Tip growth has been studied in pollen tubes, root hairs, and fungal and oomycete hyphae and is the most widely distributed unidirectional growth process on the planet. It ensures spatial colonization, nutrient predation, fertilization, and symbiosis with growth speeds of up to 800 μm h−1. Although turgor-driven growth is intuitively conceivable, a closer examination of the physical processes at work in tip growth raises a paradox: growth occurs where biophysical forces are low, because of the increase in curvature in the tip. All tip-growing cells studied so far rely on the modulation of cell wall extensibility via the polarized excretion of cell wall–loosening compounds at the tip. Here, we used a series of quantitative measurements at the cellular level and a biophysical simulation approach to show that the brown alga Ectocarpus has an original tip-growth mechanism. In this alga, the establishment of a steep gradient in cell wall thickness can compensate for the variation in tip curvature, thereby modulating wall stress within the tip cell. Bootstrap analyses support the robustness of the process, and experiments with fluorescence recovery after photobleaching (FRAP) confirmed the active vesicle trafficking in the shanks of the apical cell, as inferred from the model. In response to auxin, biophysical measurements change in agreement with the model. Although we cannot strictly exclude the involvement of a gradient in mechanical properties in Ectocarpus morphogenesis, the viscoplastic model of cell wall mechanics strongly suggests that brown algae have evolved an alternative strategy of tip growth. This strategy is largely based on the control of cell wall thickness rather than fluctuations in cell wall mechanical properties.
The prostrate filaments of the alga Ectocarpus develop via tip growth [28]. Prior to identifying the underlying mechanisms, a series of additional biophysical information was collected. The cell wall dye calcofluor-white was used in pulse-chase experiments to measure the growing region more precisely. The stain was localized in the first 3 μm distal from the tip of the cell, corresponding to roughly half of the dome (Fig 2A). To assess the direction of growth at the local level, 0. 2 μm–diameter fluorescent beads (microspheres) were loaded on the surface of the cell, and their displacement during growth was followed using time-lapse microscopy. This method was initially developed for other plant cell types [30] and recently optimized for Ectocarpus [31]. Bead trajectories were drawn, and the angles these trajectories made with the cell contour were calculated. Statistical analyses of the distribution of these angles showed a moderate deviation (relative mean difference < 10%) between the fluorescent marker trajectory and an orthogonal displacement pattern. Moreover, linear regression exhibited no systematic dependence of the angle on meridional abscissa (Pearson correlation coefficient r = −0. 03), indicating that growth can be considered orthogonal to the cell surface in the dome independently of the position along the meridional abscissa (Fig 2B and S1 Fig). In plants, turgor is the essential force for growth, whatever the mechanical properties of the wall. Although it exerts the same pressure throughout the cell wall, the resulting wall stress σe perceived locally in the cell wall varies because of fluctuation in local measurements (see below). The calculation of wall stress is independent of the mechanical features of the cell wall (e. g. , elastic, viscoplastic, plastic) and therefore independent of the biophysical model used subsequently. In addition to turgor, wall stress depends on the curvature of cell κ and on the thickness of the cell wall δ at each position of the cell surface (Fig 3A). Wall stress is partitioned into three directions: meridional (s), circumferential (θ), and normal (n) (see equation S2 in S1 Text). Because the cell wall is thin compared to cell size, the normal component of the wall stress is considered negligible compared to the two others [32]. To calculate the wall stress at many different points in an Ectocarpus apical cell, we obtained quantitative data for three components: (1) turgor (P), (2) curvature (κ) of the cell surface, and (3) cell wall thickness (δ). First, turgor in the apical cells was measured using the nonintrusive technique of incipient plasmolysis [34] on >100 cells for each of the 10 solutions of different osmolarities used in the experiment (Fig 4A). The value was subsequently corrected to take into account cell shrinking according to the protocol described in [34] (S2 Data). The calculated apical cell turgor was 0. 495 MPa, which is about 5 times the atmospheric pressure and is on the same order of magnitude as tip-growing cells from other eukaryotic groups, including the pollen tube (0. 1–0. 4 MPa, average at 0. 2 MPa; [5]). The second component of wall stress is the curvature of the cell surface (κ). To measure κ, the contour of Ectocarpus apical cells was first drawn manually. Then from 17 individual cell contours (S2 Fig), both the meridional and the circumferential curvatures as well as an average cell contour were calculated (Fig 4B). The same procedure was used for the tobacco pollen-tube contour. Compared with the pollen tube, the Ectocarpus apical cell displayed a sharper tip and a higher circumferential curvature on the shanks because of its smaller radius. The third component of wall stress is cell wall thickness (δ). Staining of Ectocarpus filaments with calcofluor-white, which labels cellulose (1–4) and callose (1–3) -beta-D-glucans [35], displayed a very clear gradient in thickness from the tip to the shanks of the apical cell (Fig 5A, also visible in 3D reconstruction from confocal microscopy). However, cellulose microfibrils are only a minor component of the brown algal cell wall (8% maximum dry weight), because they are immersed in a more abundant matrix of polysaccharides (45% dry weight) made of alginates (linear polymers of β-[1→ 4]-D-mannuronate and α-[1→ 4]-L-guluronate) and fucans (α-L-fucosyl residues) [23,36]. Therefore, we prepared longitudinal sections of apical cells. First, 300 nm–thick serial sections showed a gradient in thickness increasing from the tip to the shanks in the most meridional sections (Fig 5B, middle section), whereas the thickness appeared even throughout the cell in the most tangential sections (Fig 5B, top and bottom sections). Measurements of cell wall thickness across 70 nm–thick serial sections observed with transmission electron microscopy (TEM) further supported the presence of a cell wall thickness gradient (Fig 5C). Overall, 2,500 measurements were corrected (see Materials and methods) and plotted as a function of s. The distribution depicted a gradient that could be modeled as a Pearson-like function characterized by the lowest value δmin = 36. 2 nm at the tip (s = 0), the asymptotic maximum value δmax = 591 nm, and a midpoint at s1/2 = 16. 8 μm (Fig 5C). Cell wall thickness at the distal part of the dome (s = 8 μm) was 169 nm, i. e. , 4. 7 times the thickness at the tip (Fig 5C, close-up). The establishment of a cell wall thickness gradient contrasts with most tip-growing cells from other eukaryote groups [13,38], in which cell wall thickness is either constant (e. g. , 250 nm in pollen tube [39]) or higher at the tip (e. g. , oscillating growth in the pollen tube [40,41]). Fig 6A, 6B and 6C show a diagram of these biophysical factors in both Ectocarpus and pollen tube apices. Using this set of biological data, wall stress was calculated in both the meridional (σs) and the circumferential (σθ) directions, which ultimately allowed calculating the overall wall stress σe (Fig 3A; equation S3). Although σe fluctuates between 2. 5 and 3. 5 MPa (with the lowest value in the dome) in the pollen tube, it reaches a maximum of 25. 6 MPa in the Ectocarpus tip and decreases distally, reaching values similar to that in the pollen tube 70 μm away from the tip (Figs 6D and 7A). This stress value in the dome of Ectocarpus apical cells is remarkably high compared to other tip-growing cells, which, moreover, show the opposite stress gradient, increasing from tip to shank. Using the model, we tested the impact of the cell wall thickness gradient on tip shapes and growth rates. Steeper or gentler cell wall thickness gradients were sufficient to substantially alter the typical Ectocarpus cell shape and growth rate, suggesting that the cell wall thickness gradient must be tightly regulated in vivo (S6 Fig, central column; S4 Movie). However, cells display some significant variation in cell wall thickness (Fig 5C). In vivo observation of Ectocarpus tip growth also showed variability in growth rate and in cell shape (e. g. , displayed in S2 Fig), which may be due to transitory variation in cell wall thickness. The extremely low growth rate of this species can easily allow the activation of regulatory mechanisms that adjust the cell wall thickness gradient through modifications in cell wall biosynthesis. We performed the same experiments on cells with three different initial cell shapes (flat, typical Ectocarpus-like, and sharp). Using the Ectocarpus cell wall thickness gradient “Normal” resulted in convergence of the resulting shapes to the typical Ectocarpus shape (S6 Fig, middle row; S5 Movie). Therefore, the cell wall thickness gradient may also govern the tip resilience to deformation so that initial cell shape can be recovered after transient deformation (e. g. , due to an accident during growth). When simulations used a modified cell wall thickness gradient (“Steep” or “Gentle”) on these different initial cell shapes, all cells grew and converged to the same final shape specific to the given thickness gradient (S6 Fig, top and bottom rows; S6 Movie). These simulations supplement those by Dumais and colleagues [33], who explored various gradients in Φ and σy in a context where cell wall thickness was constant. The preponderant role of the cell wall thickness gradient in the control of tip growth raises the question of how this gradient is established and maintained. Calculations considering the cell wall extension rate and the maintenance of the cell wall thickness gradient during growth allowed inference of the level of cell wall material delivery and/or biosynthesis along the cell. According to this calculation, the overall delivery rate of cell wall material and/or synthesis in the pollen tube is much higher than in Ectocarpus (note the different scales of the x-axis in Fig 9A, left, top versus bottom). The maximum culminates 3. 0 μm away from the most distal position and drops to nil in the tube shanks (Fig 9A, top left). This calculation is in agreement with former in situ observations using FM4-64 that labels both endocytic and exocytic vesicles [52–54] (Fig 9A, top middle). This pattern is also in agreement with TEM observations in pollen tubes [55] and in other tip-growing walled cells where vesicle trafficking is concentrated in the most distal part of the tip (root hairs and green algae reviewed in [56], ascomycetes [13]). This mechanism contrasts with Ectocarpus, in which the cell wall flux is predicted to be significant in the shanks of the cell (Fig 9A, bottom left). How the cell wall is constructed in brown algae is still largely unknown. Cellulose may be synthesized from cytosolic uridine diphosphate (UDP) glucose via linear complexes of cellulose synthases localized in the plasma membrane, where they elongate cellulose microfibrils into the cell wall [57]. It is still unknown how the other main cell wall components (alginates and fucans) reach the cell wall at the tip of the Ectocarpus apical cell, but the current delivery mode is thought to be through Golgi-derived vesicles [58]. Therefore, we used FM4-64 to investigate the pattern of vesicle trafficking in Ectocarpus. FM4-64 displayed an homogeneous spatial pattern all along the cell, with no specific vesicle localization (Fig 9A, bottom middle). TEM observations provide further support because vesicles were never concentrated in any of the meridional sections of the dome of an apical cell (bottom right). Instead, chloroplasts and chloroplastic endoplasmic reticulum (CER) involved in the production of photosynthates [14] were observed in the dome as well as in the shanks of the cell (Fig 9A, bottom right). Therefore, observations of biological activity are compatible with the establishment and maintenance of a cell wall thickness gradient at an extremely slow rate, where CER can deliver the main components of the cell wall all throughout the cell with nevertheless the highest rate in the dome. To confirm this initial observation, we performed fluorescence recovery after photobleaching (FRAP) assays on Ectocarpus apical cells. We compared the fluorescence signal recovery dynamics in 5 different zones in the dome and shanks of the cell (Fig 9B, left). Considering the increase in the fluorescence signal over time, we used the normalized slope at t = 0 as a proxy for the intensity of membrane replacement by exocytosis, potentially reflecting cell wall–building activity (S7 Fig). The results showed that the exocytosis rate (Fig 9B right) reflects the cell wall flux inferred from the model (Fig 9A, bottom left). Our FRAP experiments support the prediction of the highest exocytosis activity at the base of the dome (zone C at s ≈ 5 μm) and significant activity in the shanks (zone E ≈ 10 μm from the dome end). Altogether, FRAP and TEM observations are compatible with the calculation of the cell wall flux inferred from the model.
Using a combination of serial longitudinal sections observed by TEM and optical microscopy, we first showed that Ectocarpus displays a gradient of cell wall thickness in its apical cells. Reports indicate that cells from other organisms display a cell wall thickness gradient. However, in most cases, accurate measurements could not be obtained from the methods employed such as epifluorescence microscopy of plant trichomes stained with propidium iodide [59] or bright-field microscopy of entire ghost cell walls of healing tips of the green alga Acetabularia [60]). Recently, Davì and colleagues [61] developed a technique on fission yeast enabling a resolution of 30 nm in living cells. However, this resolution is in the lower limit of Ectocarpus cell wall thickness, and TEM therefore appeared to be the most reliable technique. The accuracy of the TEM technique revealed a very steep thickness gradient ranging from 36 nm at the very tip to 169 nm at the base of the dome, corresponding to an average slope of 1. 6%. No such gradient has been reported in the growth zone of other organisms. In the diffusely growing trichome of Arabidopsis, the cell wall thickness increases in the cell with a slope of 0. 3% [59]. In the apical cell of Neurospora, cell wall thickness is constant in the dome and gradually increases along the shanks [62], a pattern similar to that observed in fission yeast [61]. In terms of biophysics, this gradient in cell wall thickness resulted de facto in a decrease in the stress from the shanks to the tip. The biological measurements specific to the Ectocarpus apical cell (turgor, dome geometry, and cell wall thickness) were integrated in the viscoplastic model initially proposed by Lockhart and further developed for tip growth by Dumais and colleagues [33]. The observed cell wall thickness gradient quantitatively compensated for the reduction of stress with an increase in curvature from the shanks to the tip. After adjusting the plasticity values, the model was able to achieve self-similar growth at the speed observed in vivo. Regarding the cell wall mechanical properties, the model inferred two main differences with the pollen tube. First, the extensibility Φ and the yield threshold σy remained constant throughout the Ectocarpus cell, in contrast to the pollen tube models in which the constant thickness of the cell wall necessarily requires modification of the cell wall mechanical properties to allow growth [63]. Plotting σy and Φ together with cell wall thickness δ clearly illustrates the different strategies developed by Ectocarpus and the pollen tube (Fig 7D): in Ectocarpus, δ is the only varying factor, but in the pollen tube, both σy and Φ vary, whereas δ remains constant. Using a Lab-on-a-Chip platform, Shamsudhin and colleagues [64] confirmed that the pollen tube displays an apparently increasing elastic modulus from the tip to the shanks, which is correlated with the presence of methyl-esterified pectins [65]. Secondly, compared with the pollen tube, the overall value of strain rate is approximately 100 times lower, but stress is approximately 10 times higher in Ectocarpus (Fig 6), suggesting that the Ectocarpus cell wall is generally more resilient to yielding during growth. Experimental work is clearly still needed to refine the values of the yield threshold and extensibility, but our calculation of wall stress offers a solid basis on which their order of magnitude can be inferred. Nano-indentation of Ectocarpus cell wall produced values of elastic modulus much lower (approximately 1–4 MPa, [27]) than those reported in the pollen tube (approximately 20–400 MPa, [64]). However, the different nano-indentation experimental procedures used in these studies (depth of indentation, shape of the indenter, osmotic conditions, physical model) make comparisons questionable [47]. Nevertheless, the elastic modulus assessed using nano-indentation and the cell wall mechanical properties inferred from the growth model together suggest that Ectocarpus is more elastic but less prone to expansion during growth than the pollen tube. The distinction between cell wall elasticity and growth has already been made in the green alga Chara [66] and has since been reported in other plant cells (reviewed in [67]). The inverse relationship observed in Ectocarpus is fairly compatible with the dual role of the cell wall in brown algae, i. e. , coping with frequent environmental changes in osmotic pressure (tides), which requires a high elasticity, and resistance to yielding in the face of high wall stress due to the thin cell wall and high turgor. The lack of a functional relationship between intrinsic elasticity and cell wall extensibility has already been reported in land plants [67]. Likewise, the presence of stiff or soft polysaccharides—as assessed in vitro—does not correlate with the expansion of the cell wall during plant growth (e. g. , [68,69] and reviewed extensively in [67]), nor apparently during growth in brown algae [70]. Another puzzling question is how Ectocarpus controls the cell wall thickness gradient necessary to ensure the maintenance of cell shape. It is unknown whether cell wall thickness fluctuates during growth, as recently reported in the fission yeast [61], but this fluctuation may account for the variation in cell shape and growth rate observed in living organisms. Nevertheless, the gradient in thickness requires regulation of cell wall biosynthesis, which in brown algae like in land plants involves both in muro cellulose synthesis and the delivery of other components (fucans and alginates in brown algae) through vesicle trafficking (Golgi and flat cisternae respectively in Fucales; [58]). FRAP data showed that the highest exocytosis activity was localized in the basal region of the dome, just before the cell adopts its cylindrical shape. This coincides with the highest cell wall flux computed from the model and with the pattern described in the pollen tube [71,72]. How exocytosis vesicles are targeted to these positions is unknown. In yeast and land plants, mechanosensors localized in muro control cell wall biosynthesis enzymes to modulate cell wall thickness and respond to cell wall damage [60,73]. The Ectocarpus genome codes for several mechanosensor proteins (integrins, transmembrane proteins containing the WSC carbohydrate-binding domain [25,74]), and these proteins may well be key regulatory factors in this process. The palette of tip-growing strategies among species is not restricted to the control of cell wall thickness and of cell wall mechanical properties through pectin methyl-esterification. Other molecular mechanisms, including pectate distortion cycle in Chara [75], secretion of glucanases and chitinases in fungi [13], and intussusception in prokaryotes [76], have been proposed to account for the differential cell wall mechanics along the cell. Therefore, distinct key cell wall biophysical factors, and potentially a combination of them [61], appear to have been selected during evolution to achieve cell wall growth. The evolutionary history of brown algae is short (approximately 250 My) and distinct from that of land plants. The marine environment characterized by high physical pressure and ionic concentrations, low gravitational forces, and high drag forces due to tremendous sea currents may have promoted the development of specific, singular strategies in these peculiar organisms. In the case of tip growth, although we cannot formally exclude the possibility that a gradient of wall mechanical properties exists and contributes to morphogenesis in Ectocarpus, our results suggest that this organism has favored a singular approach based on cell wall thickness and hence on control of wall stress. The question remains whether the particular features of this organism, including its slow growth, make the control of cell wall thickness more efficient than the control of cell wall mechanical properties.
Parthenosporophyte filaments of Ectocarpus sp. (CCAP accession 1310⁄4) were routinely cultivated in natural seawater (NSW) as described in [77]. For microscopic observations and time-lapse experiments, early parthenosporophytes were obtained from gamete germination on sterile coverslips or glass-bottomed petri dishes. Ectocarpus prostrate filaments were treated with 1,10, and 50 μM IAA (Sigma-Aldrich I3750) prepared in 2,20, and 100 μM NaOH, respectively (final concentration). Growth rates were measured for each concentration 24 h post treatment (n = 10), using NSW supplemented with 2 μM NaOH as a control. Turgor was measured in 1 μM IAA using 2 μM NaOH as the control (see Measurement of turgor in the apical cell and correction for details). Ectocarpus filaments were immersed for 1 min in a range of sucrose concentrations (diluted in NSW), and the proportion of plasmolyzed apical cells was measured by counting apical cells (n > 100) under an optical microscope. The rate of plasmolysis was plotted against external osmolarity (ce). The limit plasmolysis (cpl) corresponds to the value of ce at which 50% of apical cells were plasmolyzed. The mean cpl value was calculated from 3 independent experiments. Solution osmolarities were measured with an osmometer (Osmometer Automatic, Löser, Germany). Because the cell wall of Ectocarpus is partly elastic, plasmolyzed cells have a reduced volume that must be taken into account to calculate the real internal osmolarity (ci) and thus the real internal turgor (P). To do so, the coefficient of apical cell volume shrinking (x, equal to the ratio of the cell volume upon plasmolysis to the cell volume in normal growth conditions) was measured on apical cells (n = 9), and the corrected internal osmolarity was calculated as ci = x. Cpl. The difference between internal and external osmolarities is Δc = ci − 1,100 with the seawater osmolarity = 1,100 mOsm L−1, and the turgor is P=ci–ce410, in MPa. Apical cell contours were drawn manually from confocal images of meridional plans of apical cells immersed in NSW. Similar procedure was followed for tobacco pollen tubes from photos given by Gleb Grebnev (B. Kost’s lab, Erlangen Univ. , Germany). We devised a Python 3 script to compute the average contour for a series of images and used it on Ectocarpus (n = 17; S2 Fig) and tobacco pollen tubes. The program starts with a hand-drawn contour for each cell, from which it computes a smoothed cubic spline curve. A set of equidistant points (we used a point-to-point distance of 50 nm) are extracted from the spline, and the meridional curvature κs is computed at each point (S2 Fig). To obtain average symmetrical curvatures, a pair of windows starting from the tip point and sliding in both directions was used (window width = 200 nm, sliding step = 50 nm). The discrete values of the κs = f (s) function were used to iteratively compute the position of cell wall point coordinates as values of x (the axial abscissa) and r (the distance to the axis), together with the meridional abscissa s, the curvatures κs and κθ, and φ the angle between the axis and the normal to the cell wall. In particular, the circular symmetry of the dome imposes at the tip (where s = 0), that κθ = κs and thus σθ = σs, whereas in the cylindrical part of the cell κs = 0 and thus σθ = 2σs. Ectocarpus filaments were prepared for TEM. Filaments grown on sterile glass slides were fixed with 4% glutaraldehyde and 0. 25 M sucrose at room temperature and washed with 0. 2 M sodium cacodylate buffer containing graded concentrations of sucrose. The samples were post-fixed in 1. 5% osmium tetroxide, dehydrated with a gradient of ethanol concentrations, and embedded in Epon-filled BEEM capsules placed on the top of the algal culture. Polymerization was performed first overnight at 37 °C and then left for 2 d at 60 °C. Ultrathin serial sections were cut tangentially to the surface of the capsule with a diamond knife (ultramicrotome) and were mounted on copper grids or glass slides. Two types of sections were produced. Serial sections (300 nm thick) were stained with toluidine blue to show the main cellular structures, including the cell wall, and mounted on glass slides. Sections (70 nm thick) were stained with 2% uranyl acetate for 10 min and 2% lead citrate for 3 min, mounted on copper grids (Formvar 400 mesh; Electron Microscopy Science), and examined with a Jeol 1400 transmission electron microscope. A compilation of the sections for the 15 cells is shown in S3 Fig. Original photos are available at https: //www. ebi. ac. uk/biostudies/studies/S-BSST215. From TEM pictures obtained on fixed Ectocarpus apical cells, only longitudinal sections with the thinnest walls were considered to avoid bias due to misaligned sections (all images are shown in S3 Fig). Measurements were carried out every 386 nm along 15 different cells, at the meridional abscissa from the tip (s = 0) up to s = ±70 μm using Fiji image analysis software. Altogether, 2,500 measured values of apparent thickness w were corrected, making the assumption that actual cell radius was R = 3. 27 μm (but was seen as apparent radius a) and applying the following formula: δ=R-a2+R2- (a+w) 2 (S4 Data). As askew sectioning results in cell walls looking thicker, the only remaining bias is expected to cause overestimation of the thickness at the tip. Corrected values δ for cell wall thickness were plotted as a function of the position s along the cell. As the relationship δ = f (s) displayed the aspect of an inverted bell, we designed 3 functions with this shape, derived from classical functions, to match them with the experimental values— (1) “Gauss”: δ=δmax- (δmax-δmin) exp (- (s/s1/2) 2log (2) ); (2) “Lorentz”: δ=δmax- (δmax-δmin) (1+ (s/s1/2) 2) -1; and (3) “Pearson”: δ=δmax- (δmax-δmin) (1+3 (s/s1/2) 2) -1/2. The values δmin, δmax, and s1/2 were adjusted for each of these functions, with a respective residual standard error of 0. 08,0. 05, and 0. 04. Therefore, we used the Pearson model with its optimized values δmin = 36. 2 nm, δmax = 591 nm, and s1/2 = 16. 81 μm for further modeling (Fig 5C). Ectocarpus cells were boiled twice in 1% SDS, 0. 1 M EDTA and then treated with a solution of 0. 5 M KOH at 100 °C. Pellet was rinsed extensively with MilliQ water and dried on a glass slide. Imaging was performed on dried samples. A Veeco Bioscope catalyst atomic force microscope coupled with a Zeiss inverted fluorescent microscope was used for imaging. RTESP probes (Bruker) were used in Scanasyst mode. The protocol was adapted from [30] and is described in detail in [31]. Young sporophyte filaments grown in glass-bottom petri dishes were covered with sonicated 0. 1% (w: v NSW) of FluoSpheres amine, 0. 2 μm, red (F8763, Molecular Probes), washed with NSW and mounted under a TCS SP5 AOBS inverted confocal microscope (Leica) controlled by the LASAF v2. 2. 1 software (Leica). The growth of 25 apical cells growing parallel to the glass surface was monitored, and bright-field and fluorescent pictures of median planes for each apical cell were acquired at several time points. Cell wall contours were hand-drawn on time-lapse images using GIMP, together with their respective indicator points. The position of the extreme tip (s = 0) was fixed for each meridional contour, and the drawing of cell contours and microsphere positions were aligned during the time course by using steady microspheres attached on fixed positions. A spline was adjusted on each contour and on each series of indicator points. The angle at each possible intersection between these trajectories and the cell contour splines were computed, making use of their first derivatives. Further analysis performed using R [78] consisted of (1) determining the distribution of angles, their mean, and standard deviation and (2) testing the hypothesis of dependence between the angle and the meridional abscissa. From the 156 measured angles between the tangent to cell wall and the trajectory, we computed the mean value m = 1. 71 = π/1. 83 radian (or π/2 − 9. 16%) and the standard deviation s = 0. 52 = π/6. 09 radian. To test independence between the angle and the position in the dome, we computed the Pearson correlation coefficient between the angle and the absolute value of the meridional abscissa. It was r = −0. 031. Staining of Ectocarpus filaments with calcofluor-white was carried out as described in [28]. FM4-64FX (F34653, Invitrogen) stock solution was diluted to 385 μM in DMSO and then diluted to 7. 7 μM in NSW. Coverslips with Ectocarpus filaments were immersed in 50 μL of 7. 7 μM cold FM4-64FX on ice and immediately mounted on a confocal microscope. Endocytosis and further trafficking of the fluorochrome was followed for 1 h at room temperature. The fluorochrome was excited with a 561 nm neon laser, and emission was observed with a 580–630 nm PMT. For the FRAP assay, filaments were stained with 100 μM FM4-64FX for 10 min at 4 °C and rinsed 4 times with cold, fresh seawater. Photobleaching was performed on about 25 μm (s) along the cell from the tip, and recovery was monitored using an inverted Nikon Ti Eclipse Eclipse-E microscope coupled with a Spinning Disk (Yokogawa, CSU-X1-A1) and a FRAP module (Roper Scientifics, ILAS). Images were captured with a 100x APO TIRF objective (Nikon, NA 1. 49) and an sCMOS camera (Photometrics, Prime 95B). For the detection of the FM4-64FX stained samples, we used a 488 nm laser (Vortran, 150 mW) for the excitation and the bleaching steps and collected the fluorescence through a 607/36 bandpass filter (Semrock). Image acquisition using the MetaMorph software 7. 7 (Molecular Devices) was as follows: 1 image/s, displaying 6 images before bleaching, 1 image at the precise time of bleaching, 50 images during the recovery phase, for a total of 57 images by cell. Images for one given cell were processed as a stack using Fiji [79] and R [78]. For each time point t (with bleaching occurring at t = 0), the background signal Z (t) was averaged from 4 separate square regions of approximately 1 μm2; the spontaneous fluorescence decrease was estimated by monitoring the signal U (t) in an unbleached region; the local signal was recorded in regions A–E as defined in Fig 9B. Note that all zones, including E, are sufficiently far from the edge of the photobleached zone to be unaffected by homogenization due to membrane lateral flux in the considered timescale. Following [80], the corrected signal for region A (and similarly for regions B–E) was computed as: Ac (t) = (A (t) -Z (t) - (A (0) -Z (0) ) ) U (0) -Z (0) U (t) -Z (t). The recovery activity was estimated by matching the measured Ac (t) values to the function Y (t) = Y (0) + α (1 − exp (−t/τ) ), where Y (0) and α and τ are free parameters. We computed the normalized slope at t = 0 as (1/α) (dAc/dt) (0) = 1/τ, for 9 observations in each of the 5 (A–E) zones selected (see S7 Fig). The meridional contour of 6 tobacco pollen tube apices were traced from photos given by Gleb Grebnev (B. Kost’s group, Erlangen University, Germany), and the curvature was computed as described for Ectocarpus cells. Turgor and cell wall thickness were obtained from the literature [38]. In the absence of precise determination of their respective values, we derived a working hypothesis from previous literature reports showing that variations of Φ and σy occur simultaneously in opposite directions [49–51]. This intuitive relationship is consistent with molecular models of the cell wall [50]. Given that our model can derive the value of the expected strain rate ε˙* from other values (S1 Text), we propose to partition this product equally between its two members. Thus, we computed Φ=ε˙* and (σe-σy) =ε˙*, leading to σy=σe-ε˙*. These arbitrary values were useful for giving an example of what could be a possible state (Fig 6F and 6G; Fig 7B right) and performing simulations. Programs developed as part of this work were written in Python 3. 6 [81], making use of NumPy [82] and Matplotlib [83] libraries, in a GNOME-Ubuntu environment (laptop and workstation). The source code is available at https: //github. com/BernardBilloud/TipGrowth. Modeling is described in S1 Text. The simulation program performed a simple simulation with graphic output or an array of simulations within a range of Φ and σy values. The input was a list of cell wall point coordinates and values from, for instance, computations of average contours (ad hoc generated data were also used for simulations starting with geometrically designed profiles). For each point, the stress was computed from turgor, curvature, and cell wall thickness values. Then, using Φ and σy, the strain rate and the normal velocity were computed. The velocity and displacement direction (normal to the cell wall) gave the new position of the point, calibrated for a tip growth of 1 nm at each step. After computing new positions for all points, the program designed a cubic spline (without smoothing) from which a new sample of points was extracted, thus keeping a constant distance between points throughout the simulation. Accuracy of the simulation was evaluated by averaging point-to-point distances between the simulated profile and the initial profile translated at the expected speed. Values of Φ and σy were progressively optimized using a steepest-descent approach. As starting values, we used the coefficients of the linear model derived from the points (σe, Φ (σe − σy) ) for which Φ (σe − σy) ) > 1: Φ = 2. 5 × 10−3 min−1 MPa−1 and σy = 11 MPa. These values were used to simulate growth up to 25 μm, and divergence with the expected behavior was evaluated by comparing them to the initial points translated by 25 μm in the axial direction. As a numerical value, we took the logarithm of rD (residual distance), which was the weighted average point-to-point distance, where the weight was exp (s2log (2) ), to maintain the dome shape. Optimized values Φ = 2. 51 × 10−3 min−1 MPa−1 and σy = 11. 18 MPa gave a simulation with a log (rD) of −3. 0. As a comparison, the mean log (rD) between the initial average contour and the 17 experimental contours used to build it was −4. 41, with a standard deviation of 0. 35. To assess the robustness of the results, we performed a bootstrap analysis. Three thousand samples were constructed by drawing with replacement 17 cell contours and 15 cell wall TEM images out of their respective datasets. For each sample, the average contour and the cell wall gradient were computed as explained above. The stress σe and expected strain rate ε˙* were computed as functions of the meridional abscissa s. To test consistency with the model, the (σe; ε˙*) points were fitted a Lockhart equation by adjusting values Φ and σy and computing the Pearson correlation coefficient (r2) for the increasing part of the function, i. e. , σe > σy; ε˙*>0.
Tip growth is known in organisms with filament-like structures, such as fungi (e. g. , hyphae), plants (e. g. , root hairs, moss protonemata), and algae (e. g. , filamentous thalli). The driving force for growth in these organisms is the difference in osmotic pressure (turgor) between the inside of the cell and the external medium, a force contained by the cell wall. Physical laws imply that the higher the curvature of the cell, the lower the pressure (stress) perceived by the cell wall. Paradoxically, growth takes place at the dome-shaped cell apex, which has high curvature. Tip-growing cells studied so far (mainly plants) compensate the low wall stress in the apex by chemically loosening their cell wall. We studied Ectocarpus, which is a representative of brown algae, a eukaryotic branch very divergent from land plants, fungi, and green algae. We carried out a series of quantitative measurements at the cellular level and showed that the cell wall is thinner at the tip (36 nm) than on the shanks (170 to 500 nm). Using a viscoplastic model of cell wall expansion, we showed that the cell wall thickness gradient, together with dome curvature, generates sufficient wall stress to account for the observed growth pattern.
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The brown algal mode of tip growth: Keeping stress under control