TECHNICAL ADVANCE
KymoRod: a method for automated kinematic analysis of
rod-shaped plant organs
Renaud Bastien1,2, David Legland1,3, Marjolaine Martin1, Lucien Fregosi1, Alexis Peaucelle1, Stephane Douady4, Bruno Moulia5,6,* and Herman H€ofte1,*
1Institut Jean-Pierre Bourgin, INRA, Centre National pour la Recherche Scientifique, AgroParisTech, Universite Paris-Saclay, RD10, 78026, Versailles Cedex, France,
2
Department of Collective Behaviour, Max Planck Institute for Ornithology and Department of Biology, University of Konstanz, Konstanz, Germany,
3Biopolymeres Interaction et Assemblages, INRA, UR1368, Nantes F-44316, France, 4Matiere et Systemes Complexes, Universite Paris-Diderot, Paris Cedex 13 75025, France, 5
INRA, UMR 547 PIAF, Clermont-Ferrand F-63100, France, and
6Clermont Universite, Universite Blaise Pascal, UMR 547 PIAF, Clermont-Ferrand F-63100, France Received 26 February 2016; revised 21 June 2016; accepted 24 June 2016; published online 8 September 2016.
*For correspondence (e-mails herman.hofte@versailles.inra.fr; bruno.moulia@clermont.inra.fr).
SUMMARY
A major challenge in plant systems biology is the development of robust, predictive multiscale models for organ growth. In this context it is important to bridge the gap between the, rather well-documented molec-ular scale and the organ scale by providing quantitative methods to study within-organ growth patterns. Here, we describe a simple method for the analysis of the evolution of growth patterns within rod-shaped organs that does not require adding markers at the organ surface. The method allows for the simultaneous analysis of root and hypocotyl growth, provides spatio-temporal information on curvature, growth aniso-tropy and relative elemental growth rate and can cope with complex organ movements. We demonstrate the performance of the method by documenting previously unsuspected complex growth patterns within the growing hypocotyl of the model species Arabidopsis thaliana during normal growth, after treatment with a growth-inhibiting drug or in a mechano-sensing mutant. The method is freely available as an intu-itive and user-friendly Matlab application called KymoRod.
Keywords: kinematics, image analysis, hypocotyl, root, elongation growth, mechano-sensing, cellulose synthesis inhibitor, Arabidopsis thaliana, technical advance.
INTRODUCTION
A major challenge in plant biology is to understand organ growth in terms of the underlying molecular processes. All known hormones, nutrition and many environmental fac-tors influence organ growth. Despite major advances in our understanding of the underlying signalling networks, little is known on how these networks are connected to observed changes in organ growth (Wolf et al., 2012). Advances in this area require efficient methods of kine-matic analysis that bridge the gap between molecular and organ scales by providing information at cellular and supra-cellular scales. Indeed, growing organs show an intra-organ growth flow that is similar to the flow of a vis-cous fluid, putting it at the frontier between the mechanics
of fluids and solids (Moulia, 2013). In recent years many tools have been developed to measure the kinematics of growing organs. Some of them only measure part of the process, focusing on apical hook angle (Geng et al., 2013; Russino et al., 2013; Slovak et al., 2014), while others only measure length and length variation (French et al., 2009; Wang et al., 2009; Geng et al., 2013), the relative elemental growth rate (REGR; van der Weele et al., 2003; Walter et al., 2002) or curvature (French et al., 2009; Wang et al., 2009). Recent advances have been made in measurement of the full kinematics, mainly in roots (Basu et al., 2007; Chavarria-Krauser et al., 2008; Basu and Pal, 2012), inflo-rescence stems (Hall and Ellis, 2013) or a variety of organs
(Barbier de Reuille et al., 2015). One method relies on a structure tensor and optical flow to track the speed of each pixel (Chavarria-Krauser et al., 2008). In order to retrieve the REGR on a curving organ, the velocity due to the curva-ture of the organ needs to be subtracted from the overall velocity of the organ in space. The previously developed Kineroot software (Basu et al., 2007) solves this problem by constant reference to the midline of the organ. However, it requires more user intervention, and interpolation that can modify the morphometric measurements. The more recently developed MorphoGraphX method works directly with curved surface images extracted from three-dimensional microscopic data (Barbier de Reuille et al., 2015). This method is very precise but of limited through-put, in particular for larger organs. Some other methods require contrast-enhancing manipulations (Basu et al., 2007; Hall and Ellis, 2013), have relatively low throughput (Iwamoto et al., 2013) and/or are not well adapted for the analysis of organs with more complex shapes and growth patterns. Here we propose a method that encompasses all the processing steps of the kinematic analysis. The median axis of seedlings is computed using Voronoi skeletonisa-tion and local velocity along seedlings is measured with a correlation algorithm, Rod-PIV, combining the classical algorithm to measure fluid velocity, namely particle image velocimetry (PIV; Sveen and Cowen, 2004), with the kine-matics of the theory of structure mechanics (Lautrup, 2011). This algorithm is able to track the displacement of elements directly on the midline of the organ and thus can cope with complex organ movements. The method is fully automated and is provided as the intuitive and user-friendly Matlab application KymoRod, which also provides an interface for the quality control of the successive operations.
Using this method, we studied the evolution of the intra-organ growth pattern during normal growth of the Arabidopsis hypocotyl, upon treatment with the cellulose synthesis inhibitor isoxaben or in the mechano-sensing mutant feronia (fer)-4 (Shih et al., 2014). This revealed the unsuspected complexity of the growth patterns and con-firmed the role of mechano-sensing in the supra-cellular coordination of cell expansion. KymoRod should greatly facilitate the analysis of molecular mechanisms underlying growth processes in plants.
RESULTS AND DISCUSSION
Overview of the KymoRod algorithm and software To elucidate the full kinematics of a growing organ, multi-ple geometrical traits need to be measured in parallel. First, following the Euler–Bernoulli description of rod/beam theory (Bastien et al., 2014) and the fact that plant cells do not experience shear growth (Silk, 1992), the organ can be described by its midline and its cross-section, respectively by the curvilinear abscissa, s, and the radius of the cross
section, R(s, t). Thus the measurements of the angle of each element along the midline, A(s,t), and of the curva-ture, C(s, t), are then sufficient to get a full description of the shape (Bastien et al., 2013). As the organ is growing, it is necessary to measure the REGR _Eðs; tÞon the midline, as well as on the cross-section, _ERðs; tÞ. Furthermore, since
these three quantities can vary over time while the ele-ments of the growing organ are moving with respect to each other, the material derivative D/Dt also needs to be taken into account (Bastien et al., 2014). Finally, the dimen-sionless differential growth term, D(s, t), expresses the pro-portion of differential growth in total organ growth (Bastien et al., 2014) can be obtained from the ratio between DC(s,t)/Dt, the material derivative of the curvature, and _Eðs; tÞ, the REGR on the midline.
An overview of the first processing steps of the KymoRod software is shown in Figures S1 and S2 in the Supporting Information. A more detailed description of the algorithms underlying each step is provided in the Experimental Procedures and a description and screen-shots of the interface for each step are shown in the KymoRod user guide (File S1). The software was tested on time-lapse photographs of etiolated Arabidopsis seedlings grown vertically under infrared light (Figure S1, Video S1). After importing the image stack, the contour of the seed-lings is computed from a threshold value that can be deter-mined manually or automatically. The skeleton of the contour is identified using the Voronoi diagram of the con-tour as described in Clement et al. (2012). The midline is extracted automatically as the longest branch within the skeleton (Figure S1). If the cotyledons have touched the hypocotyl we first manually separated the two organs with a black line using IMAGEJ to make sure the midline followed
the entire apical hook. The midline terminated at the cotyledon tip in a way similar to that of a previously described method (Miller et al., 2007). As a result the cotyledons are included in the REGR profiles. Variations in the choice of the endpoint between successive images in some cases created instabilities in the total length of the midline. These variations, however, did not interfere with the analysis of the intra-organ growth patterns and could be smoothened digitally. To obtain a measure of the length of the hypocotyl without the cotyledons, it is conceivable to terminate the midline, for instance at the position of maximal curvature as described (Wang et al., 2009)
The curvilinear abscissa along the midline is calculated as the cumulative length to the origin (Figure 1a). For each segment, the angle with respect to the vertical, A(s) and the curvature C(s)= dA(s)/ds are also calculated (Fig-ure 1b,c). The radius of the organ, R(s) is computed as the distance between each point on the skeleton and its closest neighbour on the contour (Figures 1 and S3). The transi-tion between root and hypocotyl is identified automatically as the point presenting the largest radius (Figure S3). This
point is first identified on the last image, and then in the close neighbourhood of each preceding image. This point is used for defining the origin of curvilinear abscissa, s= 0. The root is then defined as s < 0 and the hypocotyl as s> 0 (Figure S3).
To calculate the median elongation, we developed a novel algorithm, Rod-PIV. PIV is commonly used in fluid dynamics and has been introduced in biology to measure displacement fields (Py et al., 2005; Adrian and Wester-weel, 2011). The principle of PIV is to measure a dense velocity field in the image using sub-image correlation. However, PIV only measures the global velocity of the ele-ments in the image. With this technique, the velocity due to the REGR can be masked by the velocity due to the movement of the organ in space (e.g. tropism). We adopted an alternative, calculation-efficient approach based on rod theory, in which moving sub-windows are centred on the midline (Figure S2). Here the measured velocity directly reflects the velocity due to the REGR along the organ. A sub-window on the image at time t is corre-lated with a moving sub-window on the midline on the image at time t+ dt. The position of the correlation peak along the curvilinear abscissa gives the displacement, delta, on the midline between two images (Figure S2). Then by derivation of this displacement, the REGR on the midline, _Eðs; tÞ, can be obtained. The kinematics is per-formed independently of noticeable shifting of the stem axis in space and without any interpolation. This allows detection of the REGR even during complex organ move-ments such as tropisms.
This Rod-PIV algorithm depends on the presence of a clear pattern of contrasting pixels at the organ surface, which was mostly the case with the illumination conditions and the image resolution (around 5 lm/pixel) used. The window of correlation should then be taken as large as possible while remaining within the organ boundaries. For the images of the Arabidopsis hypocotyl, the diameter of the organ was around 20 pixels. The pattern produced by the cells on a sub-window of 16 pixels showed a standard deviation for the light intensity of 10 (8 bits between 0 and 255). This value was sufficient to properly discriminate images through image correlation.
The time step between two consecutive measurements is also critical. If it is too short, no displacement can be measured, but if it is too long the deformation of the pat-tern may prevent a successful correlation. A strong upper limit is given by the stretching of the pattern over a dis-tance of 1 pixel. For the sub-window of 16 pixels under our growth conditions, a deformation of 1 pixel is observed after 100 min. A weak lower limit is given by the inability to measure a displacement of less than 1 pixel. This dis-placement, however, is integrated over the whole length of the growth zone. In our current setup, a growth zone of 5 mm is represented by around 1000 pixels. A difference of 1 pixel within the growth zone is detectable within 2 min. The time step of 20 min used in this study is therefore a good compromise for the measurement of the local REGR. It should be noted that small spatial fluctuations corre-sponding to experimental noise may have to be smooth-ened. The limits of the PIV method have been discussed in s = LH t (h) s (mm) –2 0 2 4 6 –2 0 2 C (mm–1) t (h) –10 –5 0 5 10 t (h) 0 10 20 30 –2 0 2 4 6 10–5 0 2 4 6 8 10 0 10 20 30 A –2 0 2 4 6 0 10 20 30 (a) (b) (c) (d) E (sec. –1) s = 0 s = LR A
Figure 1. Kinematics of hypocotyl and root growth of Arabidopsis thaliana.
(a) Contour of a seedling showing the midline of the plant (in red) obtained through Voronoi skeletonisation of the contour. The curvilinear abscissa s is defined along this line from the transition between the root and the hypocotyl (s= 0). Positive and negative values of s account for the hypocotyl (with length LH) and root (with length LR) respectively.
(b) Angle (A) with respect to the vertical along the midline for all time points. The high values towards the top of the hypocotyl correspond to the position of the hook.
(c) Curvature (C) along the midline in mm1for each time point.
(d) The relative elemental growth rate (REGR) ( _E) along the midline in sec1for each time point. The growth is localized towards the apex for the hypocotyl and the root, with a higher REGR in the root compared with the hypocotyl.
more detail in previous studies (Adrian, 1997; Kahler et al., 2012).
Establishment of growth gradients in developing seedlings
Figure 1(d) shows the distribution of the REGR along the hypocotyl and root during early seedling development. The emerging seedling first grew slowly, but accelerated its growth first in the root, then in the hypocotyl, with the lat-ter starting from the base as described previously (Pelletier et al., 2010). Steady-state growth was reached at a hypoco-tyl length of around 5 mm. Subsequently, the length of the growth zone remained fixed, and as a result the growing part of the hypocotyl became progressively smaller. Con-cerning the root, in the growth conditions used (dark, no sucrose in the medium) growth slowed down after about 15 to 20 h to stop at 40 h. Additionally we were able to characterize the trends of the radial REGR, _Erad , and the
logarithm of the growth anisotropy, logðaÞ ¼ logð _E= _EradÞ
(Figure S4). At around 15 h, seedlings showed a strong radial expansion along their length, after which radial expansion primarily occurred below the apical hook. Fluctuations in the spatial distribution of growth within the organ
Intra-organ growth patterns were analysed for the hypoco-tyls of 18 seedlings, which showed a REGR of 59 106 1 9 106sec1 (1.8 0.4% h1) (Figure S5). These seedlings served as a control for the experiment in
Figures 3 and S6 and were therefore transferred to a con-trol medium 5–7 h after the start of the observation (indi-cated by vertical black bars in Figures S5 and S6). Figure 2 shows the analysis of two representative seed-lings without the smoothing step in Figures S5 and S6 to better reveal the intra-organ fluctuations in REGR. The length of the growth zone, Lgz, increased with total organ length up to a length of 5–6 mm and then remained more or less constant. Interestingly, the imaging method pro-vided insights into the cellular processes that underlie organ growth. Indeed, at the apical side of the growth zone, the REGR ( _E) abruptly reached a maximum of over 3% h1(19 105sec1) This reflects the growth accelera-tion coinciding with the breaking of the growth symmetry of the cells that leave the apical hook (Peaucelle et al., 2015). Further towards the base of the growth zone, the REGR decreased more gradually, corresponding to the gradual cessation of cell elongation. Interestingly, despite the overall steady-state growth rate of the organ, large fluctuations in REGR ( _E) were observed within the growth zone. This presumably reflects the growth pulses of indi-vidual cells that appear to be coordinated through a mechanical feedback mechanism (Uyttewaal et al., 2012), as shown below.
Kinetics of growth inhibition by a cellulose synthesis inhibitor
We next studied the kinetics of the response of growth to the cellulose synthesis inhibitor isoxaben (Figures 3 and
5 0 10 15 s (mm) 0 5 t(h) 10 15 0 1 2 10 15 Lgz (mm) 0 2 4 6 v (mm h –1 ) 0 0.05 0.1 0.15 0.2 10 15 0 5 t (h) 5 0 5 0 5 0 5 0 5 0 10 15 E (sec –1) 0 1 2 10 15 Lgz (mm) 0 2 4 6 v (mm h –1 ) 0 0.05 0.1 0.15 0.2 0 0.5 1 × 10–5 × 10–5 × 10–5 × 10–5 0 0.5 1 (a) (b) E ( sec –1) Figure 2. Kinematics of wild-type (WT) hypocotyl growth.
Growth kinematics of two different hypocotyls are displayed. The relative elemental growth rate ( _E) of each WT hypocotyl is measured along the midline. In contrast to Figure 1, the curvilinear abscissa is measured from the apex (position 0) to the base of the hypocotyl.
S6). Isoxaben triggers internalization of the cellulose syn-thase complex (Gutierrez et al., 2009), causing inhibition of cellulose synthesis and cell elongation (Desprez et al., 2002). The link between reduced cellulose synthesis and growth inhibition is not exactly understood, but depends at least in part on the receptor kinase THESEUS1, which appears to act as a sensor of cell wall integrity (CWI; Hematy et al., 2007). Understanding this signalling mecha-nism requires information on the kinetics of the response to isoxaben. To investigate this, seedlings were grown for 4 days in the dark before being transferred (at a hypocotyl length between 4 and 7 mm) to a medium supplemented or not with 100 nM isoxaben. In control seedlings
(Fig-ures 3A and S5) overall REGR was maintained [a 4.4 0.6 mm growth zone (Lgz) with an average REGR ( _Eav) of 5.49
10–6 0.6 9 10–6sec1, 1.9 2% h1]. (Note that in some cases a small transient reduction in _Eav and Lgzcould be observed, presumably as a result of the handling during the transfer.) Upon transfer to isoxaben, the overall hypo-cotyl growth rate (v) as well as the average REGR ( _Eav)
diminished gradually, with the first observable effects occurring at between 2 and 3 h, before reaching zero between 7 and 10 h after transfer (Figures 3 and S6). Inter-estingly, the maximum REGR (Figure 3, bottom panel, red line) already showed a drop, starting from the top, after less than 1 h. This indicates that, firstly, the lag-time for growth inhibition is only slightly longer than that for cellu-lose synthase complex internalization (about 30 min;
Gutierrez et al., 2009) and, secondly, that the drug primar-ily prevents growth acceleration rather than the growth of cells that had already undergone this transition. In conclu-sion, the analysis of intra-organ growth behaviour pro-vided valuable insights into the mode of action of the drug, which could not be obtained by the study of overall organ growth.
Growth oscillations in a mechano-sensing mutant
To investigate the role of mechano-sensing in supra-cellular growth coordination, we compared hypocotyl growth between the wild type and fer-4 (Shih et al., 2014) (Figures 4 and S7). As observed previously (Li et al., 2015), the fer-4 mutant showed a reduced overall hypocotyl growth rate. Interestingly, the maximum REGR ( _E) (99 106 2 9 106sec1, 3.2 0.7% h1) was compara-ble to that of the wild type (89 106 2 9 106sec1, 2.8 0.7% h1), indicating that in the mutant the cells have a normal ability to expand. The shape of the growth zone however, was different, it was narrower (2.6 0.8 versus 3.2 0.8 mm) with a more gradual increase and a steeper decrease in REGR ( _E) (Figure 4). This indicates a less abrupt growth acceleration and a more rapid deceleration of the cells in the mutant. In addition, the growth behaviour of the cells was more chaotic, as shown by the REGR oscillations. This confirms that mechanical feedback signalling is required to coordinate cell growth within the organ (Shih et al., 2014). 10 15 s (mm) 0 5 10 15 0 1 2 t (h) 10 15 Lgz (mm) 0 2 4 6 v (mm h –1 ) 0 0.05 0.1 0.15 0.2 5 0 10 15 0 5 t (h) 5 0 5 0 5 0 5 0 5 0 10 15 0 1 2 t (h) 10 15 Lgz (mm) 0 2 4 6 v (mm h –1 ) 0 0.05 0.1 0.15 0.2 (a) (b) E (sec−1) E (sec˙ −1) 0 0.5 1 × 10–5 × 10–5 × 10–5 × 10–5 0 0.5 1 E (sec –1 ) E (sec –1 )
Figure 3. Kinematics of wild-type hypocotyl growth before and after application of isoxaben.
The relative elemental growth rate (REGR) ( _E) mea-sured along the midline, from apex to base (upper panels of parts a and b), the overall hypocotyl growth rate (v, red line) and the length of the growth zone (Lgz, blue line) (middle panels of parts a and b) and the maximum REGR ( _E, red line) and average REGR inside the growth zone ( _E, blue line) (lower panels of parts a and b) over 15 h. After 5.3 h, plants were transferred to a new plate with-out (a) or with (b) 100 nM isoxaben. The vertical black line indicates the moment of the transfer.
In conclusion, we have presented a simple and robust method for the analysis of sub-organ growth patterns in plant seedlings. Its performance exceeds that of previous methods that are mostly too laborious for the analysis of large numbers of samples. The method is packaged in KymoRod, a user-friendly application freely available on internet (http://github.com/ijpb/KymoRod), which should facilitate the study of the cellular basis of organ growth for non-specialist users.
EXPERIMENTAL PROCEDURES
Plant growth, image acquisition and pre-treatment Arabidopsis seeds (genotypes Col-0 and fer-4; Duan et al., 2010) were surface sterilized (Santoni et al., 1994), plated on Arabidop-sis medium (Santoni et al., 1994) without sucrose and 0.6% agar-ose and stratified in the dark at 4°C for 2 days. Next, seed coats were removed using tweezers and a fine needle and six embryos were positioned on the agarose surface of a round 5-cm Petri dish with all the roots oriented to the same side. The seed coat was removed to facilitate the subsequent automated image analysis. Plates were exposed to 200 lMm2sec1white light for 3 h to induce germination, then wrapped in two layers of aluminium foil and incubated at 24°C vertically with the roots oriented to the bot-tom for the indicated time period. Next, plates were positioned vertically under green safe light with the bottom of the plate ori-ented towards a camera (Nikon D7000 with camera lens AF-S micro nikkor 60 mm f/2.8g ed) and illuminated for the indicated time period by infrared light emitted by LEDs (Belux, RS compo-nents, http://www.rs-components.com/). Pictures were taken every 5 min. For the isoxaben treatments, seedlings were transferred
individually to a new plate with 50 nMisoxaben or a control plate using tweezers under green safe light. Critical factors for the image analysis are the contrast (for the thresholding) and the res-olution (to have sufficient surface features for the measurement of the elongation) of the images and the stability of light source (to avoid fluctuations in light intensity between images). Seedlings should be well separated on the plate. Image stacks were cropped into stacks corresponding to individual seedlings using IMAGEJ. If the cotyledons touched the hypocotyl, a black line was introduced manually to separate the two organs on the image.
Contour detection and skeletonisation
The first step is the identification of the midline of the organ. This step is critical and its success depends primarily on the quality of the image, in particular the contrast between the organ of interest and the background. A threshold value is obtained from grey-scale images using automatic or manual thresholding on each image. Automatic threshold algorithms comprise the Otsu method, which consists of maximizing the variance of grey levels between classes (Otsu, 1979), or the maximum entropy method, which consists of maximizing the sum of entropies of the grey levels in each class (Kapur et al., 1985). Contours are obtained using the Matlab sub-pixellar ‘contour’ function, resulting in a more accurate and smoother contour than the results obtained with simple binarised images (Figure S1). The largest contour is identified as the contour of the organ of interest.
The midline of the contour is then identified using the Voronoi diagram of the contour (similar to the technique used in Clement et al., 2012). The Voronoi diagram of a set of points, called the germs, associates each germ with a polygon corresponding to the region of influence of the germ. The vertices of the Voronoi dia-gram are therefore equidistant to three or more germs. When the germs constitute the contour of a shape, the vertices of the
5 0 10 15 s (mm) 0 5 t (h) 10 15 0.5 1 10 15 Lgz (mm) 0 2 4 v (mm h –1 ) 0 0.05 0.1 0.15 0.2 5 0 5 0 5 0 5 0 5 0 10 15 0 5 10 10 15 0 0.5 1 10 15 Lgz (mm) 0 2 4 6 v (mm h –1 ) 0 0.05 0.1 0.15 0.2 (a) (b) ˙E (sec−1) ˙E (sec−1) 0 0.5 1 × 10–5 × 10–5 × 10–5 × 10–5 0 0.5 1 E (sec –1) E (s e c –1) Figure 4. Kinematics of hypocotyl growth of the wild type and fer-4 mutant.
The relative elemental growth rate (REGR) ( _E) mea-sured along the midline, from apex to base (upper panels of parts a and b) the overall hypocotyl growth rate (v, red line) and the length of the growth zone (Lgz, blue line) (middle panels of parts a and b) and the maximum REGR ( _E, red line) and average REGR inside the growth zone ( _E, blue line) (lower panels of parts a and b) over 15 h.
Voronoi diagram located within the shape represent the skeleton of the shape.
The beginning of the midline can be defined as a particular point of the skeleton inside the contour, e.g. the tip of the root can be defined as the lower point of the image in a vertically growing hypocotyl. However, this method can induce lateral branches if contours present bumps. In order to avoid this, the midline of the contour is defined as the longest path on the skeleton starting from the lower point of the skeleton. The midline ends at the end of the cotyledons.
Curvilinear abscissa, angle and curvature
The curvilinear abscissa along the midline, s, is computed from the cumulative sum of distances between points:
ds2¼ dx2þ dy2
So for each point of the midline i, its curvilinear abscissa si is
obtained by si¼ si1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxiþn xinÞ2 þ ðy yinÞ2 q
where n is the number of points used to compute the curvilinear abscissa. One of the benefits of the Voronoi diagram is that it associates each point of the midline with its closest neighbour on the contour. The distance between these two points is the radius of the organ, R(s).
From this it comes directly that the angle A(s) of each segment respective to the vertical is given as:
AðsÞ ¼ arcsin dxðsÞ dyðsÞ
Finally, the relation that gives the curvature C(s) is CðsÞ ¼dAðsÞ
ds
Alignment of the curves and detection of the transition between the root and the hypocotyl
In order to uncover some of the errors that can be found in the rel-ative position of each element, it is necessary to ensure that curvi-linear abscissae are aligned between two successive images. Correlations are performed between two different measurements of the radius, R(s), as the radius should vary between two consec-utive images. The maximum correlation gives the relative dis-placement between two midlines.
Measurement of the elongation
At each point along the skeleton at a given time t, we can define a sub-window W1 (s0) centred on the curvilinear abscissa
at position s0. On the image at time t + dt, a sub-window W2
(s0) centred on the midline is moved between the curvilinear abscissa s0= s0+ Δs and s0= s0 Δs. The peak reflecting the
maximum of correlation between W1 and W2gives directly the
displacement d along the midline of an element, as compared with s0, at time t during a time interval dt. From this
displace-ment we calculated the REGR ( _E) (Figure 1d) by derivation along the curvilinear axis, where
_EðsÞ ¼ dd ds
1
dt
Growth zones in the root and hypocotyl could thus be identified. The kinematics are performed independently of noticeable shift-ing of the stem axis in space and without any interpolation. This
allows the detection of the REGR when a displacement in space of the organ is observed (e.g. tropism) (Figure S2).
Material derivative and elongation in the radial direction As the organ is expanding, the position of the material elements relative to each other is modified. In order to get a meaningful measurement, the material derivative needs to be computed. This can be easily obtained with the REGR (Silk, 1984; Merret et al., 2010; Bastien et al., 2014) D DT¼ d dtþ v d ds
where v represents the velocity of an element as a position, and is given by integrating _E from the base, s= 0, to the position s:
v¼ Z s
0 dl _EðlÞ
The REGR in the radial direction is then defined by _ERðsÞ ¼ R1
ðsÞ
DRðsÞ
Dt
The anisotropy, a, between the longitudinal REGR, _E, and the radial REGR, _Erad, is given by (Figure S4):
a¼ _E _ Erad
Length of the growth zone and median REGR
The growth zone of elongating organs is known to be finite and to move with the position of the apex (Peters and Tomos, 2000; Wal-ter et al., 2002; Bastien et al., 2014). The length of the growth zone, Lgz (Zegzouti et al., 2006), is then a relevant and simple
descriptor of organ growth. Due to noise in the measurement, this length is defined as the distance from the tip of the organ that accounts for 80% of the elongation. Finally, the average REGR ( _Eav ) of the organ is defined as the average of the REGR inside the growth zone.
Computation of a kymograph
The results of kinematic analysis can be visually represented using a kymograph. A synthetic image is created, with the x-axis repre-senting time and the y-axis reprerepre-senting a curvilinear abscissa. For each time frame, the measure of the REGR ((\dot{E}(s))along the medial axis is projected onto the vertical line corresponding to the time frame.
ACKNOWLEDGEMENTS
Part of this work was financed by ANR/FWF grant ‘Pectosign’ (to HH) and the Laboratoire d’Excellence ‘Saclay Plant Science’. RB thanks Etienne Couturier and Rapha€el Clement for their help with the skeletonisation function.
CONFLICT OF INTEREST
The authors have no conflict of interest. SUPPORTING INFORMATION
Additional Supporting Information may be found in the online ver-sion of this article.
Figure S1. Overview of the skeletonisation algorithm. Figure S2. Displacement algorithm.
Figure S4. Radial relative elemental growth rate and growth aniso-tropy.
Figure S5. Kymographs of the relative elemental growth rate _Eðs;tÞ for wild-type hypocotyls.
Figure S6. Kymograph of the relative elemental growth rate _Eðs; tÞ for wild-type hypocotyls before and after the application of isox-aben.
Figure S7. Kymographs of the relative elemental growth rate _Eðs;tÞ for fer-4 (a) and wild-type (b) hypocotyls.
Video S1. Intra-organ growth patterns in an etiolated Arabidopsis seedling.
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