicated that upon Akt inhibition, both untransformed and cancer cells showed a global bias towards CCPs with lowered TfnR intensities. This conclusion could not have been reached by considering only the LI, which increased for normal and decreased for transformed cell lines.

The DeBias procedure is depicted in Figure 2A. The marginal distributions X and Y are estimated from the experimental data, $\forall i,x_i,y_i\in \left[0,90°\right]$. The experimentally observed alignment distribution (denoted observed) is calculated from the alignment angles θ_i of matched (x_i,y_i) paired variables, for all i.

The resampled alignment distribution (denoted resampled) is constructed by independent sampling from X and Y. N random observations (where N = |X| is the original sample size) from X and Y are independently sampled with replacement, arbitrarily matched and their alignment angles calculated to define the resampled alignment. This type of resampling precludes the local dependencies between the originally matched (x_i,y_i) paired variables.

The uniform alignment distribution (denoted uniform) is used as a baseline for comparison between distributions. This is the expected alignment distribution when neither global bias (reflected by uniform X, Y distributions) nor local interactions exist. The Earth Mover's Distance (EMD) (Peleg et al., 1989; Rubner et al., 2000) was used as a distance metric between alignment distributions. The EMD for two distributions, A and B, is defined as follows:

${\displaystyle EMD\left(A,B\right)=\sum _{j=1,\dots ,K}|\sum _{j=1,\dots ,i}{a}_j-\sum _{j=1,\dots ,i}{b}_j|}$, where $a_j$ and $b_j$ are the frequencies of observations in bins $j$ of the histograms of distributions A and B, respectively, each containing K bins.

The global index (GI) is defined as the EMD between the uniform distribution and the resampled alignment:

GI = EMD(uniform,resampled)

The local index is determined by subtraction of the global index from the EMD between the uniform distribution and the experimentally observed alignment distribution:

LI = EMD(uniform,observed) - global index

The following adjustments to this procedure are implemented to allow DeBias to quantify protein-protein co-localization:

1. Levels of fluorescence are not comparable between different channels due to different expression levels and imaging parameters. Thus, each channel is normalized to [0,1] by the fifth and 95th percentiles of the corresponding fluorescence intensities to achieve a stable and robust distribution.

2. The alignment angle θ_i of the matched observation (x_i,y_i) is calculated as the difference in normalized fluorescence intensities x_i - y_i and the alignment distribution is thus defined on the interval [−1,1].

The number of histogram bins for the alignment distributions (observed, resampled and uniform) was K = 15 for orientational data, 19 for PKC and 40 for CME co-localization data.

The Freedman-Diaconis rule (Freedman and Diaconis, 1981) was used to automate the selection of histogram bin width: ${\displaystyle binsize=\frac{Q_3\left(x\right)-Q_1\left(x\right)}{\sqrt[3]{n}}}$, where Q_i is the i^th quartile of the empirical distribution x and n is the number of data points contained. A function to calculate K is included in our publicly available source code and this functionality was also integrated to the web-server implementation. Importantly, GI and LI across experiments can be compared only when evaluated with the same K value and this is enforced by the web-server. It is the responsibility of the source-code user to validate using the same K values when comparing different experimental conditions.

Coupled measurements of velocity direction and stress orientation were taken from the data originally published by Tamal Das et al. (Das et al., 2015). Particle image velocimetry (PIV) was applied to calculate velocity vectors, and monolayer stress microscopy (Tambe et al., 2011) was used to extract stress orientations. Velocity and stress measurements were recorded 3 hr after collective migration was induced by lifting off the culture-insert in which the cells have grown to confluence. Validated siRNAs were used for gene screening. Detailed experimental settings can be found in Das et al., 2015.

hTERT-RPE-1 cells (ATCC, RRID: CVCL_4388) expressing GFP and mCherry (for the control experiment) or C kinase activity reporter (CKAR, for the PKC activation experiment) (Violin et al., 2003) were plated with DMEM/F12 medium containing 10% fetal bovine serum and 1% penicillin-streptomycin in fibronectin-coated 35 mm MatTek plates (P35G-0–10 c). The cell line has been tested negative for mycoplasma contamination. Cells were incubated overnight and imaged with a custom-built DeltaVision OMX widefield microscope (GE healthcare life sciences, Chicago, IL) equipped with an Olympus PLAN 60 × 1.42 N.A. oil objective and CoolSNAP HQ2 interline CCD cameras. FRET experiments were performed with a 445 nm laser, and control experiments were performed with 488 and 561 nm lasers. 478/35, 541/22, 528/48 and 609/37 emission bandpass filters were used for the CFP, YFP, GFP and mCherry channels, respectively. The output powers of the lasers were set to 10% the maximal output (100 mW). The exposure time was 200 ms per frame for both channels. 10 µM of the PKC inhibitor HA-100 dihydrochloride (Santa Cruz Biotechnology, Dallas, TX) was added to the media after the first image was recorded and a second image was recorded 20 min later.

Single cells were manually selected for analysis. In particular, cells with higher intensities in the CFP channel were found to provide reproducible changes in their FRET intensity. Each cell was manually annotated and analyzed with the Biosensor Processing Software 2.1 to produce the ratio images (Hodgson et al., 2010). Briefly, the field of view was corrected for uneven illumination, background was subtracted, the image was masked with the single cell annotation, and the ratio image was calculated as CFP/YFP_CFP. Statistics was determined using the non-parametric Wilcoxon signed-rank test.

The DeBias code was implemented in Matlab, compiled with Matlab complier SDK and transferred to a web-based platform to allow public access for all users at https://debias.biohpc.swmed.edu. The graphical user interface (GUI) was designed to be simple and easy to use. The user uploads one or more datasets to the DeBias webserver and selects the mode of operation (co-localization/orientation). GI and LI values are calculated and the results are displayed and emailed to the user. ‘DeBias Analyst’ enables to group experiments into two experimental conditions (usually control versus treatment), visualizes and outputs statistics on the alterations of GI and LI. The software’s flow chart and a detailed user manual are available in the online user manual. Source code is publicly available, https://github.com/DanuserLab/DeBias Danuser, 2017 (with a copy archived at https://github.com/elifesciences-publications/DeBias).

To assess the performance of DeBias we tested its ability to retrieve a pre-determined local interaction parameter ${\displaystyle \zeta }$ (see Figure 1B) from simulated synthetic data. ${\displaystyle \mathrm{X}}$ and ${\displaystyle \mathrm{Y}}$ were modeled as truncated normal distributions on (−90, 90), with $\mu$=0 and changing ${\displaystyle {\sigma }_{\mathrm{x}},{\sigma }_{\mathrm{y}}}$. Pairs of (${\displaystyle {\mathrm{X}}_{\mathrm{i}},{\mathrm{Y}}_{\mathrm{i}}}$) were sampled from ${\displaystyle \mathrm{X},\mathrm{Y}}$ and shifted towards each other by ${\displaystyle \zeta }$ degrees (similar to Figure 1B, but with a constant cumulative ${\displaystyle \zeta }$) to construct the observed alignment angles. To avoid confusion we denote ${\displaystyle \mathrm{X},\mathrm{Y},{\sigma }_{\mathrm{x}},{\sigma }_{\mathrm{y}}}$ as the observed values post-simulation. For a given constant ${\displaystyle \zeta }$, we exhaustively explored the ${\displaystyle {\sigma }_{\mathrm{x}},{\sigma }_{\mathrm{y}}}$ space. For each ${\displaystyle {\sigma }_{\mathrm{x}},{\sigma }_{\mathrm{y}}}$, we performed 20 independent simulations with N = 1600 observations (${\displaystyle {\mathrm{X}}_{\mathrm{i}},{\mathrm{Y}}_{\mathrm{i}}}$). For each simulation we constructed the resampled distribution 10 times based on 400 observations drawn from the marginal ${\displaystyle \mathrm{X},\mathrm{Y}}$ distributions, and used the mean GI, LI. The final recorded GI, LI were averaged over the independent simulations.

The expected mean alignment when neither global bias nor local interactions exist is 45°. We begin by examining the deviation of the mean observed alignment from this value (45° - θ_mean). Better alignment is reflected by higher 45° - θ_mean values implying a larger deviation from the unbiased and no-interactions scenario. Low standard deviations ${\displaystyle \sigma }$, correspond to better alignment, improving with growing ${\displaystyle \zeta }$, as expected (Appendix 2—figure 1A). The GI follows a similar pattern and remains relatively stable for small changes in ${\displaystyle \zeta }$ (Appendix 2—figure 1B). The similar patterns between Appendix 2—figure 1A and B indicate that the global bias has a prominent role in determining the observed alignment.

The LI grows with ${\displaystyle \zeta }$ (Appendix 2—figure 1C) and its relative contribution to the observed alignment grow with increasing ${\displaystyle \zeta }$ (Appendix 2—figure 1D, quantified by LI/(LI+GI)), as expected. This relative contribution can be harnessed to restore an estimated ${\displaystyle \zeta }$ as the corresponding fraction from (45° - θ_mean) (Appendix 2—figure 1E). The estimated ${\displaystyle \zeta }$ is a lower bound for the actual value (Appendix 1, Theorem 4). Estimation is more accurate for larger ${\displaystyle \zeta }$ and for large ${\displaystyle \sigma }$ (Appendix 2—figure 1F). These results again highlight the importance of exploiting the GI for better interpretation of the LI (first introduced in Figure 2D–E).

Appendix 2—figure 1

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We also investigated the effect of the choice of the number of bins K, used for sampling and computation of the EMD between distributions. Increased K induces linear growth in LI and GI values, as expected (Appendix 1 Theorem 5, Appendix 2—figure 2A–B) and stabilized its accuracy in predicting ${\displaystyle \zeta }$ for K ≥ 11 (Appendix 2—figure 2C–E). Large K will require more observations to estimate the true distribution. Using a constant K for a specific application assures fair comparison between different cases. Varying N, the number of observations, did not have a major effect on these measurements (Appendix 2—figure 3A–D), but increasing N reduced the noise which increased the accuracy in predicting ζ (Appendix 2—figure 3E).

Appendix 2—figure 2

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Appendix 2—figure 3

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Co-localization of molecules is commonly used to predict potential local interactions under the assumption that the stoichiometry of interacting molecules remains constant across all observations. We demonstrated this by simulating a random variable ${\displaystyle \mathrm{X}}$, representing molecular counts, and a random variable ${\displaystyle \mathrm{Z}}$, representing the local interaction. Pairs of samples ${\displaystyle {\mathrm{x}}_{\mathrm{i}}}$ from ${\displaystyle \mathrm{X}}$ and ${\displaystyle {\zeta }_i}$ from ${\displaystyle \mathrm{Z}}$were drawn, and ${\displaystyle y_i=x_i{\zeta }_i}$ represented the molecular counts of the co-localized molecule (Appendix 3—figure 1A). The joint distributions of ${\displaystyle \mathrm{X},\mathrm{Y}}$ for four simulations are shown in Appendix 3—figure 1B. X is normally distributed with mean ${\displaystyle \mu }$ = 0.5 and standard deviations ${\displaystyle \sigma }$ = 0.2, truncated to [0,1]. Z is normally distributed with mean ${\displaystyle \mu }$ = 1 and different standard deviations, simulating gradually increasing local interactions (Appendix 3—figure 1B, left-to-right). Applying DeBias on these data demonstrated that increased local interactions (reduced ${\displaystyle {\sigma }_{\zeta }}$) are translated to increased LI (Appendix 3—figure 1C).

Appendix 3—figure 1

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We next simulated a scenario, in which a partial subset of observations ${\displaystyle {\mathrm{y}}_i}$ interacted with ${\displaystyle {\mathrm{x}}_i}$; the remaining ${\displaystyle {\mathrm{y}}_i}$ were drawn independently from the distribution ${\displaystyle \mathrm{Y}=\mathrm{X}}$. We found that LI values increased with the fraction of interacting observations (Appendix 3—figure 1D).

Finally, we assessed how variation in the quantization parameter (K) and number of observations (N) alter GI and LI (Appendix 3—figure 1E-F) and this was also demonstrated in our experimental data (Figure 6—figure supplement 1G–J).

Many methods have been developed to quantify protein-protein co-localization. Pixel-based methods measure pixel-wise correlation coefficients (Adler and Parmryd, 2010; Bolte and Cordelières, 2006; Costes et al., 2004; Manders et al., 1993; Pearson, 1901), exploiting the notion that fluorescence levels of co-localized proteins are correlated. They suffer from background signal where no co-localization exists. Object-based methods first detect objects of interest and then assess co-localization based on second-order statistics of the spatial distributions of the detections (Helmuth et al., 2010; Kalaidzidis et al., 2015; Lagache et al., 2015; Rizk et al., 2014; Semrau et al., 2011). Object-based methods remove the background pixels but lose the information contained by the fluorescence levels at the detected objects. Thus, object-based methods are best applicable for the co-localization of binary signals (Lagache et al., 2015), but less suited for applications in which co-localization accounts for coupling of molecular counts on a continuous spectrum. Moreover, object-based methods require detection of objects in both channels, which often limits their applicability. Pros and cons for using either of these methods are presented in Appendix 4—Table 1. In the examples of receptor-CCP co-localization we implemented a hybrid of the two approaches: co-localization analysis by DeBias focused on the intensity of fluorescent readouts within detected CCPs to decompose the coupling of the two intensity variables into LI and GI (Figure 6). The same decomposition was demonstrated for a diffuse signal without objects in the PKC-FRET example (Figure 5).

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank the BioHPC team at UTSW and especially Liqiang Wang for the help in implementing the DeBias web-server and making it freely available for public use. We are grateful to Tamal Das and Joachim Spatz for providing us with the motion and stress data from their tight-junction screen. We thank Andrew Jamieson for helping to package the source code and creating the repository. We thank Dana Reed for assistance with cell characterization and mycoplasma testing. We thank Sangyoon Han, Claudia Schaefer, Meghan Driscoll, Erik Welf, Phillippe Roudot and Marcel Mettlen for critically reading the manuscript and Marcel Mettlen for fruitful discussions and advice. This work was supported by the Cancer Prevention and Research Institute of Texas (CPRIT R1225 to GD), by NIH P01 GM103723, P01 GM096971 (to GD) and by GM713165 (to SLS and GD). UO was supported by an EMBO Long-Term fellowship. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

© 2017, Zaritsky et al.

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