Journal of Structural Biology 178 (2012) 19–28

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Journal of Structural Biology

journal homepage: www.elsevier .com/locate /y jsbi

Actin filament tracking in electron tomograms of negatively stained lamellipodiausing the localized radon transform

Christoph Winkler a,⇑, Marlene Vinzenz b, J. Victor Small b, Christian Schmeiser a,c

a Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Apostelgasse 23, 1030 Vienna, Austriab Institute of Molecular Biotechnology, Austrian Academy of Sciences, Dr. Bohr-Gasse 3, 1030 Vienna, Austriac Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, Austria

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 September 2011Received in revised form 2 February 2012Accepted 16 February 2012Available online 1 March 2012

Keywords:Actin filamentLamellipodiumRadon transformFilament tracking

1047-8477/$ - see front matter � 2012 Elsevier Inc. Adoi:10.1016/j.jsb.2012.02.011

⇑ Corresponding author. Fax: +43 (0) 732 2468 521E-mail address: christoph.winkler@ricam.oeaw.ac.

The aim of this work was to develop a protocol for automated tracking of actin filaments in electrontomograms of lamellipodia embedded in negative stain. We show that a localized version of the Radontransform for the detection of filament directions enables three-dimensional visualizations of filamentnetwork architecture, facilitating extraction of statistical information including orientation profiles. Wediscuss the requirements for parameter selection set by the raw image data in the context of other,similar tracking protocols.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Lamellipodia are cell protrusions functioning as essential partsof the machinery for crawling of many different cell types (Smallet al., 2002). They are thin, sheet-like structures densely filled withfilamentous actin. Actin filaments are flexible double helices with adiameter of approximately 7 nm that can form networks andbundles regulated by a large number of different proteins,performing the tasks of nucleation, branching, and cross-linking.The lamellipodia network interlinks to the extra-cellular matrixby transmembrane proteins and undergoes a permanent remodel-ing process by polymerization and depolymerization of filaments.These characteristics of actin networks allow cells to exert a direc-ted protrusive force (Pollard and Cooper, 2009).

In their pioneering studies, Medalia et al. (2002) showed thatelectron tomography may be used to visualize actin filaments inthin regions of amoeba in vitreous ice, opening the way forthree-dimensional analysis of actin networks in motile cells. Morerecently, tomograms were obtained from lamellipodia in cytoskel-etons of vertebrate cells contrasted by the negative stain procedure(Urban et al., 2010). Owing to a higher specimen contrast than invitreous ice, actin filaments in the stained preparations were visu-alized with particular clarity, such that they could be manuallytracked in three dimensions along most of their length. For the firsttime, mapping of the entire actin network architecture in a

ll rights reserved.

2.at (C. Winkler).

cytoplasmic compartment became feasible. However, since man-ual tracking is potentially prone to errors or bias, it was necessaryto develop an automatic tracking protocol to complement themanual analysis.

The problem of automatically extracting curvilinear structuresfrom images has previously received much attention. A first groupof algorithms uses mainly local information for their tracking, suchas open active contour models (Li et al., 2009), fast marching(Sethian, 1999), and exploiting differential geometry properties(Steger, 1998). Another class of approaches relies on filteringimages with the help of Gauss kernels as a preprocessing step(Sargin et al., 2007). This procedure requires selection of the rightfilter scale for gradient analysis. Although the diameter of an actinfilament is known, an appropriate Gauss kernel does not seem toexist. Either the filter mask is too small, such that the single mono-mers obscure the axis, or it is too large and individual filamentsmeld with one another or the background clutter. An alternativepreprocessing procedure could be image segmentation (Volkmann,2002), which as yet has not been combined with filament tracking.Anisotropic diffusion (Weickert, 1998) is not readily useable in anenvironment with particles of other material having the same oreven higher levels of intensity, but in combination with tubeenhancing filtering (Jiang et al., 2006) has been shown to work wellfor the detection of microtubules. Whether this approach is alsosuitable to track the much thinner actin filaments has yet to beestablished.

An interesting approach avoiding background problems usestemplate matching (Mayerich and Keyser, 2009; Rigort et al.,

Table 1List of parameters.

Ltrack Length of line segments in the localized Radon transform for thetracking iteration

d Maximal angular deviation between two iteration stepsl Stepsize of the tracking iterationItrack Intensity threshold for filament pieces in the tracking iterationLgap Maximal length of gaps that can be bridgedh Mesh size of the grid of possible starting pointsLs Length of line segments in the localized Radon transform for the

starting point selectionIs Intensity threshold in the starting point selectione Radius of the tube around identified filaments that excludes starting

points

ps

ds

−ds

Fig. 1. Scheme to determine whether a point ps is a starting point for the tracking.Among the segments centered at ps with various directions that with the highestintensity (and direction ds) is highlighted.

pjdj

a

pj

dj

dj+1b

pj+1pj

dj+1

c

ppp

pp p p p p p p

pn

d

Fig. 2. The tracking algorithm. (a) Last identified point pj and search cone for thenew direction around the old direction dj. (b) Identification of the new directiondjþ1. (c) Computation of the new point pjþ1 and search cone for the following step.(d) piece of identified filament after n steps.

Fig. 3. Scheme for endpoint detection; (a) Improving the old direction looking back;(b) search cone for the new direction.

Fig. 4. 0.746 nm thick layer from a tomogram of a lamellipodium, contrasted by thenegative stain procedure, showing a section through the network of actin filaments(Vinzenz et al., 2012). Bar, 200 nm.

Table 2Parameter values used for moving mouse embryofibroblast.

Ltrack 37.3 nmd 5 degreesLgap 7.46 nmItrack 0.55l 7.46 nmLs 52.22 nmIs 0.71h 3.73 nme 11.19 nm

20 C. Winkler et al. / Journal of Structural Biology 178 (2012) 19–28

2012). However, in the high-resolution images of negativelystained actin networks cross-sectional profiles of filaments andbranch sites are too variable for the purpose of template matching.The Radon transform (Radon, 1917; Gelfand et al., 1966), has beenused in digital image processing for the detection of lines alreadyfor many years (Duda and Hart, 1972). Very recently Sandbergand Brega (2007) developed a segmentation protocol for thin

structures, based on a Radon transform approach. We indepen-dently adapted the Radon transform approach for the special caseof negatively stained preparations (Urban et al., 2010) which fea-ture markedly different filament profiles, contrast and backgroundconditions as compared to cryo-EM tomograms.

2. Methods

In the present study we use a localized version of the Radontransform to determine the local direction of filaments. The meth-od is implemented in three steps as follows: (1) selection of start-ing points; (2) filament tracking; (3) filament end determination.The parameters used in the algorithm are listed in Table 1 anddetails of the algorithm are given in the Appendix.

2.1. The Mean Localized Radon Transform

The essence of this tracking method is the identification ofpoints that lie on a filament axis. Mathematically speaking we callsuch an average gray value of a line segment of length L, starting at

Fig. 5. (a) Tracking results for the tomogram depicted in Fig. 4 The manually tracked filaments are depicted in blue, the automatically tracked ones in red. The area for theactin filament density measurement is indicated by the green square. (b) Combined layers. Both bars, 200 nm.

Fig. 6. Detail of Fig. 5 showing a close match of automatically and manually trackedfilaments. The manually tracked filaments are depicted in blue, the automaticallytracked ones in red (the model is tilted to illustrate the overlap).

Table 3The number of starting points is determined by the parameters Is for the minimalintensity and Ls for the integration length. High values in both parameters lead to thedetection of few starting points on rather stiff and prominent filaments, whereassmall values result in many seed points, increasing the possibility of false positives.Taking a long Ls and a medium Is ensures the detection of starting points on filamentsand excludes noise with locally high intensity.

Is ¼ 0:67 Is ¼ 0:71 Is ¼ 0:75

Ls ¼ 37:5 59869 15337 3053Ls ¼ 44:76 43598 10702 1984Ls ¼ 52:22 32952 7958 1377

C. Winkler et al. / Journal of Structural Biology 178 (2012) 19–28 21

point p, with direction d the Mean Localized Radon Transform(MLRT)

MLRTðp;d; LÞ :¼ 1L

Z L

0Iðpþ kdÞdk;

where I is the intensity (or gray value) function.

2.2. Starting point selection

A crucial ingredient for the algorithm is the creation of a grid ofstarting points with a mesh size h, chosen with the intention thateach filament contains at least one grid point. A grid point psisaccepted as a starting point for tracking a filament if the maximal

MLRT of line segments of length Ls and centered at psis above athreshold Is (Fig. 1). Tracking from all starting points is then initi-ated in both directions ds and �ds. In contrast to Sandberg andBrega (2007), the criteria for choosing a starting point are notbased on analysis of all voxels of the image, which significantlyreduces the computational cost. The starting point selectionalgorithm is fully parallelizable.

2.3. Tracking filaments

The essence of one tracking step is to find the optimal directiondjþ1 from a point pj on the filament towards the next point pjþ1 onthat filament (Fig. 2), given an approximation dj of the direction ofthe filament at pj, using the parameters defined in Table 1. For thispurpose we compute the MLRT with pj, a fixed length Ltrack and a setof directions (Fig. 2a). We constrain the search to directions thatdiffer from dj by at most an angle d and determine the directiondjþ1 with the biggest MLRT (Fig. 2b). We complete one trackingiteration by making a step of length l in direction djþ1 to pointpjþ1 (Fig. 2c). We refer to this as a forward search, since gray valueslying ahead on the tracked filament are used. We continue to re-peat this step until a boundary is reached or the MLRT along theoptimal direction is below a threshold Itrack. The tracking is re-sumed if, after a small gap of length Lgap, the filament continues.

To prevent different starting points initiating the tracking of thesame filament, seed points that lie closer than � to a trackedfilament are removed. Because of this procedure, the trackingalgorithm cannot be parallelized.

2.4. Determination of filament ends

If the tracking has stopped at a point pj close to a filament end,the forward search for the optimal direction is not useful any more.Inevitably the line segments overshoot the end of the tracked fila-ment and other material codetermines the tracking direction. Toavoid the loss of the last part of the filament a Janus-faced versionof the algorithm is used in this situation. As a first step, the approx-imative direction dj at pj is improved by looking backwards alongthe filament instead of forwards (Fig. 3a). Then the iteration is re-started, but now we only look forward as far as one iteration step l,with the remaining length Ltrack � l of the control segment extend-ing backwards (Fig. 3b). When, during this new iteration, the MLRT

Fig. 7. Examples of typical, irregular features of the raw data shown in subsequent tomogram sections, with the feature centered. (a–d) Unknown background material; (e–h)branch; (i–l) two filaments closely at overlapping; (m–p) weakening of signal along a filament. Bar, 50 nm.

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of a new piece becomes too small, the last identified point is con-sidered the filament end.

2.5. Choice of parameters

Optimal results depend on an appropriate choice of parameters.The differences between lamellipodial pictures in magnification,contrast, and staining require some tuning of the parameters (Ta-ble 1) by hand.

The use of the localized Radon transform relies on the assump-tion that the filaments are not strongly bent; more precisely: theirdiameter d (approximately 7 nm for actin filaments) has to besmall enough compared to their typical curvature radius. Fromin vitro experiments, minimal curvature radii of actin filamentsbetween 150 and 200 nm have been reported (Arai et al., 1999;Taylor et al., 2000). However, to accomodate short distance inho-mogeneities in filament trajectories we allowed a local curvatureradius rmin ¼ 50 nm. During tracking, a diversion to filamentsbranching off the one being tracked must be avoided. The maximalangle d ¼ p=36 between adjacent pieces of tracked filaments corre-sponds to 5 degrees. With the typical values for rmin and d statedabove, this gives l � 4:4 nm for the stepsize of the tracking itera-tion (see Appendix E).

The maximal length L of a line segment contained in a thin cir-cular tube with diameter d and curvature radius r is L � 2

ffiffiffiffiffiffiffiffi2rdp

. Thevalue L � 53 nm, computed with r ¼ rmin ¼ 50 nm and with thediameter of an actin filament is, in practice, a reasonable choicefor the lengths Ltrack and Ls of the line segments in the localized

Radon transform. The maximal length Lgap of a low intensity gap,which can be bridged, is smaller than Ltrack (see below).

The choice of the mesh size h of the grid for starting points is atrade off between computational cost and accuracy. We typicallychoose several nm. To avoid multiple identification of the same fil-ament, starting points lying closer than a distance e from a trackedfilament, corresponding to the diameter of an actin filament, areexcluded.

The intensities Itrack and Is are chosen according to the bright-ness and contrast of the tomogram.

For the starting point selection we use values of Ls and Is a littlelarger than Ltrack and Itrack, in order to reduce the identification offalse positives.

3. Results and discussion

The method is applied here to two electron tomograms oflamellipodia in different stages of movement. The first exampleis a tomogram of a part of the lamellipodium of a moving mouseembryo fibroblast (Fig. 4).

Most of the filaments have been tracked manually (Vinzenzet al., 2012), which gives us the opportunity for a direct comparison.The edge length of a cubic voxel 0.746 nm was used as an internalreference length in the implementation, leading to non integervalues for the length parameters. The image intensity is scaled tothe interval [0,1], with 0 representing black, 1 encoding whiteand filaments are brighter than the background. The parametervalues listed in Table 2 have been used.

Fig. 8. Success of tracking protocol for problem cases shown in Fig. 7. Only selected tracks are shown in green to demonstrate: (a–d) no false positives inside the clutter; (e–h)both filaments forming the branch are identified; (i–l) both closely overlapping filaments are tracked correctly; (m–p) a low intensity gap is bridged. Bar, 50 nm.

C. Winkler et al. / Journal of Structural Biology 178 (2012) 19–28 23

Fig. 5 (Supplementary video 1) shows a projection of the 3Dtracking results (a) and the layers of the tomogram combined inz (b). Automatically and manually tracked filaments are repre-sented as cylinders with a constant circular cross section. A detailof the tracked tomogram in Fig. 5 is shown in Fig. 6 and reveals aclose correspondence between the manually tracked filaments(blue) and the automatically tracked filaments (red). It was possi-ble to track individual filaments over lengths exceeding 1 lm. Theeffects of parameters on the number of starting points is indicatedin Table 3. The grid spacing h of the search grid has not been varied,since it is typically chosen very fine, such that the detection of allfilaments is guaranteed. Irregularities in filament contours andnon-filamentous background material (Fig. 7) are typical hurdlesin filament tracking. However, as shown in Fig. 8 the method dealswith these quite successfully. Local curvatures along the automat-ically tracked filaments have been computed (see Appendix E). Themedian and mimimal curvature radii are rmedian ¼ 410 nm andrminimal ¼ 196 nm. The latter is in the range of minimal curvaturesof actin filaments reported in the literature and well above thethreshold value used in the tracking, which shows that our choiceof parameters is not too restrictive.

From the tracked filament contours the total filament length ina tomogram as well as the distribution of filament orientations arereadily computed. The total filament length in the projected areaindicated in Fig. 5 was estimated as 14.2 lm by manual trackingand 15.8 lm by automatic tracking. Correlation of both trackingsets with the raw data indicated that the difference was mainlydue to incomplete manual tracking.

Computation of the mean angles of the two-dimensionalprojections of the filaments (Fig. 9) shows the expected bimodal

distribution for a moving cell (Koestler et al., 2008). The distanceof the peaks of approximately seventy degrees corresponds tothe branching angle of actin filaments. As shown in Fig. 9 the angu-lar distributions are essentially insensitive to the chosen value ofItrack. The effects of changing tracking parameters using the rawdata from Fig. 5 are illustrated in Fig. 10. The paramters chosenby correlation with manual tracking are shown on the left of thefigure and the effects of varying these parameters on the right.The variations of the stepsize l (a–c), the length of the line seg-ments Ltrack (d–f), the maximal angle deviation d (g–i) and the max-imal length of gaps Lgap (m–o) have little effect on the outcome,whereas deviations in the intensity threshold Itrack (j–l) and theexclusion radius � (p–r) significantly change the result.

An example of filament tracking in the lamelliopodium of a 3T3fibroblast spreading on polylysine (Urban et al., 2010) is shown inFig. 11. For this tomogram the parameter values listed in Table 4were chosen. The automatically tracked filaments are shown inFig. 11a and the angular distribution in Fig. 12.

The total filament length in the highlighted area is 30.3 lm andthe filament length density is 342 lm�1. The angle distributionindicates that the cell was not organized for persistent protrusion(Fig. 12; Koestler et al., 2008).

Tracked filaments superimposed onto the original data are de-picted in Fig. 13 (Supplementary video 2), which shows sequential2D sections of a detail.

4. Conclusions

An approach has been introduced for automatic identificationand tracking of actin filaments in three-dimensional EM tomograms

0 15 30 45 60 75 90 105 120 135 150 165 1800

0.5

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x 104

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0 15 30 45 60 75 90 105 120 135 150 165 1800

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x 104

α

λ

b

0 15 30 45 60 75 90 105 120 135 150 165 1800

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α

λ

c

0 15 30 45 60 75 90 105 120 135 150 165 1800

0.5

1

1.5

2

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3

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α

λ

d

Fig. 9. Automated tracking highlights bimodal angular distribution. Total filaments lengths k in nm vs. horizontal angle to the cell edge in degrees a. An angle of 90 degreescorresponds to a filament orthogonal to the leading edge. (a) Automatically tracked filaments. (b) Manually tracked filaments. Bottom: automatically tracked filaments withaltered parameter values; unmentioned parameters are unchanged. (c) Itrack ¼ 0:51. (d) Itrack ¼ 0:63.

24 C. Winkler et al. / Journal of Structural Biology 178 (2012) 19–28

of negatively stained lamellipodia. The method combines a system-atic search for starting points and a tracking iteration based on achoice of the continuation direction by a localized Radon transform,which eliminates problems with different types of imperfectionsand noise. Apart from three-dimensional visualisations, the resultsenable the computation of statistical properties of the meshworksuch as anglular distributions and total filament length.

Future work will concentrate on extending the systematicapproach for determining suitable parameter values, possiblyincluding adaptive thresholds for the filament intensity levelsbased on the brightness and contrast. This should speed up thewhole process significantly. In addition, the tracking protocolshould be further developed to automatically identify branchjunctions in the network (Vinzenz et al., 2012). Efforts in thisdirection, including analysis also of cryo-electron tomograms ofcytoskeletons are underway.

An ambitious project is the combination of the tracking resultswith mathematical models for the nucleation, polymerization, anddepolymerization dynamics as well as the mechanics of the net-work. This will require the identification of branches (Smallet al., 2011), the computation of forces between filaments (e.g.,caused by cross links), and of forces between filaments and the cellmembrane by solving mathematical inverse problems for bendingfilaments. In particular, for a continuum modeling approach (Oelzet al., 2008; Oelz and Schmeiser, 2009) the computation of statis-tical properties in localized regions will be important.

5. Technical aspects

The introduced automatic tracking protocol for actin filamentswas implemented in MATLAB, commercially distributed by TheMathWorks, Inc. The determination of the starting points for NIH3T3 fibroblast takes �20 min on a conventional dual core notebookwith 6 GB RAM, parallelized to 3 workers, while the serial trackingof the filaments then takes �1.5 min. The visualizations were pro-duced with the help of MATLAB, IMod (Kremer et al., 1996), Para-view by Kitware Inc. and InkScape. The protocol may be obtainedfrom the corresponding author upon request.

Acknowledgment

This work has been supported by the Vienna Science and Tech-nology Fund under Grant no. MA09-004.

Appendix A. The Mean Localized Radon Transform

The tomogram is represented as a list of integer gray values be-tween 0 and 255 for each pixel. After scaling, we shall interpret itas a real valued function I : X! ½0;1�, given on the picture domainX. We define the Mean Localized Radon Transform as the averagegray value of a straight line segment given by the starting point po-sition p 2 X, the direction d, out of all directions in the 3d space ðS2Þa three-dimensional vector of length one, and the length L > 0:

Fig. 10. The effects of changing the tracking parameters on the result (for details see text).

C. Winkler et al. / Journal of Structural Biology 178 (2012) 19–28 25

MLRTðp;d; LÞ :¼ 1L

Z L

0Iðpþ kdÞdk:

It is well defined, as long as the whole line segment lies in X. For theimplementation the integral has been discretized.

Appendix B. Starting point selection

A grid point is accepted as a starting point, if

MLRT ps �Ls

2ds;ds; Ls

� �P Is

holds, where ds is the choice of the direction d0 maximizingMLRTðps � Lsd

0=2;d0; LsÞ.

Appendix C. Tracking algorithm

The search cone is defined by

UdðdjÞ :¼ fd0 2 S2 : \ðdj;d0Þ < dg;

where \ðdj;d0Þ 2 ½0;p� denotes the angle between dj and d0. The

direction for the continuation step is djþ1, which is the choice of

d0 2 UdðdjÞ maximizing MLRTðpj;d0; LtrackÞ. The tracking step is com-

pleted by

pjþ1 :¼ pj þ l djþ1; ð1Þ

In the implementation, the computation of pjþ1 includes a shift tothe nearest voxel center. This iteration is carried out until one ofthe following stopping criteria is satisfied:

� The boundary of the picture is reached, i.e., not all the line seg-ments in the search cone lie in X.� The intensity of the newly added piece of filament is too low,

i.e.,

MLRTðpj;djþ1; lÞ < Itrack; ð2Þ

However, we continue the tracking with (1) if it bridges a gap, i.e.,

MLRTðpj þ Lgapdjþ1; djþ1; Ltrack � LgapÞ P Itrack:

Abortion of the tracking indicates that we arrived close to a fila-ment end.

Fig. 11. Swiss 3t3 mouse fibroblast. (a) The automatically tracked filaments of the second example in red. The area for the actin filament density measurement is surroundedby the blue square with 298.4 nm edge length; (b) the collapsed image; (c and d) individual sections through the lamellipodium; bar, 200 nm. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)

Table 4Parameters used for Swiss 3t3 mouse fibroblast.

Ltrack 37.3 nmd 5 degreesLgap 14.92 nmItrack 0.76l 7.46 nmLs 37.3 nmLs 0.8h 3.73 nme 7.46 nm

0 15 30 45 60 75 90 105 120 135 150 165 1800

0.5

1

1.5

2

2.5

3

3.5

4

x10 4

α

λ

a

Fig. 12. Angular distribution and mean length of the filaments corresponding to Fig. 11. Tthe cell was not organized for persistent protrusion.

26 C. Winkler et al. / Journal of Structural Biology 178 (2012) 19–28

Appendix D. End point detection

The direction of the filament at the last identified point isimproved by looking backwards to obtain dj, which is the choice ofd0 maximizing MLRTðpj � Ltrackd0; d0; LtrackÞ. Now the tracking is re-started by looking in a search cone centered at dj for the new direc-tion djþ1, the choice of d0 maximizing MLRTðpj � ðLtrack � lÞd0; d0;LtrackÞ, followed by (1) until (2) happens.

0 15 30 45 60 75 90 105 120 135 150 165 1800

50

100

150

200

250

300

α

μ

b

he number of filaments perpendicular and parallel to the leading edge indicate that

Fig. 13. (a–d) Sequential z-layers of tomogram in Fig. 11 showing tracking details. Bar, 50 nm.

C. Winkler et al. / Journal of Structural Biology 178 (2012) 19–28 27

Appendix E. Choice of parameters

A reasonable choice for the search length L can be computed asthe length of the longest straight line inside a filament of curvatureradius r and diameter d (Fig. 14):

L ¼ 2ffiffiffiffiffiffiffiffi2rdp

:

Estimates for the local curvature radius can be computed as the ra-dius of circumference of the triangle built by consecutive pointspj�1; pj;pjþ1 on a tracked filament. In terms of the edge lengths a,b, and c of the triangle, the radius is given by

r ¼ abcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ bþ cÞð�aþ bþ cÞða� bþ cÞðaþ b� cÞ

p :

The minimal curvature radius computed from 3 consecutivelytracked points with a maximal deflection angle d is then given by

rmin ¼l

2 sinðd=2Þ :

Noise on the filament points (i.e., wiggling inside a filament, round-ing) strongly affects the curvature determination. As a regulariza-tion we do not take consecutive points, but a triple of points with

Fig. 14. The longest straight line (with length L) inside a filament with diameter dand curvature radius r.

a distance along the filament of 36 nm, which is on the other handsmall enough compared to typical values of the curvature radius.Since the distances between manually tracked points can be signif-icantly larger, the computation of local curvature is only feasible forthe highly resolved automatically tracked filaments.

Appendix F. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.jsb.2012.02.011.

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