Growth rate distribution in the forming lateral root of arabidopsis

Abstract

Background and Aims

Microscopic observations of lateral roots (LRs) in Arabidopsis thaliana reveal that the cross-sectional shape of the organ changes from its basal to its apical region. The founder cells for LRs are elongated along the parent root axis, and thus from the site of initiation the base of LRs resemble an ellipse. The circumference of the apical part of LRs is usually a circle. The objective of this study was to analyse the characteristics of changes in the growth field of LRs possessing various shapes in their basal regions.

Methods

The LRs of the wild type (Col-0) and two transgenic arabidopsis lines were analysed. On the basis of measurements of the long and short diameters (D_L and D_S, respectively) of the ellipse-like figure representing the bases of particular LRs, their asymmetry ratios (D_L/D_S) were determined. Possible differences between accessions were analysed by applying statistical methods.

Key Results

No significant differences between accessions were detected. Comparisons were therefore made of the maximal, minimal and mean value of the ratio of all the LRs analysed. Taking into consideration the lack of circular symmetry of the basal part, rates of growth were determined at selected points on the surface of LRs by the application of the growth tensor method, a mathematical tool previously applied only to describe organs with rotational symmetry. Maps showing the distribution of growth rates were developed for surfaces of LRs of various asymmetry ratios.

Conclusions

The maps of growth rates on the surfaces of LRs having various shapes of the basal part show differences in both the geometry and the manner of growth, thus indicating that the manner of growth of the LR primordium is correlated to its shape. This is the first report of a description of growth of an asymmetric plant organ using the growth tensor method. The mathematical modelling adopted in the study provides new insights into plant organ formation and shape.

INTRODUCTION

The basic functions of lateral roots (LRs) are in anchoring the plant in the soil and in the absorption of water and nutrients. LRs have an endogenous origin (Cutter, 1971), which means that their primordia are initiated internally within the parent root to which the LR base always remains attached. LRs originate in the pericycle – a layer of cells encircling the vascular tissue, and they are initiated opposite the xylem poles. In diarch roots of Arabidopsis thaliana, in which vascular tissue is composed of two strands of xylem alternating with two strands of phloem, new LRs are formed acropetally in two rows, in a left–right alternating pattern (Fig. 1A; Dubrovsky et al., 2006). The LR initiation takes place in the region located between the basal meristem (or transition domain; Ivanov and Dubrovsky, 2013) and the distal differentiation zone (Benková and Bielach, 2010), and involves three adjacent pericycle cell files. Usually two neighbouring cells of each file participate in the process (Dubrovsky et al., 2001; Goh et al., 2012). These cells, designated the founder cells, are elongated along the parent root axis, which results in the primordium base assuming an oval shape (Fig. 1B) from its earliest stages (Lucas et al., 2013). Initial anticlinal divisions of the founder cells are followed by periclinal divisions, resulting in the formation of two cell layers (Casimiro et al., 2003). Subsequently, after a more or less co-ordinated series of both periclinal and anticlinal divisions (Malamy and Benfey, 1997; Białek et al., 2014) a meristem resembling that of the parent root is formed. The protrusion grows through tissues of the parental root and eventually emerges through its surface (Malamy and Benfey, 1997). The primordium develops in the direction perpendicular to the parent root axis.

Fig. 1.

(A) Schematic representation of lateral root positioning in arabidopsis. (B) Typical oval shape of the primordium base. LRP, lateral root primordium; per, pericycle.

Lateral root initiation implies the formation of a new meristem. For this to occur, cells of the parent root locally dedifferentiate and start growing. From a physical point of view, growth may be considered an irreversible deformation (Nakielski and Hejnowicz, 2003). This also means that an individual local field of growth arises dedicated to the new LR primordium (Szymanowska-Pułka and Nakielski, 2010). This field is thought to become independent of the growth field of the parent root (Hejnowicz and Hejnowicz, 1991) as early as the occurrence of new oblique cell division walls (Szymanowska-Pułka et al., 2012; Białek et al., 2014), resulting in the establishment of new principal directions of growth (direction in which growth rates attain extreme values; Hejnowicz and Romberger, 1984) within the primordium.

Plant organ growth may be described with the use of the growth tensor (Hejnowicz and Romberger, 1984). This requires a basic assumption of the continuous character of growth. Indeed, due to the fact that walls of neighbouring plant cells are joined by middle lamellas, the manner of growth of plant organs is co-ordinated and may be considered as continuous (Erickson, 1986). Mathematically, at the organ level, there exists a field of the displacement velocity, V, of points represented by the continuous and differentiable function of position (Gandar and Chalabi, 1989; Silk, 2006). Knowledge of V is necessary in order to determine growth rates at points within the organ and on its surface. The linear relative elemental rate of growth (RERG_l) at a given point along particular directions can be calculated from the following equation (Hejnowicz and Romberger, 1984):

$${\text{RERG}}_{\mathrm{l}}(\text{s})=[(\text{grad}\mathbf{V})\cdot {\mathbf{e}}_{\mathrm{s}}]\cdot {\mathbf{e}}_{\mathrm{s}}$$ (1)

where e_s is a unit vector of the direction s, and each dot represents a scalar product. As gradV represents the second rank operator (Spiegel, 1959), RERG_l(s) is of a tensor type and is called the growth tensor (Hejnowicz and Romberger, 1984).

Graphical representations of RERGs in three-dimensional (3-D) form are the so-called indicatrices, which are regular closed surfaces surrounding a point for which the RERGs in various directions are calculated (Nakielski and Lipowczan, 2013). In Fig. 2, plots representing indicatrices for different types of growth are shown. In the case of isotropic growth, the indicatrix is a sphere (Fig. 2A, left), because the values of RERGs in all directions are equal. For the case of anisotropic growth (Fig. 2A, middle), the indicatrix dimension is largest along the direction of the maximal rate of growth (largest value of RERG) and smallest in the direction of the minimal rate of growth (smallest value of RERG). In the case of negative anisotropy (Fig. 2A, right), its dimension is also largest along the direction of the maximal rate of growth, but negative along the direction of the minimal rate of growth, which means shrinkage (negative growth) in this direction. Values of RERGs may vary for different points and thus they may change with direction (Erickson, 1966). This indicates that there is a field of RERGs (or growth tensor field) within the growing plant organ. By calculating values of RERGs for a number of points of the plant organ, we are able to obtain maps of growth rates for this organ (Hejnowicz and Karczewski, 1993; Szymanowska-Pułka, 2007).

Fig. 2.

(A) Indicatrices (bottom) calculated for different types of growth using the example of a cube (top); isotropy (left) – growth in each direction is the same; anisotropy (middle) – stronger growth in the horizontal direction than in the vertical direction; negative anisotropy (right) – growth in the horizontal direction and shrinking in the vertical direction. (B) Principal directions of growth in the parent root, P, periclinal; A, anticlinal; L, latitudinal.

In applying this method to the case of the plant root apex, the organ is considered to form a dome. In its axial section, cell walls are arranged in lines parallel and perpendicular to the organ's surface (Fig. 3A). These mutually orthogonal lines are designated periclines and anticlines, respectively. A growing LR also has the shape of a dome, from its early stages. Because the cellular organization of the LR apex resembles that of the main root, a similar system of periclines and anticlines may be drawn in the LR's axial section (Fig. 3B). From the point of view of kinematics, a growing point changes its position in time. In this regard, it was shown previously (Hejnowicz, 1984) that displacement velocities of particular points of the organ may be defined in the so-called natural co-ordinate system, lines of which approximate the pattern of periclines and anticlines. Such a natural co-ordinate system is the curvilinear, orthogonal system, defined in such a manner that the components of displacement velocity are functions of only this co-ordinate, to which the given velocity component pertains (Hejnowicz, 1984). For the root apex, such a system is the logcosh system (Hejnowicz and Karczewski, 1993). In Fig. 3B this is shown in two dimensions or, more specifically, in a (x, z) Cartesian plane. By comparing the arrangement of the u and v co-ordinate lines visible in this plane with the pattern of periclines and anticlines in the axial plane of the root apices (Fig. 3A, B), we can understand why this system may be called ‘natural’ for these organs. The 3-D system is obtained by revolving the plane ϕ = 0 (the plane of Fig. 3B) around the z-axis. Transformation from the logcosh co-ordinates to the Cartesian co-ordinates is given through the following equations:

$$\begin{array}{rl}& x=\frac{2}{\pi }\arctan [\tanh (u)\tan (v)]\cos (\varphi )\\ & y=\frac{2}{\pi }\arctan [\tanh (u)\tan (v)]\sin (\varphi )\\ & z=\frac{1}{\pi }\log [{\cosh }^2(u)-{\sin }^2(v)]\end{array}$$ (2)

The system was applied to the description of LR growth in radish (Szymanowska-Pułka, 2007; Szymanowska-Pułka and Nakielski, 2010) and arabidopsis (Szymanowska-Pułka et al., 2012).

Fig. 3.

Natural co-ordinate system for the root apex. (A) An axial section of the main root apex and the pattern of periclines and anticlines (white curves) overlapped. (B) Axial section of an LR at an early stage of development with the pattern of periclines and anticlines (white curves) overlapped; straight lines represent the pattern of periclines and anticlines of the parent root which is cylindrical in this region. (C) A logcosh co-ordinate system in 2-D; a 3-D system is obtained by revolving this co-ordinate system around the z-axis. Straight lines v = ±π/2 are asymptotes.

Recently it has been shown that the cross-section of the LR primordium near its base is elongated along the axis of the parent root; however, while its length (the dimension of the cross-section parallel to the root axis) is more or less preserved in the course of development, the width (the dimension of the cross-section perpendicular to the root axis) slightly increases with time (Lucas et al., 2013). This means that the LR primordium does not expand in the axial (periclinal, P; Fig. 2B) direction, with little growth in the circumferential (latitudinal, L) direction and extensive growth in the radial (anticlinal, A) direction. During growth, the LR primordium undergoes radialization (Lucas et al., 2013), which means that its cross-section changes from oval-like at the base to round in the apical region. Such a geometry determines a characteristic shape of the external surface of LR, whose apical part demonstrates a radial symmetry whereas its basal part preserves a bilateral symmetry (Lucas et al., 2013). Our objective was to describe this surface, establishing a specific co-ordinate system dedicated to an LR, and to analyse the character of changes in the growth field of LRs with basal regions of various shapes. In our description we considered morphological differences between LRs as well as the changing shape of the LR cross-section along its axis. Finally, maps showing the distribution of growth rates on the external surface of LRs of various geometries were elaborated and the character of their changeability around the organ circumference analysed.

MATERIALS AND METHODS

Microscopic observations

The shapes of the bases of 16 LRs at various stages of development of wild type (Col-0) and two transgenic arabidopsis lines were analysed: Col-0 three cases; DR5::GFP (green fluorescent protein) three cases; and AUX1::YFP (yellow fluorescent protein) ten cases. The small number of observations results from the fact that laterals, when put on a microscopic slide, tend naturally to lie on their sides, so it is difficult to observe them in front view. Seedlings for microscopic observations were grown on MS medium (Murashige and Skoog, 1962) for 10–14 d at 21 °C and 16 h of light, 8 h of dark. Seedlings were either used when fresh or fixed in aceto-alcohol, then cleared in chloral hydrate solution, water and glycerol. Observations were made using epifluorescence microscopy, phase contrast and Nomarski differential interference contrast microscopy.

In the case of epifluorescence microscopy, the GFP or YFP fluorescence signal enabled identification of the shape of the LR (Fig. 4A). The LR base was determined at the site of attachment to the parent root, so in the case of contrast microscopy a vascular bundle of the parent root was clearly visible (Fig. 4B). The long (D_L) and short (D_S) diameters of the ellipse-like figures representing the LR bases were measured directly from microscopic images (Fig. 4A), and for each analysed case an asymmetry ratio D_L/D_S was determined. The approximate age of the primordia was estimated on the basis of their size. In a young primordium, the distance from its apical part to the base was short, with the base considered as the level of the parent root vascular bundle. In the images of young primordia, both the apical part of the primordium and the vascular bundle of the parent root are visible within the depth of field (see Fig. 6A, B). In the case of older primordia, either the apical part (Fig. 6C, E) or the base was visible (Fig. 6D). All the microscopic photographs were taken with the use of the × 40 objective lens.

Fig. 4.

Determination of the LR base and its geometry. (A) GFP fluorescence signal indicating the shape of the LR primordium. White lines are the long (D_L) and short (D_S) diameters of the LR base. (B) The site of the LR attachment to the parent root considered a base of the LR; a vascular bundle of the parent root (arrow) is visible along a long diameter of the LR base. The analysed accessions of arabidopsis: (A) DR5::GFP; (B) AUX1::YFP. Scale bar = 20 μm.

Fig. 6.

A change of the shape of the LR cross-section along its axis. The basal (A, B, D) and apical (C, E) part of LRs in arabidopsis at the (A, B) early, (C) intermediate and (D, E) late stage of development. (D) is the same as Fig. 4A. Ellipses in (A–D) indicate the oval shape of the LR base, the circle in (C) indicates a regular circular cell arrangement in the apical region, and a circle in (E) shows a circular shape of the LR cross-section in the apical part. The analysed accessions of arabidopsis: (A) AUX1::YFP; (B, E) wild type Col-0; (C) DR5::GUS; (D) DR5::GFP. The primordium in (C) was not taken into consideration in the estimation of the asymmetry ratio as the basal portion was not clearly visible. Scale bar = 20 μm.

Statistical analysis

Non-parametric (because of the small number of observations) analysis of variance was applied to detect significant differences between the asymmetry ratios of the three analysed accessions. The statistical analysis was performed using the software Statistica 10·0 (StatSoft Inc, Tulsa, OK, USA).

Modelling

For the purpose of visualization the RERG_l(s) distributions on the outermost surfaces of young arabidopsis LRs, the velocity components, neglecting the rotation of the root around its axis, were defined in the logcosh co-ordination system (u, v, φ) (Hejnowicz and Karczewski, 1993) in the following way (Szymanowska-Pułka et al., 2012):

$$\begin{array}{rl}& \frac{\mathrm{d}u}{\mathrm{d}t}=c(u-u_0)\\ & \frac{\mathrm{d}v}{\mathrm{d}t}=-k\sin \left(\frac{\pi v}{v_0}\right)\\ & \frac{\mathrm{d}\varphi }{\mathrm{d}t}=0\end{array}$$ (3)

where u₀ = 0·175 and v₀ = 0·280 are constant lines referring to geometrical features of the root, while c = 1·20 and k = 0·25 are parameters determining the rates of a given point position in the direction u and v, respectively. Both u₀, v₀ and c, k are adjusted for a specific plant species. For LR in arabidopsis, these terms were specified and described (Szymanowska-Pułka et al., 2012). Equations (3) are valid for u > u₀ and v > v₀, respectively.

In the study we considered an example of cylindrical symmetry with a circular LR base (symmetrical case), and three examples of non-cylindrical symmetry with oval LR bases (asymmetrical cases), although in the latter there exist two planes of reflectional symmetry of the axial view of the organ. In the presented model, the diameter of the LR base of the symmetrical case is equal to the shorter diameters (D_S) of the LR bases of the three asymmetrical cases. The surfaces representing LRs with both cylindrical and non-cylindrical symmetry are defined by eqns (2), in which for the case of the circular LR base the v co-ordinate has been assumed to be a constant, in our case v = 0·56 (according to Szymanowska-Pułka et al., 2012), while for the case of the oval LR base the v co-ordinate has been defined through sigmoids and trigonometrical functions in the following way:

$$v=\frac{v_{min}}{1+e^{-au+b}}\left(\frac{1}{6f}+\frac{\cos \varphi }{5f}\right)+v_{min}$$ (4)

where v_min = v(ϕ = π/2), in our case v_min = 0·56; a, b are parameters of the sigmoid, determining the slope and the initial point of the sigmoid function, here a = 3, b = 3·5; f is the parameter chosen on the basis of the empirical observations and dependent on the asymmetry ratio D_L/D_S. For the three asymmetrical cases considered in the study, f was equal to 1·15, 0·6 and 0·35, decreasing with the increasing value of D_L/D_S. Thus for the asymmetrical cases, the logcosh co-ordination system [eqns (2)] was modified by introducing the sigmoid function for the v co-ordinate. Such a modification guarantees the oval shape at the base of the LR, and the circular shape at the top, as observed in real cases. For u = 0 (the top of the LR), v is almost constant around the axis of the LR apex; when u increases (towards the base of the LR), the asymmetry arises, which means that v changes around the axis. For u = 2 (at the base of the LR), the highest asymmetry is obtained. In the neighbourhood of ϕ = π/2, the v function [eqn (4)] is slightly modified (to v = 0·56) to eliminate cavity formation on the surface.

As for the asymmetrical surfaces, the points on the LR circumference are not equidistant from the organ axis, and defining the analytical form of the RERG field in such a modified co-ordinate system would be complicated. However, we applied a simple method to obtain the maps of RERG in these cases; namely, for a given point on a surface of the established co-ordinates (u, v, ϕ), RERG was calculated as for the symmetrical case, according to eqn (1). Then using the calculated RERGs, a map was developed for each asymmetrical surface. Such a solution could be adopted because RERG depends exclusively on the point co-ordinates (position in the co-ordinate system) and, according to our assumption, the apex grows with no rotation around its axis. This approach is very convenient because it simplifies calculations.

RESULTS

In Table 1 the measured values of the long and short diameters (D_L and D_S, respectively) and the calculated values of the asymmetry ratios D_L/D_S of 16 analysed LR bases are presented. No significant differences between the asymmetry ratios of the three analysed accessions were detected (Fig. 5), so in the subsequent procedure the mean value of all 16 cases was taken into consideration.

Table 1.

Quantitative description of morphological features of the bases of the analysed LRs and LR primordia in the three arabidopsis accessions

Line	No.	DL	DS	DL/DS
AUX1::YFP	1	6·77	3·23	2·10
	2	3·71	2·56	1·45
	3	9·74	5·71	1·71
DR5::GFP	4	7·42	3·83	1·94
	5	5·34	3·96	1·35
	6	10·98	5·71	1·92
	7	7·08	4·86	1·46
	8	8·79	5·28	1·66
	9	6·55	4·57	1·43
	10	10·26	4·94	2·08
	11	9·21	5·73	1·61
	12	8·57	4·91	1·75
	13	5·46	3·94	1·39
Col-0	14	8·8	4·87	1·81
	15	7·98	4·66	1·71
	16	5·83	3·42	1·70

D_L long and D_S short diameter of the ellipse-like figure representing the base; D_L/D_S asymmetry ratio.

Fig. 5.

Mean values ± standard errors of the asymmetry ratios of the three arabidopsis lines. No significant differences between the accessions have been detected (indicated by the letter a, P > 0·05).

The mean value ± standard error of the asymmetry ratios D_L/D_S of 16 analysed LR bases was 1·69 ± 0·06, with a maximal observed ratio equal to 2·10 and a minimal ratio equal to 1·35. An oval shape of the basal portion of the LR is present both in young primordia (Fig. 6A, B) and in more developed organs (Fig. 6C, D). Moreover, their cell pattern (in the front view) undergoes radialization towards the central part of the forming apex (Fig. 6C), which indicates a change of shape of the organ circumference along its axis. Interestingly, in older LRs, the pattern in the apical part is usually circular, as shown in Fig. 6E.

The maps of RERGs for LRs of various asymmetry ratios of the basal portion are presented in Fig. 7. The map in Fig. 7A represents the RERG distribution for the case of the hypothetical LR with a regular circular base, for which the asymmetry ratio is 1. The three remaining maps (Fig. 7B–D) refer to the cases observed in anatomical studies, i.e. in LRs with bases of ellipse-like shape. Thus Fig 7B–D represents the maps for the LRs with the least observed asymmetry ratio (1·35), with an average asymmetry ratio (1·69) and with the largest observed asymmetry ratio (2·10), respectively. Such a choice of parameters allows us to best consider the model in terms of the real geometry of the LR base.

Fig. 7.

Distribution of growth rates for the LR in arabidopsis with (A) cylindrical symmetry and (B–D) non-cylindrical symmetry. The asymmetry ratios calculated as (D_L/D_S) are the following: (A) 1·00, (B) 1·35, (C) 1·69 and (D) 2·10. u and v are curvilinear co-ordinates of points for which RERGs are calculated (ϕ changes along the circumference from 0 to 2π). Arrows point to indicatrices calculated at the points localized to the end of the long diameter. The inset in the central part shows a comparison of the LR shapes in the case of cylindrical symmetry and the non-cylindrical symmetry; the long (D_L) and short (D_S) diameters of the LR base are indicated.

The maps differ mostly in the region of the LR bases, while towards the apical part, where the organ shape is the same in all cases, growth distribution is similar. For the whole LR surface, growth rates in the periclinal direction (P in Fig. 2B) predominate, while growth rates in anticlinal and latitudinal (in Fig. 2B, A and L, respectively) directions are smaller. In reference to the case of an LR with a circular base (Fig. 7A), RERGs along the P direction change only slightly with increasing asymmetry (Fig. 7B–D), and do so independently of the position of the points for which they are calculated. Obviously, for the points localized to both ends of the short diameter, RERGs attain the same values as in the symmetrical case. However, for the points localized to both ends of the long diameter (Fig. 7, arrows), growth rates in the A direction are larger in the case of both the minimal (Fig. 7B) and average (Fig. 7C) asymmetry, whereas in the case of the maximal asymmetry (Fig. 7D) they are evidently smaller. There are some – but not significant – differences in the values of growth rates in the L direction for all analysed cases of asymmetry.

It is rather difficult to indicate differences in the values of RERGs along the circumference of the LR base on the basis of Fig. 7. However, in the asymmetrical cases, the RERGs change with the co-ordinate ϕ. In Fig. 8, the four plots are shown directly referring to the four above-described cases of various asymmetry ratios of the LR base, namely for the asymmetry ratio equal to 1, 1·35, 1·69 and 2·10 (Fig. 8A–D, respectively). The three lines on each plot represent values of growth rates calculated in the principal directions of growth: periclinal (RERG_P), anticlinal (RERG_A) and latitudinal (RERG_L), as functions of the co-ordinate ϕ. In Fig. 8A, referring to the LR of the cylindrical symmetry, RERGs do not change around the axis, as the points on the circumference are at an equal distance from the axis and for this case the map of RERGs is of a symmetrical character (Fig. 7A). As the asymmetry is introduced (Fig. 8B–D), values of RERGs change with the position on the circumference, and this changeability becomes more and more pronounced as the asymmetry increases.

Fig. 8.

Values of RERGs vs. the angular co-ordinate ϕ for the LRs of various types of asymmetry of the base (u = 2·0) directly referring to the cases shown in Fig. 5: (A) cylindrical symmetry D_L/D_S = 1; (B–D) non-cylindrical symmetry; (B) D_L/D_S = 1·35; (C) D_L/D_S = 1·69; (D) D_L/D_S = 2.10. The three curves in each plot represent RERGs calculated in the three principal directions of growth specified in Fig. 2B: periclinal, RERG_P; anticlinal, RERG_A; and latitudinal, RERG_L (see the key in A). Vertical dashed lines indicate values of ϕ referring to the points where D_L and D_S end; the inset in (A) shows the positions of points at the ends of the long and short diameter of the circumference at the base (for ϕ = 0 and ϕ = π/2, respectively). Notice the increasing changeability of the RERG values with the accruing asymmetry ratio.

The values of RERGs at the points on both ends of the short diameter (ϕ = π/2 and ϕ = 3π/2) are equal to the constant values of the relevant RERGs for the case of cylindrical symmetry (Fig. 8A), namely RERG_P = 1·26, RERG_A = 0·68 and RERG_L = −0·22. Interestingly, the value of RERG_P at the point referring to D_S is maximal, while the values of both RERG_A and RERG_L at this point are minimal (or locally minimal, as in Fig. 8D). The curves pertaining to RERG_P and RERG_L reveal a rather simple periodic character for all cases of the asymmetrical LR base (Fig. 8B–D), while the curves pertaining to RERG_A are more complex. In the case of a slight asymmetry (Fig. 8B), the course of the curve RERG_A is simple and resembles that of RERG_L; namely for ϕ = 0, referring to the point where D_L ends on the circumference, the rate of growth in the A direction is maximal, then it decreases with decreasing diameter to attain the minimal value for ϕ = π/2, referring to the point where D_S ends on the circumference. Again, with increasing diameter, the value of RERG_A increases to the point where ϕ = π, at which point the diameter is maximal. However, as the asymmetry increases, the curve RERG_A changes more rapidly, resulting in the appearance of some additional local minima and maxima (Fig. 8C, D). Thus for the cases shown in Fig. 8C and D, for ϕ = 0 and ϕ = π (both referring to the points where D_L ends), as well as for ϕ = π/2 and ϕ = 3π/2 (referring to the points where D_S ends), the rate of growth in the A direction attains a locally minimal value, while the maximal value of RERG_A appears for ϕ = π/4, ϕ = 3π/4, ϕ = 5π/4 and ϕ = 7π/4, the points in the vicinity of the point where D_S ends. The new minima on the curve RERG_A (for ϕ = 0 and ϕ = π) result from a decrease in the rate of growth in the A direction at the points localized to both ends of the long diameter. Such a situation seems natural in the case of strong asymmetry, because a strongly elongated base of the LR is usually observed when the LR is at the developmental phase of leaving the elongation zone of the parent root, when the rate of growth along the parental root axis (the A direction) actually decreases. This finally leads to a decrease of the asymmetry ratio. Consequently the LR base tends to cylindrical symmetry.

DISCUSSION

Plant organs such as the root and shoot apices usually reveal a rotational symmetry with reference to their axes. Microscopic observations of the LRs in arabidopsis provided evidence for an oval shape of the LR base in cross-section, which indicated the lack of rotational symmetry of the organ in this region. Values of the asymmetry ratios for various LRs were in the range 1·35–2·10 and were always larger than 1. This means that the bases of all analysed LR primordia or LRs were elongated along the axis of the main root. A significant range of the ratios may result from significant differences in rates of elongation for individual parent roots.

Such a case of atypical geometry has been reported recently by Lucas et al. (2013) who showed that the radialization of the LR primordium takes place at early stages of its formation through tangential divisions of the cells flanking the central group of rapidly dividing cells (fig. 1B in Lucas et al., 2013; fig. 3D in Szymanowska-Pułka et al., 2012). Lucas et al. (2013) managed to reconstruct the LR primordium in 3-D based on confocal images. Spatial analysis of the reconstructed structure revealed an elliptical shape of the basal part of the primordium and a round shape of the apical part. Here, we confirm these results and introduce a mathematical description of the growth of an organ undergoing radialization in the course of its development.

Calculating the RERGs in many (infinite number) directions at chosen points on the surface of the LR allowed for the construction of maps of the distribution of growth rates. The maps of RERGs drawn for LRs of various values of the asymmetry ratio revealed significant differences in the distribution of growth rates for the analysed cases. This suggests that the manner of growth of the plant organ depends on its geometry. The above conclusion has also been reached from mathematical functions describing growth rates of an organ whose geometry (size and shape) changes during development (Hejnowicz and Hejnowicz, 1991), for example, LR (Szymanowska-Pułka, 2007; Szymanowska-Pułka and Nakielski, 2010; Szymanowska-Pułka et al., 2012) or leaf (Lipowczan et al., 2013). Namely, these functions are of an non-steady (time-dependent) character in contrast to the case of the organs which preserve their geometry during growth, such as the mature main root (Nakielski, 2008; Nakielski and Piekarska-Stachowiak, 2013) or shoot (Nakielski, 2000) apices, for which the functions for growth rates are of a steady (time-independent) character.

Thus, based on these premises, we may consider the manner of growth of a plant organ to be dependent on its geometry. Now, a question arises of whether there is a feedback loop between the two. Recent studies (Lintilhac, 2014) show that changes in shape of an organ surface impose a re-organization of the principal stress direction, that next affects the re-orientation of the principal directions of growth (Bałek et al., 2014). As the principal directions of growth determine the orientation of division cell walls (Szymanowska-Pułka et al., 2012), this leads to the reshaping of the growing organ (Lintilhac, 2014). So the answer to our question is, yes, the manner of growth of a plant organ and the organ geometry mutually affect each other. It is worth noticing that in this co-relationship a mechanical aspect plays an important role, i.e. a geometrical signal (shape) is transferred to a directional signal (principal directions of growth) through redistribution of mechanical stress within the organ (Szymanowska-Pułka, 2013).

The growth rates in the latitudinal (RERG_L) direction may attain negative values, as shows in Fig. 8 (blue lines). It is worth mentioning that a negative value of RERG does not mean that there is a ‘negative’ growth in a given direction, but rather that there is a negative rate of growth (change of velocity) in this direction (Nakielski and Lipowczan, 2013). In other words, growth slows down in this area. This may happen in the outermost region of the root apex due to natural deformation of the root cap cells, that may become thinner with time. Nevertheless, when we refer to the other lines in Fig. 8, representing growth rates in periclinal (red) and anticlinal (green) directions, we see that their values are always positive, which means that growth in these directions speeds up. To summarize, it is necessary to take into consideration the RERGs in the three principal directions to obtain a full description of growth at a point.

In our research we observed basal parts of LRs coming from various transgenic lines. However, it was not our objective to find differences in the geometry of LRs in these lines, but to illustrate differences in growth rate distribution in LRs of various asymmetry ratios. The main novelty of the present study is a description of growth of a plant organ locally lacking a rotational symmetry in terms of the growth tensor – a mathematical tool previously applied only to description of the organs of rotational symmetry. The results show a direct correlation between the mannery of growth of the LR primordium and its shape. The method employed for determining the field of growth rates for LRs presenting different shapes appeared sufficient and satisfactory for visualizing the asymmetric RERG distribution on the surfaces of the LR apices of various geometries. This new approach may provide new insights into plant organ formation and shape and may be helpful in studies on developmental processes in plants. In order to determine precisely the shape of the LR base, further studies are necessary with regard to the localization of the LR boundaries. Another challenge will be to analyse changes in the asymmetry of the LR base during organ development in order to determine a possible time dependence of the growth rate distribution.