Principal growth directions in development of the lateral root in Arabidopsis thaliana

Abstract

Background and Aims

During lateral root development a new meristem is formed within the mother root body. The main objective of this work was to simulate lateral root formation in Arabidopsis thaliana and to study a potential role of the principal directions in this process. Lateral root growth is anisotropic, so that three principal directions of growth can be distinguished within the organ. This suggests a tensorial character of growth and allows for its description by means of the growth tensor method.

Methods

First features of the cell pattern of developing lateral roots were analysed in A. thaliana and then a tensorial model for growth and division of cells for this case was specified, assuming an unsteady character of the growth field of the organ.

Key Results

Microscopic observations provide evidence that the principal directions of growth are manifested at various developmental stages by oblique cell walls observed in different regions of the primordium. Other significant features observed are atypically shaped large cells at the flanks of young apices, as well as distinct boundaries between the mother root and the primordium. Simulations were performed using a model for growth. In computer-generated sequences the above-mentioned features could be identified. An attempt was made to reconstruct the virtual lateral root that included a consideration of the formation of particular tissue types based on literature data.

Conclusions

In the cell pattern of the developing lateral root the principal directions of growth can be recognized through occurrence of oblique cell divisions. In simulation the role of these directions in cell pattern formation was confirmed, only when cells divide with respect to the principal directions can realistic results be obtained.

INTRODUCTION

The process of the lateral root (LR) development in Arabidopsis thaliana is – to a high degree – repeatable and predictable (Malamy and Benfey, 1997; Laskowski et al., 1995; Dubrovsky et al., 2006, 2008; Casimiro et al., 2001, 2003). Laskowski et al. (1995) stated that the new LR primordium is unable to function as an independent root meristem until it consists of at least three to five cell layers, and then it becomes able to develop into an LR. According to Malamy and Benfey (1997) LR formation begins with anticlinal divisions of a group of neighbouring pericycle founder cells. Subsequently, periclinal divisions of the cells occur and thus two layers of a new primordium are formed: the inner layer (IL) and the outer layer (OL). In subsequent stages both periclinal and anticlinal divisions of cells take place, resulting in the formation of new layers and in an increase in the number of cells in each layer, respectively. Soon a protrusion is formed that grows through tissues of the mother root. Finally a new LR emerges through the surface of the mother root. Before the emergence both the cellular and growth organization of the LR apex become similar to that in the mother root (Dolan et al., 1993); i.e. the apex consists of the root proper and the root cap (RC), with the pole of the root proper occupied by a quiescent centre (QC) – a zone of slow growth and rare divisions, which is crucial for proper root growth and organization (Feldman and Torrey, 1976; Jiang and Feldman, 2005). In the root proper, cells elongate and are displaced towards the base (proximal end) of the root, while in the columella – a central part of the RC – cells elongate and are displaced towards the tip of the root, and in the peripheral RC, towards the flanks.

Applying molecular methods Malamy and Benfey (1997) managed to predict the lineages of the cell type in a forming LR, from initiation until the moment when the cell pattern of the young LR became stable (fig. 8 in their paper). After the first periclinal division in the pericycle of the mother root, the IL maintains the character of the tissue while the OL reveals features typical of epidermal cells. In subsequent stages, produced through periclinal divisions of the OL and IL, new cell layers manifest characteristics of the cortex and vascular tissue, respectively. Soon, in the vicinity of the cortex cells, near the tip of the primordium, a group of cells is present whose tissue type cannot be characterized; these cells are thought to form the future QC and initials (Malamy and Benfey, 1997). Finally, inside the layer of cortex a new layer of endodermal cells and their initials become visible adjacent to the undifferentiated QC cells near the tip. Thus in the process of the LR formation in arabidopsis a vascular cylinder and pericycle derive from the cells of IL, while the endodermis, cortex, epidermis and the RC arise from the cells of OL.

In a plant organ growing symplastically (Erickson, 1986) growth is usually anisotropic, which means that at each point within the organ and on its surface three specific, mutually orthogonal principal directions of growth (PDG) can be distinguished. In two of the PDGs growth is maximal and minimal and in the third direction, corresponding to the saddle point, growth is in-between (Hejnowicz and Romberger, 1984; Szymanowska-Pułka and Nakielski, 2010). These three directions are longitudinal, transversal and tangent to the organ circumference (periclinal p, anticlinal a and latitudinal l, respectively; Fig. 1A, B) and they change continuously from one to another point of the organ forming the PDG trajectories. In a steady-growing organ, the PDG trajectories are preserved and adjusted to the overall geometry of the organ. In the case of the root apex, trajectories of the directions p and a are curvilinear in the apical part of the organ (Fig. 1A) and straight lines (parallel and perpendicular to the root axis, respectively) in the cylindrical part (Fig. 1B), while trajectories of the direction l are circular in any part. The PDG trajectories are manifested in cell wall system of the growing organ – two types of continuous mutually orthogonal lines known as periclines and anticlines (von Sachs, 1887) which are used to describe this system in longitudinal section represent the PDG trajectories (Hejnowicz, 1989). Hejnowicz postulated (1984) that the PDGs play a significant role in morphogenesis; namely, during growth, cells are able to detect the principal directions and they divide in the planes defined by two PDGs, while the third PDG is perpendicular to the division plane. It is worth noting that the organ's surface is also defined by two of the PDGs: p and l, and the direction a is oriented along the organ's radius (Fig. 1A). In this way the cell pattern fits the geometry of the organ.

Fig.1.

Principal directions of growth (periclinal, p; anticlinal, a; latitudinal, l) (red) of two chosen points on the surface of the root apex (A) and their influence on the cell pattern in the axial section during formation of the virtual lateral root apex (B). In (A) the apical and cylindrical parts of the organ are shown. In (B) division walls are inserted with respect to p and a; the lines in the background represent trajectories of these directions. Notice that new division walls are arranged along the trajectories (green). (B) Modified after Szymanowska-Pułka and Nakielski (2010).

The divisions with respect to PDGs result from tensorial properties of growth. The concept of growth tensor (GT), proposed by Hejnowicz and Romberger (1984) assumes that the field of growth rates of the plant organ is of the tensor type. Such a field, defined mathematically, is a basic element for the model of growth and cell divisions. The model specified for the root apex allows for the simulation of growth of this organ by computer (Nakielski, 1991, 1992, 2008; Nakielski and Hejnowicz, 2003). The method was also used to simulate LR formation in radish (Szymanowska-Pułka, 2007; Szymanowska-Pułka and Nakielski, 2010). A developing LR was presented in the axial section of the organ, and thus in a plane in which two principal directions, p and a, were visible. As a consequence of the cell divisions in PDGs, the cell walls were aligned along trajectories of either a p or an a direction (Fig. 1B). Resulting from this model was a realistic sequence of events showing the subsequent changes in the cell pattern taking place during the LR formation in radish (Szymanowska-Pułka and Nakielski, 2010).

This paper is an attempt to verify the significance of the PDGs in LR development in Arabidopsis thaliana. The PDGs manifestation in the cell pattern of the forming LR is analysed. The results of anatomical observations have provided data for modelling. In simulation a fundamental role of the PDGs in the pattern formation has been confirmed through comparison of two variants for cell division: in PDGs and along a direction perpendicular to the nearest existing wall (von Sachs, 1887). An important confirmation of the PDGs significance is a realistic sequence of the steps in LR development obtained from simulation, illustrating plausible lineages of the cell types according to the model proposed by Malamy and Benfey (1997).

MATERIALS AND METHODS

Anatomy

Arabidopsis thaliana seeds of wild type Col-0 (WT) and transgenic lines pPIN1::PIN1:GFP, DR5::GFP, DR5::GUS, AUX1::YFP were surface sterilized in 20 % bleach and 0·5 % Triton X-100 for 5–10 min, washed three times with sterile water and incubated at 4 °C in darkness for 3 d. Seeds were then plated on MS medium (Murashige and Skoog, 1962) supplemented with 1 % sucrose and 0·8 % agar (pH 5·8). Plates were oriented vertically. Growth conditions were 16 h of light, 8 h of dark, and 21 °C. Ten to twelve days after germination, seedlings were either taken fresh for microscopic observations or fixed in aceto-alcohol (1 : 6, v/v), rinsed three times in 100 % ethanol and once in 70 % ethanol, then cleared in 8 : 3 : 1 (w/v/v) solution of chloral hydrate, water and glycerol for at least 1 h. For histochemical analysis of glucuronidase (GUS) activity, the seedlings were prefixed in ice-cold 90 % acetone, washed twice with washing buffer (50 mm NaPO₄ buffer, pH 7·0, 2 mm K₃Fe(CN)₆, 10 mm EDTA) and incubated in staining buffer at 37 °C for 30 min (50 mm NaPO₄ buffer, pH 7·0, 1 mm K₃Fe(CN)₆, 1 mm K₄Fe(CN)₆, 10 mm EDTA and 1 mg mL⁻¹ X-Gluc). The samples were then fixed in aceto-alcohol and cleared as described above.

Observations of the fresh roots were made with epifluorescence microscopy while observations of the cleared roots were accomplished using phase contrast and Nomarski differential interference contrast microscopy. Pictures of LRs in different stages of their development were taken with a digital camera Nikon DS-Fi1. Changes in the cellular pattern of an LR apex were analysed in the axial plane of the organ.

Modelling

To produce a sequence representing the LR development in arabidopsis, the model previously applied to LR development in radish (Nakielski, 2008; Szymanowska-Pułka and Nakielski, 2010) was adopted. The model considered growth in the axial section of the organ, and thus in a plane defined by the directions a and p. There were three basic elements of that model: (1) a polygon meshwork representing data at the input, (2) the GT field generating growth and (3) the algorithm for cells division; they were defined as follows (Fig. 2).

• (1) The polygon meshwork (Fig. 2A), taken from Malamy and Benfey's 2D initial cell pattern of the LR primordium, consists of two cell layers: outer (OL) and inner (IL) (Malamy and Benfey, 1997). The cells were represented by polygons, whose vertices formed at most three-way junctions (see Nakielski, 2008). Co-ordinates of the vertices were the input data for the computer program.

• (2) The GT field (Fig. 2B) came from previous papers (Szymanowska-Pułka, 2007; Szymanowska-Pułka and Nakielski, 2010), except for the fact that values of the parameters were modified for the case of arabidopsis (for details, see Supplementary Data S1). Defined in the logcosh co-ordinate system (u, v, ϕ = const; Hejnowicz and Karczewski, 1993; see also Supplementary Data S1) the field consisted of six zones (1, 2, 2a, 3, 4, 4a) responsible for growth of various parts of the LR apex (Fig. 2C; see also Szymanowska-Pułka and Nakielski, 2010). It was of unsteady type (Hejnowicz and Hejnowicz, 1991), meaning that the parameters defining borders between the zones u₀, v₀ and u_b changed with time (Fig. 2D) according to the equations:

where u₀(t₀) = 0·175, v₀(t₀) = 0·28, u_b(t₀) = 0·35, b = 0·5, c = 1·2.

• (3) Two variants of the algorithm for cell divisions were applied. In variant I cells were divided in one of the two PDGs, a or p. In variant II a division wall was perpendicular to the nearest existing cell wall (Nakielski, 2000, 2008). In both variants the procedure began with determining a small circle around the geometrical centre of the cell, and then a point was randomly selected within the circle. Then in variant I the PDGs (p and a) were determined for the point and, of the two directions, one was chosen for which the division wall appeared shorter. In variant II the distance from the point to the nearest wall was determined and the new wall was set in perpendicular to this wall. In both variants the division wall was inserted as a segment joining two opposite walls of the cell. Finally the segment was slightly shrunk to make the angle between the inserted wall and the older wall more realistic (for details, see Nakielski, 2008).

Fig. 2.

Modelling of the LR formation: (A) the polygon meshwork representing two layers (inner, IL; outer, OL) of pericycle cells; (B) the GT field generating growth assumed at the initial time; (C) the field applied to the polygon meshwork at the initial time of the simulation – the field consists of six zones (1, 2, 3, 4, 2a, 4a) separated by the lines u₀, v₀, u_b that correspond to the PDG trajectories; (D) the unsteady character of the field – dimensions of the zones change with time, the field superimposed on the LR outline at the initial (t₀) and final (t₅) time. The scale in (D) is smaller in comparison to (A–C)

Applying the three elements in the model resulted in a temporal sequence visualizing the LR formation. The manner of growth of a single cell was determined by its position in the GT field. The initial position of each cell was known (Fig. 2C) and, on this basis, a new position was calculated after a time-step Δt = 0·05 (43 time-steps completed in the whole simulation). With time each cell, and consequently the whole primordium, enlarged. The cells divided when a critical value of the cell area equal to 150 % of the average area of the cells was exceeded (see Nakielski, 2008). The cell pattern at the input consisted of 12 cells, and at the final time point there were about 180 cells.

RESULTS

Microscopic observations

In the LR apices of Arabidopsis thaliana, characteristic oblique cell walls are present at various stages of organ development (Fig. 3). New walls that do not fit the orientation of the older cell walls are inserted in the newly formed organ's principal directions. The anticlinal divisions resulting in the oblique wall's formation occur in the pericycle cells before their first periclinal division (Fig. 3A, B, arrowheads). In very young primordia consisting of two layers of cells, the oblique walls are formed mostly anticlinally either in the OL (Fig. 3C, arrowhead) or in the IL (not shown). Also at these early stages, latitudinal division walls are visible on the surface of the young primordia (Fig. 3D, arrowhead). When a small protrusion is formed, the oblique walls inserted in periclinal direction become more frequent (Fig. 3E, F, arrows), although oblique anticlinal walls still occur (Fig. 3F, arrowhead). At subsequent stages the periclinal divisions predominate in the whole apex so a typical pericline pattern soon becomes evident (Fig. 3G, H, arrowheads). Some cell walls in the central part of the primordium form a strongly acute angle in relation to the axis of the apex (Fig. 3G, arrows); when the protrusion develops the acuteness becomes more and more distinct (Fig. 3H, I, J, arrows). On the flanks of some very young primordia, large, atypically shaped cells are formed of which periclinal walls are distinctly curved (Fig. 3C, asterisk). Such cells are also present in the more-developed apices (Fig. 3G, H, asterisks). In the outermost cell layers of some primordia at advanced stages, the anticlinal walls of larger cells happen to be obliquely oriented in relation to the organ surface (Fig. 3I, J, empty arrowheads).

Fig. 3.

Oblique walls leading to the new PDGs establishment within the cell pattern during the LR development in the axial (A–C, E–J) and surface (D) views of the apex; anticlinal (A–C, arrowheads) and latitudinal (D, arrowhead) divisions at the early stages followed by periclinal divisions (E, F, arrows) at subsequent stages, when anticlinal oblique walls (F, arrowhead) are also observed; periclines (G, H, arrowheads) formed resulting from a series of periclinal divisions; sharply pointed cells near the primordium axis at the intermediate stage (G, arrows) and later stages (H–J, arrows); a large cell with curved walls occurring in the very young (C, asterisk) and older primordia (G, H, asterisks) on their flanks; oblique anticlinal walls of the surface cells (I, J, empty arrowheads) at the lateral parts of the young LRs. (A, E) DR5::GFP; (B, C, D, F, H) AUX1::YFP; (B, D) the YFP signal in the plasmalemma adhering to the cell wall; (G) DR5::GUS; (I, J) WT. Scale bars = 20 µm.

In the lateral parts at the base of the LR primordia some characteristic features of the cell pattern can be observed from the very beginning of organ development. Numerous young primordia appear sharply separated from the mother root body through distinct borders caused by (bolder) anticlinal cell walls (Fig. 4A–C, arrowheads). At later stages the borders are still visible (Fig. 4D, E, H, arrowheads) and, moreover, in their vicinity a cell arrangement is observed which does not fit a regular pattern of the periclines and anticlines present within the primordium in its axial view (Fig. 4E, F, H, I, arrows). The significantly disturbed cell arrangement is often accompanied by the occurrence of abnormally large cells of various shapes located at the bases of the apices. In relatively young primordia such cells are usually situated on the mother root side (Fig. 4E, F, asterisks), while in the mature LRs, in which a vascular system has been initiated, the abnormally large cells occur on the flanks of the organ (Fig. 4G, I, asterisks). These non-dividing cells, usually occurring in the area where the LR outline forms the strongest curvature, very often are curved outward as a consequence of compression (Fig. 4F, I, asterisks).

Fig. 4.

Cell arrangement at the bases of the forming LRs in the axial view; distinctly marked boundaries between the primordium and the mother root at the early (A–C, arrowheads) and subsequent stages (D, E, H, arrowheads); a distinct pericline formed by bolder cell walls separating complexes of cells deriving from the IL and OL (C, D, empty arrowheads); a disturbed pattern of periclines and anticlines on the flanks occurring at the middle stages (E, F, arrows) and maintained in a developed LRs (H, I, arrows); large cells on one of the flanks (E, F, G, asterisks) or on both flanks of the LR (I, asterisks) in various stages. (A, H) WT; (B) pPIN1::PIN1:GFP; (C, D, F, G, I) AUX1::YFP; (G) the YFP signal in the plasmalemma adhering to the cell wall; (E) DR5::GUS. Scale bars = 20 µm.

There are few relatively young primordia in which an especially distinct pericline is observed (Fig. 4C, open arrowhead). The pericline formed by the bolder periclinal cell walls separates the cell complexes derived from the cells of the IL and the OL of the previously divided single layer of the pericycle cells of the mother root. In more developed primordia this ‘separation’ is hardly ever observed, although in some primordia at intermediate developmental stages, the pericline can still remain visible (Fig. 4D, open arrowhead).

These features noted above have been observed either in WT or in the analysed transgenic lines with comparable frequency.

Model and simulation

Results of applying the model for growth and cell division to a description of LR development in arabidopsis are shown in Fig. 5 (see also Supplementary Data Videos 1 and 2). Two sequences consisting of six time points, with the first time point (t₀) common to both sequences, are presented. The upper sequence relates to the case of divisions with respect to PDGs (variant I), and the lower sequence to the case of divisions perpendicular to the nearest wall of the parent cell (variant II).

Fig. 5.

The computer-generated sequences of the LR formation in arabidopsis (axial section): in variant I, cells divide with respect to PDGs and, in variant II, division walls are perpendicular to the nearest wall. The initial t₀ and final t₅ times correspond to Fig. 2D, the cell wall meshwork at t₀ is common for both sequences. In both variants at t₁ some of the newly inserted division walls (short arrows) and at t₃ regular arrangement of the central cells (asterisks) are indicated. In variant I at t₄ periclinal walls (bold) aligned along PDG trajectory (dashed line) and a local lack of the pattern regularity (short arrow) are indicated. The small schema at the top shows the GT field with its zones separated by border lines (red). The complexes of cells originating from two individual cells of IL (dark grey) and OL (light grey) are indicated. Notice the differences in the orientation of the newly inserted cell walls in both variants: in variant I the walls are arranged in regular periclines and anticlines, whereas they are not so arranged in variant II.

Derived from the anatomical data by Malamy and Benfey (1997), the meshwork at t₀ represents the cell pattern with two cell layers, IL and OL, of the pericycle of the mother root. The pattern refers to the stage in which each cell has divided periclinally, so that there are 12 cells in the meshwork, six in each layer. All the zones of the GT field are very small, so none but zone 4a contains even one whole cell. The central area of the cell pattern with fragments of the cells from either IL and OL happens to zone 1, where there is no growth. At the subsequent time points, the cell meshwork grows and changes its overall geometry in a similar manner for both variants of the division of cells. Growth of the apices is accompanied by the increase of the GT field zones which is the same in both sequences. However, there are noticeable differences in the cell pattern formation. At t₁ the meshwork has expanded due to growth of the cells of either IL or OL and, as a consequence, a slight protrusion has formed. Both in the upper and lower sequence almost all cells of the primordium have undergone divisions, although in the PDGs variant the new walls are distinctly oblique in reference to the primordium outline. At the next time point (t₂) the protrusion has enlarged and the whole apex has grown. In both sequences new walls have appeared resulting from divisions of cells of the IL and OL. In the upper sequence some of the new walls are oblique in relation to the older walls, especially to those forming the outline, yet, they are aligned along the PDG trajectories. The new division walls present in the lower sequence are more or less perpendicular to the hitherto existing walls and the cell pattern still appears regular.

At t₃ the primordium has enlarged further and now it has assumed a dome-like shape. The central files of cells in both the distal and proximal regions of the apex are parallel to the organ axis and the cell walls are perpendicular to their neighbours; these features appear to be the only similarities of the patterns between the two. The walls of the cells in the upper sequence form periclines and anticlines typical of the young LR apex. Only the cells localized in the side parts at the base of the LR seem not to belong to the regular pattern. At the base of the primordium, some walls are arranged along the periclines, but there are more oblique walls, especially in periphery of zone 4a. In the lower sequence the cell arrangement is relatively regular only near the axis in zones 2 and 3; it is difficult to indicate a typical pericline and anticline pattern elsewhere. The time points t₄ and t₅ show the final stages of the LR formation. The apex is elongated as a consequence of strong growth of the cells in the zones 2 and 3. Zone 1 has increased as well and at t₅ it includes nearly two whole cells in the upper sequence and one cell in the lower sequence. In both primordia, at t₅ the pattern present at the previous time point has been maintained and developed, namely, the apex in variant I displays a regular cell arrangement with walls forming periclines and anticlines, while the apex in variant II lacks this regularity.

A comparison of the marked complexes of cells derived from the IL and OL (Fig. 5, dark grey and light grey, respectively) indicates differences in the cell pattern formation resulting from application of two variants of the cell division. At t₁ the dark grey cell has grown and developed a three-cell and a two-cell complex in variants I and II, respectively. Interestingly, in variant I this cell divided transversally (along the a direction) first, and then its derivative divided longitudinally (along the p direction) while in variant II the first division of this cell was longitudinal. The light grey cell has also divided, forming a multi-cell complex. In the upper primordium, where the complex comprises five cells, the new walls are inserted along the PDG trajectories, thus obliquely to the older walls, while in the lower primordium, in which the complex is composed of four cells, the division walls remain perpendicular and parallel to the outermost wall. At t₂ the cell (dark grey) of the IL has developed into the elongated complex of six cells either in the upper or the lower sequence. In both variants the division walls are oblique in relation to the outline of the primordium. In the light grey complexes of both apices the subsequent divisions resulted in a similar nine-cell complex formation, and only the wall orientations are different. In variant I, the periclinal walls (along the p direction) form acute angles, and anticlinal walls (along the a direction) form obtuse angles with respect to the apex outline, while in variant II these walls are mostly parallel and perpendicular to the apex outline, respectively. At t₃, t₄ and t₅ the relationship appears to be opposite: the newly inserted walls in the light-grey complex of the upper primordium are perpendicular and parallel to the outline while in the lower primordium they are oblique. In both complexes the outermost cells are larger than cells localized closer to the central part. At these time points, the dark-grey complexes of either the upper or the lower sequence have elongated, forming the central files of the LR apex. The cells of the complex in the upper sequence are regularly arranged and their walls form smooth periclines. In the lower sequence strongly oblique walls are present which do not fit any pericline.

In Fig. 6A possible ways of formation of particular tissues are indicated, based on the cell pattern development and on the fates of individual cells of the virtual LR. The sequence comes directly from Fig. 5 (variant I). At the initial time point t₀ when the primordium consists of two layers, the IL is assumed to preserve the pericycle character and the OL to acquire characteristics of the epidermis. Next (t₁), a new layer is formed in the middle of those cells which may be identified as cortical. At t₂ a group of cells at the base (proximal end) of the primordium, near its axis, is postulated to form the pre-vasculature of the young primordium, and at t₃ another group of cells localized near the tip of the apex may form the future initial centre of the organ. The cells at the tip of the apex (distal end) are supposed to develop into the RC. As a result of periclinal divisions of the cortex cells at t₄, a new layer appears that is assumed to consist of endodermal cells. At this time point the cortex and endodermis initials can be distinguished. The last time point t₅ shows a more developed apex with all particular tissues formed. The outermost layers of cells forming the epidermis, cortex and endodermis are derived from the OL while the central layers comprising the pericycle cells and the vascular tissues are derived from the IL.

Fig. 6.

(A) A sequence of the LR development from Fig. 5, variant I and (B) experimental results by Malamy and Benfey (1997, modified) in the aspect of a putative derivation of the tissues. Subsequent time points t₀–t₄ of (A) correspond to the stages shown under them in (B), time point t₅ predicts possible further tissue formation. In t₁ and t₄ of (A) characteristics of the cell pattern of the virtual LR apex with reference to the features observed in real apices are indicated, oblique division walls in young (t₁) and older apex (t₄, bolder walls), the former related to establishing the new PDG pattern, the latter associated with large cells (asterisks) occurring at the sides, a clear line separates complexes of cells derived from the IL and OL (bold black). There are distinct boundaries between the main part of the primordium and its sides (arrowhead), ‘disorder’ in the pattern near its basis (arrow), sharply pointed cells (circles) near the axis at the early stage (t₁) and its derivative at the later stage (t₄). Colour coding after Malamy and Benfey (1997).

DISCUSSION

This paper shows computer modelling of the LR formation in the Arabidopsis thaliana root. To initiate a protrusion, an unsteady growth field defined by the GT was applied to a group of cells of the pericycle of the main root. Successive developmental changes in these cells, via growth-induced cell extension and divisions with respect to PDGs resulted in formation of the LR. The used method of modelling is not new. More than 20 years ago, Hejnowicz and Hejnowicz (1991) demonstrated how a new root apex arises from an originally rectangular uniform grid of marks. The apex developed under a steady GT field with a boundary condition, which restricted growth in the basal part of the protrusion. Similar field, but of an unsteady type (Szymanowska-Pułka, 2007), where redefinition of the field zones had been introduced, was used to simulate the LR formation ‘on cells’ (Szymanowska-Pułka and Nakielski, 2010). Taking radish as an example, realistic sequences of the LR formation were obtained, from the stage of early protrusion to maturity. Such instability of the field, necessary to generate early protrusion, was successively reduced during development, so that with time the field became steady and similar to that occurring in the main root. The present study applies all previously verified tools to model the case of arabidopsis in close correspondence to anatomical data. It is focused on the problem, how PDGs and, particularly cell divisions, oriented with respect to these directions affect developmental changes in cell structure during the LR formation. Three aspects of the results of the simulations are discussed: (1) how principal directions of growth are manifested in the cell pattern of the virtual LR; (2) what ‘cellular’ effects observed in simulation of PDGs are not taken into account; and (3) what the symptoms of PDG manifestation observed in real LRs are.

The PDGs are manifested during cell pattern formation

The oblique walls inserted into the cell pattern of very young primordia (Fig. 3A–C) illustrate how new PDGs are introduced in relation to the original pattern of walls that are primarily parallel and perpendicular to the axis of the mother root. Although during the LR initiation the orientation of new walls introduced as a result of the first divisions are in accordance to the principal directions of the mother root, remaining parallel or perpendicular to the older walls (see fig. 2A, B in Malamy and Benfey, 1997), soon new directions are initiated and obliquely oriented walls become more frequent (Fig. 3, early stages). Hence the PDGs are introduced with some delay in relation to the initiation itself. This observation may relate to the results of Laskowski et al. (1995) who determined the stage at which the LR primordia in arabidopsis were able to function as separate root meristems. In the cited work, primordia at various stages were excised and cultured on the MS medium. The primordia consisting of two cell layers failed to develop while the primordia which consisted of at least three to five layers of cells developed independent meristems capable of LR formation. Therefore one may hypothesize that gaining independency is related to the ‘switching on’ of the new field of growth, with individual principal directions specific for the developing young organ. The moment at which a newly formed LR primordium becomes an independent meristem capable of self-controlled development may relate to the activation of the new field responsible for the LR formation.

Based on the limited literature about oblique walls in the LRs we can conclude that their occurrence in young primordia is not very unusual. An interesting case was reported by Blakely et al. (1982) who observed frequent oblique anticlinal walls in the single cell-layer LR primordia in radish (fig. 7 in the cited paper). In slightly older primordia consisting of two cell layers, new division walls occurred, ‘adjacent to the oblique anticlinal walls and perpendicular to them’, which resulted in the formation of ‘cuneiform cells’. These authors suggest that both the oblique walls and the cuneiform cells ‘may have some developmental significance’ (Blakely et al., 1982) in the process of the protrusion formation at these early stages. We would be inclined to link their evidence with the independent GT field activation.

The rules for divisions

In the simulations two variants for cell division were applied. Variant I, in which the division walls are inserted with respect to the PDGs, derives directly from the Hejnowicz's (1984) ideas of the mathematical description of plant organ growth. It is worth pointing out that, although this rule is based on theoretical assumptions, it considers a mutual relationship between the organ shape and its cell pattern. Thus in this approach both cellular and organ level of the growth organization are taken into account. Variant II, in which the wall is inserted perpendicular to the nearest existing wall, may be related (1) to the proposition of von Sachs (1887) about new walls inserted perpendicular to the old ones, or (2) to the view of Errera (1888) that new wall orientation is determined by the minimal-area condition, or (3) to the concept of Hofmeister (1863) about the division wall arising perpendicular to the strongest direction of growth. All the above-mentioned hypotheses, drawn directly from anatomical observations (Lloyd, 1991), are well satisfied in the central part of the virtual apex in the vicinity of its axis. This is so because in this region the wall of the minimal area, i.e. the shortest in the 2D view, would overlap the wall perpendicular to the hitherto existing, and also with the wall perpendicular to the direction of the maximal growth, here along the organ axis. In this region two of the principal directions of the GT are parallel (direction p) and perpendicular (direction a) to the LR axis. This is why both in variant I and in variant II of the simulation (Fig. 5), the cells near the axis undergo divisions in such a way that a new wall is inserted perpendicular to the old walls.

In variant I (Fig. 5, upper sequence) in the early stages of the LR development, new walls inserted in accordance to PDGs are oblique to the existing ones (mostly at a distance from the axis). However, in later stages such obliqueness is not observed. In variant II (Fig. 5, lower sequence) the situation is different, in the early stages, new walls fit the existing walls, whereas in later stages they do not, especially near the outline of the apex. Thus variant II delivers acceptable results in the whole apex but only in the very early stages, except near the axis of the organ where it also applies in later stages. However, applying the two variants shows that only variant I, with the algorithm for divisions of cells along the PDG, gives realistic results. Satisfying support for this conclusion may be presented in Fig. 7 by comparison of the cell pattern of the real LR apex in advanced stages of development (coming from Fig. 3J), with the pattern of the PDG trajectories superimposed. Both the periclinal and anticlinal cell walls are arranged in a regular way along the trajectories of the p and a directions, respectively. Such accordance between the two may indicate that the cell pattern has gained a state close to stability.

Fig. 7.

A pattern of the PDG trajectories (white) superimposed on the LR apex from Fig. 3J; notice a significant regularity of the cell pattern with respect to the trajectories of the p and a directions (red).

The results suggest that the cell arrangement in a developing organ like the LR, for which geometry continuously changes until its maturity, is dynamically adjusted to the PDGs pattern. From other studies (J. Nakielski, unpubl. res.) we know that the process of regeneration of an excised RC may be well simulated with application of a steady GT field only slightly displaced compared with an intact apex. Hence, determining and introducing into simulations a precisely specified GT and the PDGs guarantees a proper reconstruction of a growing plant organ, even in some atypical conditions. Thus the above conclusions are good confirmation of the previous observations (Hejnowicz, 1984; Nakielski, 2008) concerning a steadily growing plant organ, in which the GT field with its principal directions generates growth of cells and affects their divisions, thus ensuring the preservation of the cell pattern of the whole organ.

Anatomy and simulation

The above-presented results of the microscopic observations (Figs 3 and 4), as well as the results by other authors (Malamy and Benfey, 1997), provide knowledge of specific and timely events that occur in the developing LR. Many of the events can be realized in the virtual apex obtained from the simulation. For example, in Fig. 6A, possible ways of formation of particular tissues in the virtual apex are shown. The sequence is an attempt to reconstruct, based on experimental results, the model presenting the putative derivation of tissues in a developing LR primordium in arabidopsis, as proposed by Malamy and Benfey (1997). In Fig. 6B, selected stages of the cited model are shown to make comparisons to the above-mentioned sequence. The subsequent time points of the simulation (Fig. 6A, t₁–t₄) appear to mirror quite accurately the situation obtained from the experiment (Fig. 6B). The order of development of each tissue type in both sequences (Fig. 6A, B) is practically the same. Also the number of cell layers is similar. The analogy applies even to the position of individual cells, such as the cells marked in dark blue and identified as the initials for endodermis and cortex, or a group of cells of unknown identity supposed to form the future QC. The final time point (Fig. 6A, t₅), having no equivalent in experimental data, may serve as a kind of prediction for the future fates of tissue development. All in all, particular tissues of the virtual apex can be easily identified and related to those observed in anatomical studies.

Some of the anatomical features found in the real LR primordia are also present in the virtual apex as indicated in Fig. 6A. Oblique division walls inserted at the early stages are a manifestation of the PDGs within the forming organ, and are also observed in the real apices (Fig. 3A–F). In the older primordium, oblique walls are present mostly in the outermost cell layers near the bases of both the virtual apex (Fig. 6A, t₄) and real apices described in this paper (Fig. 3I, J), as well as presented in the work by Malamy and Benfey (1997, fig. 2I–L). Also, in both virtual and real primordia at the flanks, near their bases, large, irregularly shaped cells occur (Figs 3G, H, 4E–G, I and 6A, t₄; also fig. 2J, L in Malamy and Benfey, 1997). A sharply pointed cell in the central region of the virtual apex close to its axis occurs relatively early in the simulated LR (Fig. 6A, t₁). As in the real apices (Fig. 3G–J; also fig. 2G, H in Malamy and Benfey, 1997), a derivative of this cell forms an even more acute angle with the axis (Fig. 6A, t₄).

In the virtual apex in Fig. 6A (t₄) a pattern of periclines is easily recognizable, with the pericline separating the groups of cells derived from the IL and OL especially distinct (t₁, t₄, bold line). In apices observed in anatomical sections this separation is not always clear, although in some very young primordia it may be seen (Fig. 4C, D). The sharply marked boundaries between the LR and the mother root body that is so evident in the real apices at their flanks (Fig. 4A–E, H) are also visible in the virtual apex (Fig. 6A, t₄). In this regard it is worth noting that the orientation of the separating wall of the apex derived from the simulation corresponds with observations from the real apices, but at later stages (Fig. 4H, arrowheads). And finally, a remarkable disorder in the cell pattern observed at the bases of the real primordia (Fig. 4E, F, H, I) is also present in the virtual apex (Fig. 6A, t₄). Interestingly, features similar to this perturbed cell pattern were also described by Hirota et al. (2007) and attributed to a mutation in the auxin-regulated AP2/EREBP PUCHI gene. Their results showed that PUCHI was responsible for the occurrence of a restricted area of cell proliferation in the regions near the base of the primordium-to-be. As the authors report, ‘recessive mutation in PUCHI’ resulted in a pronounced disturbance in the cell division pattern, viz. the LRs bases became swollen due to redundant divisions and the cell pattern in the region is remarkably changed. Nevertheless, the mutation was observed only in primordia consisting of three and more layers. But, in the younger primordia the mutation did not cause any observable changes. This may have a connection with what was described above, as ‘activation’ of the individual GT field for the primordium, which at this stage of development gains a status of an independent meristem (Laskowski et al., 1995).

Hirota et al. (2007) suggest that, among other agents involved in the process of the organ formation, PUCHI may control morphology of the LR. Although no specific agents are suggested, we posit that the GT field may be one of them. This assumption finds good support in the above-mentioned results (J. Nakielski, unpubl. res.) concerning modelling of RC regeneration, as well as in sequences showing LR formation in radish in a situation of a changed position of the cell pattern in relation to the GT field (fig. 7 in Szymanowska-Pułka and Nakielski, 2010). In the studies cited, the displaced GT field caused a change in the identity of some cells. For example, the cells which before the displacement had been situated within the QC, where no growth had occurred, after the displacement were found in one of the non-zero growth zones of the field. This resulted either in a changed morphology of the root apex or in a new pattern formation. Both of these examples provide evidence of the sensitivity of the cell pattern to the GT field that corresponds both with morphology and – through divisions along the PDGs – with the cell arrangement.

Finally, let us gather together the new results presented in this paper. (1) We have applied the model for growth and division of cells (Nakielski, 2008), based on the GT concept of Hejnowicz and Romberger (1984), to the formation of the LR in arabidopsis. (2) Our simulation of LR formation in arabidopsis includes both changes in organ size and in cell pattern development. (3) A direct reference to empirical data allowed us to indicate characteristic features of the cell pattern not previously described. (4) Biophysical or, more precisely, tensorial aspects of growth are taken into consideration, and their role is confirmed in the simulation: only the variant of division of cells, in accordance to PDGs, has given a realistic result, suggesting that oblique walls resulting from divisions in the early stages of the LR formation are not accidental, but they are a manifestation of the establishment of new principal directions, attributed to the newly formed organ.

SUPPLEMENTARY DATA

Supplementary data are available online at www.aob.oxfordjournals.org and consist of the following. S1: the logcosh curvilinear orthogonal co-ordinate system u, v, ϕ = const, and the field velocities (V field) for the lateral root apex. Video 1: simulation of the lateral root formation, variant I; in the simulation new division walls are inserted along one of the principal directions of growth. Video 2: simulation of the lateral root formation, variant II; in the simulation new division walls are inserted perpendicular to the existing wall.