(American Journal of Botany. 2004;91:1277-1293.)
© 2004 Botanical Society of America, Inc.
Structural integration at the shoot apical meristem: models, measurements, and experiments1
Dorota Kwiatkowska2
Institute of Plant Biology, Wroc
aw University, Kanonia 6/8, 50-328 Wrocaw, Poland
Received for publication April 1, 2003.
Accepted for publication April 29, 2004.
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ABSTRACT
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The shoot apical meristem (SAM) produces stem and initiates leaves. Its structure is maintained despite a continuous flow of cells basipetally from the distal portion of the meristem. The apoplasm and symplasm are the obvious means of cell integration, and their role in chemical cell-to-cell signaling is known. However, the cell wall apoplasm is most likely also involved in a mechanical integration mode, in which mechanical stress and strains (elastic and plastic strain, i.e., growth) are putative signaling factors. Shoot apex cells grow symplastically and their growth is in general anisotropic. Therefore tensor of growth rates that depends on the displacements caused by growth is the most suitable physical entity to describe growth. The tensor approach introduces the concept of principal directions of growth, i.e., the directions in which growth rates attain extremal values. Because of the symplastic mode of growth, the cell wall pattern within the shoot apical meristem informs us about the sequence and planes of cell divisions and about the deformation of existing walls. In consequence, within the meristem, periclines and anticlines can be recognized, both representing the principal directions of growth.
Key Words: geometry • growth • mechanical integration • mechanical stress • shoot apical meristem
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INTRODUCTION
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The functions of the shoot apical meristem (SAM) are the formation of the shoot axis and the initiation of lateral organs, such as leaves. To maintain this ability the SAM has to be self-perpetuated, i.e., general features of the SAM geometry and size have to be maintained or change only slowly. The self-perpetuation takes place despite the continuous flow of cells basipetally from the distal SAM portion and the cyclical initiation of lateral organ primordia. In this process variously defined zonations are also maintained in the SAM interior, such as the cytohistological zonation (Newman, 1965
) or zones of specific gene expression (Bowman and Eshed, 2000
; Brand et al., 2001
; Traas and Doonan, 2001
). While SAM structure and zonations have been thoroughly studied in various plant species, the mechanisms integrating cell behavior to maintain these spatial features (Erickson, 1976
; Silk and Erickson, 1979
) are still poorly understood.
Plant cell protoplasts are connected by plasmodesmata throughout the plant body, including the SAM, and form a continuum called the symplasm. Another basic component of the plant body is the apoplasm (Erickson, 1986
; Romberger et al., 1993
). It is the system of cell walls glued side by side with middle lamellas. The apoplasm also embodies the "water free-space" of older literature that includes lumens of tracheary elements filled with water. From our perspective the most important system is the cell wall apoplasm. Another fraction of non-symplasmic (extraprotoplasmic) space is an "inner gas space," which is usually excluded from the apoplasm. The two continuum systems, apoplasm and symplasm, are the obvious means of integration of cell activities over the SAM. Their role in chemical cell-to-cell signaling is well documented (as recently reviewed by Traas and Doonan, 2001
), and this mode of signaling is undoubtedly one of the ways in which plant cell activities are integrated. However, the symplasm and the apoplasm are also "conductors" of electrophysiological signals (action and variation potentials; e.g., Tr
bacz et al., 1997
). Moreover, the cell wall apoplasm, as a load-bearing mechanical structure, may play a role in mechanical integration of cell activities. Mechanical stress and strain (the elastic, i.e., reversible, or irreversible deformation) are putative signaling factors in this integration mode. The fact that meristematic cells are stuck together and therefore that a growing cell may force the expansion of its neighbor walls (Trewavas and Knight, 1994a
) can itself be a mechanical signal informing the cell of its neighborhood behavior. Other putative signals of a physical character are mechanical stresses in meristematic cell walls (Hussey, 1971
, 1973
; Selker et al., 1992
; Dumais and Steele, 2000
). Mechanical stress is unique among other mechanical factors, because it is strongly affected by the geometry of a surface in which it operates (Wainwright et al., 1976
; Niklas, 1992
). Stresses in cell walls result from the superposition of stresses caused by the turgor pressure, which can be regarded as primary stresses, and the secondary tissue stresses, which are a function of turgor and an overall plant body structure and growth (Hejnowicz et al., 2000
). Tissue stresses are on the one hand a type of pre-stresses, i.e., stresses that exist before external force is applied. On the other hand they are a special type of tensegral stresses (Stamenovi
et al., 1996
; Ingber, 2003
), which is a term coined for internal stresses within the cell (the cytoskeleton). Tissue stresses are transmitted via apoplasm and participate in the regulation of plant organ growth (Sinnott, 1960
; Hejnowicz and Sievers, 1992
, 1996
). The significance of the mechanical integration is supported by the observation that plant cellular structures are sensitive to mechanical signals like stress or strain (e.g., Trewavas and Knight, 1994b
; Fisher and Schopfer, 1998
). For example, mechanical stress or elastic strain affect the orientation of cortical microtubules (Hejnowicz et al., 2000
), which tend to align in the direction of maximal stress operating in cell walls. The significance of mechanical factors in morphogenesis has been documented and modeled in the development of animals (e.g., Murray et al., 1988
; Held, 1992
; Harrison, 1993
; Beloussov, 1997
; Beloussov et al., 1997
). However, as pointed out by Green (1996)
, the possibility of mechanical integration of cells building the SAM has been only infrequently addressed.
The objective of the present paper is to review mainly mechanical aspects of SAM growth and to discuss the possible role that mechanical factors could play in the integration of cell activities in the SAM. Chemical (molecular) aspects of integration are only briefly discussed as these topics have been treated thoroughly in a number of reviews (Bowman and Eshed, 2000
; Brand et al., 2001
; Traas and Doonan, 2001
). The paper starts with a description of the structure and geometry of SAM. Then the nature of shoot apex growth and of putative growth regulating factors in the physical sense is discussed. Finally, mechanical aspects of the two fundamental processes taking place at the SAM are addressed, i.e., the self-perpetuation of the meristem and the initiation of leaf primordia. A broad definition of the SAM is used throughout the text and includes the apical dome and the region where the youngest leaf primordia emerge.
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STRUCTURE AND GEOMETRY OF THE SHOOT APICAL MERISTEM
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Cyclical and directional changes in SAM shape
Two fundamental processes contribute to the SAM growth. In the self-perpetuation process the expansion of the apical dome is such that its size and shape are generally maintained. At the same time lateral organs are cyclically initiated as bulges at the SAM dome flanks, giving rise to phyllotactic patterns of fascinating regularity. As a result the shape and size of the SAM cyclically change. The fundamental cycle is a plastochron, i.e., the time interval between the initiation of consecutive leaves, during which the dome size oscillates between minimal and maximal area. The plastochronic changes are generally most pronounced in apices with relatively large leaf primordia. The radial expansion of the SAM dome during one plastochron can be assessed by means of the plastochron ratio (Richards, 1948
; Erickson, 1959
), which is the ratio of radial distances from the dome vertex to two successively initiated leaves (the distance from the older leaf primordium over the younger). Changes in SAM shape and size can occur also in seasonal cycles (Gifford, 1950
; Romberger, 1963
; Steeves and Sussex, 1989
). Spectacular seasonal changes are characteristic of some conifers, such as spruce (e.g., Owens and Molder, 1976
). In their apices the two SAM activities (the self-perpetuation and the leaf initiation) are to some extent separated in time (Figs. 1–6). The apical dome enlarges greatly at first, but virtually no leaf primordia are initiated (Figs. 2–4). Afterwards, a large number of primordia are initiated using up the undifferentiated surface produced earlier (Figs. 5–6).
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Figs. 1–6. Developmental phases of the vegetative shoot apical meristem (SAM) of spruce. The shape and size of the SAM are shown in outlines of SAM longitudinal sections. Figure 1 is drawn in smaller scale than the remaining figures (after Owens and Molder, 1976
, changed). 1. The embryonic shoot in a dormant terminal bud. 2. The SAM in the early spring, at the bud scale initiation stage. 3. The same stage but in late spring. 4. The most rapid apical growth in summer. 5. Less rapid apical growth and the beginning of leaf initiation in autumn. 6. The SAM of the dormant bud in winter
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In numerous shoot apices, directional size and/or shape changes are overlaid over those occurring in cycles. In spruce (Gregory and Romberger, 1972a
), Linum usitatissimum L. (Linaceae) (Williams, 1975
), or Sedum maximum Sut. (Crassulaceae) (Kwiatkowska and Florek-Marwitz, 1999
), the apex dome gradually increases in size during the early shoot ontogeny. A main switch-point in shoot apex development, i.e., the change from the vegetative to reproductive phase of development, is accompanied by additional changes in meristem size and shape, which are in general more abrupt than the previous ones. The changes may take place both when the SAM becomes the inflorescence meristem, as in Nerium oleander L. (Apocynaceae) (Williams et al., 1982
), and when it changes into the floral primordium, as in Ranunculus repens v. glabratus DC (Ranunculaceae) (Meicenheimer, 1979
).
Cytohistological zonation and domains of gene expression within the SAM
The SAM interior is usually divided into zones of cells differing in the wide range of their properties, including rates of growth (Clowes, 1961
; Romberger et al., 1993
). Widely accepted cytohistological zonation (Fig. 7) applies to the SAM of seed plants. The central meristem zone comprises slowly growing cells located at the distal portion of the apical dome. The fast growing rib meristem zone is located proximal to the central meristem zone. Because its cells grow predominantly in an axial direction, they form characteristically elongated ladder-like cell files. The peripheral meristem zone surrounds the central meristem zone and the rib meristem. It is also composed of fast growing cells.
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Fig. 7. Cytohistological zonation of the SAM dome. The central meristem zone (I), comprising the apical-initial zone (Ia) and central mother cells (Ib), the peripheral meristem zone (II), and the rib meristem zone (III) are shown in the outline of the median longitudinal section
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The cytohistological zonation in which the term "central" appeared was introduced by Foster (1943)
for the SAM of cycads. Foster (1943)
recognized the zonation around the central mother cells (Fig. 7). These cells are more vacuolated and larger than other SAM cells. Their walls are thickened. The most distal portion of the dome is the initiation or apical-initial zone (Fig. 7).
A number of genes have been shown to control the maintenance and size of the SAM dome and its cytohistological zones (Bowman and Eshed, 2000
; Brand et al., 2001
; Traas and Doonan, 2001
). The borders of gene expression domains within the SAM are clear-cut. Mutual inhibition of expression of these genes in adjacent domains contributes to the meristem zonation (Brand et al., 2001
). The maintenance of the indeterminate character of cells of the Arabidopsis thaliana (L.) Heynh. (Brassicaceae) apical dome is controlled by the SHOOTMERISTEMLESS (STM) gene, which is expressed throughout the SAM except for the sites of leaf initiation. The STM gene is negatively regulated by ASYMMETRIC LEAVES1, which in turn is expressed at primordium initiation sites. Similar relationships have been reported between the KNOX genes and PHANTASTICA or ROUGH SHEATH2 for Antirrhinum majus L. (Scrophulariaceae) and Zea mays L. (Poaceae) apices, respectively (Tsiantis et al., 1999
; Byrne et al., 2000
).
The expression pattern of some genes within the STM expression domain is related to the cytohistological zonation (Traas and Doonan, 2001
). The most distal part of the central meristem zone is the expression domain of CLAVATA3 (CLV3) (Brand et al., 2000
). Cells expressing CLV3 are regarded as stem cells (Mayer et al., 1998
; Brand et al., 2000
). The CLV3 domain most likely coincides with the apical-initial zone, while the concept of the ultimate stem cells closely corresponds to the older concept of initial cells or initials (Foster, 1943
; Clowes, 1961
). CLAVATA1 (CLV1) and WUSCHEL (WUS) are expressed in other portions of the central meristem zone. The maintenance and the size of this zone are collectively regulated by CLV1, 3, and WUS genes. The WUS gene is expressed in the proximal portion of the central meristem zone, possibly coinciding with what Foster (1943)
would call the central mother cells. The WUS gene expressing cells are often referred to as an "organizing center" of the SAM (Mayer et al., 1998
; Lenhard and Laux, 1999
). This center is believed to specify the overlying cells (the putative apical-initial zone) as the stem cells. Lenhard and Laux (1999)
draw an analogy between the organizing center defined by WUS and the quiescent center of the root apical meristem (Clowes, 1961
), which also specifies the surrounding cells as stem (initial) cells.
The UNUSUAL FLORAL ORGANS (UFO) gene is expressed in a cup-shaped domain that corresponds to the peripheral meristem zone and the rib meristem (Lee et al., 1997
; Brand et al., 2001
). Within the UFO domain, the CUP-SHAPED COTYLEDON2 (CUC2) domain is located at the putative leaf primordia boundaries (Brand et al., 2001
; Traas and Doonan, 2001
). The site of the leaf primordium initiation is in turn distinguished by the expression of genes encoding a number of enzymes involved in carbohydrate metabolism (Pien et al., 2001a
) and by LeExp18 gene encoding expansin (Reinhardt et al., 1998
). The earliest specification of the future primordium initiation site comes from the expression pattern of PINHEAD/ZWILLE (PNH) (Lynn et al., 1999
). The PNH gene is expressed in provascular tissue before the STM is down-regulated at the initiation site. This expression is regarded as a source of positional information for the site of leaf primordium initiation.
Initial cells and tunica/corpus organization of the SAM interior
The ultimate stem cells of the SAM, i.e., the initial cells, have to be located at the pole of the apical dome. The SAM of the majority of ferns and horse-tails is distinguished by the apical cell (apical initial) located at the dome pole (Clowes, 1961
; Bierhorst, 1977
). This cell divides unequally (Fig. 8) and as a consequence its unique tetrahedral shape (Fig. 9) and location within the dome are preserved. Divisions in the progeny of the apical initial are equal and result in daughter cells that do not differ in size or shape (Fig. 9).
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Figs. 8–13. Schematic representation of divisions typical of initial cells in SAMs from various taxonomic groups (Figs. 8, 10, 12), accompanied by patterns of cell walls in the SAM interior (Figs. 9, 11, 13). Both outlines of the initial cell and the cell wall patterns are shown as they appear in the median longitudinal section of the SAM. In Figs. 8, 10, 12 dashed lines represent division planes; arrows represent the directions in which the initial cell progeny is displaced during growth (Figs. 8, 10, 12 based on Newman, 1965
, changed). 8. The apical initial cell in the SAM of a fern. 9. Cell wall pattern within the SAM of a fern Osmunda L. (Osmundaceae) (based on a micrograph of Bierhorst, 1977
, changed). 10. An initial cell in a gymnosperm. 11. The SAM of Chamaecyparis Spach (Cupressaceae) (based on Hejnowicz, 2002
, changed). 12. Initial cells in the SAM of an angiosperm with a single tunica layer. 13. The SAM of pea with two layers of tunica (based on a micrograph of Lyndon, 1998
, changed)
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In seed plants, initial cells cannot be identified by their shape or size but only by the position they occupy in the apical dome (Figs. 10–13). Moreover, the pole of the dome cannot be localized precisely, especially in domes with no rotational symmetry (not axisymmetric). Clonal analysis of the cell packets present on the surface of the dome in seed plants shows that the number of initial cells (ranging from one to four) may change over time and that initial cells are impermanent (Zagórska-Marek and Turza
ska, 2000
). The impermanence of initial cells is supported also indirectly by the study of sectorial chimeras (Ruth et al., 1985
; Klekowski, 1988
). In angiosperms, initial cells are arranged in two or more layers. This is related to the tunica/corpus organization that is typical for this taxonomic group (Fig. 13). The tunica comprises one or more layers of protodermal or subprotodermal cells (denoted as L1, L2, etc., respectively), while corpus cells are not arranged in layers. Each tunica layer and corpus have their own initial cells (Fig. 12). The tunica/corpus organization is superimposed and independent of the cytohistological zonation (Lyndon, 1998
).
The protodermal tunica layer (L1) of the Arabidopsis SAM is distinguished by the expression of Arabidopsis thaliana MERISTEM LAYER1 (ATML1) and PROTODERMAL FACTOR2 (PDF2) genes (Lu et al., 1996
; Abe et al., 2003
). Expression of these genes is restricted to the L1 and epidermis throughout the shoot development (vegetative and reproductive phases). The ATML1 and PDF2 genes are involved in the maintenance of L1 cells and in the regulation of expression of cell layer specific genes (Abe et al., 2003
).
Quantification of the SAM shape
To study morphogenesis, i.e., changes and perpetuation of shape (Thompson, 1942
; Sinnott, 1960
), one needs tools to quantify the shape. In the case of the apical dome, the first approximation of the shape quantification is the measurement of its various dimensions, such as height, width (diameter), or basal dome area (Gregory and Romberger, 1972a
, b
; Laufs et al., 1998
). However, because a specification of shape can be a subject of optical illusion (Silk, 1984
), it is advisable to quantify the shape. The dome outline as it appears in a median longitudinal section can often be approximated by a parabola (Meicenheimer, 1979
). This approximation allows one to compare outlines of various domes based on the coefficients that specify the parabola. The outline of the apical dome has been described also in terms of polar coordinates (Kelly and Cooke, 2003
). Both the parabola approximation and polar coordinates are useful for nearly axisymmetric domes (those with rotational symmetry). However, if the dome is not axisymmetric, we need to compute local shape variables instead of approximating the overall shape with a single equation (Dumais and Kwiatkowska, 2002
). The quantification of the local shape variables is necessary also if a change in shape during the leaf initiation is to be measured.
Besides the curvature radius and the curvature proper (the reciprocal of the curvature radius), two important variables quantifying the surface shape are curvature directions and Gaussian curvature (Struik, 1988
). The curvature directions are the directions in which the curves lying on the examined surface attain their extremal (maximal or minimal) curvatures (Figs. 14–16). Gaussian curvature in turn measures the overall surface curvature and shows to what extent the surface is different from a plane. It is the product of extremal curvatures. Gaussian curvature is positive if the surface is convex (Fig. 14) or concave, i.e., the two curvatures are of the same sign. Such a surface would tear if forced to be flat without folds, and edges formed during the tearing would become separated. If the two curvatures are of opposite signs the surface curvature is negative. Such a surface folds during flattening, and if it tears the edges are superimposed, as in the case of the saddle shape surface (Fig. 15). Every surface that can be flattened without any tearing or folding has a zero Gaussian curvature (at least one curvature is zero). Such is the curvature of a cylinder (it can be flattened when cut along its axis), or a ridge-like surface (Fig. 16) like that of a leaf midrib. Data on curvature can be provided by the reconstruction of SAM shape from serial cross sections (Williams, 1975
), a stereoscopic reconstruction from scanning electron micrographs (Dumais and Kwiatkowska, 2002
; Fig. 20), or by confocal laser scanning microscopy (Laufs et al., 1998
).
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Figs. 14–16. Surfaces differing in curvature. Line segments point to curvature directions. Their length is proportional to the curvature. If the surface is concave in a given direction, the line is dotted. 14. A convex surface with positive Gaussian curvature. 15. A saddle-shaped surface with negative Gaussian curvature. 16. A portion of a cylinder, the Gaussian curvature of which is zero
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Figs. 17–20. Curvature plots for the vegetative SAM of Anagallis arvensis (based on Kwiatkowska and Dumais, 2003
, changed). 17. Scanning electron micrograph of the SAM surface, top view. Leaf primordia (Ln+1–Ln+3) are numbered from the oldest primordium to the youngest. 18. Curvature directions on the surface of the same SAM. Line segments pointing to the curvature directions have length proportional to the given curvature. Red lines indicate the directions, in which the surface is concave. Primordia Ln+1–Ln+3 are outlined. 19. Gaussian curvature plot for the same surface. The Gaussian curvature on the scale bar is given in 10–4 µm –2. 20. Side view of the reconstructed SAM surface, on which the Gaussian curvature is plotted
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The shape has been quantified for the vegetative SAM of Anagallis arvensis L. (Primulaceae) (Kwiatkowska and Dumais, 2003
; Figs. 17–20). In this species, leaf primordia are large in comparison with the apical dome and the plastochronic changes in shape and size of the SAM are prominent. New leaf primordia can be recognized and delineated at the earliest stages of their formation when they first appear as regions of increased Gaussian curvature (like Ln+3 in Fig. 19). If the curvature quantification were not employed, this place would be an incipient leaf primordium, meaning the region where the next primordium will be formed (e.g., Snow and Snow, 1931
; Lyndon, 1968
).
Curvature quantification in A. arvensis shows that the shape of the SAM dome flank (the surface of the peripheral meristem zone) depends on the age of an adjacent leaf primordium (Figs. 18, 19). The youngest leaf primordia are not separated from the dome by a leaf axil. Instead, the surface between the primordium and the dome summit is ridge-shaped (e.g., between Ln+2 and the dome summit in Fig. 18). The curvature along the ridge is nearly zero (Fig. 18). Older leaf primordia are separated from the apical dome by a leaf axil (e.g., Ln+1 in Figs. 17–20). The axil is saddle-shaped and has a negative Gaussian curvature. It is concave in the meridional direction, and convex in the latitudinal direction (Fig. 18). Sectors of dome flanks contacting this saddle-shaped region have positive Gaussian curvature (Fig. 19).
It is worthwhile to note that surfaces between layers of meristematic cells composing the SAM have the Gaussian curvature different from that of the outer SAM surface. In particular, the Gaussian curvature of the surface between the first tunica layer (L1) and deeper SAM cells is higher in the convex regions of SAM dome with the positive curvature, like the dome summit. In turn, the curvature of such an inner surface at saddle-shaped regions, like the leaf axil, is still lower than that of the outer SAM surface. Because the curvature of both the inner and outer surfaces is negative, the latter has a higher absolute value. These facts cannot be overlooked because the deeper SAM layers contribute to the shoot morphogenesis (Lyndon, 1998
) and the distribution of mechanical stresses depends on the curvature of the surface under consideration (Ugural, 1999
; Dumais and Steele, 2000
; Steele, 2000
).
Relationships between the histological zonation of the dome and its geometry
Some features of the cytohistological zones, such as the cell shape, arrangement (the cellular pattern), and division rates, may be a direct consequence of the dome shape (Niklas and Mauseth, 1980
). The cellular pattern can be described in geometric terms like cell volumes and ratios between two of the three cell dimensions (e.g., length to width), as well as the numbers of longitudinal files of cells radiating from the top of the SAM dome. The geometry of the dome is in turn described by its surface curvature. All these parameters are related by several equations. Niklas and Mauseth (1980)
showed that various cellular patterns can be modeled with only two assumptions: (a) growth induces a basipetal flow of cells from the dome top and (b) the number of cell divisions necessary to generate the specific morphology is minimized. In particular, computer simulations show that if the number of cell files radiating from the dome top is allowed to change while other cellular pattern parameters are kept constant the pattern resembling histological zonation is generated. In the model of the SAM dome in longitudinal section, the zone of central mother cells, the peripheral zone, and the rib meristem become apparent as regions differing in cell shape and "cell density." The cellular parameters in these regions are close to empirical data known for cacti (Mauseth and Niklas, 1979
; Niklas and Mauseth, 1980
). Moreover, the shape of zones depends on the dome surface curvature, as in real apices. Therefore the geometric constraints of the dome and its cells are postulated to work in combination with genetic factors (Niklas and Mauseth, 1980
).
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TENSORIAL NATURE OF THE SHOOT APEX GROWTH
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Symplastic growth and the continuum condition
Growth is a change in plant body dimensions (usually increase in dimensions, i.e., expansion). It can be interpreted as an irreversible deformation, i.e., a plastic strain, of cell wall system (Green, 1962
). Meristematic plant cells grow predominantly in a symplastic3 mode (Priestley, 1930
), "observing" the rules of continuum mechanics. Symplastically growing cells preserve their physical contacts. When the cells expand and divide they do not slide along their walls with respect to each other (Fig. 21) (in contrast to animal cells) because adjacent cells are stuck along their common middle lamella. The only departure from the rule of "not sliding" is the intrusive growth occurring exclusively on the cell edges, where local sliding occurs (Figs. 22–23). This growth is typical for cambium initials and some of their derivatives (Romberger et al., 1993
), such as differentiating vessel members of early wood (Fig. 22) or secondary xylem fibers (Fig. 23).
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Figs. 21–23. Schematic representation of growing plant cells. In Figs. 21, 23 letters A–C point to imaginary contacts between cells, which are preserved or not during growth. In Fig. 22 letters are used to label cells (Figs. 22, 23 after Hejnowicz, 2002
, changed). 21. Symplastic and anisotropic growth. A group of cells before and after the growth is shown. Cell contacts are preserved. 22. Schematic representation of differentiating vessel member of early wood, shown in transverse section. Its longitudinal edges grow intrusively. During the intrusive growth, first the middle lamella in front of the edge is locally disintegrated. Then due to the tensile stress across this lamella (arrows), a free space is formed between adjacent cell walls and used by the intrusively growing edge of a cell. Cells surrounding the future vessel member are labeled with letters. Due to the intrusive growth, the future vessel member contacts different cells at each growth stage. 23. Schematic representation of an apical portion of the differentiating xylem fiber, as seen in tangential longitudinal section. Its apical edge grows intrusively. Not all the contacts between cells are preserved during growth. Arrows point to the direction of tensile stress
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Symplastic growth is a unique feature of plant organs. It makes plants exceptional in comparison with animals, as only in plants can the history of cell divisions be reconstructed on the basis of a single examination (Schüepp, 1926
, 1966
; Bierhorst, 1977
; Puawska, 1986
; Silk et al., 1989
; Zagórska-Marek and Turzaska, 2000
). The symplastic mode of growth is also crucial for the quantification of growth of the SAM, as we can interpret the symplastic growth as a deformation of the cell wall system in which the continuum condition is observed (Niklas, 1977
; Silk and Erickson, 1979
; Gandar, 1983a
, b
; Silk, 1984
). In the course of such a continuum deformation the system is "kept in one piece," which is exactly as it happens during the symplastic growth of a plant organ. During the continuum deformation or growth, material points within the cell wall system are displaced. The displacement rates of neighboring points are a continuous and differentiable function of position. They are the basis for the computation of growth rates. Growth rates in a region must accommodate the displacement of cells caused by growth in other regions. For example, there must be a latitudinal growth in the surface cell whenever there is a radial growth in cells located more centrally at the same level (this latitudinal growth has to compensate for the increase in radius if cells move away from the center).
Although in the case of growth the empirical data are finite, the symplastically growing cells observe the continuum condition and the displacement due to growth can be treated as differentiable. Thus in principle, any finite growth may be thought of as a result of the integration of the differentiable growth (Skalak et al., 1982
).
The tensor of growth rates
The continuum deformation and plant organ growth have two significant features in common. They observe the continuum condition and they may also be anisotropic, meaning that the rates of deformation are different in different directions (e.g., Fig. 21). The continuum and anisotropic deformation and growth are both fully described by a tensor (Silk and Erickson, 1979
; Hejnowicz and Romberger, 1984
; Silk, 1984
), which is a mathematical operator performing such a function. To characterize an anisotropic growth at a given point we need to know displacement rates in any direction. The tensor operates on the field of these rates. The information about growth could not be provided by a number only (scalar) as could a volumetric growth rate at a given point. Even if a number were assigned to a direction, as in vectors, this would specify growth only in this particular direction and not in others. Tensor as a physical entity, not an operator, is a generalized vector in the sense that it assigns a number (e.g., a growth rate) to any given direction at a given point. While vector implies the existence of a direction in which the considered physical quantity acts, tensor specifies directions to which extremal values of a quantity described by the tensor are assigned. These are the principal directions of the tensor. They are always mutually orthogonal. The specification of principal directions is the most significant feature of the tensor, and the principal directions could not be explained without using the tensor.
For the study of plant organ growth the tensor of growth rates (for simplicity further referred to as growth tensor [GT]) has been defined (Hejnowicz and Romberger, 1984
). This concept applies in particular to shoot and root apices. Like the strain rate tensor, which describes the deformation of a beam, plate, or shell, GT describes local changes in linear dimension of a plant organ, which take place due to growth. They are expressed as relative elemental rates of expansion of a line segment: RERG(l) (Erickson, 1966
, 1976
; Silk, 1984
). The growth tensor is the special kind of the derivative in curvilinear coordinates (covariant derivative) of the velocity (displacement rate) vector field.
Principal directions of growth
Principal directions of GT are called principal directions of growth (PDGs) (Hejnowicz and Romberger, 1984
). These are the mutually orthogonal directions in which growth rates attain their extremal values, i.e., principal values of GT. Obviously, PDGs are defined only if growth is anisotropic. For isotropic growth all the directions are "principal." For an anisotropic growth in three dimensions, generally three principal directions exist (Hejnowicz, 1989
). One is the direction of the highest growth rate of all the directions in space (PDGmax). Another is the direction of the lowest growth rate in space (PDGmin). If the distribution of growth rates is not axisymmetric (i.e., not transversely isotropic), the third principal direction (saddle type extreme) appears. It is the direction of the highest growth rate in a plane normal to the PDGmax, and simultaneously the direction of the lowest growth rate in a plane normal to the PDGmin. If the distribution of growth rates is axisymmetric, growth is isotropic in one of the planes. Then only two extremal values exist. If the principal values of GT are known, the relative growth rate in area or volume, as well as the growth anisotropy, can be computed. Other important information we can obtain from PDGs is the change in the angular distance between two line segments, like cell wall edges, taking place during growth. Such a change has to take place if growth is anisotropic and the segments are not oriented along PDGs (Fig. 24). If the growth is steady in time, line segments, which are aligned in these particular directions, preserve their relative orientation (Fig. 25).
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Figs. 24–25. Changes in a square shape due to the anisotropic growth. Arrows point to principal directions of growth (PDGs). In the PDGmin (vertical) the length of a line segment increases twofold; in the PDGmax (horizontal), it increases four times. 24. The square edges do not co-align with PDGs, and the square changes into a parallelogram. 25. The square edges co-align with PDGs, and the square changes into a rectangle
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Growth rates in various directions at a point can be visualized by means of the GT indicatrix. It is a surface plotted around the point in such a way that a distance from this point to the surface in a chosen direction is proportional to the growth rate in this particular direction (Figs. 26–30). If growth at a point is isotropic, the indicatrix has a shape of a sphere because no principal direction is distinguished (Fig. 26). For a point at which RERG(l) is different in different directions (positive for enlargement and negative for contraction), the indicatrix is of various shapes illustrated in Figs. 27–29. The indicatrix for pure elongation growth looks like that in Fig. 30.
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Figs. 26–30. Indicatrices for various growth at a point located at the origin of the coordinate system. In Figs. 26–28 this point is inside the indicatrix, while in Figs. 29–30 it is at the contact of the two (Fig. 30) or four (Fig. 29) indicatrix portions. 26. Isotropic growth. 27. Anisotropic and axisymmetric growth with PDGmax along the z-axis, PDGmin in the horizontal plane xy. 28. Anisotropic and non-axisymmetric growth. PDGmax is along the z-axis; along x, the saddle extreme occurs; along y, the PDGmin is oriented. 29. Another case of anisotropic and not axisymmetric growth. Along the z-axis the extension takes place (PDGmax); along y, growth is zero; along x, contraction occurs (PDGmin). The black portion of indicatrix points to the directions in which contraction takes place. 30. Indicatrix for pure elongation along the z-axis
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Another graphical visualization of PDGs is a cross, the arms of which point to principal directions. The cross has been used for the illustration of growth in two dimensions, like the expansion of the SAM surface. The lengths of the cross arms are proportional to principal strains or strain rates (e.g., Fig. 41). The cross illustrating the strain, a "strain cross," has been used by Green and collaborators (Goodall and Green, 1986
; Hernández et al., 1991
; Tiwari and Green, 1991
). Dumais and Kwiatkowska (2002)
used the cross to represent the principal directions of strain rate.
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Figs. 37–43. Strain rate plots for the vegetative SAM of Anagallis arvensis (based on Kwiatkowska and Dumais, 2003
, changed). Leaf primordia (Ln+1– Ln+4) are numbered from the oldest primordium observed in the sequence to the youngest. 37. Top view of the SAM at the beginning of the observation. 38. The same SAM, but 23 h later. 39. The same, 24 h later than in Fig. 38. 40. The plot of areal strain rates for the SAM surface cells as they looked at the beginning of the observation (Fig. 37). Only cells the progeny of which can be recognized in the next stage (Fig. 38) are shown. Areal strain rates in the scale bar are given in units per hour. 41. Principal directions of the strain rate for the same cells. Line segments point to the principal directions and have length proportional to the strain rate. The slowest expanding region at the dome summit and leaf primordia are outlined. 42. The same as in Fig. 40 but for the later period, i.e., between stages shown in Figs. 38 and 39. 43. The same as in Fig. 41 but for the later period
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Most significantly, two of the three PDGs are always on the surface of a growing organ, i.e., tangent to the surface. Therefore the third PDG is perpendicular to the surface. The two PDGs on the surface are called periclinal, while the third (perpendicular) one is called anticlinal.
Periclines and anticlines
Trajectories of periclinal PDGs are called periclines. In the case of an axisymmetric dome-like surface two kinds of periclines can be distinguished— meridional (longitudinal) and latitudinal (transverse) (Fig. 31). For an organ with bilateral symmetry longitudinal and transverse periclines are distinguished. Anticlines are trajectories perpendicular to periclines. They cross the organ surface at right angles. The course of anticlines within the organ depends on the course of periclines. Because the whole organ surface is nonplanar (although locally it may be planar), the arrangement of periclines and anticlines resembles a curvilinear orthogonal system of coordinates (Figs. 32–34).
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Fig. 31. Theoretical periclinal surfaces of an SAM dome. Latitudinal and meridional periclines are shown on the surfaces. A wedge-shaped portion of the dome has been removed to show the dome interior
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Figs. 32–34. Exemplary arrangements of periclines and anticlines as seen in the axial plane of a dome. 32. A confocal arrangement. 33. Prolate spheroidal arrangement. 34. Coaxial arrangement
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Principal directions of growth define also plane elements of a surface. Two periclinal PDGs define a periclinal (tangential) surface element, meridional periclinal and anticlinal PDGs define a longitudinal radial element, while latitudinal periclinal and anticlinal PDGs define an anticlinal transverse element.
Natural coordinate system for SAM domes
In a median longitudinal section of the dome, meridional periclines and anticlines can be recognized (the reason periclines and anticlines are manifested in the cell wall arrangement will be explained further below in "Tensorial factors ...: Principal directions of growth [PDGs] and planes of cell divisions"). If the dome growth is nearly steady in time periclines approximate the outlines of cell clones, while anticlines approximate walls separating groups of cells within a cell clone (Figs. 35– 36). Such a network of periclines and anticlines was used as a framework to describe directions of SAM growth already by Julius Sachs (see Romberger et al., 1993
). Schüepp (1926
, 1966)
also used the periclines/anticlines network in his analysis of the SAM dome growth performed on median longitudinal sections. Based on the increase of "patches" delineated by periclines and anticlines (Fig. 35) and the changes of patch shapes as one moves along a chosen pericline, Schüepp deduced directions of maximal growth over the dome section.
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Figs. 35–36. The confocal natural coordinate system and a corresponding cellular pattern as seen in the longitudinal section of the SAM dome (Fig. 36 based on Hejnowicz, 2002
, changed). 35. An arrangement of periclines and anticlines (u, v). Arrow points to the focus of the coordinate system. An exemplary patch delineated by periclines and anticlines has been shaded. 36. An SAM of a gymnosperm with anticlinal divisions dominating in superficial cells. Real cellular pattern is shown in the right side of the section, outlines of cell clones are shown in the left
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Meridional periclines and anticlines have been used as coordinate lines (u and v for periclines and anticlines, respectively) for an axial plane of a growing plant organ. These lines represent the natural coordinate system for this organ (Hejnowicz, 1984
; e.g., Fig. 35). The third dimension can be added to the system by rotation around the axis u = 0, v = 0. This coordinate system is orthogonal and curvilinear. It has been defined for various growing plant structures, like a SAM dome, a root apical meristem, a growing fruit (e.g., an apple), or a stem segment (Hejnowicz, 1984
). The natural coordinate system generally applicable to SAM domes is confocal (Hejnowicz, 1955
; Romberger et al., 1993
). In the confocal coordinate system (Fig. 35) all the periclines and anticlines surround the focal point (focus), which is the system singularity (arrow in Fig. 35). Periclines form arches above it, while anticlines form arches below. The dome axis is composed of two overlapping coordinate lines: the narrowest and innermost pericline v = 0 (the u portion of the axis from the base of the dome to just above the focal point); and the innermost, "folded" upward, anticline u = 0 (the v portion of the axis from beneath the focal point upward). The root apical meristem with the quiescent center (Clowes, 1961
) is the clearest example of the apical meristem with confocal coordinates. In this case the focal point is located in the quiescent center, where growth rates are the lowest in all directions (Nakielski and Barlow, 1995
). In the case of the SAM dome, the focal point is located at the putative position of the organizing center.
The natural coordinate system has been a framework for growth modeling with the aid of the GT operating on the velocity (displacement rate) field V(u,v). Adopting the natural coordinate system greatly simplifies the modeling by using the GT (Hejnowicz, 1984
, 1989
). It is sufficient to specify displacement rates du/dt and dv/dt along the axis. For instance if in a paraboloidal coordinate system du/dt > 0 (along the u portion of the axis), the dome increases in length (or along periclines). If simultaneously dv/dt = 0 (along the v portion), the diameter (specified by vsurf) remains constant. It means that the distance from the top of the dome to the focal point is constant, and the width of the dome at the level of focal point is constant. Then the dome will not only remain paraboloidal but its diameter will be constant (maintained) during growth, analogously to the elongation of a cylinder keeping its diameter. If in turn dv/dt > 0 and it is a function of v only, then the dome remains paraboloidal but increases its diameter. If du/dt is a function of both u and v, the dome not only increases in length but also changes its shape in the sense that it departs from a paraboloidal geometry. Displacement rate du/dt may be various functions of u, which allows us to model different distribution of growth along periclines. For instance, if du/dt = cu (where c = constant) the periclinal RERG is constant. If in turn we assume that along the axis, from the focal point to u = a, du/dt = 0, then the portion of the dome above the anticline u = a is not growing in a periclinal direction. Instead of the specification along the axis, the specification of du/dt may be done for the surface. It is possible because the growth along the u portion of the axis is strictly related to the periclinal growth on the surface. This, however, is not possible for the v portion of the axis.
Growth simulations have been performed mostly for root apical meristems (e.g., Hejnowicz, 1989
; Nakielski and Barlow, 1995
). For SAM domes only preliminary results have been obtained (Nakielski, 1982
, 1987
).
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TENSORIAL FACTORS IN THE REGULATION OF THE SHOOT APEX GROWTH AND CELL DIVISIONS
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Mechanical stress and strain
Stress is a tensor entity similar to strain. Stress and strain are related to one another. Two types of strain, however, have to be considered separately. The value of the elastic (reversible) strain is related to the mechanical stress. The simplest relation is given by the generalized Hooke's law (Ugural, 1999
). In the case of linear reversible deformation, strain is proportional to the applied stress. In three dimensions and in the case of anisotropic materials, however, this simple proportionality relationship becomes especially complex. Then the strain is related to stress by means of the higher (fourth) order tensor, which characterizes mechanical properties of a material ("elasticity tensor"). There is no relationship between the value of an irreversible strain (growth) and stress, but the value of strain rate is a function of stress (Ugural, 1999
). An analogue of Hooke's law for the irreversible strain is the proportionality of strain rate and stress value exceeding a yield threshold such as in the Lockhart equation (Lockhart, 1965
).
Classical rules of cell division
Despite the expansion of cells in all the SAM regions, the overall shapes, sizes, and arrangement of cells are maintained. This is so because adequate cell divisions compensate the effect of cell expansion, although the sequence of divisions is not determined. Cell sizes are maintained as the frequency of cell divisions is related to cell expansion rates in the process of "cell size homeostasis" (Jacobs, 1997
). The characteristic cell geometry is maintained due to the proper orientation of cell division planes. This orientation is to some extent predictable, which has been recognized long ago by the plant biologists Hofmeister, Errera, and Sachs (Sinnott, 1960
). Hofmeister's rule states that if an organ grows in different directions, cell division planes are perpendicular to the direction of the fastest growth. According to the Errera's rule, new walls follow the shortest path that will halve the parental cell. The Sachs' rule states that the new cell wall meets side walls at a right angle. It has also been recognized that during the formation of new cell walls the four-way junctions are usually avoided, i.e., cell plates tend to avoid aligning with cross walls between neighbors (Sinnott and Bloch, 1941
; Korn, 1980
; Lloyd, 1991a
, b
; Cooke and Lu, 1992
). As a result three walls typically meet at a cell edge. Mechanical stress in cytoskeleton elements building the phragmosome may play a role in the avoidance of four-way junctions in vacuolated cells, as tensile elements were shown to seek a minimal path (Flanders et al., 1990
; Lloyd, 1991a
, b
). Similar mechanism may function also in meristematic cells (Lloyd, 1991b
). It will be shown below that Hofmeister's, Errera's, and Sachs' rules represent a single rule that relates the division planes to the GT.
Principal directions of stress and planes of cell divisions
According to Linthilhac (Lintilhac, 1974
; Lintilhac and Vesecky, 1980
, 1984
; Lynch and Lintilhac, 1997
) meristematic cells divide in shear free planes, which are the planes defined by two principal directions of stress (i.e., normal to the third principal direction4). The relationships between the principal directions of stress and division planes have been shown in the experiment, where isolated plant protoplasts were embedded in an agar-solidified medium and the medium was brought under compressive stress. The protoplasts divided in planes perpendicular to one of the principal directions of the applied stress (Lynch and Lintilhac, 1997
). Earlier support for the Lintilhac's hypothesis has been provided by photoelastic modeling (Linthilhac and Vesecky, 1980
). In this modeling implemented for the study of the initiation of the lateral shoot meristem, the stress was applied to the gelatin models of apical shoot portions as they appear in median longitudinal sections. Trajectories of principal stress directions mimicked the orientation of cell walls known for axillary shoot meristems at the earliest stages of their formation (the "arcuate shell zone"). This observation led to the conclusion that the characteristic cellular pattern in a shell zone may be understood as a response to the stress distribution in the leaf primordium axil.
Lintilhac postulates that cells recognize principal stress directions by measuring slight dimensional changes, i.e., measuring strain rather than the stress itself, just as the known stress transducers measure strain (Lynch and Lintilhac, 1997
). This is in agreement with the earlier postulate of Green (Gertel and Green, 1977
). Both reversible and irreversible strains have been proposed also as means by which stresses in cell walls can be sensed (Hejnowicz et al., 2000
). One should, however, recognize that stress can be measured directly in the same way that a piano-tuner "measures" sounds.
Principal directions of growth (PDGs) and planes of cell divisions
Hejnowicz (1984)
postulated that meristematic plant cells divide in planes defined by PDGs, i.e., normal to one of the PDGs. A surface element normal to the PDG is in a principal plane. The support for Hejnowicz's postulate comes from the observation that in the case of pronounced anisotropy of growth, cell division planes are perpendicular to anticlines or periclines that are PDG trajectories. This is further supported by comparison with computer-simulated orientation of PDGs (Hejnowicz, 1989
; Nakielski and Barlow, 1995
). The tendency to orient new cell walls in a plane perpendicular to one of the PDGs is more pronounced when growth anisotropy is high (Hejnowicz, 1984
). If growth is nearly isotropic, planes of cell divisions are no longer related to PDGs (because they are not obvious) and appear to be oriented more randomly.
The relationship between the planes of cell division and PDGs is a generalization of the rules of cell divisions formulated by Hofmeister, Errera, and Sachs (Hejnowicz, 2002
). The Hofmeister's rule, which states that cell division planes are perpendicular to the direction of the fastest growth, captures exactly the relationship between PDGmax and division planes. However, such a generalization explains also the cases in which division walls are opposite to what the Hofmeister's rule implies, i.e., the cell walls are perpendicular to the minimum rather than the maximum PDG. This is observed in the formation of cortex initial cells in the Arabidopsis root. Also if division planes are normal to PDGmax, new walls usually follow the shortest path that will halve the elongating parental cell, which is the Errera's rule. There are, however, exceptions to this rule, in which the new walls follow the longest of possible paths, such as in cambial fusiform initials. This case can again be explained by the fact that new walls are normal to the PDGmax, but the cell is elongated in the direction of PDGmin. Finally, the rule that the new cell wall meets side walls at right angle (the Sachs' rule) may also result from the fact that division planes are normal to PDGs when the existing walls follow the PDGs. The PDGs are mutually orthogonal. Thus consecutively formed walls that are normal to different PDGs will also meet at right angles.
The alignment of cell walls normal to PDGs is apparent in meristems that maintain their structure (the shape and cellular pattern), i.e., grow steadily in time, such as some apical domes or the root apical meristem. If the growth is not steady, which means that the orientation of PDGs at a given point may change in time, the newly formed cell walls, but not necessarily the older walls, should be perpendicular to PDGs. Moreover, an alignment of walls at their formation does not have to be preserved in the pattern of older walls, even if growth is steady. The two parts of a parental cell wall oriented along the PDG are located on the opposite sides of a newly formed edge. In the course of growth following the division, they tend to reorient to equilibrate angles between the three walls meeting at the edge (Thompson, 1942
; Lloyd, 1991a
; Cooke and Lu, 1992
). Therefore a zigzag line is formed along a periclinal cell file. Cooke and Lu (1992)
studied cell wall pattern in planar structures. In fern prothalli if growth is nearly isotropic (isodiametric expansion) the reorientation of walls is such that eventually they tend to meet at equal angles. The cells are supposed to attain a state of local mechanical equilibrium (Cooke and Lu, 1992
). In regions of strongly anisotropic growth (preferential elongation) this tendency is much less pronounced. Cooke and Lu (1992)
explain this phenomenon by the tendency of cell walls to align with the global pattern of mechanical stresses. The tendency is realized during growth after the cell divisions.
There is an evidence that the relationship between PDGs and division planes postulated by Hejnowicz (1984)
is not always observed (Green and Poethig, 1982
). At the base of a detached mature leaf of Graptopetalum paraguayense E. Walther (Crassulaceae), a shoot is formed de novo from a residual meristem. This process involves considerable changes in growth and symmetry. As a consequence groups of protodermal cells of the residual meristem are highly sheared. In some of these cells the new wall is aligned at approximately 45° to both the principal directions of strain (strain cross arms). The latter correspond to the PDGs. This is the maximum possible departure. Therefore, Green and Poethig (1982)
ruled out the principal directions of strain as a direct factor in orienting new walls. However, in the Green and Poethig study (1982)
the strain crosses were calculated for 24 h, i.e., for a time much longer than that needed for a cell division. It is not obvious that actual PDGs at the moment of the cell division were the same as those indicated by the strain cross. One should remember that the region studied shows a dramatic change of growth pattern.
The relationship between principal directions of stress and principal directions of growth rate
Whether the alignment of new cell walls is stress- or growth-mediated remains an open question, but the two concepts are not necessarily mutually exclusive. If the mechanical properties of a material are isotropic, the principal directions of strain (growth) are the same as the principal directions of stress (Ugural, 1999
). Plant cell walls are usually anisotropic. However, because isotropy is a "special case" of anisotropy, the difference between the principal directions of stress and strain in the wall plane cannot be large, especially if the wall is in the plane normal to one of the PDGs (Hejnowicz, 1984
). This does not necessarily mean that the direction of maximum stress is the same as the direction of maximum strain. In particular, it is postulated that the maximal principal growth rate direction may be the direction of the minimal principal stress (tension) in the wall, i.e., perpendicular to the direction of the maximal tension. The direction of maximal tension is the direction in which cell walls are reinforced. The reinforcement of primary cell walls (Green and Brooks, 1978
; Green, 1988
) is the major physical constraint of the SAM growth. The anisotropic distribution of cellulose microfibrils in primary cell walls leads to anisotropic growth: maximal cell wall extension is normal to the wall reinforcement. It seems to be particularly significant in the outer walls of protodermal SAM cells (Green and Selker, 1991
), as these walls are the thickest of all the SAM cell walls.
If principal directions of stress and strain in the SAM indeed nearly coincide, the rules of cell division could be derived from the relationships between cell division planes and either stress or growth (strain) principal d
TOP
ABSTRACT
INTRODUCTION
