Jerzy Nakielski

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.

The description of growth of plant organs: a continuous approach based on the growth tensor

Our developing systems are growing plant organs, Plant organs grow symplastically, i.e., in a co- ordinated way. In our approach we treat the symplastic growth of a plant organ as an example of irreversible deformation (plastic strain). Specificity of the organ in comparison to elasto-plastic or elasto-viscous solids is expressed mainly in cell divisions. Symplastic growth leads to the concept of a growth tensor. Cell divisions occur in the principal planes of this tensor.

Effect of mechanical stress on Zea root apex. I. Mechanical stress leads to the switch from closed to open meristem organization

The effect of mechanical stress on the root apical meristem (RAM) organization of Zea mays was investigated. In the experiment performed, root apices were grown through a narrowing of either circular (variant I) or elliptical (variant II) shape. This caused a mechanical impedance distributed circumferentially or from the opposite sides in variant I and II, respectively. The maximal force exerted by the growing root in response to the impedance reached the value of 0.15 N for variant I and 0.08 N for variant II. Significant morphological and anatomical changes were observed. The changes in morphology depended on the variant and concerned diminishing and/or deformation of the cross-section of the root apex, and buckling and swelling of the root. Anatomical changes, similar in both variants, concerned transformation of the meristem from closed to open, an increase in the number of the cell layers at the pole of the root proper, and atypical oblique divisions of the root cap cells. After leaving the narrowing, a return to both typical cellular organization and morphology of the apex was observed. The results are discussed in terms of three aspects: the morphological response, the RAM reorganization, and mechanical factors. Assuming that the orientation of division walls is affected by directional cues of a tensor nature, the changes mentioned may indicate that a pattern of such cues is modified when the root apex passes through the narrowing, but its primary mode is finally restored.

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.

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.

Mechanics of the Meristems

In this chapter, the structure, function, and growth of apical meristems and cambium are discussed from a perspective of mechanics. We first characterize the meristems and point to implications of the symplasm, apoplasm, and organismal concepts for our understanding of plant morphogenesis. Then we discuss the symplastic (coordinated) growth and a putative role of principal directions of growth and mechanical stress tensor in the meristem function, also explaining how the principal directions are manifested in cellular pattern and cell behavior. The present knowledge on the mechanics of meristems, in particular on the distribution of mechanical stress and on the mechanical properties of the meristems, is to a large extent speculative. Our objectives are to present and discuss the available empirical data and hypotheses on the meristem mechanics, and the evidence on the role of mechanical factors in plant morphogenesis.

The simulation model of growth and cell divisions for the root apex with an apical cell in application to Azolla pinnata.

In contrast to seed plants, the roots of most ferns have a single apical cell which is the ultimate source of all cells in the root. The apical cell has a tetrahedral shape and divides asymmetrically. The root cap derives from the distal division face, while merophytes derived from three proximal division faces contribute to the root proper. The merophytes are produced sequentially forming three sectors along a helix around the root axis. During development, they divide and differentiate in a predictable pattern. Such growth causes cell pattern of the root apex to be remarkably regular and self-perpetuating. The nature of this regularity remains unknown. This paper shows the 2D simulation model for growth of the root apex with the apical cell in application to Azolla pinnata. The field of growth rates of the organ, prescribed by the model, is of a tensor type (symplastic growth) and cells divide taking principal growth directions into account. The simulations show how the cell pattern in a longitudinal section of the apex develops in time. The virtual root apex grows realistically and its cell pattern is similar to that observed in anatomical sections. The simulations indicate that the cell pattern regularity results from cell divisions which are oriented with respect to principal growth directions. Such divisions are essential for maintenance of peri-anticlinal arrangement of cell walls and coordinated growth of merophytes during the development. The highly specific division program that takes place in merophytes prior to differentiation seems to be regulated at the cellular level.

Spatial and Directional Variation of Growth Rates in Arabidopsis Root Apex: A Modelling Study

Growth and cellular organization of the Arabidopsis root apex are investigated in various aspects, but still little is known about spatial and directional variation of growth rates in very apical part of the apex, especially in 3D. The present paper aims to fill this gap with the aid of a computer modelling based on the growth tensor method. The root apex with a typical shape and cellular pattern is considered. Previously, on the basis of two types of empirical data: the published velocity profile along the root axis and dimensions of cell packets formed in the lateral part of the root cap, the displacement velocity field for the root apex was determined. Here this field is adopted to calculate the linear growth rate in different points and directions. The results are interpreted taking principal growth directions into account. The root apex manifests a significant anisotropy of the linear growth rate. The directional preferences depend on a position within the root apex. In the root proper the rate in the periclinal direction predominates everywhere, while in the root cap the predominating direction varies with distance from the quiescent centre. The rhizodermis is distinguished from the neighbouring tissues (cortex, root cap) by relatively high contribution of the growth rate in the anticlinal direction. The degree of growth anisotropy calculated for planes defined by principal growth directions and exemplary cell walls may be as high as 25. The changes in the growth rate variation are modelled.

A method to determine the displacement velocity field in the apical region of the Arabidopsis root

In angiosperms, growth of the root apex is determined by the quiescent centre. All tissues of the root proper and the root cap are derived from initial cells that surround this zone. The diversity of cell lineages originated from these initials suggests an interesting variation of the displacement velocity within the root apex. However, little is known about this variation, especially in the most apical region including the root cap. This paper shows a method of determination of velocity field for this region taking the Arabidopsis root apex as example. Assuming the symplastic growth without a rotation around the root axis, the method combines mathematical modelling and two types of empirical data: the published velocity profile along the root axis above the quiescent centre, and dimensions of cell packet originated from the initials of epidermis and lateral root cap. The velocities, calculated for points of the axial section, vary in length and direction. Their length increases with distance from the quiescent centre, in the root cap at least twice slower than in the root proper, if points at similar distance from the quiescent centre are compared. The vector orientation depends on the position of a calculation point, the widest range of angular changes, reaching almost 90°, in the lateral root cap. It is demonstrated how the velocity field is related to both distribution of growth rates and growth-resulted deformation of the cell wall system. Also changes in the field due to cell pattern asymmetry and differences in slope of the velocity profile are modelled.

Growth and cellular patterns in the petal epidermis of Antirrhinum majus: empirical studies

BACKGROUND AND AIMS Analysis of cellular patterns in plant organs provides information about the orientation of cell divisions and predominant growth directions. Such an approach was employed in the present study in order to characterize growth of the asymmetrical wild-type dorsal petal and the symmetrical dorsalized petal of the backpetals mutant in Antirrhinum majus. The aims were to determine how growth in an initially symmetrical petal primordium leads to the development of mature petals differing in their symmetry, and to determine how specific cellular patterns in the petal epidermis are formed. METHODS Cellular patterns in the epidermis in both petal types over consecutive developmental stages were visualized and characterized quantitatively in terms of cell wall orientation and predominant types of four-cell packets. The data obtained were interpreted in terms of principal directions of growth (PDGs). KEY RESULTS Both petal types grew predominantly along the proximo-distal axis. Anticlinal cell walls in the epidermis exhibited a characteristic fountain-like pattern that was only slightly modified in time. New cell walls were mostly perpendicular to PDG trajectories, but this alignment could change with wall age. CONCLUSIONS The results indicate that the predominant orientation of cell division planes and the fountain-like cellular pattern observed in both petal types may be related to PDGs. The difference in symmetry between the two petal types arises because PDG trajectories in the field of growth rates (growth field) controlling petal growth undergo gradual redefinition. This redefinition probably takes place in both petal types but only in the wild-type does it eventually lead to asymmetry in the growth field. Two scenarios of how redefinition of PDGs may contribute to this asymmetry are considered.