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Developmental Biology and Developmental Genetics |
Department of Botany, Miami University, Oxford, Ohio 45056 USA
Received for publication December 16, 2004. Accepted for publication October 2, 2005.
ABSTRACT
How overlapping nodes in shoot apical meristems become vertically separated on mature stems was investigated via analysis of spatial and temporal characteristics of growth rate fields of Linum usitatissimum stem units. Linum grown under constant environmental conditions exhibited constant spiral phyllotaxis for nodes 35–80, during which time the stem unit was delimited vertically and tangentially by the boundaries of four successive leaf primordia along the 3-, 5-, and 8-contact parastichies and radially by these boundaries extended to the centroid of the stem. Stem unit age was assessed by a stem unit plastochron index (SUPI) based on a 5 mm reference length. Growth characteristics of –32–0 SUPI stem units were investigated using cylindrical coordinates. Vertical growth velocity was uniform within stem units, increasing basipetally as the units expanded. Vertical strain rate of nodes was nonlinear and consistently lower than the linear rate of subjacent stem units. Radial and tangential stem unit velocity fields were steady and uniform up to –25 SUPI; thereafter, these fields were characterized by uniform spatial and temporal patterns. Stem unit analysis suggests that the stem is a steady-state growth structure up until –25 SUPI, but thereafter it should be viewed as a repetitive growth structure.
Key Words: internode formation • Linaceae • repetitive growth • steady state growth • stem unit
The subdivision of mature stems into node and internode regions is well known. Surprisingly, we know little about how internodes actually develop from the shoot apical meristem (SAM). There are numerous analyses of stems at the level of the shoot apex, where exponential growth predominates (Meicenheimer, 1981
, 1982
, 1987
) and numerous analyses at the level where leaves and internodes have clearly emerged from the shoot bud (Garrison and Briggs, 1975
; Silk and Abou Haidar, 1986
; Orkwiszewski and Maksymowych, 1997
). Although a spatial and temporal continuum clearly exists between the shoot apex and the emergent areas of the stem, little is known about the growth of this intermediate region, in part because young leaves surround this region and prevent direct observation of it. This situation necessitates indirect methods of study on this portion of the stem, which are described herein. This study investigates how the leaf primordia and associated stem units at the level of the SAM develop into the node internode arrangement characteristic of mature stems. To the best of this author's knowledge, this is the first known report of the growth kinematics in this intermediate region.
There are conceptual problems with the units of analysis that are often used to examine older stems at the level of the shoot apex and the intermediate region (Meicenheimer, 1992
). Plant stems have often been subdivided into node and internode regions, where a node is defined as a point on the stem at which one or more leaves arise and an internode is the part of the stem lying between two adjacent nodes (Blackmore and Tootill, 1984
). A related subdivision of the stem is the phytomere, defined as a developmental unit consisting of one or more leaves, the node to which the leaves are attached, the internode below the node, and one or more axillary buds (Taiz and Zeiger, 2002
). Node, internode, and phytomere concepts do not lend themselves well to analyses of growth and differentiation processes of the shoot apex and intermediate region between the SAM and the mature stem (Meicenheimer, 1992
). There are obvious regions of stem tissue between areas of leaf primordia insertion in these regions of the stem, but the term internode and phytomere cannot be applied to these regions by definition due to the vertical overlap of nodes (cf. Fig. 1).
|
). In stems exhibiting k (m, n) contact parastichy phyllotaxis, the stem unit is delimited vertically and tangentially by the boundaries of four successive leaf primordia along the m-, n-, and (m + n)-parastichies and radially by these boundaries extended to the centroid of the stem (cf. Fig. 1). In this notation, k = jugacity, or number of leaves at the same vertical level on the stem, and m 
n are integers representing the plastochronic age differences between members of two opposed contact parastichies. With the stem unit concept, node only refers to the region of the stem where the leaf is actually inserted, rather than to the entire transverse level of the insertion. The stem unit plus the associated acropetal boundary primordium can duplicate the stem in toto through regular rotations that correspond with the divergence angle and through dilations that correspond with vertical, radial, and tangential strain rates over successive plastochrons, at the level of the SAM, of the mature stem, as well as, of the intermediate area that joins these two regions of the stem.
The plastochronic age of the stem unit is defined at the time of formation of the acropetal boundary primordium because, by definition, the stem unit is delimited by four boundary primordia and the unit does not exist until the acropetal boundary primordium is formed. Prior to the formation of the acropetal boundary primordium (e.g., 64 in Fig. 1), there is a region of the shoot apical meristem that is delimited by three primordia (61, 59, and 56 in Fig. 1), but this area cannot yet be considered a stem unit by definition. Rather, this area represents the largest portion of the shoot apical meristem that is unoccupied by leaf primordia. The stem unit (e.g., 64, 61, 59, and 56 in Fig. 1) only becomes defined at the time of formation of the acropetal primordium (64 in Fig. 1). Thus the plastochronic age of the stem unit is determined by the acropetal boundary primordium, rather than the more proximal boundary primordia. Axillary buds, when formed, occupy the basipetal-most region of the stem unit, just above the proximal boundary primordium. Stem unit length can be used to develop a plastochron index, PI, (Erickson and Michelini, 1957
) to provide an indirect measure of chronologic time for morphological and anatomical events of shoot development (Meicenheimer, 1992
). For Linum stems, a plastochron index calculated on the basis of a 5-mm stem unit reference length was convenient. The stem unit plastochron index, SUPI = PI – x, is directly analogous to the leaf plastochron index, LPI, (Erickson and Michelini, 1957
) except that stem unit length, rather than leaf length, is used in its calculation. In both SUPI and LPI, x refers to the integer sequence number of the same leaf primordium. However, because LPI has been used in studies on internode growth (e.g., Orkwiszewski and Maksymowych, 1997
), SUPI is used herein to direct focus on the stem unit.
The stem unit provides a conceptually continuous unit of analysis for the entire stem beginning within the shoot apical meristem and extending to its most mature portions. The entire stem can be subdivided into nodes and subjacent stem units, with each subdivision having a unique plastochronic age and a unique spatial position within the stem as a whole (Fig. 1). The sum of all nodes and stem units represents the entire stem in three dimensions, with no confusing spatial overlaps that are inherent to the node–internode concept of the stem. The stem unit divides the stem into regions that all have similar relationships with the leaf primordia and leaves born on the stem (Meicenheimer, 1992
).
The stem unit concept offers a significant departure from the way stem growth has been viewed in the past. Most researchers of stem growth have treated the growth of the stem as a unified organ, typically making assumptions about steady state growth and choosing the apex of the SAM as a co-moving origin in the coordinate system of analysis (Meicenheimer, 1987
; Silk and Abou Haidar, 1986
; Maksymowych et al., 1984
, 1985
). The stem unit concept subdivides the stem into repeating units, the sum total of which make up the stem. The stem is thus conceived as consisting of repeating units, rather like successive leaves on a stem, and rather unlike the unit organ of a root (Erickson and Sax, 1956
). This study investigated the growth kinematics of the stem unit in order to gain insight into whether stem growth is best modeled as a steady structure similar to a root or as a repetitive structure similar to a leaf. New insights into how overlapping nodes within the SAM become separated by internode regions on the mature stem via stem unit analysis were also obtained.
MATERIALS AND METHODS
Plant material
Linum usitatissimum cv. Culbert seed was obtained from Dr. J. Miller, USDA, Department of Agronomy, North Dakota University. Plants were grown under a 16 : 8 L : D photoperiod and a temperature regime of 20 : 15°C L : D in a walk-in growth chamber. Plants were watered with distilled water daily and full-strength Hoagland's solution (Hoagland and Arnon, 1938
) weekly.
From previous ontological studies on Linum grown under these conditions, the shoot apical meristem is known to have stable vegetative growth between initiation of nodes 35–80 prior to the onset of flowering (Meicenheimer, 1986
, 1987
). Nodes were numbered in their order of appearance, with number 1 being assigned to the first node above the cotyledonary node. For the 23 plants examined in those studies, the mean plastochron (pl) was 0.21 (SD 0.04) days. The mean divergence angle was 137.4° (SD 0.5). Mean relative rate of radial stem growth was 5.3% per pl (SD 0.5) and mean relative rate of vertical stem growth was 9.6% per pl (SD 1.5). The phyllotactic pattern was characterized by a stable 4.3 Richards phyllotaxis index (Richards, 1951
) corresponding to a 145.5° intersection angle between the (3,5) contact parastichies and a 103.9° intersection angle between the (5,8) contact parastichies. The leaf trace pattern was characterized by a stable (5; 8)/13 pattern. The stability of Linum stem growth under these conditions made 35–80 node plants suitable for more detailed characterization of the growth of nodes and stem units.
Random samples were made from a large population of plants that had at least 35 leaves visible on their stems. A plastochron index (PI) was calculated on the basis of a 5 mm stem unit index length for each stem analyzed (Meicenheimer, 1992
). Three plants at 45.4, 52.5, and 53.6 PI were selected for detailed analysis of the stem unit growth characteristics. This represented a sample pool of ca. 90 stem units, with each stem unit replicated three times at a mid-plastochron stage as measured by the SUPI. All patterns reported herein were observed to be common to all three stems analyzed unless otherwise noted.
Stem material was prepared for scanning electron microscopy (SEM) as described in Meicenheimer (1987)
. Stems examined with the SEM were mounted in an upright position on a 5 mm hexagonal nut filled with solder to create a level platform (Fig. 2). The vertex of the shoot apical meristem was sprinkled with Coprinus spores prior to sputter coating to facilitate orthogonal orientation of the vertical stem axes relative to the primary electron beam. Composite longitudinal scanning electron micrographs were generated along the entire length of the stem viewed from all six view angles fixed by the flat edges of the hexagonal mounting platform. Variation in stem diameter profiles among all six view angles was minimum, confirming the orthogonal orientation of the longitudinal SEMs (Fig. 2).
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Data collection
A cylindrical coordinate system was used in analyses of the growth of the stem units (Fig. 1). The tangential margins of the acropetal boundary primordium and of the 3-, 5-, and 8-boundary primordia (white points 1, 2, 3, 4, 5, and 6, respectively, Fig. 1) were used as natural marks in the analyses of the vertical component of stem unit growth. For each stem unit the coordinates of the tangential boundaries of the acropetal and the 3-, 5-, and 8-contact primordia were digitized (open circles 1–6, Fig. 1) to define each stem unit. All these data were translated and rotated such that the tangential boundary point of the acropetal primordium (white point 1 in Fig. 1) on the m-side of the stem unit was set at 0, and the opposite tangential boundary point of this primordium (white point 2 in Fig. 1) was horizontally level with the origin point. This fixed the m-side of the o-primordium boundary (white point 1 in Fig. 1) as the comoving origin for each stem unit for further analysis (Fig. 5). The position of the co-moving origin of each stem unit was also located relative to the SAM summit on each of the six view angles of the longitudinal composite SEMs (Fig. 4). The vertical and horizontal distances from the stem unit coordinate origin to the tangential boundaries of the 3-, 5-, and 8-primordia were delimited for each stem unit (Figs. 1 and 5). Each stem unit between –32 and –22 SUPI was subdivided into 10 µm long vertical sectors and between –21 and 0 SUPI into 50 µm vertical sectors. The coordinates of the intercepts of the stem unit boundaries at each vertical subdivision were digitized to further characterize the metrics of each stem unit. The width of the stem unit and the diameter of the entire stem were measured at each vertical subdivision using digitized images of the unrolled stem units and composite longitudinal images via Micromeasure (Universal Imaging, Media, Pennsylvania, USA) to convert Cartesian coordinates to cylindrical coordinates.
Radial length of the stem unit, r, was calculated as half the diameter of the stem associated with each vertical subdivision of the stem unit. Tangential length of the stem unit, 
, was measured in radians and calculated as the product of the ratio of the stem unit width to stem diameter and 
at each vertical subdivision. Subsequent manipulations of the (r, , z, pl) raw data arrays were performed as described using Grapher (Golden Software Golden, Colorado, USA) for two-dimensional graphics, Surfer (Golden Software) for gridding and three-dimensional graphics, Tablecurve (SPSS, San Rafael, California, USA) for regression analyses, and specially designed template spreadsheets for QuatroPro (Corel, Eden Prairie, Minnesota, USA) for numerical differentiation of these data. The minimum curvature algorithm of Surfer was utilized in all gridding operations to create 51 x 51 grids in which the upper and lower boundaries were set at constant values for each stem analyzed. Setting the boundaries to constant values facilitated subsequent matrix manipulations of the stem unit growth parameters.
Upper and lower points of leaf primordia insertion (white points 7 and 8, Fig. 1) were digitized for each acropetal node associated with each stem unit on the composite longitudinal micrographs of the stems viewed at each of the six separate view angles of the stem. These points, combined with tangential nodal boundaries (white points 1 and 2, Fig. 1), served as internal markers for the nodes associated with each stem unit measured relative to the SAM vertex in these analyses. It should be noted that the tangential boundary of a node and the acropetal tangential boundary of the associated stem unit (white points 1 and 2, Fig. 1) correspond by definition. Dimensions of the stem unit (Figs. 6–8) were used to assess the vertical (Fig. 9), radial (Fig. 10), and tangential (Fig. 11) velocity fields; as well as, the vertical (Fig. 12), radial (Fig. 13), and tangential (Fig. 14) strain rate fields.
|
|
) of each of the m-, n-, and (m + n)-natural marks were generated by plotting mark position within successive stem units as a function of SUPI (Fig. 6). Regression analyses were used to fit z vs. SUPI to generate z equally spaced in the ordinate, SUPI. This facilitated differentiation to obtain dz/dpl and plotting vs. position, z, or developmental time, SUPI. Regression coefficients from these analyses were used to generate tables of data consisting of equally spaced mark position, z, and the first derivative of mark position, dz/dpl, as functions of time measured in plastochrons. Growth velocity profiles (Silk, 1992
) associated with different vertical positions within the stem unit were generated by plotting dz/dpl as a function of z. The vertical velocity (Vz) field was then graphically displayed by three-dimensional (3D) plots of Vz as functions of z and SUPI (Fig. 9). In these 3D plots, spatial variation of the parameter of interest can be ascertained by examining the variation along the vertical axis at a fixed SUPI and the temporal variation can be ascertained by examining the variation along the horizontal axis at a fixed vertical position. Note that over time the entire size of the stem unit increases (Fig. 5), which is depicted by the boundary lines of the 3D plots (e.g., Figs. 9, 13, 14). Vertical node length (Nz) was calculated as the distance between the upper and lower boundaries of the acropetal node associated with each stem unit (Fig. 3). Equation 8092 (Asym Sig R) of TableCurve was used to regress Nz as a function of SUPI, from which equally spaced Nz was calculated for equally spaced SUPI via the TableCurve root function. Equation 8092 was then used to regress ln(Nz) as a function of SUPI and equally spaced d(lnNz/ dpl) = 1/Nz dNz/dpl was calculated. Equation 8175 (LogNorm4) was used to regress these latter data as a function of the equally spaced Nz data to examine the longitudinal strain rate of the node (1/Nz dNz/dpl; Fig. 12).
Because the radial and tangential dimensions of the acropetal node associated with each stem unit correspond with the measurements made for each stem unit, simple inspection of the acropetal-most values (lowest z values) of Figs. 7 and 8 reveal the radial and tangential dimensions of the node through time.
Radial velocity analysis
Equally spaced 51 x 51 grids were made for the radial dimensions, r, of the stem unit expressed as functions of vertical position, z, and SUPI. The first derivative of radius with respect to vertical position, dr/dz, was estimated via a three-point, second-degree numerical differentiation formula (Erickson, 1976
) from these grids. Values of dr/dz were multiplied by values of dz/dpl calculated from the regression equation coefficients described for each equally spaced z value within the stem units in order to obtain grids of dr/dpl as functions of z, r, and pl. Equally spaced 51 x 51 grids were then made for dr/dpl as functions of r and pl. The radial velocity (Vr) field was graphically displayed by three dimensional plots of Vr as functions of z and SUPI (Fig. 10).
Tangential velocity analysis
Equally spaced 51 x 51 grids were made for the tangential angular dimensions, , of the stem unit expressed as functions of z and pl. The first derivative of with respect to z, d/dz, was estimated via the three-point, second-degree numerical differentiation formula. Values of d/dz were multiplied by values of dz/dpl calculated from regression equation coefficients described for each equally spaced z value within the stem unit to obtain grids of d/dpl as functions of z, , and pl. The tangential velocity (V) field was then graphically displayed by three dimensional plots of V as functions of z and SUPI (Fig. 11).
Strain rate analysis
As fully described by Silk (1984
, 1992
), the strain rate (relative elemental growth rate) in cylindrical coordinates represents the divergence (
) of growth velocity (v) in each coordinate direction:
; Hejnowicz, 1984
):
 | (2) |
The radial velocity grids were differentiated with respect to r to obtain estimates of the radial strain rate, e11 = d(dr/dpl)/dr = dVr/dr, as functions of r and pl. Equally spaced 51 x 51 grids were made for tangential growth velocity, d/dpl, as functions of and pl. The divergence of tangential growth velocity, d(d/dpl)/d, was estimated via the three-point, second-degree numerical differentiation formula of d/dpl with respect to . Equally spaced 51 x 51 grids of stem unit radius, r, and (dr/dpl)/r were created as functions of and pl. The divergence of tangential growth velocity grid was divided by the radius grid to obtain (d(d/dpl)/d)/r grid, which was then added to the (dr/dpl)/r grid to obtain the tangential strain rate, e22 = (Vr/r + 1/r dV/d) of the stem unit as functions of and pl. The vertical growth velocity profiles were differentiated with respect to z to obtain estimates of the vertical strain rate, e33 = d(dz/dpl)/dz = dVz/dz, of the stem unit as functions of z and pl.
RESULTS
Stem unit dimensions
Expansion of the radial component of the acropetal boundary of stem units from all three plants investigated is illustrated in Fig. 4. The surface expansion of representative stem units over time is illustrated in Fig. 5. Stem unit length (solid boundary line in Figs. 7–11) increased exponentially over time. Stem unit radius (isoclines in Fig. 7 and boundary lines in Fig. 13) increased with time and with basipetal distance within the stem unit. Stem unit width, as measured by horizontal coordinates (Fig. 5) and , (isoclines in Fig. 8) narrowed in the mid-region of the stem unit between the m- and n-boundary primordia and widened in the acropetal and basipetal regions of the stem unit (Figs. 1, 5, and 8) over time. Widening of the stem unit was most extensive in the acropetal-most and basipetal-most portions of the stem unit (Figs. 5 and 8).
Stem unit vertical velocity fields
Examination of vertical growth trajectories, which are graphs of mark position as a function of time, within the stem unit (Fig. 6) indicated that the trajectories appeared uniform throughout the stem unit for a period of at least 30 plastochrons. Regression analyses of these data indicated that these growth trajectories were well fit with exponential equations; mean r2 for all growth profiles associated with three separate stems was 98.25 SD 0.50. Three-dimensional plots of vertical growth velocity, (Vz) as a function of vertical position (z) and time (SUPI), indicated that there was a uniform increase in Vz basipetally within the vertical axis of the stem unit, but Vz was steady at any given fixed vertical position within the stem unit as the stem unit elongated over time (Fig. 9).
Stem unit radial velocity fields
The radial velocity field, Vr, exhibited a steady (unchanging in time) and uniform basipetal spatial pattern up to ca. –26 SUPI; thereafter Vr increased with increasing vertical position within the stem unit. After the stem unit was 250 µm long, Vr decreased over time at any given vertical position within the stem unit (Fig. 10).
Stem unit tangential velocity fields
The tangential velocity field, V, was steady and uniform up to about –24 SUPI (Fig. 11). After this time, there was a basipetal migration of the zero V isocline located in the central portion of the stem unit over time. Acropetal to the migrating zero isocline was a region of negative V which expanded in vertical length over time. This rather complex pattern of V reflects the fact that as nodes m and n expand the tangential boundaries of the stem unit delimited by these primordia are relatively restricted (Figs. 1, 5, 8), but the stem radius continues to expand in the acropetal region of the stem unit (Figs. 4, 7). This results in a negative tangential velocity in the acropetal portion of the stem unit. In the basipetal portion of the stem unit after –24 SUPI, the (m + n) boundary expands in width (Fig. 5) but expansion of stem radius diminishes (Figs. 4 and 7) at any vertical level along the stem unit. Thus, after –24 SUPI a basipetal pattern of increasing V relative to the zero isocline became evident within the vertical axis of the stem unit (Fig. 11). It should be noted that there was a close correlation between the migration of the zero isocline for V and the vertical location of the nth primordium within the stem unit (cf. Figs. 6 and 11). Because the total tangential length of a stem is the sum of the tangential lengths of the stem unit plus node at any given vertical distance within the stem, this complex pattern of the V field indicates that the acropetal portion of the stem unit became narrower because of tangential expansion of nodes into this region of the stem unit, whereas the basipetal portion of the stem unit became broader because either tangential expansion of the nodes ceased or stem unit V exceeded nodal V.
Strain rate fields of the stem unit
Differentiation of vertical growth velocity profiles with respect to z resulted in estimates of the vertical strain rate of the stem unit. Regression analysis of growth velocity profiles indicated that these were consistently linear throughout the period of stem unit growth under study (Fig. 12). Similar analysis of vertical strain rate of nodes indicated that these were nonlinear and consistently lower than the vertical strain rates of stem units in all three plants under study (Fig. 12).
The radial strain rate of the stem unit decreased over time and, at any instant in time, decreased in a radial pattern within the stem unit (Fig. 13). That is, the relative rate of increase of the stem unit radius was highest where stem unit radius was smallest and decreased with increasing stem unit radius both with regard to time and position within the stem unit (Figs. 4, 13).
In general, tangential strain rate of the stem unit was at a minimum at intermediate angles (central region of the stem unit) and larger at minimum and maximum angular positions within the stem unit (Fig. 14). The tangential strain rate decreased asymmetrically with time in the peripheral regions. The rate decreased more rapidly over time at low angles as compared to high angles. Until –26 SUPI, the periphery of the stem unit on the m side (low angles) expanded less than the n side (high angles) (Figs. 5, 14). Tangential strain rate on the m side of the stem unit became essentially equal to the central strain rate by –15 SUPI (Fig. 14). The relatively high tangential strain rate of the n side of the stem unit persisted until ca. –10 SUPI (Fig. 14). There was a general increase in tangential strain rate in the central region of the stem unit beginning at –10 SUPI, but the pattern of increase was complex across the tangential dimension of the stem unit. There appeared to be isolated regions of enhanced tangential strain rate, which fluctuated in time (Fig. 14).
DISCUSSION
Steady vs. repetitive nature of stem growth
The tissues of roots and stems are formed by the concerted processes of division, expansion, and differentiation of cells originating from meristem populations. Through various combinations of the rates of these processes, mature organs are ultimately formed. Cells occupying tissue regions assume the properties associated with these regions in a coordinated temporal sequence as the cells and their derivatives are displaced through them (Silk, 1984
). The cellular elements comprising plant organs experience continual change, while the form of the organs remains approximately constant over long periods of time.
Powerful insights into plant growth and development can be gained by applying concepts of fluid dynamics to analyze cellular behavior within plant organs (Silk and Erickson, 1979
; Silk, 1984
). In accordance with these concepts the Eulerian specification of a growth variable defines the spatial distribution of the variable at an instant in time, whereas the Lagrangian specification of a growth variable refers to properties of material "particles" or cells forming the organ. The space comprising a plant organ has no fundamental properties at all; rather, the properties at a given position in the organ are conferred by the cells currently located at a particular position within the organ at the time of observation. Roots growing under constant environmental conditions have often been termed "steady" structures in which there is a relatively rapid change in the cellular elements comprising the root but the zones of cellular activity within the root appear unchanging in time (Silk, 1984
).
In organs with steady growth, the Eulerian and Lagrangian specifications coincide such that the same distribution of growth variables is observed regardless of the age of the organ. In organs with nonsteady growth, the Eularian and Lagrangian specifications will, in general, be different, such that one observes changes in the distribution of growth variables within organs of different ages. It is characteristic of stems and roots that if an apex is chosen as a co-moving origin of the coordinate system, steady patterns in growth variables are likely to emerge (Silk, 1984
).
Other plant growth processes, such as growth of leaves and vascular cambia, are repeated in time, but such growth cannot be said to be steady because tissue characteristics change within a plastochron or during the course of the growing season. The term "repetitive growth" has been proposed for organs or tissues that have the same developmental pattern over time, whereas "nonrepetitive growth" has been suggested for situations in which growth variables associated with the organ or tissues do not coincide in time (Silk, 1984
). These terms are analogous to steady and nonsteady distributions of growth variables.
Should stems be considered steady or repetitive structures? The answer to this question appears to be dependent on the position and age of the stem under consideration, as well as what growth parameter is under consideration. Stem unit growth velocity fields were steady up to –25 SUPI, suggesting that up to this time the stem unit could be considered as a steady structure. After –25 SUPI the radial and tangential velocity fields were characterized by uniform spatial and temporal patterns over the time course of study of the three plants investigated. This would suggest that after –25 SUPI the stem unit should be considered as a repetitive structure. Interestingly, the vertical velocity field of the stem unit was steady and uniform throughout the period of development under study. Therefore, the vertical growth of the stem unit could be interpreted as steady, at least under the constant growth conditions and stable pattern of phyllotaxis of the stems investigated. One would anticipate that the vertical growth component of the stem unit would not be steady during earlier times in stem ontogeny when the plastochron and the relative plastochron rates of vertical and radial stem expansion are decreasing and the pattern of leaf arrangement is changing (Meicenheimer, 1986
, 1987
). Studies on the vertical growth of Xanthium internodes indicated that the first formed basal internodes had shorter mature lengths and elongated at lower velocities up until the 11th internode. The average vertical strain rate of Xanthium internodes appeared to stabilize beyond the fourth internode (Orkwiszewski and Maksymowych, 1997
). It might be speculated that if Linum stem unit growth were examined during periods of ontogeny when phyllotactic patterns were changing, then evidence for nonrepetitive growth might be discovered. This would be analogous to the nonrepetitive nature of heteroblastic leaf development characteristic of many plant species. Considering the sum total of the three growth velocity fields that characterize the three-dimensional stem unit, it is concluded that the Linum stem unit should be viewed as a repetitive structure after –25 SUPI.
Answers to several basic fundamental questions concerning growth and tissue differentiation processes in stems are being sought within the context of the current research: (1) What are the domains of cellular activity that exist in the vertical axes of stems? (2) Do similar cell division and expansion rates characterize formation of analogous tissues in stems and roots of plants? (3) What influences do leaf primordia have on the cellular activities that give rise to the patterns of tissue differentiation within the vertical axes of stems? The characterization of growth velocities and strain rates of the stem unit was a necessary first step toward seeking answers to these questions. Characterization of material derivatives following tissue specific regions within the Linum stem unit are currently underway.
How internodes develop
Growth characteristics of the stem unit offer insights into the way the overlapping nodes in the shoot apical meristem and intermediate region of the stem come to be vertically separated by internodal regions on the mature stem. Expansion of the tangential form of the stem unit (Fig. 5) is rather analogous to expansion of a laboratory scissor jack (see Fisher Scientific [Pittsburgh, Pennsylvania, USA] JAK-100-070F, for example) except that the overall width of the stem unit gets broader as it gets older and elongates, rather than narrower as in the case of the scissor jack. In stem units associated with a 2(1,1) decussate pattern of phyllotaxis, the tangential boundaries of the stem unit are of equal length and angular orientation, just like the levers of the scissor jack. The Linum stem units of this study are associated with a 1(3,5) spiral pattern of phyllotaxis, which introduces an asymmetry to the boundaries (levers) of the stem unit. The shorter, more steeply inclined acropetal boundary associated with the m-boundary primordium is matched with the longer less inclined boundary associated with the n-boundary primordium (Figs. 1, 5). In the basipetal region of the stem unit, the relative length and angles of inclination of the opposing sides of the stem unit are reversed. That is, the length from the m- to the (m + n)-boundary is longer and less steep than the length from n- to the opposing (m + n)-boundary (Figs. 1, 5). This feature of the stem unit is directly related to the well-known geometric properties of phyllotactic patterns (Erickson, 1983
; Meicenheimer and Zagorska-Marek, 1989
) and opens up intriguing possibilities for theoretical studies on stem development in terms of the growth of nodes and stem units combined. It should be noted that theoretical phyllotactic studies have been confined primarily to the study of primordia patterns at the level of the shoot apex and have not addressed the issue of what happens to these patterns as the stem matures. One aspect that is quite clear in this regard is that the vertical dimension of the node is less than both the radial and tangential dimensions (Fig. 3) and that the vertical velocity and vertical strain rate of nodes is less than that of stem units (Figs. 9, 12), whereas the radial and tangential growth rates of nodes and stem units coincide along the vertical stem axis. This explains how the integrity of the stem is maintained at a given horizontal level, yet the stem can form recognizable internode components on the mature stem. In both the decussate and spiral cases, as the stem unit elongates, the stem unit boundaries expand in vertical extent and experience a decrease in their vertical angle of orientation, ultimately leading to the separation of the overlap between leaf primordia points of insertion (nodes) that exists in the shoot apex and intermediate region of the stem.
Barlow's (1994)
representation of the stem unit was taken from a leaf trace pattern map where the x-axis has been standardized at 2 (Meicenheimer, 1992
). In reality, the stem unit dimensions expand tangentially as measured by V and radially as measured by Vr. It is true that the boundaries delimiting the stem unit are oblique to vertical cell files, however, as Barlow points out, siting of leaves on angiosperm shoots are under physiological control only. Because the sum total of all stem units and associated nodes comprise the entire shoot axis, noncoincidence of the stem unit boundaries and the vertical orientation of cell files would appear to be of minor concern, at least with regard to angiosperm stems. If, as Barlow suggests, the morphogenesis of the shoot axes depends primarily upon rhythmic switches in the polarity of the cell growth and these switches are under physiological control rather than cell linage control, understanding the growth kinematics of the node and stem unit are important steps toward the ultimate elucidation of the temporal and spatial patterns experienced by cells as they "flow" though this structure. Thus, as Silk and Haidar (1986
, p. 110) point out, "The distribution of the strain rate is the basic descriptor of growth which physiologists should attempt to explain."
The most detailed analysis of vertical stem growth to date was performed on Xanthium by Maksymowych and colleagues (1993
, 1989
, 1985)
. In those studies, the vertical strain rate of internodal regions was observed to be consistently larger than subtending nodal regions. Over time the vertical strain rate of both regions decreased. Within the internodes was evidence that vertical strain rate decreased in a basipetal pattern, consistent with earlier observations on Syringa and Helianthus (Wetmore and Garrison, 1961
; Garrison, 1973
) and Phaseolus (Enright and Cumbie, 1973
).
The vertical growth rates of the current study generally agree with these previous studies. The vertical velocity field within the stem unit was steady at any given vertical position within the stem unit and increased basipetally with time. The vertical strain rate was linear in space and time for the stem units under study. The vertical strain rate of nodes was nonlinear and consistently lower than that of subjacent stem units (Fig. 12). There is also good agreement with dVs/Vs strain rate function of Silk and Abou Haidar (1986)
because in their Fig. 2 this rate appears to be constant. It should be noted that because the current study was confined to the intermediate region of the stem that lies between the shoot apical meristem and the mature stem, stem unit data did not extend to the most mature regions of the stem. Furthermore, previous studies utilized the internode subdivisions of the stem as the unit of analysis. Because the stem unit includes a radial sector of the stem that spans (m + n) internodes and is bounded tangentially by the m- and n-intermediate nodes, these previous studies only encompass 1/(m + n)th of the vertical extent of the stem unit. It would be of interest to extend the stem unit study to include the formation of mature stem units, which was beyond the scope of the current study.
In conclusion, overlapping nodes in the SAM become vertically separated by internodes on the mature stem as a consequence of lower vertical strain rates of nodal expansion compared with higher linear vertical strains of stem unit expansion (Fig. 12). The radial strain rates of both nodes and stem units diminish with time as the stem axis curvature diminishes (Figs. 4, 13). The tangential strain rate is asymmetrical relative to the m- and n-boundary sides of the stem unit (Fig. 14). The expanding tangential form of the stem unit resembles the form of an expanding laboratory scissor jack as a consequence of encroachment of the m- and n-boundary primordia via tangential expansion of these primordia flanks in the intermediate region of the expanding stem unit (Fig. 5). These observations suggest that theoretical phyllotactic models could be used to investigate stem unit and internode development if non-exponential growth functions were incorporated into their construction.
FOOTNOTES
1 The author thanks J. Kiss, N. Money, W. Kuhn Silk, N. Smith-Huerta, and anonymous reviewers for comments on the manuscript. Research supported by NSF grants DCB-8702157 and BSR-8614242. 
2 Author for correspondence (e-mail: meicenrd{at}muohio.edu
)
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