Variations of growth in shoot apical domes of spruce seedlings : a study using the growth tensor

Variations of the relative elemental rate of growth within apical domes, for the case when dome geometry changes during development, were modeled. It was ascertained that: I) the domes of spruce scedlings have a paraboloida! shape; 2) the shape is maintained during growth, but the domes become higher and widcr; 3) the relative elemcntal rate of growth in area on dome surface is isotropic, as indicated by analysis of cell packets in the surface layer. Thesc data wcre used in modeling by means of the growth tensor and natura) coordinate system. Growth of the dome was considered a superposition: I) of relatively fast steady shape growth, whcre the isotropy of growth in area on the surface of the dome, was determined, and 2) of relatively slow isogonic growth, which does not disturb the isotropy. The convergent parabolic system was selected as the natural coordinate system. Distributions of the growth rates in the form of computer-made maps for three doines differing in age, were obtained. lt appears that the growth rates within the dome are relatively high in the distal part and smaller in the cen tral and peripheral regions. This variation decreases progressively with seedling age when the dome becomes wider. The relative elemental rate of growth in volume, averaged for the whole dorne, also decreases

Distribution of linear growth rates in different directions in root opical meristems

Growth of apical meristems in plants may be wcll dcscribed by the growth tensor method. Hejnowicz ,Envir. Exp. Bot. 1989, 29) determined growth tensors for roots: one with a minimum and the other with a maximum of the relative elemental growth rate in volume and used them for the description of two types of apices: one with an apical cell and merophytes (1), and the other with files of cells converging towards a quiescent centre. CQ (Il). In the present paper the same cases are considcred from the point of view of a spatial and directional variation of the relative elemental rate of growth in length, RERG,. Maps of the RERG I in two planes: axial and tangential, the latter dctermined by periclinal-longitudinal (PL) and periclinal-tangential (PT) principal growth directions, are shown, In an apical part of apex I where there is maximum volumetriC' growth, there also occurs a maximum of RERG1 for all directions. In regions ot her than this RERG I decreases althougb RERG1 in the PL direction predominates everywhere. In apex Il RERG1 for all directions has a minimum in CQ and becomes increasingly larger with increasing distance from it - the maximum is in the PL direction in the cylindrical part of the apex. ln peripheral parts of both apices, in the place of the root/cap junction, RERG I in the anticlinal direction is significantly small

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

Spatial variations of growth within domes having different patterns of principal growth directions

Growth rate variations for two paraboloidal domes: A and B, identical when seen from the outside but differing in the internal pattern of principal growth directions, were modeled by means of the growth tensor and a natural coordinate system. In dome A periclinal trajectories in the axial plane were given by confocal parabolas (as in a tunical dome), in dome B by parabolas converging to the vertex (as in a dome without a tunica). Accordingly, two natural coordinate systems, namely paraboloidal for A and convergent parabolic for B, were used. In both cases, the rate of growth in area on the surfaces of domes was assumed to be isotropic and identical in corresponding points. It appears that distributions of growth rates within domes A and B are similar in their peripheral and central parts and different only in their distal regions. In the latter, growth rates are relatively large; the maximum relative rate of growth in volume is around the geometric focus in dome A, and on the surface around the vertex in dome B

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 spatial variations of growth within apical domes by means of the growth tensor. Pt. 1. Growth specified on dome axis

By using the growth tensor and a natural curvilinear coordinate system for description of the distribution of growth in plant organs, three geometrie types of shoot apical domes (parabolic, clliptical and hyperbolic) were modcled. It was assumed that apical dome geometry remains unchanged during growth and that the natural coordinate systems are paraboloidal and prolate spheroidal. Two variants of the displacement yelocity fields V were considered. One yariant is specified by a constant relative elemental ratę of growth along the axis of the dome. The second is specified by a ratę inereasing proportionally with distance from the geometrie focus of the coordinate systems (and the apical dome). The growth tensor was used to calculate spatial yariations of growth rates for each yariant of each dome type. There is in both yariants a elear tendency toward lower growth rates in the distal region of the dome. A basie condition for the existence of a tunica is met

Modeling of spatial variations of growth within apical domes by means of the growth tensor. Pt. 2. Growth specified on dome axis

Variations of the elemental relative rate of growth are modeled for parabolic, elliptic and hyperbolic domes of shoot apices by using the growth tensor in a suitable curvilinear coordinate system when the mode of area growth on the dome surface is known. Variations of growth rates within the domes arc obtained in forms of computer-made maps for the following variants of growth on the dome surface: (I) constant meridional growth rate, (2) isotropic area growth, (3) anisotropy of area growth which becomes more intensive with increasing distance from the vertex. In variants I and 2 a maximum of volumetric growth rate appears in the center of the dome. Such a distribution of growth seems to be unrealistic, However, the corresponding growth tensors are interesting because they can be used in combination with other growth tensors to get the expected minimum volumetric growth rate in the dome center

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. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s00425-012-1707-x) contains supplementary material, which is available to authorized users

Wpływ geometrii wierzchołka pędu na rozmieszczenie szybkości wzrostu

The distribution of the relative elementary rate of growth (RERG) in apical domes of various shapes and patterns of displacement lines can be analytically examined. The geometry of these domes may be described by parabolas of n-th order, the variant of the distribution of linear growth rate should be established along any displacement line (e.g. along the axis) and then the RERG can be studied as the function depending on the position coordinates and the parameter n. Such investigations of several aplical domes of various shapes have been performed. The results confirm the occurrence of the minimum of relative, elementary growth rate (in volume) in the subapical region of the dome independently of the type of geometry (n parabola order)

Variations of growth in shoot apical domes of spruce seedlings: A study using the growth tensor

Variations of the relative elemental rate of growth within apical domes, for the case when dome geometry changes during development, were modeled. It was ascertained that: 1) the domes of spruce seedlings have a paraboloidal shape; 2) the shape is maintained during growth, but the domes become higher and wider; 3) the relative elemental rate of growth in area on dome surface is isotropic, as indicated by analysis of cell packets in the surface layer. These data were used in modeling by means of the growth tensor and natural coordinate system. Growth of the dome was considered a superposition: 1) of relatively fast steady shape growth, where the isotropy of growth in area on the surface of the dome, was determined, and 2) of relatively slow isogonic growth, which does not disturb the isotropy. The convergent parabolic system was selected as the natural coordinate system. Distributions of the growth rates in the form of computer-made maps for three domes differing in age, were obtained. It appears that the growth rates within the dome are relatively high in the distal part and smaller in the central and peripheral regions. This variation decreases progressively with seedling age when the dome becomes wider. The relative elemental rate of growth in volume, averaged for the whole dome, also decreases