Asymmetries in leaf branch are associated with differential speeds along growth axes: A theoretical prediction
Abstract
Background: Morphogenesis, when accompanied by continuous growth, requires stable positional information to create a balanced shape in an organism. Evenly spaced branches are examples of such morphogenesis. Previously, we created a model that showed when a one-dimensional (1D) ring (a boundary of a 2D field) was periodically deformed based on a stable, doubled iterative pattern during expansion; a nested, regularly spaced, symmetrically branched structure was generated. The characteristic divaricating pattern is common in the leaves of many plant species; however, the divarication symmetry was often broken. To evaluate this type of asymmetry, we investigated several species with dissected or compound leaves. Results: Sometimes these leaves showed asymmetries in the number of lobes or segments positioned on either side of the secondary axes. The direction of the asymmetry, i.e., which side of a secondary axis has more axes, appeared to be species-specific. Conclusions: When different growth speeds along axes of a divaricating leaf were introduced into our previous model, robust and directed asymmetries were reproduced. The differences in growth speed could be predicted from the distributions of leaf segments in actual leaves. Developmental Dynamics 246:981–991, 2017. © 2017 Wiley Periodicals, Inc.
Introduction
Shapes observed in nature are an attractive feature for research, and asymmetry plays a key role in determining shapes. For example, asymmetries along the right–left axis of the body and several organs are widely conserved in plants and animals. Most asymmetries are inevitably established early in development (Wood, 1997; Mercola & Levin, 2001; Chitwood et al., 2012). Asymmetries in branched structures are more complex. Rivers, trees, lungs, and the fingers on a hand all have differing orders of asymmetries. Such branches have been analyzed to improve the understanding of their asymmetries (Horsfield et al., 1982; Prsinkiewicz and Lindenmayer, 1990; Zhu et al., 2010). However, very little is known about the mechanisms by which they are constructed.
Plant leaves provide good models for studying the generation of asymmetry with respect to the diversity of shapes found in nature. When we consider leaf formation, various peripheral structures with different complexities, regularities, and symmetries can be observed. Asymmetries are found at different orders and axes in leaves. For example, the relationship between the direction of a phyllotactic spiral and leaf shape asymmetry along the right–left axis has been studied previously (Chitwood et al., 2012). Leaves of Arabidopsis thaliana and tomato (Solanum lycopersicum) showed bilateral asymmetries in laminar outgrowth and lateral leaflet positions. The asymmetry was explained by a shift in auxin distribution in primordium, and the shift was related to the divergence angle of primordia.
In this study, we investigated the asymmetries in two-dimensional (2D) branch, or divarication of compound and dissected leaves in several plant species. Previous studies have defined leaf formation as a deformation of a boundary of the 2D field depending on a spatially periodic pattern (Bilsborough et al., 2011; Nakamasu et al., 2014). The divarication (branch) pattern of leaves can be explained by this model (Nakamasu et al., 2014). Divaricate leaves were observed in a wide range of plant species and exhibited a variety of shapes. However, the divarication patterns obtained by the model were symmetrical. To explain the asymmetries observed in actual leaves, different growth speeds along axes of a divaricate leaf were introduced into our existing model. These different growth speeds produced robust and directed asymmetries in a new model.
To evaluate the asymmetry, we investigated the divaricate leaves of several species. We focused on the asymmetries in the numbers of axes (segments) positioned on either side of a secondary axis. The direction of asymmetry, that is, the side with more segments, was considered to be species-specific. Degrees of asymmetry were theoretically estimated and compared with those observed in actual leaves. The predicted difference in growth speeds from the distribution of axes in a leaf was well matched with the observed direction of asymmetry
Results
General Asymmetries Observed in Divaricate Leaves
The 2D branches or divarications observed in both dissected and compound leaves appeared to have common characteristic. Therefore, for this study, we refer to them collectively as divaricate leaf. Several axes of different orders may be present in a divaricate leaf: primary, secondary, and tertiary (Fig. 1A); higher orders are also encountered. In a divaricate leaf, asymmetries were often present. The asymmetry primarily described in this study is the difference in the numbers of higher-orders of axes (segments) on either side of a secondary axis (Fig. 1A). We investigated this pattern in the leaves of a wide variety of species from 21 families. We then selected 20 species from 11 families and grew them in the laboratory (Table 1).
General features of divaricate leaves in several plants. A–C: Schematic of a divaricate leaf (A) and examples of asymmetries in the numbers of tertiary axes along a secondary axis of a leaf (B,C). A: Orders of axes are primary, secondary, or tertiary. The direction of asymmetry differed among the plant species shown here. B: Senecio cineraria had more tertiary axes that were basal with respect to the primary axis (numbers in filled circles) than apical (numbers in open circles). C: This type of asymmetry has a basal direction. Lavandula pinnata leaves exhibited mixed directions of asymmetry. In this type, apical and basal directions of asymmetry were observed in a single leaf. D,E: The particular directionality of asymmetry observed in divaricate leaves throughout the plant life cycle was invariable (D) or variable (E). D,E: The direction of asymmetry did not change in leaves of a single Rorippa aquatica plant (D), but did change in leaves of a Scabiosa columbaria “Nana” plant (E). Scale bars = 2 cm.
| Species | Family | Direction of asymmetry | Figure | |
|---|---|---|---|---|
| 1 | Anthriscus cerefolium | Apiaceae | Basal | |
| 2 | Anethum graveolens | Apiaceae | Basal | Figure 2A |
| 3 | Daucus carota | Apiaceae | Basal | |
| 4 | Asplenium sarelii | Aspleniaceae | Apical | Figure 2G |
| 5 | Artemisia absinthium | Asteraceae | Basal | Figure 2B |
| 6 | Cosmos sulphureus | Asteraceae | Apical | Figure 2H |
| 7 | Matricaria recutita | Asteraceae | Basal | |
| 8 | Senecio cineraria | Asteraceae | Basal | Figure 1B |
| 9 | Rorippa aquatica | Brassicaceae | Basal | Figure 1D |
| 10 | Scabiosa columbaria ‘Nana’ | Dipsacaceae | Basal > apical | Figure 1E |
| 11 | Scabiosa caucasica ‘Perfecta’ | Dipsacaceae | Basal > apical | |
| 12 | Perovskia atriplicifolia | Lamiaceae | Basal | |
| 13 | Lavandula multifida | Lamiaceae | Apical + basal | |
| 14 | Lavandula pinnata | Lamiaceae | Apical + basal | Figure 1C |
| 15 | Nandina domestica | Berberidaceae | Almost equal | Figure 5A |
| 16 | Malva moschata | Malvaceae | Basal > apical | |
| 17 | Eschscholzia californica | Papaveraceae | Basal | Figure 2C |
| 18 | Consolida ambigua | Ranunculaceae | Basal | Figure 2D |
| 19 | Nigella sativa | Ranunculaceae | Basal | Figure 2E |
| 20 | Ruta graveolens | Rutaceae | Basal | Figure 2F |
- a Twenty species from 11 families were selected to grow in the laboratory, with little duplication among families.
In most species, almost all secondary axes had the same direction of asymmetry in a single leaf, with more tertiary axes on the left or right side of the respective secondary axis. With respect to the primary axis, these asymmetries in the distribution of tertiary axes are apical or basal. In Figure 1B,C, tertiary axes that are apical distribution are indicated by a number in an open circle, while those that are basal are indicated by a number in a filled circle. Figure 1B shows a basal asymmetry, while Figure 1C shows a mix of apical and basal asymmetries in a leaf of Lavandula pinnata. This plant always shows this type of mixed phenomena. Therefore, we concluded that the directionality of asymmetries was species-specific.
Leaf shapes often change according to a plant's life stage. Heteroblasty, or maturation-related changes in plant form, is often observed in many species (Zotz et al., 2011). Rorippa aquatica, which is in the family Brassicaceae, develops more complex dissected leaves as the plant matures (Bharathan, 2002; Nakayama et al., 2014). However, the characteristic type of asymmetry direction (basal with respect to the primary axis) was approximately maintained throughout the plant's life (Fig. 1D). In contrast, some plants, such as Scabiosa columbaria “Nana” in the family Dipsacaceae, had a shift in the direction of asymmetry (Fig. 1E). In the vegetative phase, S. columbaria had secondary axes with more tertiary axes toward the basal end of the primary axis. However, in the reproductive phase, an apical direction was observed (Fig. 1E). This reinforced our conclusion that the directional distribution of asymmetries was species-specific, whether it was variant or invariant throughout the life span of the plant.
Not only were there differences in the number of segments on both sides of each secondary axis; asymmetry was also commonly observed in the positions of the tertiary axes on a secondary axis. The tertiary axes on the side with more axes tended to be positioned lower (i.e., toward the proximal end of the secondary axis) than the corresponding axis on the opposite side, as illustrated in Figure 1A and represented in Figure 1B.
Calculations of Degrees of Asymmetry in Divaricate Leaves That Were Expected to Grow Unidirectionally Toward the Apical End
We selected divaricate leaves that had the secondary axis with the highest number of segments at the basal end of the primary axes (Fig. 2). Such secondary axis was expected to be the oldest. According to our previous model, which explained the formation of a divaricate (branched) leaf (Nakamasu et al., 2014), a developmental profile of divarication could be predicted from the distribution of axes in a developed leaf. In the selected plants, the number of axes monotonically decreased toward the apical end of the leaf. Therefore, the leaves of plants were expected to have grown toward the apical end of the primary axis, adding newer secondary axes at the apical end. Because such secondary axes were hypothesized to develop unidirectionally along the primary longitudinal axis, they were effective cases for examining divarication formation.
Examples of divaricate leaves used in this study. Divaricate leaves from eight species are shown in silhouette. A–H: Leaves with a basal direction of asymmetry are indicated by Anethum graveolens (A), Eschscholzia californica (B), Consolida ambigua (C), Nigella sativa (D), and Ruta graveolens (E). Leaves with an apical direction of asymmetry come from Cosmos sulphureus (F) and Asplenium sarelii (G). Nandina domestica (H) has almost symmetrical secondary axes. Scale bars = 2 cm.
We collected data from nine species with different types of divarication (Table 1). Several of these species are shown as silhouettes in Figure 2. Anethum graveolens (Fig. 2A), Eschscholzia californica (Fig. 2B), Consolida ambigua (Fig. 2C), Nigella sativa (Fig. 2D), and Ruta graveolens (Fig. 2E) had more segments with a basal distribution, while Cosmos sulphureus (Fig. 2F) and Asplenium sarelii (Fig. 2G) had more apical segments. However, Nandina domestica (Fig. 2H) had roughly symmetrical divarications, with approximately the same number of leaflets on either side of the secondary axes.
We calculated the degree of asymmetry on either side of the secondary axis by subtracting the number of apical segments from those with basal ones on a single secondary axis. The secondary axes on a leaf had different degrees of asymmetry; therefore, we plotted them against the total number of axes along a single secondary axis. Plants that had a basal direction of asymmetry showed positive degrees of asymmetry, and broadly increasing distributions along the x-axis can be obtained. Plants with an apical direction of asymmetries showed negative and decreasing distributions along x-axes (Figs. 3A, 4G,H). Therefore, the absolute values of the degrees of asymmetry roughly increased when the total numbers of axes on a secondary axis increased.
Schematics of analyses of degrees of asymmetry in divaricate leaves. A–F: An example of degrees of asymmetry observed in Artemisia absinthium leaves (A) and theoretical explanations (B–F). A: The y-axis shows the degree of asymmetry, i.e., the number of segments with an apical distribution subtracted from the number with a basal distribution on a single secondary axis. A higher value indicates greater asymmetry. The x-axis shows the total number of segments along a single secondary axis. The z-axes show a ratio of the secondary axes with the same degree of asymmetry in each y column. B: The table shows a method to estimate the degrees of asymmetry of secondary axes, as derived from the recurrence relationship. The graph shows the theoretically predicted development of the number of axes over time (C). The dotted line indicates faster timing of divarication than the solid line. The difference (gray zone) in the timing of intermitted increases generates the specified degree of asymmetry. D: Theoretically estimated degrees of asymmetry showed an increasing zigzag pattern. E: Another graph shows the simulated increase in segments on a particular side of a secondary axis over time when the ends of a line moved at different speeds. F: Simulations also indicated increasing zigzag patterns of degrees of asymmetry at various speeds.
Degrees of asymmetry observed in divaricate leaves of several plant species. Degrees of asymmetry on either side of secondary axes, shown as graphs. The y-axes show the degree of asymmetry. A higher absolute value reflects greater asymmetry. The x-axes show the total number of segments on each secondary axis. The z-axes show a ratio of the secondary axes with the same degree of asymmetry in each y column. Plants with a basal (A–F) or apical (G, H) distribution of tertiary axes showed increasing and decreasing degrees of asymmetry, respectively, with broad distributions. A–G: The species are Ruta graveolens (A), Nandina sativa (B), Consolida ambigua (C), Eschscholzia californica (D), Anethum graveolens (E), Cosmos sulphureus (F), and Asplenium sarelii (G). The solid lines show a theoretically estimated shift in degrees of asymmetry.
Simulations of axis formation provided by differential speeds of growth along axes. A,B: A schematic of a model (A) and a generated structure (B) are presented. A: An initial 1D leaf margin composed of line segments, shown as cells expanded depending on the concentration of component u of a reaction-diffusion system, which generated a spatial-iterative pattern. B: A generated leaf started from a ring as an initial condition. The marginal segments were moved outward at the same speed for each growth point in which u was abundant. A: The segments were divided into two to grow when the length reached a threshold. B: In this example, symmetrical axes were obtained. C: A schematic of analysis for asymmetrical axis formation is presented. We considered unidirectional growth along the primary axis of a leaf primordium in the apical direction and focused on axes to the right side of the primordium. D–K: Asymmetric and symmetric axes were generated using different growth speeds along growth axes. When the apical end grew faster than other growth points, the apical side of the secondary axes had more segments (D); a close-up depiction of the asymmetry on either side of the secondary axis indicated with an arrowhead in D (E); when all growth axes grew at the same speed, symmetrical secondary axes were obtained as observed in B (F); a close-up of the axis in F indicated with an arrowhead (G). When the apical end grew more slowly than other growth points, the basal side of the secondary axis had more segments (H); a close-up of the asymmetrical axis in H indicated with an arrowhead (I); even if the apical end of a leaf grew faster than other growth points, when apical growth was stopped suddenly, secondary axes continued to grow, and the apical side of secondary axes also produced more segments (J); a close-up of the asymmetrical axis in J indicated with an arrowhead (K).
Asymmetric Axis Distributions Were Explained by an Equally Spaced Pattern With Different Growth Speeds
In the previous model for divaricate-leaf morphogenesis (Nakamasu et al., 2014), a 1D boundary domain on a 2D plane ( = leaf margin) was expanded depending on a spatially iterative pattern (Fig. 5A). A divaricate (branched) pattern was generated when new growth points were inserted at equal intervals as the leaf margin enlarged (Fig. 5B). Symmetrical structures were observed when the boundary was equally deformed at each growth point (i.e., component of the spatial pattern; Fig. 5B). In this case, the number of growth points intermittently increased with synchronous timing (Crampin et al., 2002a, 2002b). Conversely, asymmetrical divarications were generated when growth points moved at different speeds. The timings of the increments were desynchronized even if the pattern behind it was equally spaced (Crampin et al., 2002b)
To explain these asymmetric divarications, we focused on unidirectional growth during leaf morphogenesis. For example, unidirectional growth occurring in an apical direction in a R. aquatica leaf primordium is shown in Figure 5C. To represent this pattern, simulations began as straight, connected line segments (unit cells) as the primary condition. In this case, the x-axis indicated the primary axis. The apical end (on the left) moved toward the left along the x-axis (Fig. 5C), and the right end moved vertically along the y-axis to eliminate the boundary effect. The speeds of both ends were approximately equal to other growth points. In this case, a symmetrical divarication was generated again (Fig. 5F,G), similar to the isotropic growth from a ring (Fig. 5B).
When the growth speed of the apical end (i.e., unidirectional growth of the primary axis) was relatively faster (approximately two times faster) than that of the other growth points, the faster (apical) sides of secondary axes had more axes (Fig. 5D,E). Therefore, an apical direction of asymmetry can be obtained autonomously. Conversely, when the growth speed of the primary axis was relatively slower (approximately 1/2 as fast) than that of other growth points, the slower (apical) sides of secondary axes generated fewer axes (Fig. 5H,I), indicating a basal distribution of asymmetry. Moreover, simulated divarication patterns showed the asymmetric positions of tertiary axes mentioned previously (Figs. 1B, 5).
Degrees of Asymmetry in Leaf Axes Can be Theoretically Estimated
According to our previous study (Nakamasu et al., 2014), the number of higher-orders of axes on each secondary axis were predicted to increase intermittently. For example, the n th newest secondary axis would have
axes, i.e., 1, 1, 3, 5, 11… axes, so the axes on one side also increased intermittently in a
manner, i.e., the apical point was subtracted and the remainder was divided by two (Fig. 3B). The timing of increase on the faster-growing side became faster because the neighboring axis was generated at a shorter interval than on the other side (Figs. 3B,C, 5). This time lag was expected to generate specific asymmetry in the numbers of tertiary axes observed in actual leaves. The specific degrees of asymmetry could be estimated (Fig. 3D) by subtracting the theoretical number of axes on the slower side from that of the faster side (i.e., represented by the gray region in Figure 3C and C(n−2) as a formula in Fig. 3B). When the slower side caught up to the faster side, the asymmetry disappeared. Therefore, a zigzag increase in the degree of asymmetry was obtained theoretically.
The simulations produced similar results (Fig. 3E,F). In a simulation such as that shown in Figure 5, when the ends of a line moved at different speeds, the intermittent increase in higher-orders of axes on both sides of the secondary axes were out of step (Fig. 3E). The calculated degrees of asymmetry from simulation showed an increasing zigzag pattern with fluctuations (Fig. 3F). Even if the difference between speeds was varied (two or three times faster at one end), the zigzag pattern persisted as the total number of axes on secondary axes increased (Fig. 3F). When the estimated line was compared with actual data, the increasing or decreasing distributions corresponded to the zigzag pattern with fluctuations. Almost all degrees of asymmetry in divaricate leaves were included in this range (Figs. 3A, 4).
Differential Growth Speeds Were Predicted by the Distribution of Axes in Mature Leaves
The relative difference in growth speed along each axis could be theoretically predicted by comparing the distribution of axes to that obtained from the model. According to the model, the distribution of axes along the primary axis could be estimated by a recurrence formula (Nakamasu et al., 2014), as mentioned above. Nandina domestica had symmetrical secondary axes (Fig. 6A,B) and showed similar distributions of axes as predicted by the simple theory, i.e., all axes grew at the same speed (Fig. 5F). It was predicted that all axes of a leaf might grow at equal speed in N. domestica.
Distribution of axes in developed leaves. Distributions of axes in developed leaves indicated different growth speeds along axes. A: Symmetrical axes observed in leaves of Nandina domestica. A solid arrow indicates the primary growth axis; open arrows show secondary axes. B: The first graph shows a comparison of distribution curves for axes that were theoretically obtained (solid green line) and measured (dashed black line) in N. domestica leaves. If the primary axis grew faster than the secondary axes, the distribution curve would be below (solid arrow) the theoretical estimate (solid green line), and if secondary axes grew faster than the primary axis, the distribution curve would be above (open arrow) the theoretical estimate. C: The second graph shows the distribution curves of axes obtained from the leaves of several plant species. Open circles and triangles represent data from leaves with basal direction of asymmetry, which tended to be above the theoretical curve. Filled symbols represent data from leaves with apical direction of asymmetry, which tended to be below the theoretical curve.
The number of higher-orders of axes along each secondary axis in the actual mature leaves of different plant species was plotted with respect to their position relative to the primary axis (Fig. 6C). If the primary axis grew faster than the secondary axes, the distribution curve would be below the theoretical line (Fig. 6B). In contrast, if secondary axes grew faster than the primary axis, the distribution would be above the theoretical prediction (Fig. 6B).
Anethum graveolens and E. californica tended to be above theoretical values, while C. sulphureus and Asplenium sarelii tended to be below the theoretical value (Fig. 6C; Table 2). The directions of asymmetry and the expected differences in the speeds of growth axes were consistent with predictions obtained from the theoretical model.
| Position of secondary axes from apex of primary axis | ||||||||
|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| Asplenium sarelii | 1.0 | 1.0 | 2.0 | 2.3 | 3.7 | 6.1 | 7.5 | 11.0 |
| Cosmos sulphureus | 1.0 | 1.0 | 2.4 | 4.9 | 8.0 | 14.3 | ||
| Nandina domestica | 1.0 | 1.1 | 2.9 | 5.2 | 8.9 | 18.2 | 44.6 | 73.0 |
| Artemisia absinthium | 1.6 | 4.3 | 9.2 | 15.8 | 22.8 | 25.2 | ||
| Anethum graveolens | 1.0 | 1.6 | 4.8 | 8.8 | 12.8 | 19.0 | ||
| Consolida ambigua | 1.3 | 4.0 | 8.4 | |||||
| Eschscholzia californica | 1.0 | 1.9 | 3.8 | 6.7 | 13.3 | 19.0 | ||
| Nigella sativa | 1.1 | 2.2 | 4.3 | 6.0 | 8.4 | |||
| Ruta graveolens | 1.3 | 4.3 | 7.1 | 8.8 | ||||
- a Nine species with asymmetric or symmetric leaf divarication structures were examined. Table shows the average of total number of segments on each secondary axis. Blue cells show the values below the theoretical value; red cells show values above the theoretical value. Asplenium sarelii and Cosmos sulphureus had more segmented axes that were apically, not basally, distributed with respect to the primary axis. Nandina domestica had almost symmetrical axes. Anethum graveolens, Artemisia absinthium, Eschscholzia californica, Consolida ambigua, Nigella sativa, and Ruta graveolens had more segments that were basally, not apically, distributed with respect to the primary segments. The number of axes is shown according to the position from the apex of the primary axis. Values above or below the theoretical values are indicated in blue or red, respectively.
Additional Modifications Explain Other Features of Leaf Morphogenesis
In actual leaves, several variations from the simplified condition are expected. One such modification is an arrest of growth, observed particularly in simple leaves (Donnelly et al., 1999; Kazama et al., 2010; Andriankaja et al., 2012). It was assumed that the arrested timing of each growth axis might be different in an actual leaf. The asymmetry generated in the early stage of axis formation remained constant, even if the primary axis stopped growing, while the other axes continued to grow (Fig. 5J,K). In this case, the distribution of axes would not reflect the actual growth speed.
Moreover, asymmetries could also be observed along higher-orders axes (Fig. 7). For example, tertiary axes with asymmetry showed more quaternary axes toward the proximal end of the secondary axis. This asymmetry might also be explained by similar differences in growth speeds shared by higher-orders than primary axis.
Asymmetries observed in higher-orders and tertiary axes. Asymmetries existed at a higher orders than tertiary axes in Rorippa aquatica. Upper pictures show a series of nodes from a single leaf, arranged left to right from the basal to apical end. The lower right panel shows a close-up of a secondary axis. Tertiary, quaternary, and quinary axes are shown as solid, dashed, and dotted lines, respectively. The number of quaternary axes differed on each side of the tertiary axis.
Discussion
In this study, asymmetry and the distribution of axes in actual plant leaves were analyzed. Their origination was explained using a model, and the observed asymmetries in axis distributions were consistent with theoretical predictions. Thus, we provide evidence for the involvement of an equally spaced periodic pattern in 1D leaf margins during the formation of axes in leaves of several different plant species (Fig. 1). The asymmetry in the number and position of segments on secondary axes observed in leaves (Fig. 3A) could be explained by the pattern-dependent deformation of the leaf margin described in a previous model (Nakamasu et al., 2014), in combination with a difference in the growth speeds of several axes.
It is thought that branching or divarication observed in nature reflects either intentionally or randomly developed asymmetry. In leaves, the observed asymmetries are inevitable. The L-system is one of the most famous models for plant morphogenesis and explains beautifully divaricate leaf structures (Prusinkiewicz and Lindenmayer, 1990). However, directed asymmetries of each segment could not be explained without the interactions of growth points along the leaf margin.
As previously described, only the growth at each iterative point in the 1D domain will generate nonuniform growth (Crampin et al., 2002b). Nonuniform growth does not cause asymmetry of the growth point distribution; however, nonuniform growth over each peak breaks the synchrony of the growing points, which leads to an asymmetric distribution of new growth points. This desynchronization primarily works in the formation of asymmetric axes. Such spatial patterns, which are used in models for various morphogenetic events (Meinhardt, 1982; Harrison & Kolář, 1988; Bilsborough et al., 2011), must have stable positional information against disturbances, such as continuous growth. In plants, the generation of iterative patterns has been characterized at a molecular level (Hay et al., 2006; Kawamura et al., 2010; Bilsborough et al., 2011; Kasprzewska et al., 2015; Tameshige et al., 2016).
The mechanism that changes the speed of each growth point was predicted from biological studies. The progression of the “cell cycle arrest front” (Donnelly et al., 1999; Nath et al., 2003; White, 2006; Andriankaja et al., 2012) contains positional information that might change the growth speed at each point. Leaf primordia of A. thaliana have an absolute gradient of cell proliferation along the apical–basal axes. The boundary for the shift from cell proliferation to cell expansion moves from the leaf tip toward the base during development (Donnelly et al., 1999; Ferjani et al., 2007; Kazama et al., 2010; Andriankaja et al., 2012). Many other simple leaves also show growth polarity (Das Gupta and Nath, 2015). In addition to the simple leaf, the dissected leaves in R. aquatica also shows a proliferation gradient (Nakayama et al., 2014). Many candidate genes or molecules required for this mechanism have been predicted (Nath et al., 2003; Palatnik et al., 2003, 2007; Schommer et al., 2008; Rodriguez et al., 2010; Andriankaja et al., 2012; Kawade et al., 2013; Das Gupta and Nath, 2015).
Finally, we described modifications to our simple theoretical model with different growth speeds. Divaricate-leaf morphogenesis was expected to continue until a specific developmental stage. The arrest of growth was considered independently along each axis, and each growth rate was different. The growth gradients along axes other than the primary axis can be predicted from the asymmetries observed on either side of the tertiary axes. Therefore, it is difficult to accurately predict differences in growth speed only from the distribution of axes in a mature leaf. More investigation will be needed to elucidate the details of differences in growth speed. Moreover, some leaves with divarications could not be explained by this simple modification. Ferns such as Pteris excelsa var. simplicior also show asymmetric divarication that does not follow this theoretical model. Sometimes a modification of leaf shape in primary and secondary morphogenesis was observed in mature leaves (Bharathan, 2002). This might be explained by other theoretical models; however, with modifications to the theory described here, almost all divarication and observed asymmetries can be explained.
Experimental Procedures
Plant Collection and Growth Conditions
Selected plants were collected from the field in Japan and grown in the laboratory (Table 1). Senecio cineraria was collected from flowerbeds at the Ikuta Campus of Meiji University, Tokyo. Nandina domestica was collected from the backyard of a private house in the Kita-Kyusyu area. Asplenium sarelii was collected from a stone wall at the Campus of Kyoto Sangyo University, Kyoto. Rorippa aquatica was planted in the Kimura lab at Kyoto Sangyo University, Kyoto. Seeds or plants were also obtained from several local shops in Japan. Seeds of Cosmos sulphureus were produced by TOHOKU. Lavandula multifida was bought at ING-NOMORI. Matricaria recutita was produced by TANENOTAKII and bought at TANEGEN. Anthriscus cerefolium, Daucus carota, Artemisia absinthium, Consolida ambigua, Nigella sativa, and Ruta graveolens were obtained from Mikasa seeds; Lavandula pinnata and Malva moschata were plants obtained at e-tisanes. Perovskia atriplicifolia, Scabiosa columbaria “Nana,” and Scabiosa caucasica “Perfecta” were purchased at the Ogihara botanical garden. Eschscholzia californica was kindly provided by Momoko Ikeuchi at RIKEN, Yokohama. These plants were grown in a room with continuous light at 22 °C for approximately 2 months before use for experimentation and analysis.
Analysis of Divaricate Structures
Thirteen leaves from 5 Asplenium sarelii plants, 44 leaves from 4 Cosmos sulphureus, 72 leaves from 7 Artemisia absinthium, 22 leaves from 4 Anethum graveolens, 72 leaves from 4 Consolida ambigua, 78 leaves from 6 Eschscholzia californica, 39 leaves from 3 Nigella sativa, and 25 leaves from 2 Ruta graveolens plants were collected for analysis. The axes were counted and several leaves were scanned to be presented as representative examples.
Degrees of asymmetry on either side of the secondary axis were obtained by subtracting the number of basal segments from those that were apical with respect to the primary axis. These values were plotted against the total number of axes along each secondary axis. The distribution of segments with respect to the primary axis was also plotted.
Simulation of Asymmetric Divarication
The simulations were performed using the pattern-dependent expansion method described by Nakamasu et al. (2014). Briefly, the boundary propagation method (Sethian, 1996), i.e., the propagation of the leaf margin in space over time, was used. This propagation was performed iteratively by updating the connection points of arbitrary segments, regarded as a “cell” (but not a real cell). The connection points (
) of adjacent
th and
th unit cells are displaced at velocities
;
was promoted by reactant u as
, and
was a unit vector of propagation direction. The summation of the normal vectors for the margin of unit cells
points upward (or outward) with respect to line segments as the initial condition. The terms
indicated the amounts of reactant u in unit cells i and j, respectively. The coefficient
determined the effect of the reactant. A unit cell divides into two daughter cells when its length exceeds a threshold (
) by updating positions
. The two daughter unit cells inherited the reactant state of the mother. To improve the visualization of the structure of divarication, isometric growth (A, Nakamasu, unpublished 2017) was included. When
decreased as
, pointed divarication was obtained.
For the Turing pattern to maintain the spatial intervals, insertion or splitting of peaks can be selected as domain growth (Meihardt, 1982, 1995; Crampin et al., 2002a). The parameters were set as the case of insertion (Nakamasu et al., 2014). The parameters used in the simulation were
,
,
,
,
,
,
,
,
,
, and
.
To realize different speeds of growth along different orders of axes, one end of the boundary was moved in the x direction approximately 1/2, (1), 2, or 3 times faster than the other growth points. The other end was moved in the direction of the y-axis at the same speed as other growth points to realize the pseudo-infinite boundary condition. The critical parameter of “same speed” was estimated from the case in which the number of insertions was equal to an intact branch. During the calculation, each end of the reactant u was forced to zero to avoid interference from other peaks.
Acknowledgments
We thank Dr. Momoko Ikeushi, Dr. Ryo Ohtsuki, Dr. Hokuto Nakayama, Dr. Kaoru Okamoto, Saori Miyoshi, and the lab members of Prof. Takasi Miura for helpful discussions. A.N. was supported by a Grant-in-Aid for JSPS KAKENHI Grant Number 12J10320 and A.N. and N.J.S. were funded by the Meiji University Global COE program “Formation and Development of Mathematical Sciences Based on Modeling and Analysis.” In addition, S.K. was partially supported by a Grant-in-Aid for Scientific Research on Innovative Areas and a MEXT-Supported Program for the Strategic Research Foundation at Private Universities.
