Shape differences in WT and ktn1-5 papillae

Both WT and ktn1-5 papillae share an overall quasi-cylindrical shape resembling a bowling pin (Fig 1F and 1G). In our prior study [6], where we measured the total length of the papillae and the width of its head, we observed some variations between genotypes. Here, to provide a more detailed characterisation of the WT and ktn1-5 papilla morphology, we quantified more accurately their shapes using four geometric shape descriptors (Fig 1H and 1I).

Papillae display a nearly hemispherical head region, characterized by measurements and (head length and width, respectively). This head region gradually transitions into a cylindrical shaft region that narrows at the neck, characterized by the distances (from the pole to the neck) and (neck width). We found significant differences (Fig 1J and 1K and S2 Table), with WT papilla cells being more slender with a narrower head (mm) and neck (mm) than ktn1-5 counterparts (mm; mm).

Next, we wondered whether these differences in papilla morphology could impact pollen tube trajectories.

Geodesics as reference trajectories on a pin-like surface

Using a geometrical model, we first investigated how variations in shape of a pin-like structure, specifically constriction in the neck region, could influence the theoretical trajectories of an object moving along its surface.

In the absence of any forces, other than those that keep it on the surface, an object advancing locally straight on a curved surface follows a curve on this surface called a geodesic [8]. Geodesics correspond to the length-minimizing curves between pairs of points on a (smooth) surface, and possess the remarkable property that, at any given point on the surface and in any specified direction, a single, unique geodesic passes through that point in that direction [9].

To get a first insight into the role of the neck constriction in altering geodesic trajectories on nearly cylindrical surfaces, we computed geodesic paths with fixed initial positions and directions on pin-like shapes with varying neck diameters, as illustrated in Fig 2. As the neck diameter decreases (D in Fig 2), the geodesic (yellow curve) changes. For instance, decreasing the neck diameter by 38% (from 0.43 to 0.27) resulted in markedly different coiling patterns (from 1 turn to 3.5 turns before reaching the bottom region), especially around the narrowest region of the pin-like shape. In extreme scenarios, where the neck became exceedingly thin, geodesics could no longer cross the neck and remained confined to the upper part of the pin-like shape (rightmost two shapes in Fig 2). The tendency of the geodesic to coil in the region of constriction is a direct consequence of Clairaut’s theorem: This theorem states that on surfaces of revolution, like for example pin-like shapes, geodesic trajectories should follow the constraint , where R represents the distance of a point P on the geodesic curve to the axis of revolution of the surface S and represents the angle between the tangent of the geodesic and the longitudinal line of S going through that same point P [10]. As an example, on cylindrical surfaces (R is constant), geodesic lines correspond to helices with a constant pitch and slope. For our nearly cylindrical surfaces, in regions where the radius R of the pin structure decreases, the trajectory tends to orient more in the circumferential direction, i.e. increases to maintain a constant product , and coil more. This property is further discussed and illustrated in Ref [11] and in the references therein.

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Fig 2. Theoretical geodesic trajectories on pin-like surfaces of varying shapes.

This sequence displays pin-like structures, characterized, from left to right by a gradual decrease in neck mid-height diameter (D) while maintaining a constant height (H). The top row features 3D surface rendering of the shape. Bottom row: wire-frame rendering where the whole geodesic trajectory is visible (yellow curve). All the geodesics start at the same position (green dot) at point P₀ with azimuth = 0 and altitude = -0.1 with respect to the pole of the pin structure, and with an initial inclination angle of downward with respect to the circumferential direction. D is given in arbitrary units (a.u.) and H=2.0 a.u. Note that the curve can cross itself (rightmost two situations).

https://doi.org/10.1371/journal.pcbi.1013077.g002

Thus, according to this theoretical model, where the tube is assumed to grow as straight as possible of the surface, without mechanical or chemical cues, the presence of constrictions in the neck region enhances the coiling behaviour of geodesic trajectories on nearly cylindrical shapes. However, it is important to notice that substantial alteration in the neck diameter are required to induce drastic qualitative modifications of the trajectories as illustrated in Fig 2. Interestingly, the pollen tube trajectories on WT papilla (Fig 1F), despite starting in various direction, do not exhibit helical paths in the cylindrical papilla shaft nor an increased coiling behaviour in the neck region which suggests that they do not follow geodesics. In contrast, the coiling behaviour of the pollen tube trajectories on ktn1-5 papillae (Fig 1G), is reminiscent of that of geodesic lines on pin-like surfaces. To systematically explore the influence of papilla shape and possible guidance cues in pollen tube growth, we developed a minimal phenomenological model that integrates quantification of the WT and ktn1-5 papillae geometry, along with well-known features of pollen tube growth.

A minimal phenomenological mechanical model of pollen tube growth on the papilla surface

Using the previously defined papilla descriptors (Fig 1H and 1I), we fitted mathematical functions to derive analytical three-dimensional virtual WT and ktn1-5 papillae surfaces S (Fig 3A). Thereby, papilla shapes are parameterized using cylindrical coordinates , where and z respectively represent the angular direction and the distance along the papilla axis (measured from the papilla pole).

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Fig 3. Papilla prototypes and pollen tube trajectories without guidance on WT and ktn1-5 papillae.

(A) Three dimensional shapes of WT and ktn1-5 papillae. The papilla long axis is oriented along the z axis. z = 0 corresponds to the papilla pole. Scale bar = 10 m. (B) Three-dimensional view of the top region of a ktn1-5 papilla showing the initial position of the pollen grain (z₀, red dot) and the initial direction of the pollen tube upon emergence from the grain (, red arrow). The vector represents a papilla surface tangent vector pointing in the longitudinal direction. () indicates an initial tube direction towards the papilla base (pole). Scale bar = 10 m. (C) Examples of simulated pollen tube trajectories on ktn1-5 papilla surfaces, varying in initial positions z₀ of the pollen grain and initial directions of the emerging pollen tube (indicated by black arrows). The numbers in brackets denote the normalized initial positions () and the initial directions () for each trajectory. T (below each papilla) represents the number of turns the pollen tube makes to reach the papilla base . Each configuration is labelled from a to h. Trajectories on WT stigma with identical initial conditions are shown in S2 Fig. (D) Morphological phase diagrams of the pollen tube turn number T depending on the initial pollen grain position z₀ (normalised to the papilla distance head length ) and the initial pollen tube direction () on WT and ktn1-5 papillae. denotes the papilla pole, the frontier between the head and cylindrical shaft, and the pollen grain landing limit. The colour code indicates the number of turns the trajectories undergoes before it reaches the papilla base. In the white region (caged), the tube path is trapped by its own trajectory, preventing it from reaching the papilla base; such trajectories were categorised as having T>2.5. The letters a to h correspond to the example configurations depicted in (C). (E) Comparison of simulated (solid lines) and experimental (squares) cumulative distributions of pollen tube turn numbers. To calculate the cumulative fraction for experimental data, we utilized data from Ref [6], where we examined 251 WT and 327 ktn1-5 pollinated papillae; error bars represent the standard error of the mean. The label ktn1 refers to the ktn1-5 mutant.

https://doi.org/10.1371/journal.pcbi.1013077.g003

Numerous observations from our previous studies [6, 12] consistently support the finding that pollen grains tend to attach to the head region of the papilla (see as examples Fig 1F and 1G and S1 Fig). Attachment to the neck region or even below is rare and might be geometrically constrained, i.e. the pollen grain does not fit in between papillae. As a result, in our model, pollen grain attachment site and the corresponding landing position z₀ (Fig 3B) are constrained to distances ( as indicated in Fig 1H). In addition, we assumed that pollen grains attach with equal probability on any point of the papilla surface provided that (For an alternative hypothesis where pollen grains attach preferentially to the papilla pole see Materials and Methods). After emerging from the grain, the pollen tube outgrowth has an initial direction, represented by the angle made with the longitudinal papilla axis (Fig 3B). We assumed that this initial direction is devoid of any directional bias, implying that all potential initial directions are equally probable. By convention, () indicates that the tube grows along the long papilla axis towards its base (respectively pole), while an angle of indicates an initial trajectory oriented along the papilla circumferential direction.

The pollen tube, a tip-growing cell, undergoes unidirectional elongation, with its expansion restricted to the very tip of the cellular protrusion [13]. Moreover, in A. thaliana, the pollen tube progresses within the papilla cell wall, confined between the two leaflets of the cell wall (Fig 1D and 1E and Ref [12]), keeping it on the papilla surface (i.e. preventing penetration into the papilla volume or escape into the air). A precise description of the known molecular processes and mechanics governing pollen tube growth within the papilla cell wall (see e.g. Refs [7, 14, 15]) would require a numerically costly multi-scale approach with many unknown parameters. Here, in our phenomenological model, we considered the pollen tube as an inextensible filament with bending rigidity whose trajectory is maintained on the papilla surface S. This bending rigidity serves as a resistance against changes in the growth direction at the pollen tube tip, consistent with in vitro experimental observation that pollen tubes grow straight over several hundreds of micrometres in the absence of mechanical forces or chemical guiding cues [14–16].

The growth direction of the tip is obtained from the equation of momentum conservation at the filament tip, which may include an external force of mechanical and/or chemical origin. In detail, the pollen tube path is given by with s denoting the length of the trajectory from a given initial position. Denoting L(t) the length of the pollen tube at time t, the position of the elongating tip is given by the position of the filament extremity . We assume that over a small time lapse , the tip grows by a length in a direction , which results in a new tip position . To calculate the tip growth direction , the tube bending is decomposed into two components: (i) the bending in the plane defined by the surface normal and the tangent to the trajectory at the tip (i.e. to keep the tip trajectory on the papilla surface), and (ii) a bending either to the left or to the right in the local plane tangent to the papilla surface. We assume that this bending in the tangent plane is due to an external force (torque) that induces a rotation of the pollen tube and that comes from potential guidance cues (see Eq 13 in Materials and Methods). Based on our previous observations of pollinated stigma [6, 12], we noted that the advancing pollen tube tip is unable to cross its own path. This property, that we termed “self-avoidance”, is visually depicted in S1 Fig. In our model, self-avoidance is implemented by representing the pollen tube volume as a succession of spheres (at position , where i is an integer and is an small increment along the arc length s of the curve describing the tube trajectory) and cylinders (between positions and ) which cannot be penetrated by the outgrowing tip. At each simulation step, we tested whether the growing tip penetrates into the excluded volume of a previously deposited pollen tube. When a potential penetration is identified, the growth direction is not determined from the equation of momentum conservation (13). Instead, we minimised the corresponding potential function (see S1 Text) ensuring that the excluded volume is not violated.

Pollen tube growth on ktn1-5 papillae follows curves close to geodesics modulated by the property of self-avoidance

Using the above computational model, we started by simulating pollen tube trajectories on virtual papilla in the absence of guidance cues from the stigmatic side (i.e. in absence of a torque acting on the tube tip in the local plane tangent to the papilla surface).

The behaviour of pollen tube growth was assessed by calculating the number of turns T, a methodology previously employed by [6]. Experimentally, the turn number is defined as the number of revolutions a pollen tube makes around the papilla axis down to the observable papilla base (denoted as base in Fig 3A), which corresponds to a distance of approximately m. We performed simulations across a broad range of initial pollen parameters z₀ and , considering the self-avoidance property. Fig 3C provides examples of simulated tube trajectories on ktn1-5 papilla surfaces, along with the corresponding turn numbers T. For comparison, simulated trajectories on WT papillae are shown in S2 Fig.

To integrate the entire range of configurations, we represented the turn number depending on the initial pollen parameters (z₀ normalized to the papilla head length and ) using a morphological phase diagram (Fig 3D). Both morphological phase diagrams for WT and ktn1-5 papillae display only slight differences, suggesting that papilla shape heterogeneity (Fig 1H–1K) does not significantly impact pollen tube paths. This confirms our conjecture based on the coiling behaviour of reference trajectories on a pin-like surface (Fig 2), where we observed that substantial differences in geometry (e.g. neck constriction) are required to induce a change in the coiling behaviour of an object moving along geodesics. Both phase diagrams show that trajectories originating from the papilla head region, particularly close to the pole with , consistently reach the base with relatively low turn numbers whatever the initial direction of the tube (orange colour in Fig 3D). This mainly lies in the symmetrical and spherical shape of the papilla head region, where all initial angles result in similar types of trajectories, corresponding to longitudinal lines (geodesics) which either go directly to the papilla base (T) or cross over the papilla pole before going to the base (T). For trajectories starting at , the resulting paths highly depend on the initial direction of the tube. When a tube starts with an initial direction pointing close to the circumferential direction (), it exhibits a high turn number (blue color in Fig 3D, illustrated in Fig 3C, labels c and g). In specific initial conditions, the tube tip is significantly deflected by its own path (Fig 3C, label f) or self-constraints (caged in white color in Fig 3D, illustrated in Fig 3C, label e) underscoring the importance of the self-avoidance property which can lead to completely divergent trajectories even with minor changes in the initial pollen parameters. In contrast, when a tube starts with either a low () or a high () initial angle, it reaches the base with a low turn numbers, either directly (T, illustrated in Fig 3C, label a) or passing over the papilla head region (T, illustrated in Fig 3C, label d) respectively.

To conduct a comprehensive global comparison between simulated and experimentally observed tube trajectories, we computed the cumulative fraction of turn numbers F(T), which represents the fraction of papillae with turn number (Fig 3E and Materials and Methods for further details). To calculate the experimental cumulative fraction, we utilised data from [6]. We found that the cumulative fractions for simulated trajectories on both WT and ktn1-5 virtual papillae (orange and blue solid curves in Fig 3E) closely aligns, as expected from the phase diagrams (Fig 3D). Around 40–50% of the simulated pollen trajectories completed at most one turn to reach the papilla base, and a significant proportion of tubes (about 20%) reached the papilla base exhibiting a high number of turns (exceeding 2.5). Importantly, our analysis revealed that the simulated distributions reproduce fairly well the experimental distribution of pollen tube turns on ktn1-5 papillae (blue squares, Fig 3E) whereas significant disparities were observed when compared with the experimental trajectories pollen tubes make on WT papillae (orange squares, Fig 3E).

In summary, the parameters used in our simulations were sufficient to reproduce the strong coiling behaviour of pollen tube trajectories experimentally observed on ktn1-5 papillae. Given that on a curved surface, when no force is acting on its tip (apart from those maintaining surface contact), a tube advances locally along the straightest possible path following geodesic trajectories (see section “Geodesics as reference trajectories on a pin-like surface”), our comparison between simulation and experiment strongly suggests that pollen tube trajectories on ktn1-5 papillae follow geodesics or curves close to geodesics, determined entirely by their initial position (z₀) and initial direction (), with modulation from the self-avoidance property. In contrast, replicating pollen tube configurations on WT papillae, where pollen tubes reach the papilla basis by making turn, indicates a clear deviation from geodesics, and requires additional guiding factors.

Pollen tube growth on WT papillae requires guidance cues from the papilla side

We then explored how pollen tube in WT papillae deviate from geodesics paths, resulting in fewer turns. Considering that the pollen tube grows within the papilla cell wall constrained to follow the papilla surface, deviation from geodesic trajectories requires the presence of a force normal to the pollen tube growth direction in the plane tangent to the papilla surface. We therefore performed numerical simulations of pollen tube trajectories using the same framework as in the previous section (see also Materials and Methods), with the addition of an alignment force that orients the tube growth along the longitudinal axis of the papilla. This alignment force is symmetric with respect to the reorientation of the pollen tube growth direction towards the papilla base or pole and is proportional to , where denotes the angle between the direction of the advancing tip and the papilla long axis. The expression assures that tip growth already oriented towards the papilla base () will align completely with the long axis of the papilla towards its base. For growth oriented towards the papilla pole (), the growth direction will align completely with the long axis towards the papilla pole. Due to symmetry, the alignment force for growth into the circumferential direction () is zero. The magnitude of the alignment force (i.e. its maximum orientational force at and ) is determined by the adimensional constant , which quantifies the ratio of the reorientation force to the internal resistance of the pollen tube against directional changes (provided by its bending rigidity). At this stage, it is reasonable to assume that alignment forces exclusively operate in the cylindrical part of the papilla for , (Fig 4A), since the papilla head region is likely isotropic in terms of its geometry and cell wall properties due to its nearly spherical shape [17].

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Fig 4. Effect of longitudinal growth guidance on pollen tube trajectories.

(A) Two-dimensional representation of the papilla, indicating where the alignment forces act (yellow region). (B) Example of pollen tube trajectories on WT papilla surfaces under various initial conditions, the initial pollen grain position z₀ and initial pollen tube direction (indicated by black arrows), simulated without growth guidance (upper panel) and with growth guidance (adimensional guidance strength , lower panel). The numbers in brackets denote the normalized initial position () and the initial directions () for each trajectory. T represents the number of turns the pollen tube makes to reach the papilla base. Each configuration is labelled from a to f. (C) Morphological phase diagram of the pollen tube turn number T depending on the initial pollen grain position z₀ (normalised to the papilla head length ) and the initial pollen tube direction () on WT papillae. denotes the papilla pole, the frontier between the head and cylindrical shaft, and the pollen grain landing limit. The colour code indicates the number of turns the trajectories undergoes before it reaches the papilla base. The letters d, e, f correspond to the example configurations depicted in (B). Trajectories of pollen tubes on ktn1-5 papilla with identical initial conditions and the corresponding morphological phase diagram are shown in S3 Fig. (D) Comparison of simulated (solid lines) and experimental (squares) cumulative distributions of pollen tube turn numbers. Simulated cumulative distributions for turn numbers on WT papillae are calculated with a growth guidance of (dashed orange curve). Simulated cumulative distributions for turn numbers on ktn1-5 papillae are calculated without guidance (blue curve, reproduced from Fig 3E). To calculate the cumulative fraction for experimental data, we utilized data from Ref [6], where we examined 251 WT and 327 ktn1-5 pollinated papillae; error bars represent the standard error of the mean. The label ktn1 refers to the ktn1-5 mutant.

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The impact of introducing an alignment factor on pollen tube trajectories on WT papilla surface is depicted in Fig 4B. In configurations where pollen tubes previously exhibited significant coiling in the absence of growth guidance (Fig 4B, labels a–c), they now adopt straighter trajectories in the presence of a guiding cue (Fig 4B, labels d–f), reaching the papilla base with minimal turning . Notably, a guidance strength of is sufficient to direct pollen tubes towards the papilla base with few turns (see phase diagram Fig 4C; red-orange colour for most of the initial conditions). For comparison, simulated trajectories and the phase diagram for ktn1-5 papillae are shown in S3A and S3B Fig and confirm that shape differences between WT and ktn1-5 papillae have only a minor role in pollen tube growth behaviour. When comparing the simulated and experimental cumulative distributions of turn numbers, a pollen tube growth model incorporating growth alignment with the longitudinal papilla axis with a strength of 0.1 successfully reproduces the experimental turn number distributions observed on WT papillae (orange squares, Fig 4D). For completeness, cumulative distributions for various alignment strengths on WT papillae are presented in S3C Fig and confirms that an alignment strength greater than 0.025 is consistent with the distribution of turn number experimentally observed on WT papillae. In addition, S3D Fig shows the simulated cumulative distribution of turn numbers for ktn1-5 papillae in the presence of small guidance cues, which confirms that only a very small guidance cue () reproduces the experimental distributions of pollen tube turn on the mutant papillae.

In conclusion, we demonstrated that a guidance cue within the cylindrical region of the papilla, aligning the pollen tube growth direction along the longitudinal axis of the papilla with a sufficient strength (), efficiently directs pollen tube growth towards the papilla base and accurately reproduces the straight tube trajectories experimentally observed on WT papillae.

Another significant outcome of our simulations is that the shape variations observed between WT and ktn1-5 papillae (including neck and head variations, Fig 1H–1K) are not sufficient to globally impact the pollen tube trajectories, regardless of the absence (Fig 3 and S2 Fig) or presence (Fig 4 and S3 Fig) of growth guidance from the papilla side. This conclusion aligns with our previous findings in [6].

Estimation of the growth guidance through the papilla cell wall

We subsequently explored which mechanism could provide the guidance cue in the longitudinal direction. In a very simplified picture, the papilla can be seen as a quasi-cylinder and the pollen tube as a filament progressing within the elastic shell of this cylinder. Following this picture the deformation of the cylinder surface depends on the tube direction, longitudinal vs circumferential (Fig 5A). Indeed, when the tube grows in a longitudinal direction, the cylinder surface is solely strained in the circumferential direction, with no strain along the longitudinal axis (Fig 5A, insert b vs. a) while a tube growing in the circumferential direction induces a surface deformation in both directions (Fig 5A, insert c vs. a). Tube growth therefore entails different energetic costs, with longitudinal growth energetically less costly than circumferential one. We can therefore reasonably assume that this energetic difference might be at the origin of a force on the advancing tube, aligning its growth with the less costly longitudinal direction.

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Fig 5. Model for pollen tube growth guidance assuming a rigidity contrast between the two-dimensional isotropic elastic leaflets of the papilla wall.

(A) The orientation of pollen tube growth (longitudinal vs. circumferential) influences the strain experienced by the papilla cell wall. To facilitate the visualisation of the deformation generated by the pollen tube growth, the papilla surface is represented using a grid of circumferential and longitudinal lines. In contrast to the papilla without tube growth, where the grid is made of squares (square, insert a), longitudinal growth generates a deformation causing expansion of the grid in one direction (grey arrows, insert b, and dashed square for non-deformed state). Circumferential growth results in deformation causing the grid to expand in two directions (grey arrows, insert c, dashed square for non-deformed state) The colours represent schematically the local strain energy in the papilla cell wall, f_lg and f_ci, for longitudinal and circumferential growth, respectively. (B,C) Mechanical model of pollen tube growth within the WT (B) or the ktn1-5 (C) papilla cell wall. The pollen tube separates and deforms the cell wall bilayer with an outer (inner) Young’s modulus Y_out (Y_in) and exerts volume work against the papilla pressure p. The shape of the deformation cross-section is approximated by two half-ellipses with the indentation ratio corresponding to the ratio between the external (r_out) and internal (r_in) papilla deformation. for pollen tube growing within the WT papilla cell wall and for pollen tube growing within ktn1-5 papilla cell wall. (D) Relation between inner and outer cell wall rigidities for a given value of (WT) and (ktn1-5) for growth in the circumferential (circ.) direction compared to the longitudinal (long.) direction. In the shaded region, the rigidity of the outer cell wall layer Y_out is lower than the rigidity of the inner cell wall layer Y_in which corresponds to a negative rigidity contrast , otherwise the rigidity contrast is positive . Note that both tube growth directions require a similar rigidity contrast and differences in the indentation ratio are experimentally probably not detectable. For further details see S1 Text. (E) Adimensional alignment strength for WT () and ktn1-5 () papilla cells depending on the effective cell wall stiffness . The label ktn1 in (D) refers to the ktn1-5 mutant.

https://doi.org/10.1371/journal.pcbi.1013077.g005

Hence, why does a pollen tube growing on the ktn1-5 papilla, which bears a similar quasi-cylindrical geometry to WT papillae, not align with the longitudinal papilla direction? Interestingly, in our previous work [6] we found that the mechanical properties of WT and ktn1-5 papilla cell walls diverged and proposed that these differences could play a key role in pollen tube trajectory. Therefore, here, we investigated whether the mechanical properties of the papilla cell wall, in conjunction with the cylindrical papilla shape, could serve as an effective guidance mechanism for directing pollen tube growth. We considered two main hypotheses. In the first hypothesis, the papilla cell wall is supposed to be mechanically isotropic, but with different rigidities for inner and outer leaflets (Rigidity contrast hypothesis). In the second hypothesis we assumed that both wall leaflets have identical rigidity, but are mechanically anisotropic, i.e. rigidity differs depending to the direction within the papilla wall (Anisotropy hypothesis).

Rigidity contrast hypothesis.

In the first set of simulations, we assumed that the papilla cell wall is isotropic, exhibiting uniform rigidity in all directions within the wall. Thereby, we considered that the inner and outer cell wall leaflets are two-dimensional isotropic elastic sheets characterised by the Young’s moduli Y_out and Y_in, respectively. In Riglet et al., we noticed that the passage of the pollen tube deforms the two leaflets of WT and ktn1-5 papillae cell wall differently [6]. We quantified this deformation (Fig 6B–6D in [6]) and found that in WT papillae, the pollen tube growth results in an almost equal deformation of the cell wall leaflets towards the interior and exterior of the papilla, whereas in ktn1-5, the pollen tube deforms the outer cell wall layer to a greater extent than the inner one. Here, we used this deformation quantification to calculate an indentation ratio corresponding to the ratio between the external (r_out) and internal (r_in) papilla deformation. Pollen tube growth on WT and ktn1-5 papilla results in an indentation ratio (Fig 5B) and (Fig 5C), respectively. We leveraged this experimentally determined ratio to approximate the outer cell wall stiffness (Y_out) relative to the inner cell wall stiffness (Y_in) (for details see S1 Text). In the case of WT parameters (), the Young’s modulus of the outer cell wall layer is consistently higher than the Young’s modulus of the inner cell wall layer (positive rigidity contrast ; Fig 5D). Conversely, for ktn1-5 parameters (), the outer cell wall layer exhibits smaller rigidity than the inner one (negative rigidity contrast ; Fig 5D). This may explain why pollen tubes deform the soft outer layer making ridges well visible on the ktn1-5 papilla surface (Fig 1F and 1G and S4 Fig).

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Fig 6. Role of cell wall mechanical anisotropy in pollen tube guidance.

(A) Schematic representation of the papilla cell wall anisotropy and alignment strength factor . Y_lg (Y_ci) stands for the effective stiffness in the longitudinal (circumferential) direction within the papilla wall. Note that we considered Y_ci (Y_lg) to be the same in the inner and outer papilla wall leaflets. When , is negative, the longitudinal direction is softer (narrow blue arrow) compared to the circumferential one (large blue arrow). When , is positive, the circumferential direction is softer compared to the longitudinal one. A positive value of the alignment strength factor favours a circumferential pollen tube growth whereas a negative value of favours a longitudinal growth. We explored how the anisotropy can influence the alignment strength factor and hence the pollen tube growth direction (question mark). (B,C) Indentation ratio (upper panels) and alignment strength factor (lower panels) for circumferential (circ, solid line) and longitudinal (long, dashed line) tube growth depending on the papilla wall mechanical anisotropy and the effective rigidity of the papilla wall (, B: , C). The calculations were done using the ratio for ktn1-5 geometry (m/m, depicted in blue, B) or for WT geometry (m/m, depicted in orange, C). The grey lines correspond to the experimentally measured indentation ratios on ktn1-5 and on WT papillae. Additional calculations for ktn1 geometry with rigid cell wall, WT geometry with soft cell wall and intermediate wall rigidity () are provided in S1 Text. The dashed red line highlights the isotropic case ().

https://doi.org/10.1371/journal.pcbi.1013077.g006

Next, we utilised the estimated Young’s moduli Y_out and Y_in to estimate the strain energy density (f) resulting from the stretching of the cell wall due to the tube’s passage. Given that the orientation of pollen tube growth (longitudinal vs circumferential) impacts the strain encountered by the papilla cell wall (Fig 5A and as outlined above), we specifically estimated the increase in strain energy induced by the pollen tube growth in the longitudinal (f_lg) and circumferential (f_ci) direction (for details see S1 Text).

Finally, the alignment factor acting on the pollen tube tip depends on the difference in strain energies between circumferential and longitudinal tube growth. Indeed, can be estimated as follows: where signifies a typical length scale (such as the pollen tube radius), and represents the bending rigidity of the pollen tube (for more details, see S1 Text). A positive value of indicates an orientation of the pollen tube growth to align with the longitudinal papilla axis, whereas a negative value of favours circumferential tube growth. In our simulations, we found that the alignment factor for WT and ktn1-5 papillae depends on the overall cell wall rigidity ( (Fig 5E). Interestingly, in our previous study we experimentally measured the elastic modulus of the papillae cell wall for WT (MPa) and ktn1-5 (MPa) (see Fig 6F in [6]). Within this range of cell wall stiffness, this corresponds to a positive value of the (Fig 5E) for both WT and ktn1-5 papillae, indicating that the alignment of the tube growth direction with the long papilla axis is favoured in both cases. Our model also showed that the magnitude of the reorientation strength factor depends on the indentation ratio (Fig 5E). Notably, the factor for ktn1-5 papillae with an indentation is lower () compared to WT papillae (), which reaches approximately . Note that in our simulations depicting pollen tube paths with growth guidance (see previous section “Pollen tube growth on WT papillae requires guidance cues from the papilla side”; S3C Fig, a factor within the same range () reproduces the number of pollen tube turns experimentally observed on WT papilla.

Thus, our simulations showed that a alignment factor with a strength comparable to that introduced in our model to accurately reproduce the experimentally observed straight tube trajectories on WT papillae, can emerge within an isotropic papilla cell wall, provided that both cell wall leaflets are sufficiently rigid and exhibit a positive rigidity contrast (i.e. the outer leaflet is more rigid than the inner leaflet).

Anisotropy hypothesis.

In a second set of simulations, we introduced an anisotropy in the stiffness of the papilla cell wall between the longitudinal (Y_lg) and circumferential (Y_ci) direction (see S1 Text for details)

(1)

A positive value of indicates that the papilla cell was is stiffer in the longitudinal direction than in the circumferential direction, whereas a negative value of indicates that the wall is softer in the longitudinal direction compared to the circumferential one (see schematic representation of in Fig 6A). For example, a value of () corresponds to a longitudinal stiffness which is double (half) as strong as the circumferential stiffness. In comparison, a value of () indicates a very weak anisotropy where the longitudinal direction is only about 20 % stiffer (softer) than the circumferential direction. Therefore, we considered that the range of values to define a very weakly anisotropic wall material.

Next, we postulated that the anisotropy and the effective stiffness () are identical in the inner and outer leaflets of the papilla cell wall. We then, explored how the indentation ratio and alignment strength factor vary with the anisotropy for soft (MPa, Fig 6B) and rigid papilla walls (MPa, Fig 6C). See also Fig B in the S1 Text for an intermediate papilla wall rigidity (MPa). To account for the geometrical differences between WT and ktn1-5 papillae, we performed calculations using the experimentally defined ratio of the pollen tube radius to the papilla radius i.e. m/m for WT geometry and m/m for ktn1-5 geometry.

For pollen tubes growing in a soft cell wall, our simulation showed that the indentation ratio varies with cell wall anisotropy (Fig 6B, upper panel). Note that also depends on the direction of growth of the tube, the two extreme cases (circular vs longitudinal pollen tube growth) being represented in Fig 6B (dashed and solid lines, upper panel). The curves shown were obtained using a ktn1-5 geometry (blue colour); the calculation using a WT geometry gives very similar results and are shown in Fig B in the S1 Text. In every case, the indentation ratio is well above 1 meaning a greater papilla deformation towards the exterior of the cell as a soft papilla wall (MPa) poorly resists to the papilla turgor pressure pushing the pollen tube out. The experimentally measured indentation ratio for pollen tube growing in ktn1-5 cell wall () is reproduced for an isotropic or very weakly anisotropic wall, () (Fig 6B, upper panel). We also observed that for this low rigidity value of the papilla wall (MPa) the experimentally measured indentation ratio for pollen tubes growing in WT cell walls () can never be observed.

For rigid cell walls, our simulation showed that the indentation ratio varies only slightly with cell wall anisotropy and does not depend on the main tube growth direction (circular vs longitudinal, Fig 6C, upper panel). The curves shown (orange colour) were obtained using the WT geometry; calculation with the ktn5-1 geometry yielded qualitatively similar results (Fig B in the S1 Text). The calculated value, which is slightly above 1, reproduced the papilla deformation experimentally measured for pollen tubes growing in WT cell walls (), but failed to replicate the deformation measured in ktn1-5 cell walls ().

Overall, the indentation ratio for pollen tube growth observed in the ktn1-5 wall () can only be reproduced in our model assuming isotropic, or very weakly anisotropic, soft cell wall. It can be noticed that to reach the value a significant decrease of the effective stiffness of the papilla cell wall (MPa) is required (see also Fig B in the S1 Text). By contrast, the indentation ratio experimentally measured for pollen tube growth in the WT wall () can only be reproduced in our model assuming a rigid cell wall, independently of the wall anisotropy.

For both soft and rigid cell walls, our simulations showed that the alignment strength factor varies quasi-linearly with the cell wall anisotropy (Fig 6B and 1C, lower panels). For soft cell walls (Fig 6B, lower panel), a positive value, indicating that pollen tube growth aligns with the longitudinal papilla axis, is only achieved for a positive value of the anisotropy (stronger rigidity in the wall longitudinal direction, see schematic representation of and in Fig 6A). For a rigid cell wall (Fig 6C, lower panel), a positive value can also be achieved for weakly negative anisotropy values of (), indicating that a rigid cell wall with slightly greater stiffness in the circumferential direction than in the longitudinal direction can support longitudinal pollen tube growth. However, it is worth noting that for these small negative values (), the guiding factor remains below 0.025. Overall, our simulation showed that to reach a sufficient alignment strength, consistent with the straight tube trajectories experimentally observed on WT papillae (; S3C Fig, requires a positive cell wall anisotropy (), regardless of the cell wall’s rigidity.

Biological material and culture conditions

All Arabidopsis thaliana lines were in the Col-0 background and grown in growth chambers under long-day conditions (16 h light/8 h dark at 21^∘C/19^∘C with a relative humidity around 60 %). ktn1-5 (SAIL343D12) was described previously [6]. All stigmas were analyzed at stage 13 of flower development [31].

Measurement of shape parameters of papilla cells

Stigma were observed with an upright optical microscope Zeiss Axioimager Z1, under bright field, using a Plan-Apochromat 10x objective. Images processing and dimension measurements (Fig 1J and 1K and S2 Table) were performed using the segmented line and the measure tools of the ImageJ/Fiji software [32].

Geometric model of a pin-like surface

In Fig 2, we represented pin-like shapes as surfaces of revolution. These surfaces were formalized by defining a profile curve represented as a NURBS curve [33], being a free real parameter between 0 and 1, that is then rotated around a vertical axis according to the equation

(2)

represents a point at the surface, corresponding to parameters . Here, represents a normalised z coordinate - between 0 and 1 - pointing upward and represents the azimuthal coordinate varying in . The NURBS curve is defined using a set of control points and scales in width and height, respectively, denoted by scalars W and H. Different surface profiles with varying radii at mid-height (referred to as ’the neck’) of the model were obtained by changing the most central control point CP3 (see S3 Table).

Geodesics

Geodesic curves on S are defined by a set of two second order, non-linear, coupled differential equations with (see Refs [9] and S1 Text):

(3)

where are the Christoffel symbols of the second kind computed from the first and second derivatives of the surface equation (2) and from scalar products between these. Using Eqs 3, it is possible to compute geodesic trajectories on the surface from given initial conditions corresponding to some initial position and orientation on the surface. In Fig 2 we used (the point is slightly below the tip) and an initial inclination is corresponding to an angle of downwards with respect to the horizontal. To integrate these equations we used the odeint function of the Python SciPy library [34]. For the simulations of geodesics in Fig2, we used our new software package to simulate L-systems in Riemannian spaces. The information related to this package is documented in Ref [11]. The numerical code was written using the Riemannian-LPy language, available at [35]. Data were visualised using the same software. Fig 2 can be generated with the code in [36] and the control point values and parameters described in S3 Table.

Analytical expressions for virtual papilla shapes

To proceed further with the geometric model of the papilla surface and relate its parameters to geometric quantities that can be measured on the images, we designed a variant of the previous parametric expression of the surface, where the dependency of the papilla radius R on the z coordinate is expressed by an explicit formula with easily interpretable geometric parameters. Note that we introduce here a short notation, i.e. , , , and . As in the previous section, we expressed the rotational symmetry of the papilla surface S using a surface of revolution with cylindrical coordinates as:

(4)

with the papilla long axis oriented along z. However, the radius function R(z) is now given by:

(5)

where the parameters L_h and W_h can readily be measured on the images, (cf. Fig 1F and 1G). Note that the papilla head is described as an ellipsoid. At z = L_h the radius function is continuous up to the second derivative of R w.r.t. z. The shape parameters a₁ and a₂ were adapted to fulfill the following conditions (cf. Fig 1F and 1G)

(6)

(7)

and can be calculated from the following analytical expressions

(8)

(9)

Numerical details of the phenomenological model for pollen tube trajectories

The pollen tube path is discretized into equidistant points , where s denotes the arc length. In the cylindrical region of the papilla [i.e. where the surface can be conveniently parametized in cylindrical coordinates , see Eq 4] the pollen tube path is given by the following ordinary differential equation

(10)

denotes the angle between the tip tangent t and the surface tangent pointing along the long papilla axis

(11)

with denoting the first derivative of the papilla radius w.r.t. z. In the here chosen definition of , an angle () indicates a pollen tube tip oriented towards the papilla base (pole).

The second surface tangent is oriented in the circumferential direction of the papilla surface and forms an orthonormal basis with

(12)

The momentum conservation at the pollen tube tip is given by the following ordinary differential equation and determines the evolution of the angle along the pollen tube trajectory

(13)

where denotes a bending rigidity and where

(14)

denotes the external torque of magnitude m which aligns the growth direction (described by ) with the long papilla axis for z>z_c. Here we chose z_c = A, where the almost spherical cap of the papilla goes over to the cylindrical shaft. Note that expression (14) favors growth along the longitudinal direction. Depending on the angle the preferred growth direction is towards the papilla pole or the papilla base.

Eq 13 can be derived from a variational principle considering the following energy functional for the tube bending energy

(15)

where W denotes the work performed by external forces on the tube tip. Variation of w.r.t. the tube tangent vector t results in

(16)

The bulk term and the boundary term at s = 0 in (16) vanishes since the tube cannot change its position after it has been deposited. With and using cylindrical coordinates we recover Eqs 13 and (14).

To express the alignment strength m compared to the tube bending rigidity we introduce the adimensional growth guidance , with being the typical length scale (here m, corresponding to the pollen tube radius). In the absence of any external torque, i.e. , Eqs 10 and (13) describe a geodesic line on the papilla surface S.

In the head region of the papilla, i.e. the parametrization of the tube path in cylindrical coordinates will fail and we have parametrized the papilla surface in Cartesian coordinates (x,y). Further details on this alternative parametrization as well as on the the implementation of the self-avoidance are given in the S1 Text.

Eq 13 (with self-avoidance) was solved numerically on virtual papilla surfaces representing either WT or ktn1-5 papillae (see previous section) using a custom written C-code [37]. Data were visualized using the gnuplot package (version 5.2 [38]).

Calculation of cumulative distributions of turn numbers

To calculate the cumulative distributions of turn numbers T from morphological phase diagrams (Figs 3 and 4) the turn number of each initial attachment site z₀ is weighted in the cumulative distribution with a factor corresponding to the z-dependent surface area element of the papilla. Trajectories which could not reach the papilla base due to self-avoidance were always counted as trajectories with high turn numbers (T>2.5). Typically, morphological phase diagrams where calculated for increments in the starting angle and the starting position m.

To test to what extend the assumption of an equal probability for pollen grains to attach at the papilla surface up to a height z=2A could flaw the distribution of turn numbers, we tested an alternative method to calculate turn numbers from morphological phase diagrams. There, we used the assumption that pollen grains attach to the papilla surface with a probability derived from the projected area of the papilla head (for ) in the x–y plane resulting in a weight , where R(z) denotes the z–dependent papilla radius in cylindrical coordinates. This method favors attachment at the papilla pole and reduces the contribution of papilla grains attached rather to the side of the papilla head in the cumulative histogram. S3E Fig compares the cumulative turn number distributions for WT ( = 0.1) and ktn1-5 ( = 0) papillae (solid lines similar to Fig 4D) which were obtained with a method using the surface area and the projected surface area as weighing function. The curves (projected and area) are slightly different, but the choice of the weighing function has overall only a small effect on the turn number distribution.