Summary
A key question in developmental biology is how morphogenetic regulators control patterning. Recent findings have raised an important question: do morphogenetic signals carry information not only in space, as originally proposed in the morphogen concept, but also in time? The hormone auxin is an essential plant morphogenetic regulator that drives rhythmic organogenesis at the shoot apical meristem. Here, we used a quantitative imaging approach to map auxin distribution and response. We demonstrate the existence of high-definition spatio-temporal auxin distribution in the meristem. We provide evidence that developing organs are auxin-emitting centers that could allow self-sustained distribution of auxin through a structured auxin transport network converging on the meristem center. We finally demonstrate that regulation of histone acetylation allows cells to measure the duration of the exposition to auxin preceding organ initiation, providing a remarkable example of how both spatial and temporal morphogenetic information generates rhythmic patterning.
Main text
Specification of differentiation patterns in multicellular organisms is classically thought to be regulated by gradients of morphogenetic regulators (morphogens in animals) providing positional information to cells1. In plants, the hormone auxin is one of the main morphogenetic regulators2,3. This small molecule acts during embryonic development and post-embryonic development, where it is essential for the reiterative organogenesis characteristic of plants4.
Notably, plant shoots develop post-embryonically through rhythmic organ generation in the shoot apical meristem (SAM), a specialized tissue with a stem cell niche in its central zone (CZ). In Arabidopsis thaliana, as in a majority of plants, organs are initiated sequentially in the SAM peripheral zone (PZ) and at a relative angle close to 137° from the previous one, either in a clockwise or anti-clockwise spiral5. SAM organ patterning or phyllotaxis has been extensively analyzed theoretically6–8. A widely accepted model proposes that the time interval between organ initiations (the plastochrone) and the spatial position of organ initiation emerge from a combined action of isotropic inhibitory signals emitted by pre-existing organs and the SAM center5,6. Tissue growth would then self-organize organ patterning by moving organ-associated signaling centers away from the stem cells and leaving space for new ones. Biologically, evidence suggests that auxin provides the positional information that drives phyllotaxis patterning9,10. Auxin, thought to be synthesized throughout the meristem11–14, has been proposed to be transported directionally toward incipient primordia where it activates a transcriptional response leading to organ specification3,9,15. A network of PIN-FORMED1 (PIN1) efflux carriers, whose polarity determines the direction of auxin fluxes, regulates auxin spatio-temporal distribution cooperatively with other carriers9,16. The activity of this network results in accumulation of auxin that triggers organ initiation. The PIN1 network was also proposed to create an auxin depletion at the organ periphery that specifies organ boundaries and blocks organ initiation in the organ vicinity9,17–21. In addition, it has been shown that the CZ is markedly less responsive to auxin17,18. Altogether, these regional cues restrict new organ location in the growing SAM as proposed in formal models. The genetically-encoded biosensor DII-VENUS, a synthetic protein degraded directly upon sensing of auxin, recently provided a first direct experimental qualitative visualization of spatial auxin gradients in the SAM17,22.
The SAM is rather unique in that it implicates a continuous redistribution of a morphogenetic regulator in a growing tissue with helicoidal symmetry. This suggests that auxin could carry spatio-temporal morphogenetic information in the SAM. This is reminiscent of recent findings in animals that are questioning whether morphogenetic signals carry information only in space (as originally proposed in the morphogen concept23 and suggests rather a spatio-temporal nature for positional information24–26. Here, we used a quantitative imaging approach to reveal that auxin indeed provides spatio-temporal morphogenetic information, analyze the mechanisms generating auxin 4D dynamics and understand how this information is processed in the SAM to generate rhythmic patterning.
Spatio-temporal auxin distribution
In the SAM, the biosensor DII-VENUS fluorescence reports for auxin concentration with cellular resolution17,22. To extract quantitative data on auxin distribution, we generated a DII-VENUS ratiometric variant, named hereafter qDII (quantitative DII-VENUS). qDII consists of a RPS5A promoter driving stoichiometric co-expression of DII-VENUS and a non-degradable TagBFP reference27,28(Extended Data Fig. 1a-h). We also introduced in plants expressing qDII a stem cell-specific CLV3::mCherry nuclear transcriptional reporter29 that provided a functional and robust geometrical reference of the SAM center (Fig. 1a-b and Extended Data Fig. 1i-m).
All analyzed meristems (21 individual SAM, 24945 nuclei) showed qDII pattern similar to DII-VENUS, with auxin maxima locations following the phyllotactic pattern17 (Fig. 1a-b). Despite the fact that SAMs were imaged independently and not synchronized, qDII patterns appeared highly stereotypical with easily identifiable fluorescence maxima and minima. This was confirmed by image registration using SAM rotations (applying prior mirror symmetry if necessary; Fig 1c-d and Extended Data Fig. 2a-c). All images could be superimposed with limited loss of information definition (Extended Data Fig. 2a-c). This shows that auxin distribution follows the same synchronous pattern at population scale, with restricted angular and rhythmicity variability (Extended Data Fig. 2f-g, Supplementary Method 1), resulting, up to a rotation, in apparent stationarity (Fig. 1g).
To further quantify auxin distribution, we developed a mostly automated computational pipeline to extract SAM quantitative fluorescence (Fig. 1b and Supplementary Method 2). We used the spatial distribution of 1-DII:VENUS/TagBFP as a proxy for auxin distribution, named hereafter “auxin” (Fig. 1b) and focused on the epidermal cell layer (L1) where organ initiation takes place30. Primordium 0 (P0) was defined as the location of the absolute auxin maximum in the PZ and other local maxima where called Pn (Supplementary Note 1), with n corresponding to their rank in the phyllotactic spiral. The pipeline then allows quantifying nuclear signal information and aligns all the SAMs to a common clockwise reference frame with standardized x,y,z-orientation with the P0 maximum to the right. This automatic registration showed that auxin maxima are positioned in the SAM with a precision close to the size of a cell both in distance to the center and in azimuth (with a maximal standard deviation of 8.4 µm or 1.5 cells, Fig. 1f).
We then considered the temporality of auxin distribution by using time-lapses over a time span close to the system period, the plastochrone. P0 and successive auxin maxima moved radially (Extended Data Fig. 2d). Remarkably, while the average radial distance of each local maximum Pn to the SAM center progresses (Extended Data Fig. 2d), their standard deviation does not change significantly in time, reflecting the synchronized movement of local maxima, with limited meristem to meristem growth variation. After 10h, every Pn local maximum has almost reached the starting position of the next local maximum, Pn+1, but after 14h they have passed it over (Extended Data Fig. 2d). This suggests that a rotation of 137.5° (Extended Data Fig. 2e) that replaces Pn by Pn+1 corresponds to a 10 to 14 hours temporal progress (Fig. 1g). This is confirmed by error measures obtained with different rotation angles (Extended Data Fig. 2h), allowing to conclude that plastochrones have a value of 12h ± 2h. We thus derived a continuum of primordium development by placing Pn+1 time series one plastochrone (12h) after Pn time series on a common developmental time axis (Fig. 1g). Together with the developmental stationarity, this allows reconstructing auxin dynamics over several plastochrones from observations spanning only one. A 4D quantitative map of auxin distribution in the SAM demonstrates dynamic building of auxin maxima in the PZ first as finger-like protrusions (visible at P-2, P-1 and P0) from a permanent high auxin zone at the center of the meristem (Supplementary video 1), as previously predicted18. At later stages, auxin minima are progressively established precisely in between the auxin maxima and the CZ and not surrounding the auxin maximum (Extended Data Fig. 3).
We next wondered whether the motion of auxin maxima and minima could purely result from cellular growth as proposed in previous theoretical models19,21. Following a P1 maximum, we observed that cells in the auxin maximum at time 0h gradually became part of the depletion zone at time 10h (Fig. 2a-c). Using nuclear motion to estimate cell motion vectors and compare them with auxin maxima motion, we further found that the average radial speed of auxin maxima between stages P1 and P4 surpasses the average displacement of individual nuclei, by up to more than 1µm/h (or nearly 2 cells in 10h) at its peak in P2 stage (Fig. 2d-e). These results show that auxin maxima are not attached to specific cells; instead they travel as a wave in the tissue. Consequently, the SAM cellular network provides a dynamic medium in which auxin distributions move radially with their own velocity relative to the growing tissue (Fig. 2d-e). Analysis on time-courses up to 14h revealed significant auxin variations in certain cells over one plastochrone while auxin levels remained unchanged in others (Fig. 2f and g, bottom rows). However, neighboring cells always showed limited differences in their temporal auxin profiles. Altogether, we concluded from these observations that there is a high definition spatio-temporal distribution of auxin that moves faster than growth in the tissue and provides cells with graded morphogenetic information both in space and time (Fig. 2h).
The control of auxin spatio-temporal dynamics
The creation of auxin maxima first as protrusions of a high auxin zone in the CZ contrasts with the largely prevailing vision of organogenesis being triggered by local auxin accumulation at the periphery of the CZ and concomitant auxin depletion around auxin maxima9,18–21. The partial uncoupling of auxin distribution dynamics and growth led us to question our current knowledge of the spatio-temporal patterns of PIN1, given their central role in controlling auxin distribution9,18,19,21. Co-visualization of a functional PIN1-GFP3 and qDII/CLV3 fluorescence over time showed that PIN1 concentration increases from P0 and reaches a maximum at P2 before decreasing (Fig. 3a-c), consistently with previous observations15,31. To quantify PIN1 cellular polarities, we used staining with the fluorescent dye propidium iodide (PI) as a reference to position the PIN1-GFP signal in 3D relative to the L1 anticlinal cell walls32 (Fig. 3b and Supplementary Method 2). These 3D high-resolution reconstruction of PIN1 polarities demonstrated that the crescent-shape often thought to indicate polarities in cells15,31 does not always correlate with polarities and can thus be sometimes misleading (Extended Data Fig. 4). We mapped the vector fields of polarities to identify the trends in auxin flux directions at the scale of the SAM (Fig. 3d, and Supplementary Method 2). While some local converging PIN1 polarities can be seen close to auxin maxima at the PZ (Fig. 3e-f), the vector fields rather show large-scale convergence of PIN1 toward the center of the SAM that meet in front lines where auxin maxima protrude from the CZ (Fig. 3d and Extended Data Fig. 5). We detected the inversion of PIN1 polarities at organ boundaries previously observed15 and our quantifications show that this occurs only from P6, thus isolating the flower from the rest of the meristem from this late stage. P3 to P5 stages show a general flux toward the SAM that is locally deflected around the zones of auxin minima before converging back toward the meristem center (Fig. 3d and Extended Data Fig. 5). Over the course of a plastochrone, only limited changes of the PIN1 network are observed (Extended Data Fig. 5), suggesting that changes in auxin distribution at this time resolution might not require important adjustments of the direction of the auxin fluxes. Other carriers are active in the SAM16 and the role of PIN1 in auxin distribution could also have been overestimated.
The analysis of the transport network led us to question where auxin could be produced in the SAM. YUCCAs (YUCs) have been shown to be limiting enzymes for auxin biosynthesis during development11,33. We thus mapped expression of the eleven YUCs in the SAM (Extended Data Fig. 6)33,34. Only YUC1,4,6 were detected and only YUC6 in the SAM proper with a very weak expression in the CZ (Fig. 3g-h and Extended Data Fig. 6). This is coherent with genetic and expression data (Extended Data Table 1)11,35. Both YUC1 and YUC4 are expressed at the L1 layer on the lateral sides of the SAM/flower boundary from P3 for YUC4 and in P4 for YUC111 (Fig. 3g-h and Extended Data Fig. 6). From P4, YUC4 expression extends over the entire epidermis of flower primordium. Together with the PIN1 network organization, these expression patterns thus identify the primordia at P3-P5 stages as auxin production centers for the SAM.
In conclusion, there is a structured global organization of the PIN1 pump network, with high concentrations of auxin at the center of the SAM but also at P-1 and P0, all acting as flux attractors (Fig. 3i). The stem cell niche could thus play the role of a system-wide organizer of the auxin transport network, coherently with previous observations18. Auxin maxima could then first emerge from the CZ along lines where auxin fluxes converge. Convergence of the fluxes towards the auxin maxima would at the same time diverge fluxes away from areas where auxin minima appear (Fig. 3i). Our data further suggest that early flowers could provide a memory of the developmental pattern and allow self-sustained distribution of auxin by redistributing auxin back in the system through biosynthesis (Fig. 3i).
The role of time in transcriptional responses to auxin
To assess quantitatively whether and how auxin spatio-temporal distribution is interpreted in the SAM, we next introduced the synthetic auxin-induced transcriptional reporter DR536,37 driving mTurquoise into the qDII/CLV3 reporter line (Fig. 4a-d). Cells expressing DR5 closest to the CZ were robustly positioned at an average distance of 32 µM ± 7 (SD) from the center. This corresponds to a distance at which the CLV3 reporter expression is lower than 5% of its maximal value (Extended Data Fig. 7a). The distance from the center at which transcription can be activated by auxin is thus defined with a near-cellular precision.
To obtain a global vision of how auxin transcription is related to auxin concentration, we performed a Principal Component Analysis (PCA) using quantified levels of DR5, auxin and CLV3 in each nucleus of the PZ of 10h time series together with their distance from the center (Fig. 4e). With the first two axis reporting for around 75% of the observed variability, we unexpectedly observed orthogonality between auxin input and DR5 output clearly marking an absence of a general correlation in the meristem (Fig. 4e, inset). This unexpected finding was confirmed by the low numerical values of Pearson correlation coefficients between DR5 and auxin values at cell-level (Extended Data Fig. 7b). We refined our analysis by studying locally correlations between auxin and DR5 in the different primordia regions (Fig. 4d). We observed homogeneous auxin and DR5 behaviors that characterized each region (Extended Data Table 2), showing that the spatial position of a cell conditions the link between its auxin input and the state of its transcriptional response. We then assembled on a single graph all the observed couples of values (auxin, DR5) averaged over each primordium region (Fig. 4h). This evidenced that, spatially, a given auxin value does not in general determine a specific DR5 value. However, values corresponding to primordia at consecutive stages follow loop-like counter-clockwise trajectories in the auxin x DR5 space (indicated by an arrow in Fig. 4h). Such trajectories are symptomatic of hysteresis reflecting the dependence of a system on its history. In other words, DR5 expression is dependent on the developmental history of the cells.
We then used our reconstructed continuum of primordium development to study the joint temporal variations of DR5 and auxin within a local group of cells during initiation (Supplementary Method 3). This showed that the start of auxin-induced transcription follows the building-up of auxin concentration with a delay of nearly one plastochrone (Fig. 4f-g). The duration of the observed phenomenon suggests the existence of an additional process, different from protein maturation38, that creates a significant DR5 response delay of primordia cells to auxin during development. Due to this delay, DR5 is not a direct readout of auxin concentration, potentially explaining the absence of correlation between DR5 expression and auxin levels.
We wondered what could explain a time-dependent acquisition of cell competence to respond to auxin. A first possible scenario is that cells exiting the CZ proceed through different stages of activation of an auxin-independent developmental program enabling them to sense auxin after a temporal delay. A second possibility is that auxin controls this developmental program through a time integration process. In that case, cells exiting the CZ would need to be exposed to high auxin concentration for a certain amount of time to mount up an auxin transcriptional response. To test these scenarios, we treated SAMs with auxin for different durations (Fig. 5a-i). In the shorter auxin treatments (30’ and 120’), auxin output was only enhanced at P0, P1 and P2 i.e. where cells have already been exposed to auxin (Fig. 5f-g and i, Extended Data Fig. 8a). On the other hand, the longer auxin treatments (300′) activated signaling in most cells in the PZ and organs (Fig. 5h,i). Both observations are compatible with the second scenario where longer exposure allows activating signaling in more cells and are incompatible with the first one, where the capacity of the cells to respond to auxin is intrinsic and does not dependent upon auxin exposure time. Our results thus indicate that temporal integration of the auxin signal controls the activation of transcription in the SAM.
Chromatin state is one mechanism that allows for temporal integration of signals20,24,39,40. Also, ARFs and Aux/IAAs have been shown to act by modifying acetylation of the chromatin41,42. Pharmacological inhibition of histone deacetylases (HDACs) alone was able to trigger concomitant activation of DR5 at P0 and P-1 sites in the PZ (Fig. 5j-l, Extended Data Fig. 8b). This result demonstrates that the timing of auxin signaling induction at the P-1 site depends on the chromatin status. It further suggests that chromatin acetylation status of cells at the boundary of the CZ acts as a memory of their exposition time to auxin, providing a mechanism for temporal integration of auxin-based positional information.
Discussion
Through a fully quantitative analysis of auxin dynamics, our study provides a demonstration that rhythmic organ initiation at the SAM is driven by a combination of high-precision spatio-temporal graded distributions of auxin and of the use of the duration of cell exposition to auxin to differentiate temporally sites of organ initiation (Fig. 6). Such a mechanism is likely essential for rhythmic organ patterning in the SAM as auxin-based spatial information pre-specifies several sites of organ initiation and is thus insufficient (Supplementary video 1). Temporal integration of the auxin signal could occur through a chromatin acetylation mechanism. As chromatin acetylation represses auxin signaling in the CZ (Ma, Y., Miotk, A., Sutikovic, Z., Medzihradszky, A., Wenzl, C., Ermakova, O., Gaillochet, C., Forner, J., Utan, G., Brackmann, K., Galvan-Ampudia, C. S., Vernoux, T., Thomas, G. & Lohmann, J. U. WUSCHEL acts as a rheostat on the auxin pathway to maintain apical stem cells in Arabidopsis. bioRxiv 468421; doi: https://doi.org/10.1101/468421 (2018)), this provides an original mechanism tightly linking stem cell maintenance and differentiation by precisely positioning organ initiation at the boundary of the stem cell niche while allowing for sequential organ initiation.
The existence of high definition spatio-temporal auxin gradients suggests that similarly to several morphogens in animals24,43–45 the robustness of SAM patterning is at least in part due to highly reproducible spatio-temporal signal distribution. Our analysis questions how auxin transport could generate this high definition signal distribution and points again to the stem cell niche that could be crucial for organizing the transport network. The unique quantitative resource we have generated will allow addressing these questions in depth.
Methods Summary
Plant material and growth conditions
Seeds were directly sown in soil, vernalized at 4 °C, and growth for 24 days at 21 °C under long day photoperiod (16 hrs light, LED 150µmol/m²/s). Shoot apical meristems from inflorescence stems with a length between 0.5 and 1.5 cm where dissected and cultured in vitro as described in46 for 16 hrs. When required, meristems were stained with 100 µM propidium iodide (Sigma) for 5 min. Auxin treatments were performed by immersing meristems with 1 mM Indole-acetic acid (IAA) for different times. Trichostatin A (TSA – Invivogen) was added to the ACM plates to a final concentration of 5 µM. Meristems were cultured in TSA for 16 hrs prior auxin treatment. For time lapses, first image acquisition (T=0) correspond to 2 hrs after the dark period, with the exception of auxin treatments where imaging is right after lights were on.
Previously published transgenic lines used in this study are PIN1-GFP3, promCLV3:mCherry-NLS29, and promYUC1 to 11-GFP33,34. Quantitative DII (qDII), promRPS5a:DII-VENUS-N7-p2A-TagBFP-SV40, reporter line and the auxin synthetic promoter DR5rev:2x-mTurqouise2-SV40 was cloned using Gateway technology (Life Sciences), and transformed in Arabidopsis thaliana (Col-0). Stable qDII homozygous lines were then crossed with promCLV3, promDR5rev:2x-mTurqouise2-SV40 and PIN1-GFP reporter lines.
Imaging
All confocal laser scanning microscopy was done with a Zeiss LSM 710 spectral microscope. Multitrack sequential acquisition was performed using always the same settings (PMT voltage, laser power and detection wavelengths) as follows: VENUS, excitation wavelength (ex): 514 nm, emission wavelength (em): 520-558 nm; mTurquoise2, ex: 458 nm, em: 470-510 nm; EGFP, ex: 488 nm, em: 510-558 nm; TagBFP, ex:405 nm, em: 430-460 nm; mCherry, ex: 561 nm, em: 580-640 nm; propidium iodide, ex: 488, em: 605-663 nm.
Quantitative image analysis
All confocal images were pre-processed using the ImageJ software (http://rsbweb.nih.gov/ij/) for the delimitation of the region of interest. Then the CZI image files were processed in a computational pipeline developed by the authors and relying essentially on the numpy, scipy, pandas, czi_file Python libraries, as well as other custom libraries. Extensive details about the developed methods and algorithms are given in Supplementary Method 2.
Statistics and reproducibility
Confidence intervals were calculated with a confidence level of 95% in the R environment47. The boxplots displayed in the article were obtained by computing the median (central line), first and third quartiles (lower and upper bound of the box) and first and ninth deciles (lower and upper whiskers) using the R environment or numpy percentile function and rendered using the matplotlib Python library. Linear regressions were performed using the polyfit and polyval numpy functions. P-values were obtained using the scipy anova implementation in the f_oneway function. Principal component analysis was performed using the PCA implementation from the scikit-learn Python library. All data were generated with at least 3 independent sets of plants.
Data availability
All experimental data and quantified data that support the findings of this study are available from the corresponding authors upon request.
Code availability
Generic quantitative image and geometry analysis algorithms are provided in Python libraries timagetk, cellcomplex and tissue_nukem_3d (https://gitlab.inria.fr/mosaic) made publicly available under the CECILL-C license. Specific SAM sequence alignment and visualization algorithms are provided in a separate project (https://gitlab.inria.fr/gcerutti/sam_spaghetti.git).
All other custom source codes and analysis scripts are available from the corresponding authors upon request.
Author Contribution
C.G. and T.V. designed the project; C.G.-A., G.C., J.U.L., C.G. and T.V. designed experiments; C.G.-A., G.C., J.L., R.A., G.B., S.M., C.W. performed experiments; C.G.-A., G.C., J.L., R.A., G.B., S.M., C.G. and T.V. were involved in data analysis; C.G.-A., G.C., C.G. and T.V. wrote the manuscript with inputs from all authors.
Author Information
Correspondence and request for materials should be addressed to T.V. (teva.vernoux{at}enslyon.fr) or C.G. (christophe.godin{at}ens-lyon.fr).
Supplementary Method 3: Extrapolated cell motion in the developmental continuum
Throughout this work, we strongly relied on the spatio-temporal periodicity of phyllotactic systems, that we demonstrated to be a valid assumption for our considered SAMs. Indeed, the high angular precision and limited plastochrone variability make the observed systems close to a steady developing regular phyllotactic system with a divergence angle α = 137.5 ± 6.7° and a plastochrone T = 12 ± 2h (Extended Data Figure 2d-g, Supplementary Method 1).
Notably, we considered that, in the 2D cylindrical reference frame centered on the CZ of the shoot apical meristem, the dynamics of any quantifiable signal S must follow the properties of a spatio-temporally periodic function of spatial period [0, −α] and temporal period T:
This helped us consider signal dynamics on durations that largely overpass the observation range of 10 to 14 hours, by applying successive rotations of the meristems to simulate the passing of time. Notably, an aligned SAM observed at t =0h rotated of 137.5° clockwise has been shown to be the best next frame to the same aligned SAM observed at t =10h (Figure 1g, Extended Data Figure 2h). More generally, any primordium of stage p visible at time t can be use to infer information at time t + pT. This means that we were able to reconstruct trajectories of signals at a given position (r, θ) over a duration of up to 9 plastochrones (~100 hours) from observations spanning only one, but with 9 visible primordia (P-3 to P5). This was achieved only by interpolating rotated aligned information (Supplementary Movie 1).
Unfortunately, this global reconstruction heuristic could work only while we were looking at the same location in space, where the spatio-temporal property holds. To some extent, it can be generalized to robust primordia landmarks, such as auxin maxima, that we assume to be unique while moving in the course of primordium development. If they can be identified, and associated with a primordium of stage p, then they can be positioned on the same developmental axis at a time t + pT to reconstruct a developmental history at the level of this time-tracked landmark.
However, the moment we are interested in cellular processes, such as the dynamics of transcriptional response to auxin for instance, the reconstructed long-term trajectories cannot be used to draw relevant conclusions, as they reflect the dynamics either at fixed coordinates or at non-cell-specific landmark points. It is therefore necessary to find a way to access temporal cell-level information. Individual cells can be tracked in time-lapse sequences, either manually or automatically, which could be use to obtain signal trajectories over 10 to 14 hours (Figure 2g). But to achieve the mentioned 100-hour reconstruction, spatio-temporal periodicity has to be used at some point.
Extrapolated tissue area tracking
We assume the cells in the central and peripheral zones (CZ and PZ respectively) of the SAM to have essentially an outward radial motion that accelerates as cells exit the central zone. This has been confirmed by the cellular motion vectors estimated from vector fields of image deformation (Supplementary Method 2) in which the azimuthal component is in average close to 0, with a limited amplitude compared to the radial component (Supplementary Figure 5a). We note the speed of the nucleus n in the 2D polar SAM reference frame: where and are the local normal and tangential unitary vectors at P*(n). In the following, we will then assume a pure radial motion of cells in the L1 of the SAM, i.e. that:
We compute the local radial cellular speed as a 2D continuous map on the L1 (Supplementary Figure 4b-c), using the same parameters as for the other cellular signals:
Defined this way, local radial cellular speed is a tissue-level information, that is not attached to a cell but to a spatial position (r, θ). Therefore it has a spatio-temporal periodicity property and we can write:
We use this spatio-temporal periodicity property of local cellular speed to extrapolate cell motion over time from a series of acquisitions of SAMs at discrete times {t0 < ⋯ < tn < T}. Let us consider a cell with an initial position P(t0) = (r(t0), θ0), setting r(t0) = r0. Using acquisitions at t0, it is possible to estimate (r0, θ0, t0), which we use to estimate P(t1) = (r(t1), θ0) assuming a linear motion between t0 and t1:
More generally, with observations at ti−1, we derive P(ti) with:
We perform this progression until we compute the last position P(tn) for which motion can not be estimated (as there is no next image it compute image deformation) However, we can still extrapolate the lastly computed motion to reach one plastochrone, and estimate P(t0 + T) with:
To proceed further, we would like to progress in time and estimate the cell position at t1 + T. This is where we use the spatio-temporal periodicity property to derive that:
We are therefore able to estimate this next position from observations at t0 simply by rotating the radial speed map (Supplementary Figure 5d). Then iteratively it is possible to go further in time to tn + T, extrapolate motion to t0 + 2T, apply again spatio-temporal periodicity to reach t1 + 2T, and so on. Finally we obtain the two following general equations:
These are valid for positive integer values of p until reaching the maximal extent of P in the considered data (i.e. when the map used to estimate local radial speed becomes ill-defined), which determines a value pmax. This defines a radial trajectory in the 2D space that reflects the local motion of cells along several plastochrones (Supplementary Figure 5e). Note that a rigorously identical approach can be used to go backwards in time with negative values of p until t0 + pminT, using motion vectors computed using inverse image deformation (Supplementary Method 2).
In the end, from an initial position (r0, θ0), we obtain a discrete radial trajectory:
To monitor a cellular process over long time courses, the objective would be to estimate the value of the signal S along this spatio-temporal trajectory, i.e:
Using the spatio-temporal periodicity property of S, this translates into:
In other terms, we have defined a series of spatial locations in a time-series of acquisitions such that the sequence of signal values at these locations on the same meristem estimates the cell-level trajectory of the considered signal. If we represent them on a time-series of meristems, we define tissue areas that can be tracked, first in time then in space, to reconstruct the average behavior of a group of cells over time (Supplementary Figure 5f-h). This is the approach we use to reconstruct cell-level auxin trajectories (Figure 2h) and to study the relationship between auxin input and transcriptional response in a consistent group of cells (Figure 4d, Figure 4f-h).
Acknowledgements
We thank Fabrice Besnard and the members of the SIGNAL team for insightful discussions; Antoine Larrieu for helping with RNA-Seq analysis; Hélène Robert-Boisivon for the YUC transcriptional lines; We acknowledge the contribution of SFR Biosciences (UMS3444/CNRS, US8/Inserm, ENS de Lyon, UCBL) facilities PLATIM, for assistance with microscopy. This work was supported by Human Frontier Science Program organization (HFSP) Grant RPG0054-2013 to T.V. and C.G., ANR-12-BSV6-0005 grant (AuxiFlo) to T.V. and DFG FOR2581 to J.L.