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.     

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.     

MODELING OF MERISTEMATIC GROWTH OF ROOT APICES IN A NATURAL COORDINATE SYSTEM

A curvilinear, orthogonal coordinate system, which resembles the pattern of periclines and anticlines in the cellular network of root apices, is presented. The system makes possible an analysis of the dynamics of apical growth: the relationship between growth rates and tensile stresses in cell walls. In this paper the coordinate system is used in modeling the growth and cell partitioning in the apical domes. The symplastic growth is described by means of the growth tensor which is assumed to have diagonal form in the system, so its coordinate lines represent the principal directions of growth rate. The coordinate system and the growth tensor in diagonal form assure temporal stability of form and cellular structure of the modeled apex including transition from the dome to the cylindrical part of the apex. The spatial and temporal aspects of the dome part of two types of root apices—one with maximum of volumetric relative growth rate at the center region of the apex, and another with minimum of the rate (quiescent center)—are described. The maximum of the rate at the center results in the cellular pattern with an apical cell and merophytes, the minimum results in ribs of cells (inside the root proper) converging toward the quiescent center.     

The tensor-based model of plant growth applied to leaves of Arabidopsis thaliana: A two-dimensional computer model

Plant organs grow in coordinated and continuous way. Such growth is of a tensor nature, hence there is an infinite number of different directions of growth rate in each point of the growing organ. Three mutually orthogonal directions of growth can be recognized in which growth achieves extreme values (principal directions of growth [PDGs]). Models based on the growth tensor have already been successfully applied to the root and shoot apex. This paper presents the 2D model of growth applied to the arabidopsis leaf. The model employs the growth tensor method with a non-stationary velocity field. The postulated velocity functions are confirmed by growth measurements with the aid of the replica method.     

Differential growth resulting in the specification of different types of cellular architecture in root meristems

Symplastic growth of plant organs may be described by a continuous growth tensor field. In tensorial analysis of meristems, the trajectories of periclinal and anticlinal cell walls represent trajectories of the principal directions of growth (PDGs); this follows from the maintenance of mutual orthogonality between periclinal and anticlinal wall trajectories during growth. Periclinal and anticlinal cell divisions are also oriented in the principal planes of growth. The growth tensor for the root apex is specified in such a way that the principal directions of the tensor fit the pattern of periclinal and anticlinal walls in the apex, and that the grid formed by material particles aligned along PDG trajectories preserve this alignment during growth. Two growth tensors are formulated--one giving a maximum and the other giving a minimum of the volumetric relative elemental growth rate at the region of the initial cell(s). Temporal sequences of deformation of a grid formed by lines coinciding with the principal directions of growth are shown. The formation of cellular patterns in root apices is simulated. Two types of patterns are obtained: one with an apical cell and merophytes, and another with files of cells converging towards a quiescent centre.     

The source and lateral transport of growth inhibitors in geotropically stimulated roots of Zea mays and Pisum sativum

Summary The positive geotropic responses of the primary roots of Zea mays and Pisum sativum seedlings depend upon at least one growth inhibiting factor which arises in the root cap and which moves basipetally through the apex into the extending zone. The root apex (as distinct from the cap) and the regions more basal to the extending zone are not sources of growth regulators directly involved in the geotropic response. A difference in the concentration or effectiveness of the inhibitory factor(s) arising in the cap must be established between the upper and lower halves of a horizontal root. Positive geotropic curvature in a horizontal root is attributable, at least in part, to a downward lateral transport of inhibitor(s) from the upper to the lower half of the organ.     

Root Cap Mucilage and Extracellular Calcium as Modulators of Cellular Growth in Postmitotic Growth Zones of the Maize Root Apex*

Abstract: The control of maize root growth by root cap mucilage and extracellular calcium (Ca) was examined. Special attention was paid to the influence of these factors on cellular aspects of root growth, such as cell shape and organization of the microtubular (MT) cytoskeleton. Externally supplied Ca impaired the transition of early post-mitotic cells from a more-or-less apolar mode of expansion to a strictly anisotropic mode of elongation accompanied by their more rapid growth. However, this inhibitory effect of Ca was not associated with any re-arrangement of the cortical MTs, their transverse arrays, with respect to the root axis, being maintained under these conditions. Root mucilage, collected from donor root caps and placed around root tips, exerted a similar effect on cell shapes as did externally supplied Ca. In contrast, roots grown in a medium of low Ca content, or from which the root cap mucilage was continually removed, had more elongated cell shapes in their post-mitotic growth regions when compared to the control roots. These findings are consistent with a notion that Ca is present in the root cap mucilage in physiologically relevant amounts and can mediate growth responses in both the PIG region and the apical part of the elongation zone. Integrating several known effects of Ca ions on growth at the root apex, a hypothesis is proposed that a Ca-mediated and MT-independent control of cell growth in the PIG region might be involved in morphogenetic root movements (e.g. gravitropism), and that root growth responses could be initiated by an asymmetric distribution of extracellular calcium, or root cap slime, around the growing root tip.     

Principal directions of growth and the generation of cell patterns in wild-type and gib-1 mutant roots of tomato (Lycopersicon esculentum Mill.) grown in vitro

The patterns of cell growth and division characteristic of the apex of tomato roots grown in vitro were simulated by computer using a growth tensor (GT). The GT was used to clarify the basis of the altered cell patterns found within apices of roots whose gibberellin levels had been depressed by mutation (at the GIB-1 locus) or through application of the gibberellin-biosynthesis inhibitor, 2S,3S paclobutrazol. At the pole of wild-type roots, where the cell files of the cortex converge, there are commonly only one or two tiers of cortical cells sandwiched between the pole of the stele and the cap initials. By contrast, root apices of the gib-1 mutant contain additional tiers in this region. The development of these additional tiers is suppressed when roots of the mutant are grown in the presence of gibberellic acid (GA₃), but could be induced in wild-type roots when they are grown in 2S,3S paclobutrazol. The wild-type cell pattern can be simulated using the GT and by the application of appropriate rules that govern cell growth and division. The induced variations in cell pattern are interpreted as being due to displacements, within the apex, of the principal directions of growth (PDGs), which are represented, in part, by the set of periclines and anticlines seen in the cell wall network; these, in turn, are utilized in the specification of the GT. During normal (wild-type) root growth, the PDGs maintain a stable pattern and the corresponding cell pattern is also stable. However, in order to interpret the cellular behaviour found in wild-type roots grown in 2S, 3S paclobutrazol, simulation using the GT shows that, if the pattern of PDGs is destabilized and displaced distally along the root axis, the cell pattern reorganizes into that typical of gib-1 mutant roots. Conversely, the cell pattern of gib-1 roots, which reverts to wild-type upon exposure to GA₃, can be simulated if the PDGs are displaced proximally to the inside of the apex whereupon the number of cortical tiers at the root pole decreases. These results suggest a link between endogenous gibberellin level and the specification of the PDGs in the growing tomato root apex. Furthermore, the evidence of cell patterns from gib-1 roots suggests that, in order to achieve stability of PDGs with concomitant stable cellular patterning, an optimal gibberellin level is necessary. In practice, this can be attained by culturing the mutant roots in medium containing 1 M GA₃.Abbreviations GA₃ gibberellic acid - GT growth tensor - NCS natural coordinate system - PDG principal direction of growth - QC quiescent centre - RERG relative elemental rate of growth We are grateful to the former Agricultural and Food Research Council for financial support under the International Scientific Interchange Scheme to enable J.N. to work at Long Ashton Research Station, and to K. Kurczyski (Silesian University, Katowice, Poland) for help in writing a computer program for cell proliferation. Preparation of the model for growth and division was supported in part by a grant from the Committee for Scientific Research, Poland.     

Trabecular bone adaptation with an orthotropic material model.

Most bone adaptation algorithms, that attempt to explain the connection between bone morphology and loads, assume that bone is effectively isotropic. An isotropic material model can explain the bone density distribution, but not the structure and pattern of trabecular bone, which clearly has a mechanical significance. In this paper, an orthotropic material model is utilized to predict the proximal femur trabecular structure. Two hypotheses are combined to determine the local orientation and material properties of each element in the model. First, it is suggested that trabecular directions, which correspond to the orthotropic material axes, are determined locally by the maximal principal stress directions due to the multiple load cases (MLC) the femur is subject to. The second hypothesis is that material properties in each material direction can be determined using directional stimuli, thus extending existing adaptation algorithms to include directionality. An algorithm is utilized, where each iteration comprises of two stages. First, material axes are rotated to the direction of the largest principal stress that occurs from a multiple load scheme applied to the proximal femur. Next, material properties are modified in each material direction, according to a directional stimulus. Results show that local material directions correspond with known trabecular patterns, reproducing all main groups of trabeculae very well. The local directional stiffnesses, degree of anisotropy and density distribution are shown to conform to real femur morphology.     

Directional anisotropies reveal a functional segregation of visual motion processing for perception and action

Human exhibits an anisotropy in direction perception: discrimination is superior when motion is around horizontal or vertical rather than diagonal axes. In contrast to the consistent directional anisotropy in perception, we found only small idiosyncratic anisotropies in smooth pursuit eye movements, a motor action requiring accurate discrimination of visual motion direction. Both pursuit and perceptual direction discrimination rely on signals from the middle temporal visual area (MT), yet analysis of multiple measures of MT neuronal responses in the macaque failed to provide evidence of a directional anisotropy. We conclude that MT represents different motion directions uniformly, and subsequent processing creates a directional anisotropy in pathways unique to perception. Our data support the hypothesis that, at least for visual motion, perception and action are guided by inputs from separate sensory streams. The directional anisotropy of perception appears to originate after the two streams have segregated and downstream from area MT.     

Measurement of structural anisotropy in femoral trabecular bone using clinical-resolution CT images

Discrepancies in finite-element model predictions of bone strength may be attributed to the simplified modeling of bone as an isotropic structure due to the resolution limitations of clinical-level Computed Tomography (CT) data. The aim of this study is to calculate the preferential orientations of bone (the principal directions) and the extent to which bone is deposited more in one direction compared to another (degree of anisotropy). Using 100 femoral trabecular samples, the principal directions and degree of anisotropy were calculated with a Gradient Structure Tensor (GST) and a Sobel Structure Tensor (SST) using clinical-level CT. The results were compared against those calculated with the gold standard Mean-Intercept-Length (MIL) fabric tensor using micro-CT. There was no significant difference between the GST and SST in the calculation of the main principal direction (median error=28°), and the error was inversely correlated to the degree of transverse isotropy (r=−0.34, p<0.01). The degree of anisotropy measured using the structure tensors was weakly correlated with the MIL-based measurements (r=0.2, p<0.001). Combining the principal directions with the degree of anisotropy resulted in a significant increase in the correlation of the tensor distributions (r=0.79, p<0.001). Both structure tensors were robust against simulated noise, kernel sizes, and bone volume fraction. We recommend the use of the GST because of its computational efficiency and ease of implementation. This methodology has the promise to predict the structural anisotropy of bone in areas with a high degree of anisotropy, and may improve the in vivo characterization of bone.     

Root responses to soil physical conditions; growth dynamics from field to cell

Root growth in the field is often slowed by a combination of soil physical stresses, including mechanical impedance, water stress, and oxygen deficiency. The stresses operating may vary continually, depending on the location of the root in the soil profile, the prevailing soil water conditions, and the degree to which the soil has been compacted. The dynamics of root growth responses are considered in this paper, together with the cellular responses that underlie them. Certain root responses facilitate elongation in hard soil, for example, increased sloughing of border cells and exudation from the root cap decreases friction; and thickening of the root relieves stress in front of the root apex and decreases buckling. Whole root systems may also grow preferentially in loose versus dense soil, but this response depends on genotype and the spatial arrangement of loose and compact soil with respect to the main root axes. Decreased root elongation is often accompanied by a decrease in both cell flux and axial cell extension, and recent computer-based models are increasing our understanding of these processes. In the case of mechanical impedance, large changes in cell shape occur, giving rise to shorter fatter cells. There is still uncertainty about many aspects of this response, including the changes in cell walls that control axial versus radial extension, and the degree to which the epidermis, cortex, and stele control root elongation. Optical flow techniques enable tracking of root surfaces with time to yield estimates of two-dimensional velocity fields. It is demonstrated that these techniques can be applied successfully to time-lapse sequences of confocal microscope images of living roots, in order to determine velocity fields and strain rates of groups of cells. In combination with new molecular approaches this provides a promising way of investigating and modelling the mechanisms controlling growth perturbations in response to environmental stresses.     

Proton Pumping in Growing Part of Maize Root: Its Correlation with 14-3-3 Protein Content and Changes in Response to Osmotic Stress

The spatial pattern of mitotic activity, cell elongation, rate of H+ fluxes, and 14-3-3 protein content were determined in Zea mays roots. We found that the regions along the apical part of the growing root conversely differ in their proton pumping activity. Higher rate of H+ efflux coincides with higher growth rate and correlates with increased 14-3-3 protein content in membrane preparations. The segment consisting of the root cap and the apical part of the meristem exerts net inward proton pumping, which can be inverted under fusicoccin treatment or osmotic stress. In the latter case, this inversion is accompanied by accumulation of 14-3-3 protein in plasma membranes. The results obtained highlight 14-3-3 protein as an obvious candidate for the fine regulation of plasma membrane H+-ATPase in root apex.     

Growth regulators in changing apical growth at transition to flowering

Reorganization of growth in the shoot apex ofChenopodium rubrum during transition to flowering is described. Growth and morphogenic changes — a rise in cell division rate, changes in leaf and bud formation and changes in directions of cellular growth — are viewed from the aspect of a possible role of growth hormones in controlling these changes. Growth and morphogenic effects of exogenous growth regulators in the shoot apex ofChenopodium are summarized and their floral effects explained in terms of changing apical growth correlations. New evidence concerning the timing of increased cell division rate and showing the limited requirement of axillary cell division and a shift to more vertical direction of growth in the apex in the floral developmental pathway was obtained in experiments with kinetin application and by surgical treatments.     

The Root Apex of Arabidopsis thaliana Consists of Four Distinct Zones of Growth Activities: Meristematic Zone,Transition Zone,Fast Elongation Zone and Growth Terminating Zone

In the growing apex of Arabidopsis thaliana primary roots, cells proceed through four distinct phases of cellular activities. These zones and their boundaries can be well defined based on their characteristic cellular activities. The meristematic zone comprises, and is limited to, all cells that undergo mitotic divisions. Detailed in vivo analysis of transgenic lines reveals that, in the Columbia-0 ecotype, the meristem stretches up to 200 µm away from the junction between root and root cap (RCJ). In the transition zone, 200 to about 520 µm away from the RCJ, cells undergo physiological changes as they prepare for their fast elongation. Upon entering the transition zone, they progressively develop a central vacuole, polarize the cytoskeleton and remodel their cell walls. Cells grow slowly during this transition: it takes ten hours to triplicate cell length from 8.5 to about 35 µm in the trichoblast cell files. In the fast elongation zone, which covers the zone from 520 to about 850 µm from the RCJ, cell length quadruplicates to about 140 µm in only two hours. This is accompanied by drastic and specific cell wall alterations. Finally, root hairs fully develop in the growth terminating zone, where root cells undergo a minor elongation to reach their mature lengths.Key words: Arabidopsis, cytoskeleton, development, differentiation zone, elongation zone, growth, growth terminating zone, meristem, root apex, transition zone     

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.     

A biophysical analysis of root growth under mechanical stress

The factors controlling root growth in hard soils are reviewed alongside summarised results from our recent studies. The turgor in cells in the elongation zone of roots pushes the apex forward, resisted by the external pressure of the soil and the tension in the cell walls. The external pressure of the soil consists of the pressure required to deform the soil, plus a component of frictional resistance between the root and soil. This frictional component is probably small due to the continuous sloughing of root cap cells forming a low-friction lining surrounding the root. Mechanically impeded roots are not only thicker, but are differently shaped, continuing to increase in diameter for a greater distance behind the root tip than in unimpeded roots. The osmotic potential decreases in mechanically impeded roots, possibly due to accumulation of solutes as a result of the slower root extension rate. This more negative osmotic potential is not always translated into increased turgor pressure, and the reasons for this require further investigation. The persistent effect of mechanical impedance on root growth is associated with both a stiffening of cell walls in the axial direction, and with a slowing of the rate of cell production.     

Effect of Water Stress on Cortical Cell Division Rates within the Apical Meristem of Primary Roots of Maize

We characterized the effect of water stress on cell division rates within the meristem of the primary root of maize (Zea mays L.) seedlings. As usual in growth kinematics, cell number density is found by counting the number of cells per small unit length of the root; growth velocity is the rate of displacement of a cellular particle found at a given distance from the apex; and the cell flux, representing the rate at which cells are moving past a spatial point, is defined as the product of velocity and cell number density. The local cell division rate is estimated by summing the derivative of cell density with respect to time, and the derivative of the cell flux with respect to distance. Relatively long (2-h) intervals were required for time-lapse photography to resolve growth velocity within the meristem. Water stress caused meristematic cells to be longer and reduced the rates of cell division, per unit length of tissue and per cell, throughout most of the meristem. Peak cell division rate was 8.2 cells mm-1 h-1 (0.10 cells cell-1 h-1) at 0.8 mm from the apex for cells under water stress, compared with 13 cells mm-1 h-1 (0.14 cells cell-1 h-1) at 1.0 mm for controls.