Journal of Microscopy, Vol. 225, Pt 3 March 2007, pp. 214–228

Received 22 December 2005; accepted 30 August 2006

Visualization and quantification of vesicle trafficking on athree-dimensional cytoskeleton network in living cells

V I C T O R R A C I N E §, M A RT I N S A C H S E†, J E A NS A L A M E R O†§, V I N C E N T F R A I S I E R §, A L A I N T R U B U I L‡ &J E A N - B A P T I S T E S I B A R I TA §§Tissue and Cell Imaging and †Molecular Mechanisms of Intracellular Transport groups, UMR144,

Institut Curie 26 rue d’Ulm, 75005 Paris, France‡Unit e Mathematiques et Informatique Appliquees, Dommaine de Vilvert, 78352 Jouy-en-Josas,France

Key words: Kymogram, Membrane dynamics, object tracking, 3D + tvizualisation.

Summary

Recent progress in biology and microscopy has made it possibleto acquire multidimensional data on rapid cellular activities.Unfortunately, the data analysis needed to describe theobservedbiologicalprocessstill remainsamajorbottleneck.Wehere present a novel method of studying membrane traffickingby monitoring vesicular structures moving along a three-dimensional cytoskeleton network. It allows the dynamicsof such structures to be qualitatively and quantitativelyinvestigated. Our tracking method uses kymogram analysisto extract the consistent part of the temporal information andto allow the meaningful representation of vesicle dynamics.A fully automatic extension of this method, together with astatistical tool for dynamic comparisons, allows the preciseanalysis and comparison of overall speed distributions anddirections. It can handle typical complex situations, such asa dense set of vesicles moving at various velocities, fusing anddissociating with each other or with other cell compartments.The overall approach has been characterized and validated onsynthetic data. We chose the Rab6A protein, a GTPase involvedin the regulation of intracellular membrane trafficking, asa molecular model. The application of our method to GFP-Rab6A stable cells acquired using fast four-dimensionaldeconvolution video-microscopy gives considerable cellulardynamic information unreachable using other techniques.

1. Introduction

Rapid 3D + time (4D) microscopy has enormous potentialto increase our understanding of biological processes. The

Correspondence to: Jean-Baptiste Sibarita. Tel: +33 14 234 64 49; fax: +33 14 234

63 19; e-mail:JBS@curie.fr.

acquisition of 3D data over time followed by deconvolutionis a powerful tool for studying the spatial and temporalrelationships between the labelled structures. Membranetrafficking is a complex process that involves the movement ofvesicles from donor to acceptor compartments. It is responsiblefor the exchange and recycling of biological material inside thecell. The dynamics of the molecular mechanisms responsiblefor these processes can be followed by studying fluorescentproteins expressed by chimeric genes. These proteins canthen be localized over time and under various physiologicalconditions. However, in intracellular trafficking the complexityof the images obtained makes interpretation of mechanisticprocesses challenging.

As a biological model, we studied the complex dynamicsof the small GTPase protein Rab6A, which regulates specificsteps in membrane trafficking (White et al., 1999). This Rabprotein exists in three different forms: bound to vesicularstructures [membrane structures between 60 nm and 1.5 μmin diameter (unpublished data)], bound to the Golgi apparatus(a large and stable membrane structure about 12 μm indiameter) or free in the cytosol. Motor proteins can movevesicles extremely rapidly (several μm/s) along microtubules.Cells expressing Rab6A-GFP have a density of about one vesicleper μm3, heterogeneously distributed and concentrated on themicrotubule network (i.e. the cell shown in the supplementarydata has a measured mean density of one vesicle per 1.5 μm3).

The understanding of such complex cellular dynamicsrequires appropriate imaging techniques and subsequentvisualization and quantification tools. Object tracking is usedto quantify cellular structures movement. The most commonlyused technique is the connexionist approach (Anderson et al.,1992; Thomann et al., 2003; Vallotton et al., 2003). This isa two-step process: first, objects are independently detected

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in each frame, and secondly, their movement over time arecomputed by linking objects from one frame to objects inthe next frame. In the statistical framework, the two best-known approaches are the Joint Probabilistic Data-AssociationFilter (Fortmann et al., 1983) and the Multiple HypothesisTracker (Reid, 1979). In biology, such framework has beensuccessfully proposed (Genovesio & Olivo-Marin, 2004), buttheir major drawbacks are the large number of parametersand the assumption about probability distribution which doesnot necessarily hold (Cox, 1993). These general automatictracking methods did not produce satisfying results for the 4Dacquisition of Rab6A due to the high speed of moving vesicleswith respect to the 3D acquisition time (1 stack/s) and due tothe high number of fusion dissociation events. Also, manualtracking is very difficult when particle densities are high, istoo time consuming for 3D data and does not produce enoughdata to perform statistical analysis.

The visualization tools must provide the maximuminformation in a very comprehensive way. As the screenof a computer is two dimensional, 4D data are normallydisplayed by reducing the number of dimensions. In imagesacquired using fluorescence microscopy, maximum intensityprojections along the z-axis, sometimes called 2.5 dimensions,and video over time are the most commonly used visualizationtools. These methods display the global data trends withoutlosing too much information. In specific cases, the user canchoose to restrict the visualization to specific regions of interestto reduce the number of dimensions. For example, kymogramsare 2D graphs used to show intensity fluctuations over timealong a straight line defined in 2D images (Salmon et al.,2002). We report here a technique for investigating membranetrafficking, specifically vesicles moving along a filamentousnetwork. We show that this technique is a powerful tool forvisualizing and quantifying activity along this network. Itcan be used semi-automatically in user-specified regions forvisualization and qualitative purposes, or automatically forfull statistical analysis. The three-dimensional information iskept to avoid object occlusion and to monitor the 3D locationand displacement of vesicles over time.

The outline of the paper is as follows: in Section 2, wedescribe the acquisition and image processing procedures,the biological sample setup and the nature of the biologicalobjects observed in the images. In Section 3, we explain thetracking methods and propose a method to exploit the resultingtrajectories. Finally, in Section 4, we demonstrate the capabilityof this technique by simulation and we present the biologicalresults.

2. Materials

2.1. Biological materials

The following stable cell lines were used in this study: a firststable cell line expressing GFP-Rab6A described in White et al.,

(1999). Another stable cell line expressing GFP-Rab6A wasprovided by Elaine del Nery (Institute Curie, France). HeLacells were transfected with the pEGFP-Rab6A plasmid (Whiteet al. 1999). Transient transfection was carried out usingthe Lipofectamine 2000 transfection reagent (Invitrogen).Transfected cells were grown in culture medium for 3 days,treated with trypsin and then separated into 15–60-mm tissueculture dishes and cultured in DMEM supplemented with0.4 mg/mL G418. The resulting colonies were treated withtrypsin, isolated and grown in the selective medium intissue culture chambers with transparent bottoms (NalgeNunc, Naperville, IL). Potential colonies were screened usingfluorescence microscopy and the selected colonies wereexpanded. Stably transfected cell lines were resorted byfluorescent cell sorting to maintain the homogeneity of thepopulation and were maintained in medium supplementedwith 0.2 mg/mL G418.

2.2. Image acquisition

Images were collected using a rapid 4D deconvolutionmicroscope system (Sibarita, 2005). This system wasassembled on the bottom port of an inverted microscope (LeicaDM IRB2, Leica, Wetzlar, Germany) mounted on an anti-vibration table and placed in an incubator for temperaturecontrol (Life Imaging Services, Reinach, Switzerland). A 100×PL APO HC N.A. 1.4 oil immersion objective was used. Weused a cooled interline CCD detector (Roper Coolsnap HQ,1392 × 1040 pixels with a pixel size of 6.45 × 6.45 μm2),acquiring 12 bit images with a read-out speed of 20 MHz. Z-positioning was achieved using a piezo-electric driver (LVDT,10-nm precision, 40-nm repetitiveness, Physik Instrument,Karlsruhe, Germany) mounted beneath the objective lens.Illumination was provided by a DG4-Sutter instrument (Sutter,Sutter Instrument, Novato, CA, USA). The whole systemwas steered by Metamorph 6 software (Universal ImagingCorporation). An image stack was acquired in ‘z streaming’mode in which the CCD detector operated at frame rate inthe overlapped mode. With 2 × 2 binning and a z distancebetween planes of 0.3 μm, the voxel size was 129 × 129 × 300nm3, which is compatible with the Shannon sampling criteriarequired for the deconvolution process. For an exposure timeof 50 ms per image, a stack of nine images could be acquired inabout 450 ms. At the end of each cycle, the illuminating beamwas interrupted, the piezo stepper reset and the stack recordedonto disk. For time-lapse imaging, this acquisition procedurewas repeated every second for 2 min, leaving more than 500ms between acquisitions to ensure cell viability and reducephotobleaching. These extreme acquisition conditions gavenoisy images. Nevertheless, the deconvolution process usingthe PSF-based iterative constrained modified Gold algorithm(Gold, 1964; Agard et al., 1989) significantly increased thecontrast and the signal-to-noise ratio (SNR) of the data. Whenrequired, the data generated by fast 4D microscopy (up to

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4 Gigabytes per hour) are restored using distributed computing(Sibarita et al., 2002; Sibarita, 2005).

It should be noted that the goal is to capture fast3D movements without perturbing the cell activity. Inconsequence, image acquisition frame rate and image qualityare constrained by the biological sample and the wholeinstrumentation setup (illumination, optics and detectiondevices). We are operating at maximum speed allowed bythe detector in overlapping mode. Increasing the acquisitionframe rate would require reducing the region size and thereforeprecludes the acquisition of the entire cell. Increasing theillumination power in order to get better SNR would be atthe expense of cell viability. The use of novel more sensitiveCCD, based on EM-CCD technologies, for example, couldsignificantly increase the acquisition frame rate and theimage’s SNR. Nevertheless, these latter cameras have largerpixel sizes which are incompatible with the Shannon samplingcriterion when using high NA objectives. Our choice was toreach the best compromise between high 3D localization andtime resolution.

2.3. Biological model

This section summarizes the characteristics of the protein-tagged vesicles that we aim to analyze. We will use thesecharacteristics to justify our choice of method and topropose a simple realistic model for validating the technique(Section 4.1). This description is related to the general caseof membrane trafficking, in which fluorescent vesicles aremoving along a cytoskeleton network.(i) Vesicles are transported along microtubules, which are

long protein polymers that have an exceptional bendingstiffness that can be easily fit by smooth curves. Thedeconvolution process gives a good resolution of vesiclesin most parts of the cell. Therefore, their tracks in thetemporal projections can be clearly observed and easilyseparated from each other. We chose to use a kymogramto visualize and quantify the biological activity of vesiclesmoving along microtubules.

(ii) The vesicular structures do not always have an isotropicshape and their diameters range from 60 nm to 1.5 μm.Therefore, our data is unlikely to have the Gaussian shapefound in most single particle tracking analyses.

(iii) As described in the introduction, the density of vesicles incells varies and can reach one vesicle everyμm3 dependingon the shape of the cell and the region within the cell.

(iv) In cells, vesicles move in three dimensions. Although cellsgrown on coverslips are often much wider than they arethick (i.e. the cell shown in the supplementary data is 3 μmthick and 80 μm wide) and although the axial resolution(z direction) is lower than the lateral resolution (x andy directions) after deconvolution, the axial informationcannot be neglected and must be taken into account inthe analysis.

Image-based trajectory analysis for object tracking

A kymogram is a graphical representation of all objectmovements along a particular trajectory. In biology, thisrepresentation is often restricted to determining the globaltrends of object dynamics along a straight line in twodimensions (Katoh et al., 1999; Bear et al., 2002). A morequantitative 2D + time analysis introduced by Waterman-Storer et al. (1998) used the kymogram to measuremicrotubule and actin filament dynamics (Gupton et al., 2002;Salmon et al., 2002). In our approach, we use kymogramsalong 3D trajectories to visualize and quantify vesicle motionalong a not physically labelled cytoskeleton.

3.1. Kymograms and trajectories

An ‘object track’ is defined by the spatial coordinates of theobject as a function of time: (x(t), y(t), z(t)). The ‘objecttrajectory’ is the curve defined by the set of object coordinatesover all times. A trajectory can be viewed as a temporalprojection of the object track. This work is a two-step methodthat aims to recover object tracks from kymograms along theirtrajectory, in which the object trajectories are extracted fromtemporal projections of the initial 3D+t data.

The first step involves identifying the trajectories on whichthe kymograms will be computed. The initial data I (x, y, z, t)of X × Y × Z × T voxels (Fig. 1A) are projected throughthe z axis and the t axis (Fig. 1B and C). The time andspace 2D-projection I P , is defined as: I → I P with I P (x,y) = Pz(I t)(x, y) and I t(x, y, z) = Pt(I )(x, y, z), where Pz

and Pt are projection operators. In fluorescence microscopy,the maximum intensity projection operator along the Zdirection is generally used (I t(x, y, z) = maxt∈T (I (x, y, z, t))and I P (x, y) = maxz∈Z (I (x, y, z))). Unlike the average orstandard deviation projection operators, the maximumintensity projection operator keeps small and weak vesiclesfrom the background. Moreover, the geometry of the cell,which is much wider than it is thick, and the resolution ofthe microscope being much lower along the axial directionthan along the lateral direction, leads us to chose a projectionalong the Z direction. This first step makes it possible tovisualize the axial projection of all vesicle trajectories in asingle image, in which mobile structures leave a ‘trail’ on theimage I P . Let us consider one of these trails (Fig. 1C), calledL 2D. Its coordinates can be determined manually by the userdrawing out the trajectory, or either semi-automatically (seeSection 3.2.1) or automatically (see Section 3.2.2). L 2D(d )is a 1D line corresponding to the z projection of a 3D objecttrajectory, where d is the distance along L2D, such as theEuclidian distance is respected. The missing z coordinate of thetrajectory is determined by extracting a 2D curvilinear crosssection L (d , z) = I t(x(d ), y(d ), z) from the volume I t(x, y, z)(Fig. 1D and E), where (x(d ), y(d )) is the point at a distance don L 2D. The z coordinates are recovered by selecting a curve

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Fig. 1. The steps involved in tracking: Initial 4D data I (x, y, z, t) (A) are projected twice (Pt and Pz ) to obtain the 2D image I P (x, y) (C) in which theline L2D is selected. The curvilinear cross-section (D) of the data I t is used to select the line L3D (E). The kymogram data is extracted (G) and the line K (t)defines the tracking of an object (H) along L3D. The images are displayed using an inverted colour map.

on this cross-sectional image (Fig. 1E). Let L3D(d ) be the 3Dtrajectory given by this line selection process (Fig. 1E and F).In L (d , z), several continuous lines indicate several distinctpartially overlapping trajectories in the plane (x, y) but locatedat different z depths.

Trajectories in temporal projections (I P and L (d , z)) canappear as continuous or dashed lines depending on objectspeeds. When the particle density is high, dashed lines,which correspond to fast particle trajectories, are harder todistinguish than continuous lines, which correspond to slowerparticle trajectories. However, one single filament is often usedby several vesicles, which leads to better defined lines in thetemporal projections.

In the second step, the temporal information of the objecttracks is recovered from the kymograms along the trajectories.A kymogram is a 2D graph representing changes in intensity

along a line over time. An object moving along this line over agiven time period will generate a curve on the kymogram.

The kymogram is constructed from the original data alongthe 3D trajectory L 3D(d ) (Fig. 1G). It is defined as I (x(d ), y(d ),z(d ), t), where d is the distance along L 3D. All the intensityprofiles are stacked to form the kymogram, in which theith line represents the intensity profile extracted from the i thimage of the time-series.

Bright spots on the kymogram show where a fluorescentobject (Fig. 1G and H) is on the line L3D. On the kymogram, (d ,t) corresponds to a single point with coordinates (x = x(d ), y =y(d ), z = z(d ), t) in the initial stack series. An object movingalong the line L3D will generate a line on the kymogram. Asin the projection data, fast-moving objects are displayed asa dashed line, whereas slower objects generate a continuousline, depending on the temporal resolution of the acquisition.

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This time function, called K(t) (Fig. 1G), precisely defines thedistance of a given object along the line L3D over time (Fig. 1H).Its derivative dK/d t gives the speed and direction of this object.The resulting object track is defined using L 3D and K(t) as thetemporal function (x = x(K (t)), y = y(K (t)), z = z(K (t))).

K(t) can be determined semi-automatically or automaticallyusing the same techniques used for determining L 3D

(Sections 3.2.1 and 3.2.2). Several situations may be observed,each generating a different pattern on the kymogram:(i) An object moving along the trajectory L3D generates a

line on the kymogram which is always increasing overtime.

(ii) A stationary object on L3D makes a vertical line (atconstant d) on the kymogram.

(iii) A moving object crossing L3D at a particular time makesa single bright point on the kymogram.

(iv) The formation (or dissolution) of a physical structure onL3D makes a line appear (or disappear) on the kymogram.

(v) An object moving onto (or departing from) L3D creates (orstops) a line on the kymogram.

(vi) The merging of two structures on L3D generates twodistinct lines that merge at a certain time.

(vii) The splitting of a structure on L3D generates one line thatsplits in two lines at a certain time.

Only lines on a kymogram need to be taken into accountfor motion analysis. Each line represents one particular objectmoving along a trajectory during a time interval.

3.2. Line detection methods

This kymogram-based technique can be used in two differentways, each requiring line detection. First, it allows thevisualization of the 4D data in specific regions for user-definedinteractive investigations. The user can choose the trajectoriesand tracks to investigate by observing the projections andkymograms. In this case, line selection can be either manual,by clicking on the image, or computer-assisted, as describedin Section 3.2.1. Second, this tool can provide the fullstatistics on the structure dynamics by using automatic linedetection, as described in Section 3.2.2. Both approachesgive complementary information on the object dynamicsdepending on context and the problem studied.

3.2.1. Supervised trajectory detection. We propose a method toautomatically detect lines on 2D images to speed up the lineselection process, that takes into account the start and endpoints defined by the user. The nature of the data leads tosome particular problems that need to be properly addressed.First, if there are many objects within the cell, there may bemany trajectories in close proximity to each other, making pathselection difficult, especially initially (Section 3.1). Second, wewould expect continuous paths although, depending on thespeed of objects with respect to time resolution, there may bebreaks in lines that correspond to single entity motion. During

time recovery, time is forced to increase along the line by adirectionality constraint.

In our situation, a path based on global energy minimizationis useful (Cohen & Kimmel, 1996). Let us consider anenergy function E : C ∈ B → E (C) = ∫ L

0 P(C(s))d s, where Cdenotes a path belonging to an admissible set of paths B.So C : [0, L ] → R2, s is a parameterization of the path andP : R2 → R+ is a potential function. We model the pathby selecting P such that E is minimized along the pathbetween the start and end points in the image J (x). Weuse P(x) = φ1(λ−1

max(H (x))) + φ2(∇ J (x)T d (x)) where H is theHessian matrix H (x) = ∇2 J (x) and d (x) is the eigenvectorof H (x) associated with the largest eigenvalue λmax in themodule, and φ1 and φ2 are decreasing functions. The firstterm φ1(λmax(H (x))) is useful for paths that are well contrastedagainst the background; the second φ2(∇ J (x)T d (x)) comesfrom Steger’s crest detection algorithm and is useful for crestlines (See Steger, 1998, for implementation details). In practice,this semi-automatic process speeds up line selection becausethe user has only two points to select. As soon as the potentialis strictly positive everywhere, the path with global minimumenergy is obtained in a finite number of operations usingthe Fast Marching algorithm introduced in (Sethian, 1999).Nevertheless, depending on noise and clutter of tracks, onehas to regularize the image and possibly give more than twopoints per track. Herein, regularization is undertaken in afiltering step (Section 3.2.2). A possible way to achieve betterperformance in case of clutter tracks could be to use on-the-flyadaptation of potential (Gerard et al., 2002).

3.2.2. Fully automatic trajectory detection. A quantitativeanalysis requires a fully automatic method to be exhaustive, toavoid user subjectivity and to compute lots of statistics in theminimum time. Therefore, we have implemented an automaticline selection within our kymographic analysis. It comprisesimage filtering followed by automatic line detection on theprojection image, the z section and the kymogram.

For the filtering step (Fig. 2F2 and supplementary data), weuse a PDE regularization algorithm (Tschumperle & Deriche,2005) to remove noise and clutter background, and totransform the dashed lines into continuous lines. It usesisotropic smoothing in homogeneous regions and anisotropicsmoothing along edges. This clearly improves the line detectionof fast moving objects.

The automatic line detection step (Fig. 2F3, F4 andsupplementary data) is carried out using Steger’s algorithm(Steger, 1998). This method is particularly well adapted to ourimages and computes lines that follow the crest traces createdby the moving objects with sub pixel accuracy by determiningthe borders of the line.

We establish certain rules to obtain a result that is consistentwith the causality of the positioning and to avoid errors:(i) On the projection data, when crossroads are detected, lines

are split to avoid false connection links.

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Fig. 2. Simulation to validate the kymographic method. (A), (B), (C), (D) and (E) are normalized histograms representing the localization and the speederrors. Influence of the object speed (1, 3, 5, 7 voxels/frame) on the localization errors (A) and on the speed errors (B). In (C) and (D), errors in thelocalization and speed when the noise level increases (PSNR equal to 0, 0.1 and 0.2). (E) the number of the recovered objects of each speed (from 1 to 7voxels/frame) as a function of the density (20, 40, 100, 200 objects). (F) 100 beads with different speeds (from 1 to 7 voxels/frame) are simulated withoutnoise. Using an inverted colour map, (F1) (F2) (F3) and (F4) correspond to the I P projection, the regularization, the line detection and the merge of (F2)and (F3), respectively.

(ii) For the kymogram analysis, detected lines that showchanges in the temporal direction are split into severallines to satisfy the temporal increasing constraint.

(iii) Wrong detections, mainly due to noise or backgroundpatterns, are filtered out to leave only those tracksrepresenting vesicle displacement. Therefore, tracksdefined over less than four consecutive images and trackswith less than five pixels between the start and enddetection points are removed.

Our automatic line detection provides results close to thoseobtained with manual line selection. The amount of datagenerated by the automatic processing is much larger andthe process is not time consuming for the user. The resultobtained is completely user independent and reproducible andis, therefore, adapted to the blind analysis of many imaged cells.These results are complementary to those obtained manually,which are more dedicated to understanding and quantifyinglocal events.

3.3. Statistical trajectory analysis

Once the tracks are computed, the dynamics of cells underseveral conditions have to be analyzed and compared. Theproblems involved in this process are illustrated by the speeddistributions shown in the Fig. 5 in which most of the objectsmove very slowly, whereas a small but interesting part movesquickly. These curves have an ‘exponential behaviour’, so itmakes no sense to use standard estimators such as average,median value or variance, because these would be influencedby the very slow objects, and thus we cannot use the standardparametric or nonparametric statistical tests. Therefore, wehave developed a method to reveal changes in dynamicbehaviour using the shape of the speed distributions. Let usconsider two cell groups, A and B, obtained under differentconditions. The sets s A = {s Ai}i and s B = {s Bi}i are theinstantaneous speeds of each vesicle of each group A and B. Thetwo speed distribution histograms h(s A) and h(s B) are defined

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on 100 bins between 0 and 2 μm/s and are normalized suchthat

∑i h(sX )[i ] = 1. We compute the difference between the

two histograms h1 and h2 using the Bhattacharyya distance(Bhattacharyya, 1943) adapted to histogram comparisons.This is defined as d (h1, h2) = − log

∑i

√h1[i ] × h2[i ], with

d = 0 when the two histograms are equal and d = ∞ whenthe two histograms are disjoint. If we assume that the speeddistribution of B is the dilation of the speed distribution of Aby a factor α, we can express the speed changes between Aand B as h(s A) = h(α × s B) = h({α × sBi}i ). We compute α

by minimizing the Bhattacharyya distance between thehistograms: α = arg minα d (h(sA), h((α × sB)). The measured (h(sA), h(α × sB)) estimates the goodness of fit: the closer it isto 0, the better is the fit. Figure 5 shows the dilation of the redhistogram onto the green histogram, with a measured dilationfactor equal to 1.47 and a goodness of fit of 6 × 10−4.

4. Experimental results

4.1. Validation on synthetic data

This section aims to quantify the tracking precision and toshow how to interpret the results obtained by the kymogramanalysis. We have simulated objects having a Gaussian shapeof fixed size and intensity, moving in three dimensions witha constant speed and constant direction over time to fit thebiological model detailed in Section 2.3. All the units aregiven using numeric conventions for time and space. These areconvertedaccordingtotheacquisitionsetupbytakingthevoxeldimensions as 129 × 129 × 300 μm3 and one frame eachsecond. Angles are given in degrees. All the simulated data aretime-series with constant sizes of 200 × 200 × 20 × 20 unitsin x y z and time axes. The embedded objects all have the sameisotropic Gaussian behaviour with a SD of 2. The simulatedobjects are initially uniformly distributed in the 3D image. Thespeed directions in the xy plane are chosen randomly with auniform angle distribution in the interval [0◦, 360◦[ (Fig. 2F1).Speed directions in the z axis are Gaussian distributed with azero mean and a SD of 5◦. After the deconvolution process, thenoise in the images is difficult to characterize. It is correlated tothe signal, like the Poisson noise, and it is usually assimilatedto Gaussian noise for simplicity. We, therefore, studied theinfluence of Gaussian and Poisson noises on the simulateddata. The Gaussian noise is an additive white noise with aSD of σ and a peak signal-to-noise ratio (PSNR) with respectto the objects as: PSNR = maxobject/σ , where maxobject is themaximum intensity of the objects embedded in the image. Forthe synthesis of a Poisson noise, we first make a noise free image(named NFI), with a constant background (named bg). We adda Gaussian shape object as described above with a maximumintensity of maxobject, such as the maximum intensity of theimage is bg + maxobject. Poisson noise at the pixel i wassimulated in our experiments by setting a random value foreach pixel from a Poisson distribution of mean intensity equal

to the noise free image at the pixel i. In order to get a PSNRcomparable with the Gaussian noise PSNR, it is computed asP SN R = maxobject/

√bg . We set bg to 5 and 10 grey levels to

be in accordance with Cheezum et al. (2001). For higher valueof bg, the Poisson and the Gaussian distribution are similar.We quantify the error between the simulation and result for aparticular condition using the root mean square error (RMSE):E =〈(X sim − X res)2 〉1/2, where the 〈.〉 is the average computedover all objects for all the times.

All the simulated data have been analyzed using the trackingmethod presented in Section 3 with automatic line detection(Section 3.2.2), as illustrated in Fig. 2F.

4.1.1. Tracking precision. In this section, we quantify thetracking method precision as a function of the object size,shape, speed and the noise level using a series of simulationsof a single object. In our method, tracking imprecision can becaused either by lack of precision in the localization of the objector by lack of precision in the time measurement. Therefore, wehave expressed the errors in terms of localization (the distancebetween the centre of the simulated and detected objects) andspeed (the difference between speeds of the simulated andtracked objects).

For the simulation, 200 time-series trials were carried outfor object speeds of 1, 3, 5 and 7 pixels/frame. The localizationRMSE were 1.56, 1.54, 2.61 and 3.69, respectively (Fig. 2A).The speed RMSE were 0.56, 0.63, 1.29, 1.66 respectively(Fig. 2B). Since the tracking precision is found to be sensitiveto object speed, we could use this simulation in order to findthe optimal image acquisition frame rate. Unfortunately, as itis explained in the image acquisition Section 2.2, the temporalresolution is primarily dictated by both the instrumentationand the biology.

A particular object is tracked using three distinct lineselections. Lack of precision in selecting the two first projectionimages imply only localization errors, whereas line selectionmade on the kymogram K(t) implies errors in both localizationand time. Consequently, a simulated object will generate a tracecentred on the positions Xsim(t) = X.t + b on the kymogram,where X denotes the constant speed. For line detection with asimple translation error �t in time and �b in positioning, thelocalization becomes Xdetect(t) = X.t + X.�t + b + �b. Theresulting RMSE is then a function of X.�t + �b. The errors inthe term X.�t can be bigger than errors in �b, especially forhigh-speed objects, but does not affect the speed distribution asmuch. The developed technique is more precise for speed thanfor localization of the object.

We quantified the influence of the SNR for Gaussian andPoisson noise, with PSNR set to ∞, 20, 10, 6.6 and 5.Object speed was set to 5 pixels/frame and for each SNR, 200time-series trials were carried out. In the noise free image(PSNR = ∞) the localization RMSE was 2.64 and the speedRMSE was 1.29. When Gaussian noise was added, (PSNR setto 20, 10, 6.6 and 5), the localization RMSE were 2.69, 2.73,

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2.81 and 2.79, respectively (Fig. 2C), and the speed RMSE were1. 1.31, 1.34, 1.40 and 1.40, respectively (Fig. 2D). For Poissonnoise, the background value bg was first set to 10 (PSNR setto 20, 10, 6.6 and 5), the localization RMSE were 2.09, 2.72,2.96 and 3.13 and speed RMSE were 1.41, 1.42, 1.55 and1.89. For bg = 5, localization RMSE were 2.07, 2.32, 2.58 and2.78 and speed RMSE were 1.38, 1.24, 1.87 and 1.89. Thisshows that noise has a minor influence on localizations andspeeds precision due to the diffusion algorithm. Nevertheless,the automatic line detection had more and more difficulties indetecting the simulated objects as the SNR decreased.

In some biological situation, vesicles are known to exhibitdirection changes along the same microtubule. Therefore, wequantified the influence in term of RMSE of this situation.We have simulated Gaussian shape objects moving at5 pixels/frame, without noise. At each time, the speed vectorhas a probability of 1/20 (20 is the number of frames in thetime-series) to change of direction such as the new speed isthe opposite of the old one. The localization RMSE was 3.068and the speed RMSE was 1.50. In kymograms, objects tracebroken lines, with cracks when objects change of directions.The diminution of precision is due to the rigidity of theline detection process which does not precisely detect thosecracks.

In the time-series, tubulo-vesicular structures appear withvarious sizes and shapes. We have simulated objects withdifferent sizes (SD of 1, 2, 3 and 4 pixels) moving at5 pixels/frame, without noise. The localization RMSE isrespectively 2.41, 2.61, 2.57 and 2.43, and the speed RMSEis 1.26, 1.29, 1.27 and 1.26. Two effects are in competition:First, the precision is weaker when objects are bigger. Second,the precision is weaker when space between successive spots inthe projection is bigger. This competition is also present in thecase of objects with anisotropic shape. In the images, vesiclessometimes display an elongated shape in the microtubule’sdirection. For a speed of 5 pixels/frame, anisotropic gaussianshape objects with SDs of 4 pixels in the direction of the speedand 2 pixels in the perpendicular direction show localizationRMSE of 2.68 and speed RMSE of 1.33. This is worth thanfor isotropic objects with SD = 2 pixels and similar than forisotropic objects with SD = 4 pixels. Anisotropic Gaussianshape objects with SD = 2 pixels in the displacement’sdirection and SD = 1 pixel in the perpendicular direction showlocalization RMSE of 2.51 and speed RMSE of 1.247. This iscomparable to the precision on isotropic objects with SD =1 pixel and better than for isotropic objects with SD = 2 pixels.For a speed of 7 pixels/frame, these anisotropic Gaussian shapeobjects (SD = 2 and 1 pixels) show localization RMSE of 3.62and speed RMSE of 1.61, which is better than for isotropicobjects with SD = 2 pixels.

4.1.2. Object density influence. It is clear that the developedtechnique does not allow information from all the fluorescentobjects moving in a cell to be recovered. Here we quantify how

representative the subset M of detected objects is of the totalobject population, the set A.

For this study, we simulated a set of N Gaussian objectsmoving with a very simple vesicle dynamic model. The onlyheterogeneity among the synthetic objects was their speed,which remained constant and can be, in identical proportions,either 1, 2, 3, 4, 5, 6 or 7 pixels/frame (Fig. 2F). Each objectleaving the 3D image was replaced by a new object createdanywhere on the image border with the same absolute speedto keep the number of objects and the speed distributionconstant.

We determined the number of well-detected objects foreach speed as a function of N. For this, we created andanalyzed 50 time-series, each containing N moving objectswith N varying from 20 to 200 in steps of 20. The kymogrammethod only wrongly detected 1.6% of the total detected points,even for highly complex data. Moreover, the simulation neverdetected an object changing from one simulated object toanother. The reason is that, for two objects to be confused,their positions have to be confused at a particular time,which is extremely unlikely. Figure 2E shows the result ofthis analysis. It highlights that in a dense system, high-speed objects are harder to automatically detect than slowerones. This is understandable because fast moving objectsmake discontinuous lines in the temporal projections and are,therefore, harder distinguished on the image projection whenthe object density is high. Also, very slow moving objects (withspeeds less than or equal than 1 pixel/frame) are harder torecovered because their time projections show small ovoidstructures instead of straight lines.

Therefore, the subset M is not a random subset of A, butdepends on both the object speeds and density. If we know thesystematic bias function of the object density it would be usefulto correct the measurements on real data. Unfortunately, thismodel does not take into account disparities in shape andintensity, trajectory bending, abrupt direction changes, orstructure fusion and separation. Nevertheless, the analysis oftwo similar object sets (such as two different cells) A1 and A2

with identical dynamics will give two subsets M1 ∈ A1 andM2 ∈A2 with identical dynamics, assuming that the numberof detections is sufficient. This is crucial for comparing differentobject families with similar spatial and dynamic behaviours,such as for biological studies.

This simulation shows that even if object tracking is precise,the speed distributions should be interpreted with care becausethe measurement introduces a bias by underestimating verylow- and high-speed object contributions in areas with highobject densities. Nevertheless, in the next section we arguethat this tool can be used to compare dynamics by comparingthe speed distributions.

4.1.3. Influence of Brownian objects. The presented tool isdesigned to track objects moving with directional motionslike it is the case for the vesicles tagged with Rab6A proteins.

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However, it can occur that some particles follow other modelslike Brownian motion, as for example if particles move freelyin the cytosol, detached from the cytoskeleton network. Thosenondirectional particles can disturb tracking results. Then, inthis section we assess quantitatively the influence of Brownianparticles on tracking. Particles moving with Brownian motionwere simulated with different diffusion coefficients. We observethat the presented method is not well adapted for the trackingof Brownian particles because in the projection images,Brownian spots do not appear as a well-defined line butrather as tangle of trajectories not self avoiding. Between twoframes, the displacement of a Brownian object is symmetric,but its entire trajectory shows a statistical high asymmetry(Saxton, 1993). Therefore, our method can track objects alongsmall pieces of trajectories, where the objects display locally adirectional motion behaviour. Particles moving with Brownianmotion were simulated with different diffusion coefficients(called D) of 1, 2, 3, 4 and 5 pixel2/frame. 200 time-seriestrials were carried out for each diffusion coefficients. Thelocalization RMSE were, respectively, 1.71, 2.15, 2.31, 2.80and 2.87. The localization of Brownian objects is comparableto the localization of directional ones but on very restrictedpart of their trajectory.

We then studied the tracking results of time-series composedof object moving differently, some following a Brownian motionmodel while others obey to directional motion model. Each ofthe 50 time-series trials was composed of 100 objects withsame intensities, a part of them composed of r ∗ 100 objectswith a Brownian dynamic and the rest (1 − r ) ∗100 followinga directional model (where r is the ratio, r ∈ [0, 1]). For a givenr, we have studied the ratio named r

′between the number of

the detected Brownian objects and the number of the detecteddirectional objects. Respectively, forr =20%, 40%, 60%, 80%we have obtained r

′ = 11.6%, 22.8%, 30.6%, 56.9%. Usingthe method reported here, the detection of the directionaltrajectories is easier than the detection of Brownian ones.Nevertheless, Brownian objects can also be tracked whentheir trajectories are not too embroiled in the projectionimages.

4.1.4. Validation of the speed distribution comparison. In thissection, we show that differences in speed distributionsmeasured using the histogram comparison method definedin Section 3.3) correctly reflect differences in the physicalobject dynamics. We simulated two object sets, A and B, inwhich A is the reference and B has faster dynamics thanA. Two speed distribution types are tested, Gaussian andexponential distributions. Exponential distribution is closerto the experimental results displayed in Section 4.2.2 andit has been already observed in Pangarkar et al. (2005). Ina first experiment, each set contains 100 objects movingwith a Gaussian speed distribution (∝ exp(x2/2.σ 2)) centredon zero and in the second, each set contains 100 objectsmoving with an exponential distribution (∝ exp(x/σ )). In

Table 1. Table of the measured α function of the simulated values σ A andα for Gaussian and exponential distributions. The goodness of fit is shownin parentheses.

α for Gaussian distributions

σ A 1.2 1.4 1.6

1 1.196 (0.005) 1.403 (0.004) 1.620 (0.005)0.5 1.198 (0.022) 1.344 (0.019) 1.731 (0.103)

α for exponential distributions

σ A 1.2 1.4 1.6

1 1.201 (0.00052) 1.390 (0.00060) 1.530 (0.0039)0.5 1.1805 (0.00066) 1.420 (0.00080) 1.62 (0.0015)

these two experiments, the set A and the set B have samespeed distribution but are, respectively, function of σ = σ A

and σ = σ B = α × σ A, where α is the dilation coefficientbetween the dynamics of A and B. We aim to create thetwo sets, A and B, according to the values σ A and α, toanalyze the time-series and to measure the dilation factor,α, between the two speed distributions using the methoddescribed in Section 3.3 and then to finally compare α to α.The simulated values of σ A equal to 0.5 and 1 and α equal to1.2, 1.4 and 1.6 are in a good agreement with experimentaldata measured in Section 4.2.2. We created and analyzed 30samples for each condition. We found that most of the α valuesare very similar to α, except for the Gaussian distributionswhere σ A = 0.5 and α = 1.6 (Table 1). In this last case,the histograms are too different to be correctly compared(expressed as a high goodness of fit) because the low-speedobjects were underestimated by the tracking method. Thosesimulations show that the method described in Section 3.3 isrobust for measuring precise dilations between dynamics, andfor different speed distribution types.

4.2 Biological results

We have presented a method for visualizing and analyzingcomplex vesicle trafficking. We used our technique to analyzethe dynamics of membrane vesicles identified by the presence ofRab6A proteins to obtain new qualitative and quantitativeinformation on the behaviour of this protein when targetingmembranes within cells. We used two different HeLa celllines, stably transfected by GFP-Rab6AcDNA, called ‘GFP-Rab6A(1) and ‘GFP-Rab6A(2)’. The second cell line, obtainedin a subclone of HeLa cells with different morphologicalcharacteristics, was used to test the reproducibility of the dataobtained, irrespective of the cell size and shape. We monitored40 cells according to Section 2 and analyzed the imagesand extracted relevant information using the kymographictechnique.

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Fig. 3. Typical tracking result obtained from GFP-Rab6A(1) live cell imaging. (A) represents the I P projection (as defined in Fig. 1C) from 3D+t dataobtained on one representative GFP-Rab6A cell. The coloured areas (blue and green) were chosen from this projection to illustrate certain behaviourof Rab6A vesicles. In these areas, different trajectories are defined (see Section 3.1) as line 1 (blue) from area 1 and green complex and interconnectedtrajectories for area 2. The kymogram (as defined in section 3.1 and Fig. 1G) ‘kymo ’ was extracted from lines 1 in area 1. This kymogram shows thatmultiple moving structures having the same direction are routed through the same track. (B): 3D+t tracking of multiple membrane objects followingdifferent and interconnected trajectories (area 2 of part A) showing the high heterogeneity of Rab6A dynamics (for more details see Section 4.2). Original3D images of the time-series were segmented using an undecimated wavelet transform (Starck et al., 1998). The positions of the physical objects selectedon the kymograms (not shown) are matched with the 3D segmented images to give a time-defined series of segmented objects.

4.2.1 Qualitative analysis using visualization tools. We directlyobtained several new observations by visualizing theprojections and kymograms. The lines defining the trackingwere extracted semi-automatically according to Section 3.2.1to allow us to focus on specific regions of the cell. Wefrequently observed some very well-defined tracks on theprojection images. The kymograms produced from theselines displayed many different lines, which indicated thatmany vesicles were moving on the same trajectory. Fromthis, we can state that several vesicles (up to 8/min) aretransported along one microtubule or one microtubule group(Fig. 3A area 1 line 1 and kymo1). We also noticed thatvesicles moving along the same microtubule had very similarinstantaneous speeds, characterized by the same patternon the kymogram (Fig. 3 kymo1). This suggested thatthe speed of a vesicle depends more on the microtubularsupport than on the vesicle itself. This should be furtherinvestigated.

We also observed that a stationary structure may receivea moving structure from one or several different trajectories

(Fig. 3B, and the right-hand side of Fig. 3 kymo1). Ouranalysis showed an even greater heterogeneity in subsequentsteps. From these stationary vesicles, multiple elements can begenerated, each following one or many different trajectories(compare yellow and purple structures in Fig. 3B). This eithersuggests fission or budding of elements from the newly formedand relatively stable Rab6A structure or corresponds to aspecificmembraneelementdetachingfromastationaryclusterof tubulo-vesicular structures.

4.2.2 Quantitative analysis using the automatic trackingtechnique. We further analyzed Rab6 trafficking using fullyautomatic quantification. For this, all 40 cells tagged byGFP-Rab6 were automatically analyzed using the techniquedescribed in Section 3 and using automatic line detection(Section 3.2.2). The time-series comprised 120 3D images(Fig. 4A and B). These were separated into several sub-blocks comprising 20 3D images to give less compleximage projections, with each sub-block being analyzedindependently. For each cell, between 100 to greater than

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Fig. 4. (A) and (B): Two different representations obtained by automatic tracking method of the same tracking result from a cell tagged with GFP-Rab6.The arrows at a bottom of the tracks show the temporal direction. (A): 3D trajectories of the detected objects. The colour indicates the time windowwhen an object was detected. The z projection of the first 3D image of the time-series is shown below the trajectories to visualize the cell morphology.(B) 2D+t representation of the trajectories, the temporal axis is indicated by the blue arrow. The white, yellow and red colours show the directions, withvesicles moving toward the cell periphery, toward the Golgi and in the lateral directions, respectively. (C): Scheme for computing the vesicle direction. Thecentral black spot represents a vesicle. The vesicle-to-Golgi axis is shown by the blue dashed line, from which a dial is defined separating the xy planeinto four different sectors of 90◦. The black dashed arrow shows the instantaneous speed of the vesicle and determines the sector in which the vesicle isconsidered.

1000 tracks were extracted and analyzed. As a control ofthe specificity of our method, we performed similar analysison nocodazole treated cells. Figure 4 shows 812 computedtracks from a single cell, which is also shown in thesupplementary data movies. Due to the very large numberof vesicles, results were hard to represent comprehensively,although the statistics revealed very informative spatial anddynamic information. As a comparison, after microtubuledepolymerization (nocodazole treatment), tracking detects amean of 19.5 trajectories per cell (n = 11) whereas untreatedcells show a mean of 248 trajectories per cell (n =40). Togetherwith results of simulation (see Section 4.1.3) indicating that

our tracking method accurately detects both Brownian anddirectional movements, this demonstrates that Rab6 vesiclesare showing neither directional movements nor free diffusionin the cytosol, once microtubules have been disrupted (seesupplementary Fig. 1).

Rab6 vesicular structures are bound to the microtubulenetwork via molecular motor proteins (Goud, 2002) and mayalso be associated with actin cytoskeleton elements (Mataniset al., 2002). Therefore, the apparent displacement of aRab6 loaded vesicle is a mix of the vesicle motion along thecytoskeleton network via molecular motors and the inherentdynamic of the network itself. Temperature changes are also

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Table 2. Direction results.

Proportion Speed dilation ratio (fit error)

Direction Rab6 37◦C Rab6 25◦C Rab6 37◦C Rab6 25◦C

Periphery 36.7% 36.2% +8.1% (6.10−4) +0.0% (6.10−4)Golgi 24.0% 24.9% −15.2% (1.10−3) +1.4% (8.10−4)Lateral 2∗19.7% 2∗19.4% +0.0% (6.10−5) +0.0% (3.10−4)

known to affect membrane trafficking by influencing thespeed of transport intermediates. Therefore, we compared thedynamics of Rab6 vesicular structures at 37◦C and 25◦C, atemperature known to decrease the efficiency of microtubulepolymerization.

Rab6 molecules are involved in the retrograde transportfrom the Golgi apparatus toward the endoplasmic reticulum(Sannerud et al., 2003). We analyzed the direction the Rab6structures moved with respect to the location of the Golgisystem. This was done in two dimensions and coordinates ofthe Golgi was determined simply by clicking on the images. Foreach time point and each tracked object, the angle betweenthe instantaneous speed of the vesicle and the vesicle-to-Golgiaxis was measured as described in Fig. 4C. Three directionswere considered depending on whether the vesicle was movingtoward the Golgi, toward the cell periphery or in lateraldirections. For each direction, the proportion of vesicles andspeed distribution were computed. At 37◦C, we found out that36.7% of the vesicles were moving toward the cell periphery,24.0% toward the Golgi apparatus and 39.4% in the lateraldirections (Table 2). As the lateral direction is twice as largethis latter percentage should be considered as 2∗19.7%. At25◦C, the values was less than 1% difference to the values at37◦C.

For speed distributions, we computed the dilation coefficientas defined in Section 3.3. We compared the speed distributionsof the two different cell lines GFP-Rab6A(1) and (2) at 37◦Cas a primary validation test. We obtained a dilation coefficientof 0.990 with a goodness of fit of 1.3 × 10−3, showing nosignificant difference between the speed distributions of thetwo cell lines.

For the effect of temperature (Fig. 5), we measured a dilationcoefficient of 1.393 between 25◦C and 37◦C, meaning thatthe vesicles at 25◦C were 39% slower than vesicles at 37◦C,which is consistent with expected values. We also comparedthe speed distribution of the different directional sub-familiesof vesicles. We found that vesicles have different dynamic indifferent directions (Table 2). At 37◦C, vesicles moving towardthe cell periphery were 8.1% faster that the global vesiclepopulation, whereas vesicles moving toward the Golgi were15.2% slower, and vesicles moving in the lateral directionshad similar speeds to the global vesicle population. At 25◦C,the speeds were significantly affected by the decreasing of the

Fig. 5. Speed distributions of the vesicles at 25◦C and 37◦C and dilateddistribution. The histograms ‘Rab6 25◦C’ and ‘Rab6 37◦C’ are thenormalized speed distributions at 25◦C and 37◦C of the moving objectsdetected in the cells tagged by GFP-Rab6. The dilation coefficient betweenthese two distributions is 1.47 and the goodness of fit is 6 × 10−4.The histogram ‘Rab6 25◦C dilated’ shows the result of dilation of thedistribution ‘Rab6 25◦C’ by this coefficient.

temperature as mentioned above, although we observed nospeed differences in the different directions.

5. Discussion and conclusion

We here presented a novel method for monitoring biologicalstructures moving in a specific cellular context. It analyzescomplex data with a high density of objects movingheterogeneously along a filamentous network. We haveextended the kymographic approach in three dimensions toaid visualization of objects moving along a three dimensionalpath. We have used this representation to recover temporalinformation and track objects. We have implemented twocomplementary methods: an interactive tool in which theuser can define trajectories of interest directly on time andspace image projections. It allows the user to focus on specificregions for understanding complex local behaviour. The

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second automatically tracks objects by detecting their tracefrom image projections and kymograms. From the trackingresults, we have extracted relevant descriptions of objectdynamics, in terms of speeds and direction. We have shownthat these descriptions can be used to quantify and comparemolecular dynamics within cells in a robust manner.

The direct application of both complementary techniques tothe dynamic behaviour of vesicles labelled with Rab6A allowsa new understanding of this complex molecular system. Firstof all, it revealed a speed distribution of moving structurescompatible with movement driven by motor proteins alongmicrotubules. However, our analysis of the movement of theRab6A vesicles revealed a far more complex pattern thanexpected. It is mainly characterized by the export of themajority of Rab6 positive vesicles and tubules from the Golgialong microtubules in + end direction while statistics alsorevealed vesicles that were moving from the periphery ofthe cell towards the Golgi. Several + end directed transportintermediates can use the same track but we did not observemany + and − end directed vesicles on the same track. Thesefindings suggest that Rab6 vesicles would move unidirectionalto either the + or − end of a given microtubule. Progresson this question depends on the identification of the motorproteins that mediate transport of Rab6 vesicles (Echard et al.,2000; Fontijn et al., 2001; Gross et al., 2002; Matanis et al.,2002; Short et al., 2002). A number of other particularand unexpected events were revealed by our approach suchas relatively stable structures. While the identification ofthese structures will require further dedicated analysis, wecan assume nonetheless that at least some of them maycorrespond to sites where Rab6 dissociates from membranesand recycles back to the cytosol, being prepared for anotherround of function. Future experiments based on the presentinvestigation should give insight into the nature of themembrane compartment defined by these sites. This will helpto better understand the role of Rab6 in the Golgi to ERtransport. As a more general consideration, the dynamics ofmany different transport-intermediates, such as Rab-bearingmembranes, and the mechanisms involved in this traffickingdepend on the dynamic architecture of large molecularcomplexes. Our work provides an important tool to tackle sucha molecular complexity by using advanced light microscopyand multidimensional image analysis.

Acknowledgements

We would like to thank Francois Waharte, Yohanns Bellaıche,Bruno Goud, Charles Kervrann and Maxime Dahan forhelpful discussion and critical reading of the manuscript. Wealso acknowledge Sebastien Huart for his expert technicalassistance. We wish to thank Elaine del Nery for providingus with Rab6A cell lines. V. Racine is funded by the FrenchMinistry of Research. M. Sachse is funded by Fondation pourla Recherche M edicale. This work benefits from the constant

support of the Curie Institute and the CNRS and was inpart funded by the European Community (EAMNET-Europeanadvanced microscopy network). We also would like to thankthe GDR 2588 of the CNRS.

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Appendix

In this section, we express precisely the mathematicalformalism used to build and analyze kymograms. We alsoshow how object positions over time are recovered from theinitial time-series. At several steps, we use linear interpolations

to estimate values in images between pixel centres. Somefunctions (e.g. f (x)) are estimated using the line detectionmethod; We use the symbol to differentiate the initial function(f (x)) from its approximation ( f (x)).

The initial data are a 3D time-series I: I (x, y, z, t) wherex ∈ [x1, . . . , x2], y ∈ [y1, . . . , y2], z ∈ [z 1, . . . , z 2] and t ∈[t1, . . . , t2] according to the spatiotemporal sampling of themicroscope. Embedded in this series, an object called O moveswith a position defined as a discrete temporal function givenin the same referential than the initial data:

O (t) = (O x(t), O y(t), O z(t)).

During the temporal projection we define

I t(x, y, z) = Pt(I (x, y, z, t)) = maxt∈[t1, ..., t2]

I (x, y, z, t).

The function O projected over time can be expressed as the setof positions O t :

O t = {(O x(t), O y(t), O z(t)) , t ∈ [t1, . . . , t2]}.As this set displays continuous lines in the temporal projectionI t , it can be expressed as a parametric function O t(s) usinga common parameterization with the curvilinear abscissa s ∈[s 1 = 0, . . . , s 2]:

O t(s) = (O x(s), O y(s), O z(s)).

During the z projection we define

I p (x, y) = Pz(I t(x, y, z)) = maxz∈[z1, ..., z2]

I t(x, y, z).

The function O t(s) projected over the z axis can be expressedas O p (s):

O p (s) = (O x(s), O y(s)).

O p is a detection of the line formed by O p (or a part of O p ) inthe image I p (called L 2D in the text):

O p (s ′) = (O x(s ′), O y(s ′)),

where s ′ ∈ [s ′1 = 0, . . . , s ′

2] is a parameterization defined by thedetection that can be different from the parameterization s. Theobject O moved in the curvilinear cross-section CS defined fors ∈ [s 1 = 0, . . . , s 2] and z ∈ [z 1, . . . , z 2]:

C S(s, z) = I t(O x(s ′), O y(s ′), z).

This cross-section is estimated by the cross-section C S definedfor s ′ ∈ [s ′

1 = 0, . . . , s ′2] and z ∈ [z 1, . . . , z 2]:

C S(s ′, z) = I t(O x(s ′), O y(s ′), z).

The objectO has traced a line in CS but as O p is close to O p ,Ohas also traced a line in C S. The detection of O in C S is aparametric function L C S defined by

L C S(s ′′) = (O z(s ′′), s ′(s ′′)),

C© 2007 The AuthorsJournal compilation C© 2007 The Royal Microscopical Society, Journal of Microscopy, 225, 214–228

2 2 8 V I C T O R R A C I N E E T A L .

where s ′′ is a new parameterization defined by the detection. Anapproximation of the function O t named O t can be thereforedefined (called L 3D in the text):

O t(s ′′) = (O x(s ′(s ′′)), O y(s ′(s ′′)), O z(s ′′)).

The object O moved in the kymogram Kymo defined for s ∈[s 1 = 0, . . . , s 2] and t ∈ [t1, . . . , t2]:

K ymo(s, t) = I (O x(s), O y(s), O z(s), t).

This kymogram is estimated by the kymogram ˜K ymo definedfor s ′′ ∈ [s ′′

1 = 0, . . . , s ′′2] and t ∈ [t1, . . . , t2]:˜K ymo(s ′′, t) = I (O x(s ′(s ′′)), O y(s ′(s ′′)), O z(s ′′), t).

TheobjectOhastracedalineinKymobutas O t iscloseto O t,Ohas also traced a line in ˜K ymo. The detection of O in ˜K ymo isconstrained to be a function of t named L ˜K ymo (called K in thetext). It gives the relationship between t and s ′′. This functioncan be expressed as L ˜K ymo (t) = s ′′(t). We can therefore definethe approximation of O named O as:

O (t) = (O x(s ′(L ˜K ymo (t))), O y(s ′(L ˜K ymo (t))), O z(L ˜K ymo (t))),

or

O (t) = (O x(s ′(s ′′(t))), O y(s ′(s ′′(t))), O z(s ′′(t))),

where Ox(t) is approximated by O x(s ′(s ′′(t))), O y(t) byO y(s ′(s ′′(t))) and O z(t) by O z(s ′′(t))).

Supplementary data

Cell.mov

Cell tagged by GFP-Rab6 showing the trafficking activityaround the Golgi apparatus. Each image of the movie is a zmaximum intensity projection of the 3D image.

CellProcess.mov

Different steps of the automatic tracking computed on themovie ‘Cell.mov ’. The initial time-series is decomposed into setsof 20 images. For each set, the first image shows the temporalprojection (Section 3.1), the second the regularization, thethird the automatic line detection (Section 3.2.2) and the lastimage the final resulting trajectory detection (also presentedin the Fig. 4).

Supplementary Figure 1

Z and temporal projections of GFP-Ra6A cell time-series with(A) or without (B) nocodazole treatment (10μm for 1 h at37 ◦C).

C© 2007 The AuthorsJournal compilation C© 2007 The Royal Microscopical Society, Journal of Microscopy, 225, 214–228