Coordinated circadian timing through the integration of local inputs in Arabidopsis thaliana

To investigate the coordination of clock rhythms, we analyzed rhythms across entire seedlings under different entrainment regimes. To do this, we monitored promoter activity of the core clock gene GIGANTEA (GI) [30] fused to the LUCIFERASE (LUC) reporter gene, for multiple days at near-cellular resolution (Materials and methods). This reporter line was chosen because of its strong expression level and its similar spatial expression to other clock components [5].

In order to observe the endogenous component of the rhythms, we first imaged seedlings under LL, having previously grown them under LD cycles (LD-to-LL; Fig 2A and Materials and methods). Under the LD-to-LL condition we observed phase differences of GI::LUC expression between organs (Fig 2B and 2C). The cotyledon and hypocotyl peaked before the root, but the tip of the root peaked before the middle region of the root (Fig 2C, S1 Fig, and S1 Video). Furthermore, we observed a decrease in coherence between regions over time, with a range between the earliest and latest peaking region of 4.92 ± 3.79 h (mean ± standard deviation) in the first and 18.36 ± 5.67 h in the final oscillation. This is due to the emergence of period differences between all regions (Fig 2D). The cotyledon maintained a mean period of 23.82 ± 0.60 h, whereas the hypocotyl and root ran at 25.41 ± 0.91 h and 28.04 ± 0.86 h, respectively. However, the root tip ran slightly faster than the middle of the root, with a mean period of 26.90 ± 0.45 h, demonstrating the presence of endogenous period differences across all regions. We verified that our results were not specific to the GI::LUC reporter, as we observed similar differences in periods and phases across the plant using luciferase reporters for promoter activity of the core clock genes PSEUDO-RESPONSE REGULATOR 9 (PRR9) [31], TIMING OF CAB EXPRESSION 1 (TOC1) [32], and EARLY FLOWERING 4 (ELF4; S2 Fig) [33]. These observations are also qualitatively similar to the periods and phases previously observed in isolated organs [6,22,24], and at the cellular level across the seedling [5], validating our whole-plant assay for the circadian clock.

The phase at which a rhythm entrains to the environment can depend on the mismatch between its endogenous period and the period of the entraining signal [34–36]. We therefore tested the consequence of endogenous period differences between organs on the entrainment of the plant, by monitoring GI::LUC rhythms under LD cycles (LD-to-LD; Fig 2A and Materials and methods). Under the LD-to-LD condition, we observed robust and entrained rhythms (Fig 2E). However, closer inspection of the timing of the peaks of the oscillations revealed significant differences in clock phase between organs (Fig 2F). The cotyledon and hypocotyl consistently peaked earlier than the root regions, but the root tip peaked earlier than the middle of the root (Fig 2F, S3 Fig and S2 Video). This is qualitatively similar to the pattern observed under LL (Fig 2C). However, under the LD-to-LD condition, the organs showed a more stable phase relationship than under LL, with a range between the earliest and latest peaking region of 2.08 ± 1.56 h in the first oscillation and 1.10 ± 1.44 h in the final oscillation. This is due to the fact that all organs oscillate with a period of approximately 24 h (Fig 2G).

Spatial waves of clock gene expression have been previously reported in plant leaves [18,19,25,26] and roots [5,20,27] under LL. However, their relation to one another and the relevance under LD cycles remained unclear. We analyzed our LD-to-LL and LD-to-LD data set of whole, intact seedlings at the sub-tissue level in order to address these questions. We extracted the phase of the luminescence signal across longitudinal sections of seedlings (S4 Fig, Materials and methods) and present phase plots and time-lapse videos of single seedlings representative for each light condition (Fig 2H and 2I, S1 and S2 Videos). The clearest waves of expression could be observed in the LD-to-LL condition, as phase differences increased with time. In the cotyledon, a wave of GI::LUC expression propagated from the tip to the base (Fig 2H, top), and downwards into the hypocotyl (Fig 2H, middle). In the hypocotyl, we observed a second wave traveling from the root junction upwards into the hypocotyl (Fig 2H, middle). Finally, within the root we observed two waves: one propagating down from the hypocotyl junction and the second from the root tip upwards into the root, as we have reported previously (Fig 2H, bottom) [5]. Evidence of waves of clock gene expression could also be observed under the LD-to-LD condition. Although they are less pronounced, small phase waves could be discerned within the cotyledon (Fig 2I, top), hypocotyl (Fig 2I, middle), and root (Fig 2I, bottom) of the phase plots and time-lapse videos (S2 Video).

Previous work has proposed that spatial waves of clock gene expression are driven by local cell-cell coupling [5,18–20]. However, plant cells can communicate through both local and long-distance, interorgan pathways [29], and the root clock has been proposed to be driven by long-range signals from the shoot [9,22,24]. To investigate whether rhythms and spatial waves are driven by long-distance communication, we blocked signal transmission between organs by cutting the seedling into sections. We cut the root at either the hypocotyl junction, the root tip, or both the hypocotyl junction and the root tip, and then monitored the rhythms under LL (Fig 3A). We found that sectioning the plant did not substantially affect the phase of the rhythms (Fig 3B–3D). Some minor phase differences were observed between cut and uncut controls after cutting, but these were no longer apparent after 6 d (S5 Fig). Period differences across the plant also persisted after cutting (Fig 3E–3G). Next, we focused our analysis to within the hypocotyl and root, where the simple geometry means the wave patterns can be most easily observed. Strikingly, after all cuts we observed the persistence of waves propagating both from the hypocotyl down into the root and from the root tip upwards (Fig 3I–3K and S3 Video). Our results show that in all organs excised, rhythms are autonomous, and the spatial waves that travel between them are not dependent on a long-distance signal.

The persistence of rhythms and spatial waves in the absence of long-distance communication suggests clocks may instead be coupled through local interactions. We extended the mathematical framework we employed in previous work [5] to investigate whether local coupling can explain the entrainment behaviors that we observe under LD and LL. As before, we used a Kuramoto phase oscillator model [37]. In this framework, each pixel (which in fact represents multiple cells) on our seedling template is an individual oscillator with an intrinsic period and is weakly coupled to its nearest neighbors (S6 Fig). The intrinsic period of each pixel was set according to its location in the seedling. Pixels from the cotyledon, hypocotyl, root, and root tip were drawn from distributions centered around the mean periods that we observed experimentally in each region under LL (Fig 4A, S6 Fig, Materials and methods). These period estimates were made from in vivo experiments and therefore include the effects of coupling. They are, however, as good an estimation of the cell autonomous periods as possible in a physiologically relevant context. In our LD-to-LL simulations, because of the differences in intrinsic periods, and coupling, we saw increasing phase shifts between organs (Fig 4C) and two increasingly large waves in the root (Fig 4E), as observed in experiments (Fig 4G). We observed these spatial waves in simulations under a range of coupling strengths, K (S7 Fig). As the coupling strength was increased, the distribution of periods and phases of the oscillation became tighter, as previously reported in models of coupling in the SCN (S8 and S9 Figs) [38].

In our model, the amount that each oscillator phase is shifted is set by the mismatch of its intrinsic period and the period of the entraining rhythm [34–36]. This prediction is supported by experimental evidence in various organisms, including plants [39], although dawn can also reset the phase of some clock genes in bulk Arabidopsis experiments [40,41]. We tested whether the phase differences that we observe between organs in Arabidopsis under our LD conditions can be reproduced in our model by this mismatch with the entraining rhythm. In our simulations, organs were forced to oscillate with a period of approximately 24 h, due to entrainment to the external rhythm (Fig 4B). However, because of the mismatch between the intrinsic period and the entraining rhythm, organs entrained with different phases, matching those observed experimentally (Fig 4D). Phase shifts could also be observed at the sub-tissue level; two short waves could be observed in the root (Fig 4F), as in experiments (Fig 4H).

In a set of coupled oscillators, variation in period causes a decrease in synchrony, whereas coupling and external entrainment maintain or increase synchrony [42–45]. In order to make predictions about the presence of local coupling in seedlings, we simulated our model in the absence of LD entrainment. We simulated the duration of the experiment without entraining the oscillators, and thus assume that the phases are initially random (LL-to-LL; Fig 5A, Materials and methods). In contrast to the LD-to-LL condition, in which oscillators begin synchronous but become less synchronized whilst under LL, in LL-to-LL simulations, oscillators began less synchronous but maintained their order over the time course (Fig 5B). Interestingly, in the root, the model predicted a complex spatial pattern, with multiple phase clusters and spatial waves in a single seedling (Fig 5C and S4 Video). These patterns of gene expression were similar to the zigzag patterns previously reported by others, when roots are grown on sucrose supplemented media [20,27,46]. We found that these zigzag patterns emerged with, but not without, local coupling (S10 Fig).

An alternative mechanism to explain the spatial patterns of rhythms that we observed can also be envisaged. If, within the tissue, there exists a gradient in the periods of oscillators, spatial waves may be seen in the absence of cellular coupling. Simulations of this plausible alternative model, without coupling but with a gradient of the intrinsic periods within the root, were indeed sufficient to generate simple waves similar to those we observed under the LD-to-LL condition (), but not the complex zigzag waves predicted in the LL-to-LL condition (). S10C and S10D Fig S10E and S10F Fig

In order to test our model and validate the assumption of local coupling, we experimentally tested the LL-to-LL model prediction. We both grew and imaged seedlings under LL conditions (LL-to-LL; Fig 5A), so that seedlings never see an entrainment cue beyond germination [47,48]. Roots maintain their coherence over the full time course (Fig 5D) and display a zigzag expression pattern (Fig 5E and S11 Fig) as predicted by the model, supporting the hypothesis of weak, local coupling.

To test our model further, we attempted to manipulate the periods in specific organs to determine whether we could modulate the spatial waves of gene expression. In the most severe case, removing all period differences across the plant should result in perfectly coherent rhythms. We found mutations to the core clock network to have little effect on the organ specificity of periods in organs scored as rhythmic (). However, there were some organ-specific effects on rhythmicity (). We also note that we cannot rule out that mutations to other clock components could have a larger effect on the organ specificity of periods. S12 Fig S1 File

Next, we tested whether we could alter periods in an organ-specific manner by modulating inputs to the clock. We first tested the effect of light input by growing seedlings under LD cycles before imaging seedlings under constant darkness (DD). Under DD, we observed a drastic slowing of periods in the cotyledon and hypocotyl, whereas the middle region of the root maintained its speed compared with LL (Fig 6A). This is in contrast to previous lower-resolution work that found the period of the root as a whole increased under DD [28]. The lengthening of periods in the aerial organs reduced the phase differences between the aerial organs and the root (Fig 6B and S13 Fig), resulting in the loss of the spatial wave traveling from the hypocotyl down the root (Fig 6C and S5 Video). Inversely, in the root tip we observed a decrease in period compared with LL (Fig 6A), causing larger phase shifts between the root tip and the root (Fig 6B and S13 Fig) and resulting in a longer spatial wave traveling from the root tip upwards into the root (Fig 6C and S5 Video). We observed the same effect when seedlings were grown so that the roots were not exposed to light during entrainment or imaging (S14 Fig). Additionally, a qualitatively similar but lesser effect was observed under monochromatic red or blue light (S15 Fig). Together, our experiments show that light input can set the periods of oscillators across the plant, either increasing or decreasing the speed, depending on the region. These differences are sufficient to drive spatial waves of gene expression between organs.

We next tested whether the effect of light on organ specificity is direct, through known light signaling pathways. We imaged GI::LUC expression in the phyb-9 background, a null mutant for the primary red light photoreceptor in A. thaliana, PHYTOCHROME B (PHYB) [49,50]. Because PHYB has a tissue-specific expression pattern in the plant [51–54], we reasoned that its period-shortening effect under red light [55] may be organ specific. Under red light, in the phyb-9 mutant we observed the loss of period differences between the cotyledon, hypocotyl, and root (Fig 6D). This caused the loss of phase shifts between the aerial organs and the root (Fig 6E and S16A Fig) and the loss of a distinct spatial wave traveling down the root (Fig 6F and S6 Video). We also observed a decrease in rhythmicity across the seedling (S1 File). The effect was particularly large in the root tip, with only 24% of root tips classed as rhythmic compared with 96% in the wild type. In the root tips classed as rhythmic, the period ran approximately 3 h slower, at approximately the same speed as the middle of the root (Fig 6D). Therefore, after 6 d under constant red light, the phase shifts between the root tip and root (Fig 6E and S16A Fig), and the spatial wave traveling from the root tip upwards, were attenuated (Fig 6F and S6 Video). The phyb-9 mutation, however, does not abolish the faster periods observed in the root tip under DD (S16B and S16C Fig).

In addition to the external environment, the circadian clock is exposed to biochemical signals from within the cell [56]. We investigated whether these endogenous signals could also alter periods in an organ-specific manner, modulating the spatial waves of clock gene expression. First, we imaged seedlings under LL in the presence of 3-(3,4-dichlorophenyl)-1,1-dimethylurea (DCMU), a specific inhibitor of photosynthesis. During inhibition, we observed a slowing of periods specifically in the cotyledon and hypocotyl (Fig 7A), causing a loss of phase shifts between the hypocotyl and root (Fig 7B and S17 Fig) and the loss of the spatial wave down the root (Fig 7C).

Photosynthesis modulates the clock through the production of sugars, which feed into the oscillator [57–59]. We next tested whether the application of sucrose to part of the plant could locally reduce clock periods and generate spatial waves. This is a direct test of the hypothesis that local period differences drive spatial waves of gene expression. We designed a protocol that allowed us to rest only the top portion of the root on sugar-supplemented media and observe the effect throughout the root. We did this with roots cut at the hypocotyl junction to minimize developmental effects, and under DD, where we ordinarily observe no spatial waves down the root (Fig 7D). In comparison with mannitol (a poorly metabolized sugar that acts as an osmotic control), contact with sucrose-supplemented media caused a larger decrease in period length (Fig 7E). This caused larger phase shifts from the top to the middle of the root (Fig 7F and S18 Fig). Within the root, a clear spatial wave of clock gene expression propagates down from the top of the root when in contact with the sucrose (Fig 7G and S7 Video) but not mannitol (Fig 7H and S8 Video) supplemented media. Together these results show that altering the speed of clocks locally, either via modulating light perception or the addition of photosynthetic sugars, can drive spatial waves of clock gene expression.

The wild-type GI::LUC, PRR9::LUC, and TOC1::LUC lines are in the Col-0 background and as described previously [80,81]. To produce the wild-type ELF4::LUC line, the promoter region of ELF4 was amplified from A. thaliana genomic DNA using the primer pairs ELF4_prom_Fwd/Rev (ELF4_prom_Fwd, 5′- AGAGAATTCCTAAATCATCAAAGCCAACC-3′ and ELF4_prom_Rev, 5′-CTTTCTAGAAATAATTTTTAATTGTGTTTTTCTCTC-3′). Unique restriction sites were designed at the ends of the amplicons to facilitate cloning in the pPCV-LUC+ binary vector [82] via EcoRI (5′) and XbaI (3′) sites. The cloned ELF4 promoter fragment was of 1,546 base pairs and contained the full 5′-untranslated region, but not the ATG. The construct was transformed into the Col-0 background by means of agrobacterium-mediated transfection [83]. Homozygous third generation lines were used for experiments.

The phyb-9 GI::LUC line is in the Col-0 background and as described previously [80]. The cca1-11 (TAIR:1008081946; Ws background back-crossed with Col-0 three times) [84], prr9-1 (TAIR:3481623) [85], prr7-3 (TAIR:3662906) [85], toc1-101 (TAIR:6533848449) [86], and lux-4 (TAIR:1008810333) [87] alleles are loss-of-function mutations that have been previously described, and were transformed with the GI::LUC [80] construct by means of Agrobacterium-mediated transfection [83]. Homozygous third generation lines were used for experiments.

Seeds were surface sterilized and placed in the dark at 4 °C for 3 d. Seeds were sown at dawn of the fourth day on full-strength Murashige and Skoog (MS), 2% agar, pH 5.7 media, without sucrose unless otherwise specified. Seeds were then grown inside of plant growth incubators (MLR-352; Panasonic, Japan) for 4 d under 80 μmol m² s⁻¹ cool white light at a constant temperature of 22 °C. Seedlings were grown under 12-h light–12-h dark cycles unless otherwise specified. Plates were orientated vertically during growth.

For experiments in which roots are grown in the dark (S14 Fig), seedlings were grown in full-strength MS liquid solution as described previously [88]. After 4 d of growth, working under green light only, seedlings were transferred to MS 2% agar plates and transferred to imaging cabinets.

At dusk of the fourth day of growth, seedlings were sprayed with a 5 mM D-Luciferin (Promega, Madison, WI), 0.01% Triton X-100 solution (Sigma-Aldrich, St. Louis, MO). At dawn of the fifth day, 6–8 seedlings were transferred into a 3- by 3-cm area of the media plate in order to fit inside of the camera’s field of view. Plates were orientated vertically during imaging.

Imaging was performed inside of growth incubators (MIR-154; Panasonic, Japan) at a constant temperature of 22 °C and under an equal mix of red and blue light–emitting diodes (40 μmol m⁻² s⁻¹ total), unless specified as red light only (40 μmol m⁻² s⁻¹ red) or blue light only (40 μmol m⁻² s⁻¹ blue). For experiments under LD cycles, lights were switched on to full intensity at dawn and completely off at dusk. Images were taken every 90 min for 6 d, with an exposure time of 20 min. Images were taken using a LUMO charge-coupled device camera (QImaging, Canada) controlled using Micro-Manager (V2.0; Open Imaging) as previously described [89,90]. The camera lens (Xenon 25 mm f/0.95; Schneider, Germany) was modified with a 5-mm optical spacer (Cosmicar, Japan) to increase the focal length and decrease the working distance.

For cut experiments, seedlings were cut approximately 3 h after dawn of the fifth day of growth, immediately prior to the commencement of imaging. For “hypocotyl cut” experiments (Fig 3B, 3E and 3I), seedlings were cut in the root as close to the hypocotyl junction as discernible by eye; for “root tip cut” experiments (Fig 3C, 3F and 3J), seedlings were cut approximately 100–200 μm from the root cap. Cuts were made with a pair of Vanna’s type microdissection scissors (Agar Scientific, UK). Following all excisions, the organs were gently separated with a pair of forceps to ensure no physical contact.

DCMU (Sigma-Aldrich, St. Louis, MO) was added to the media at a final concentration of 20 mM. Seedlings were transferred to the DCMU-containing media at dusk of the fourth day of growth. For sugar application experiments (Fig 7E–7H), media was added in 8-well rectangular dishes (NUNC; Thermo-Fisher Scientific, Waltham, MA) so that one well contains media supplemented with MS and sugar whilst the adjoining well contains media supplemented with MS only. Wells were filled with equal volumes to the brim of the wells so that the two agar pads form a continual flat surface but do not touch. Sucrose or mannitol (Sigma-Aldrich, St. Louis, MO) was added at a final concentration of 90 mM (3% w/v). Seedlings were cut at the hypocotyl junction (as described above) and laid across the adjoining agar pads so that approximately the top 1 mm of the excised root rests on the sugar-supplemented media, and the remainder of the root rests on the non-sugar-supplemented media. Seedlings were cut and transferred to the media at dawn of the fifth day of growth, immediately prior to the commencement of imaging.

For the organ-level analysis of the period and phase, organs were first tracked manually in Imaris (BitPlane, Switzerland) using the “Spots” functionality. We use a circular region of interest (ROI) of approximately 315-mm diameter and track the center of a single cotyledon, hypocotyl, root, and the root tip from each seedling. As the root grows, we maintain the root ROI a fixed distance from the hypocotyl junction. A small number of cotyledons and hypocotyls were not trackable due to their orientation or their overlap with each other. These organs were excluded from the analysis. The median of the ROI was extracted to give the time series. Prior to the analysis of period and phase, the time series were first background subtracted. Very low expression rhythms with a minimum intensity value of less than zero after background subtraction were then removed. All time series were inspected by eye after preprocessing steps and prior to analysis.

Period analysis was conducted in BioDare2, a data server for the analysis of circadian data (biodare2.ed.ac.uk) [91]. All period estimates were performed on non-normalized data between 24–144 h from dawn of the day imaging began using the fast Fourier transform nonlinear least squares (FFT-NLLS) algorithm [92,93]. Data were first baseline detrended by subtraction of a polynomial of degree three from the data. Oscillations were classed as rhythmic if the FFT-NLLS algorithm returned a period in the range of 18–36 h with a confidence level (as defined in [91]) below 0.6.

For the analysis of the times of peaks of expression, peaks were identified using the MATLAB (MathWorks, UK) “findpeaks” function. This was done after the application of a third-order Butterworth filter to remove high-frequency noise. Only peaks in which all organs complete the full cycle within 24–144 h from dawn of the day imaging are used. Additionally, peaks were discarded if they are closer than 18 h or further than 36 h apart.

In all figures, data points, measure of error, statistical test used, N, and the range of n are reported in the figure legend. Exact p-values, exact n, percentage rhythmicity, and other test statistics are reported in S1 and S2 Files. When values are described in the text, they are quoted as mean ± standard deviation of the mean. For the comparisons of period estimates, one-way ANOVA (with Tukey post hoc method) was used for comparisons of more than two groups, and the t test (with Welch correction) for comparison of two groups. For comparison of times of peaks of expression, the distribution is often skewed, therefore the Kruskal-Wallis one-way ANOVA (with Dunn post hoc method) was used for multiple comparisons and the Wilcoxon rank-sum test for comparison of two groups. An alpha level of 0.05 was used for all ANOVA tests.

To analyze spatial patterns within the organ, we first created space-time intensity plots of the luciferase images before obtaining a phase representation of the plots using a wavelet transform (henceforth called “phase plots”). These phase plots allowed interpretation of the space-time dynamics of the signal across the length of the organ independent of amplitude fluctuations.

Space-time intensity plots of the luciferase data were created as described previously [5], although with some modifications—most importantly of which, we include a modification that allowed us to better section curved roots. The method including modifications is outlined here in its entirety. Unless otherwise specified, steps are implemented via custom-developed MATLAB scripts.

By looking at the all-to-all synchrony between pixels within the hypocotyl and root, the synchrony of oscillators in these tissues can be estimated. We excluded the cotyledons from the analysis because their orientation and movement make phase extraction difficult. For each time point, the order parameter [43] R at time t was obtained as

where N is the total number of pixels in the hypocotyl and root combined and θ_j the phase of the j-th pixel. R values range from 0 to 1, with a value of 1 indicating a set of completely synchronized oscillators and a value of 0 a set of completely desynchronized oscillators.

As in [5], we used the Kuramoto phase oscillator model to describe the dynamics of GI::LUC in each pixel (here, a pixel represents a set of individual, neighbor cells). We viewed the plant in two dimensions with positions in horizontal and vertical (longitudinal) directions described by index positions i and j, respectively, so that every pixel, P(i,j), has an associated position, (i,j). The phase at the pixel P(i,j) was represented by θ^(i,j), where its dynamics in time, t, are governed by the following equation

Here, the first term is the intrinsic frequency of the pixel, ω^(i,j). The second term is the coupling contribution from the nearest-neighbor pixels in positions (m,n) that are closest to (i,j), namely m = i−1, i, i+1, while n = i−1, i, i−1. We assumed a plant template that is symmetric and resembles the shape of a seedling (S6 Fig). For the sake of simplicity, we assumed that the coupling constant, K, is the same across all pixels, and set it arbitrarily to K = 1 unless otherwise specified. The final term represents the coupling of the oscillator to the external force—in this case, the light force. Here, K_LD is the constant for the intensity of the light forcing, in which all oscillators are subject to 24-h forcing. Note that when the clocks are not entrained to the LD cycles, K_LD = 0. Because GI tends to peak at the onset of dusk in 12-h light–12-h dark cycles and shorter photoperiods [100], we assume that the phase of GI will be antiphase to light, hence the negative sign in front of K_LD. In our simulations of the LD-to-LD model, we set K_LD = 1.

Intrinsic periods are different across different sections of the plant. Intrinsic periods of the pixels in each section were taken from normal distributions with means of 23.82 h, 25.41 h, 29.04 h, and 26.90 h for cotyledon, hypocotyl, root, and root tip pixels, respectively, with the standard deviation at 10% of the mean value. The root tip is 5 pixels long and wide.

Initial values of all phases in the LD-to-LL and LD-to-LD simulations were at the time of the start of measurement identical, with first peaks occurring approximately 11 h after the first measurement. In the LL-to-LL model, because we had no information about the initial phases, we set them to be uniformly distributed across a cycle (i.e., random). We note that in the LL-to-LL model, setting the phases to be in phase or close to in phase (e.g., approximately 11 h after the first measurement ±2 h [standard deviation]), we could not obtain the results seen. ODEs were solved using the Euler method and simulations performed in MATLAB.

Because the seedlings in our experiments grow, we introduced growth to the template seedling: we allowed the root to grow by one pixel every 5 h. Every newborn cell (and hence the new pixel) had the same phase as the closest set of cells (pixels) in the template, namely new pixels P(i, j), P(i+1, j), and P(i+2, j) will inherit the phases from P(i, j−1), P(i+1, j−1), and P(i+2, j−1), respectively. Their periods were taken from the normal distribution, with the mean of 26.90 h and the standard deviation 10% of the mean value.

After root growth, the root tip should stay fixed in size (of 5-by-5 pixels), so the previous most upper set of root tip pixels at the root/root tip junction were, from then on, considered as root tissue instead. This means that their periods lengthen and they were chosen from a normal distribution with the mean of 28.04 h and the standard deviation 10% of the mean value.

The expression of GI for each pixel, GI^(i,j), was calculated from the phase model as GI^(i,j)(t) = cos(θ^(i,j)(t)) + 1. It follows that the total sum of the luminescence for every longitudinal position j is GItot(j)=∑i=1njGI(i,j), where the total number of cells measuring across that section of the plant is n_j. The total luminescence was normalized so the maximum peak of expression in every longitudinal position is 1. The phases were extracted from the luminescence using the wavelet transform, as described above for the experimental data in ‘Phase space-time plots’.

To calculate the periods of the tissues as shown in Fig 4A and 4B, we took regions of 5-by-5 pixels in each tissue (S6 Fig) and calculated the median GI expression level for each region. Periods were calculated as the mean of the peak-to-peak periods of the median trace. To observe the distributions of periods and phases within a single simulation of a seedling (S8 and S9 Figs), we analyzed GI expression for all pixels on the plant template individually. Periods were calculated as the mean of the peak-to-peak periods, and phases were taken at t = 96 h. Due to growth of the seedling template, there were a small number of pixels with a short time series containing less than two peaks of expression. These pixels were excluded from the analysis.

An alternative model that could give rise to the LD-to-LL spatial wave behaviors observed is one where there is no coupling but periods increase towards the middle of the root. This means that K = 0, and we set periods in the root to increase linearly from 25.41 h at the hypocotyl/root junction to 28.04 h in the middle of the root, and then decrease linearly again to 26.90 h at the root/root tip junction. All other previous assumptions were adopted. Here, although a bow-shaped wave of expression could be obtained in the LD-to-LL simulations (S10C and S10D Fig), the model failed to reproduce the behavior observed in the LL-to-LL condition (S10E and S10F Fig).

Underlying quantitative data and model code are available from the project GitLab page (www.gitlab.com/slcu/teamJL/greenwood_etal_2019↗).