Application of a changing field of growth rates to a description of root apex formation

Topological traits of a cellular pattern versus growth rate anisotropy in radish roots

The topology of a cellular pattern, which means the spatial arrangement of cells, directly corresponds with cell packing, which is crucial for tissue and organ functioning. The topological features of cells that are typically analyzed are the number of their neighbors and the cell area. To date, the objects of most topological studies have been the growing cells of the surface tissues of plant and animal organs. Some of these researches also provide verification of Lewis's Law concerning the linear correlation between the number of neighboring cells and the cell area. Our aim was to analyze the cellular topology and applicability of Lewis's Law to an anisotropically growing plant organ. The object of our study was the root apex of radish. Based on the tensor description of plant organ growth, we specified the level of anisotropy in specific zones (the root proper, the columella of the cap and the lateral parts of the cap) and in specific types of both external (epidermis) and internal tissues (stele and ground tissue) of the apex. The strongest anisotropy occurred in the root proper, while both zones of the cap showed an intermediate level of anisotropy of growth. Some differences in the topology of the cellular pattern in the zones were also detected; in the root proper, six-sided cells predominated, while in the root cap columella and in the lateral parts of the cap, most cells had five neighbors. The correlation coefficient rL between the number of neighboring cells and the cell area was high in the apex as a whole as well as in all of the zones except the root proper and in all of the tissue types except the ground tissue. In general, Lewis's Law was fulfilled in the anisotropically growing radish root apex. However, the level of the applicability (rL value) of Lewis's Law was negatively correlated with the level of the anisotropy of growth, which may suggest that in plant organs in the regions of anisotropic growth, the number of neighboring cells is less dependent on the cell size.

Growth rate distribution in the forming lateral root of arabidopsis

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 (DL and DS, respectively) of the ellipse-like figure representing the bases of particular LRs, their asymmetry ratios (DL/DS) 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.

Form matters: morphological aspects of lateral root development

BACKGROUND

The crucial role of roots in plant nutrition, and consequently in plant productivity, is a strong motivation to study the growth and functioning of various aspects of the root system. Numerous studies on lateral roots, as a major determinant of the root system architecture, mostly focus on the physiological and molecular bases of developmental processes. Unfortunately, little attention is paid either to the morphological changes accompanying the formation of a lateral root or to morphological defects occurring in lateral root primordia. The latter are observed in some mutants and occasionally in wild-type plants, but may also result from application of external factors.

SCOPE AND CONCLUSIONS

In this review various morphological aspects of lateral branching in roots are analysed. Morphological events occurring during the formation of a typical lateral root are described. This process involves dramatic changes in the geometry of the developing organ that at early stages are associated with oblique cell divisions, leading to breaking of the symmetry of the cell pattern. Several types of defects in the morphology of primordia are indicated and described. Computer simulations show that some of these defects may result from an unstable field of growth rates. Significant changes in both primary and lateral root morphology may also be a consequence of various mutations, some of which are auxin-related. Examples reported in the literature are considered. Finally, lateral root formation is discussed in terms of mechanics. In this approach the primordium is considered as a physical object undergoing deformation and is characterized by specific mechanical properties.

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

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.

The tensor-based model for growth and cell divisions of the root apex. II. Lateral root formation

In this work, the formation of the virtual lateral root (VLR) is shown. The VLR is formed using the 2D simulation model of growth and cell divisions based on the concept of growth tensor, specified for radish. Growth is generated by the field of growth rates of an unsteady type (GT field). Principal directions of growth (PDGs) are assumed to define the orientation of cell divisions. Temporal sequences of the VLR formation are a result of an application of the GT field to the polygon meshwork representing cell pattern of already initiated primordium. The computer-generated lateral root (LR) develops realistically, and its cell pattern is vivid and similar to that observed in anatomical sections. The real and virtual LRs show similar cellular organization, both originate from a small group of cells situated in two-cell layers of the pericycle and both layers are engaged in the LR development. The LR formation seems to be controlled at the tensor level and individual cells presumably detect PDGs and obey them in the course of the cell divisions. PDGs are postulated to affect the cellular organization of the LR. Using the method of computer simulations, cellular aspects of the LR morphogenesis are discussed.

The tensor-based model for growth and cell divisions of the root apex. I. The significance of principal directions

Plant organs grow symplastically, i.e. in a continuous and coordinated way. Such growth is of a tensor nature, which is manifested in the property that at every point of the organ three mutually orthogonal principal growth directions (PDG) can be recognized. The PDGs are postulated to affect orientation of cell divisions. This paper shows for the first time the 2D simulation model for growth in which cells divide taking into account the PDGs. The model, conceptually based on the growth tensor (GT), is applied to the root apex of radish, having a quiescent centre (QC). It shows the simulation of how exemplary cell pattern of the real root apex develops in time. The results provide satisfactory description of the root growth. The computer-generated cell pattern is realistic and more or less steady indicating that PDGs are important for growth. Presumably cells detect PDGs and obey them in the course of cell divisions. Computer generated division walls, perpendicular to PDGs, form periclinal and anticlinal zigzags as regular as those observed in microscopic sections. Growth tensor defines a field of growth rates at the organ level. QC, fundamental in this field, determines the group of quiescent initial cells which is, in turn, surrounded by active functional initials, from which all tissues of the root apex originate. The present simulations have shown that stability of generated cell pattern depends on whether the group of the functional initials is permanent; if this is not the case, the cell wall pattern changes in accordance with PDGs.