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Annals of Botany 89: 409-417, 2002
© 2002 Annals of Botany Company

Maintaining Apical Dominance in the Fern Gametophyte

DAVID M. HOLLOWAY*,1 and MARIA LANTIN{dagger},2

1Department of Chemistry, University of British Columbia, Vancouver, B.C. Canada V6T 1Z1 and 2School of Computing Science, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6

* For correspondence. Fax 00 1 604 822–2847, e-mail david{at}pepe.chem.ubc.ca. Also at: Mathematics Department, British Columbia Institute of Technology, Burnaby, B.C., Canada V5G 3H2
{dagger} Present address: IDELIX, Suite 400–1122 Mainland St, Vancouver, B.C., Canada V6B 5L1

Received: 3 August 2001; Returned for revision: 12 November 2001; Accepted: 17 December 2001.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 THE MODEL
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX
 LITERATURE CITED
 

A kinetic model is developed for cell differentiation in the fern gametophyte to test hypotheses on the role of spatially patterned plasmodesmata networks in development. Of particular interest is the establishment and maintenance of apical cell type in a single cell, with concurrent suppression of this character in all other cells (apical dominance). Steps towards understanding apical cell localization in geometrically simple gametophytes may shed light on the establishment and maintenance of apical meristems in higher plants. The model, based on the plasmodesmata maps of Tilney and colleagues and involving kinetics for a requisite minimum of two morphogens, successfully produces the apical/non-apical cell differentiation patterns of normal development, and redifferentiation due to cell isolation, in six stages from 0–30 d of development. Our results indicate that increasing apical cell plasmodesmata number, as development progresses, is not required for effective transport across apical cell walls in maintaining apical dominance.

Key words: Fern, Onoclea sensibilis, gametophyte, apical dominance, plasmodesmata, intercellular transport, meristem, cell differentiation, redifferentiation, pattern formation, mathematical model, plant development.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
THE MODEL
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX
 LITERATURE CITED
 
The shape of a multicellular plant is chiefly determined by actions in its meristematic tissues throughout development. In higher plants, apical meristems are distinguished by rapid and continual division (providing a source of new cells), and by uniquely expressed molecules. The spatial patterning of these molecules, within and surrounding the meristem, provides architectural cues for lateral structures (Jackson et al., 1994; Ferrandiz et al., 2000; Dengler and Tzukuya, 2001; Reinhardt et al., 2000). Understanding the workings of meristems will require not only further identification of the molecules associated with these regions, but also elucidation of the molecular interactions (reactions and transport) which regionalize their expression patterns, necessary for normal development. Ultimately, a complete picture of how meristems affect plant shape must combine this chemical pattern formation with the effects of cell division orientation, cell growth and cell wall mechanics.

To form a pattern across a multicellular tissue, molecules must not only react intracellularly, but must be transported intercellularly. In plants, a general pathway for this is through the plasmodesmata: gateable channels that connect a common cytoplasm through cell walls. [Transport of meristematic transcripts through plasmodesmata has been verified, e.g. KNOTTED1 (Lucas et al., 1995).] The distribution of plasmodesmata is not generally uniform. The potentially strong effect on patterning of non-uniform channelling, in both the meristem and its neighbouring tissues, should be considered.

Plasmodesmal distributions have been mapped in root meristems (Gunning, 1978). However, complexities of geometry and tissue layering in higher plants complicate the issues of how the meristematic region is defined by chemical patterning, and what effect plasmodesmal distribution may have. The fern gametophyte is a much simpler model system for addressing these issues. Following spore germination in ferns, this two-dimensional (cellular monolayer) structure develops through cell division and growth, eventually leading to development of the sexual organs, reproduction and generation of the sporophyte.

In the fern gametophyte, meristematic activity is greatest in a single apical cell (AC). As with multicellular meristems, events occurring in this AC (chemistry, orientation of cell divisions) are crucial to proper development of the gametophyte. While non-uniform channelling is not an issue within the AC, intercellular transport throughout the gametophyte is extremely important for proper development of the fern: if the AC is isolated from the rest of the gametophyte, non-apical cells will redifferentiate to take on apical character, commencing AC-type divisions and producing new gametophytes (e.g. Nagai, 1914; Albaum, 1938a; Ito, 1962; Korn, 1974). Normal development depends on apical dominance, i.e. the localization of AC type to a single cell, with concurrent suppression of AC type in all other cells. The fern gametophyte may serve as a model for apical dominance in higher plants, and the role intercellular transport might play in this.

Tilney et al. (1990) mapped the plasmodesmal network of gametophytes of the fern Onoclea sensibilis at six developmental stages (0–30 d). These authors found this network to be quite non-uniform, with the AC always having the greatest number of plasmodesmata. Over the course of development, this apical number increases dramatically (at 30 d the density of plasmodesmata in the AC is comparable with that found in secretory structures of higher plants). These observations led Tilney et al. (1990) to propose that changes in plasmodesmata number are necessary for effective export of a ‘disciplinary substance(s)’ from the apical region to give normal development.

The aim of the current work is to develop a kinetic model to test such hypotheses on the role of plasmodesmata in development. Explicit treatment of plasmodesmal function bridges the cell vs. organism debate in plant morphogenesis (e.g. Kaplan and Hagemann, 1991; Cooke and Lu, 1992) by investigating how a collection of cells can operate organismally (or, how an organism develops with cellular boundaries). Our model simulates the kinetics of an activator of AC type and its inhibitor, along with intercellular transport of the inhibitor based on the plasmodesmata maps of Tilney et al. (1990). High levels of activator define the AC in simulations. If understanding meristematic function requires knowledge of the relevant molecules, of the reactions contributing to their spatially patterned expression and of how intercellular transport may affect this, then the present work aims directly at understanding the latter two processes. This may shed light, indirectly, on molecular identity, through limits found on reactivity or size. In addition to building our model from published experiments, we can also conduct new ‘computer experiments’, e.g. altering plasmodesmata number, which serve to test existing hypotheses where biological experiments would be difficult or impossible.


   THE MODEL
TOP
 ABSTRACT
 INTRODUCTION
 THE MODEL
RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX
 LITERATURE CITED
 
The principal features of reaction and transport mechanisms in the model are presented here. Mathematical and computational details are given in the Appendix.

Reaction
The aim in developing a reaction mechanism is to capture the kinetics necessary to generate and maintain apical dominance in the fern gametophyte, and especially to account for redifferentiation following disruption of intercellular communication. Such experiments indicate that apical dominance is dynamically maintained, and that AC and non-AC types are not terminally differentiated, but retain the competency to form either cell type. A kinetic model accounting for these observations must likewise be based upon chemically equivalent and multipotent (having the ability to form either type) cells.

Models with a single spatially patterned chemical species specifying cell differentiation state (the term morphogen will be used in this paper for such chemicals) have traditionally been used for apical dominance in gametophytes (reviewed in Miller, 1968). However, these models are only capable of defining one cell type by concentration. The other cell type must be assumed to be in a terminal, static (not dynamically maintained) differentiation state (Fig. 1A and B). Defining two cell states (and the switching between these occurring with redifferentiation) by concentration requires two morphogens: the activator to AC-type must have an inhibitor. A two-morphogen model (Fig. 1C) can produce spatial concentration pattern if intercellular transport is present, but relaxes to a uniform steady-state concentration (corresponding to horizontal dashed line in Fig. 1C) in its absence. To model gametophyte patterning, the AC state is defined by having activator concentration at or above this steady-state concentration (black cells in Figs 2–8)345678, and the non-AC state is defined by having activator concentration below this level (grey cells in Figs 2–8). This representation of the differentiation states of the cells defines all cells as multipotent, with nothing special about the chemistry of the AC. As in the gametophyte, cell type is dynamically maintained, with intercellular transport playing a crucial role.



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  		Fig. 1. Schematic plots of morphogen (see text) concentration vs. position for mechanisms of apical dominance in the fern gametophyte. A and B, One-morphogen mechanisms [after Linsbauer (1926) who discussed meristematic activity, without distinguishing cell type]; C, a two-morphogen mechanism. Apical position is to the left, basal (or peripheral) position is to the right. A, Apical cell (AC) is a local source of AC-type inhibitor (solid line). Horizontal dashed line represents an inhibitor threshold, above which cells are non-AC type. However, the concentration of the AC-type inhibitor is highest in the AC, which should suppress AC character there, unless the AC operates under unique chemical rules (is terminally differentiated as AC-type), in contradiction with redifferentiation experiments. B, AC is a local sink of AC-type activator (solid line). The horizontal dashed line represents an activator threshold, below which cells are non-AC type. In this case, the AC activator is at its lowest level in the AC, which should again suppress AC character there. (Switching the roles of activator and inhibitor in A and B allows the AC to be defined by concentration, but not the non-apical cells.) In A or B, removal of the AC could lead to a partial activation of non-apical cells, but there is no mechanism to turn these cells into new sources (A) or sinks (B) of morphogen (necessary for full AC character in directing the development of new gametophytes). One-morphogen models can only succeed in defining one cell type by concentration. Switching between two cell types requires two morphogens. C, Two-morphogen model of the type used in this paper (solid line, AC-type activator; heavy dashed line, its inhibitor). Horizontal dashed line as for B, with value set by the uniform steady-state concentration of the model (see text).

 


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  		Fig. 2. Cell outlines for protonema, adapted from Tilney et al. (1990, Fig. 4A). In this and all subsequent figures, cells are shaded dark grey for AC activator at or above its uniform steady-state concentration, and light grey for activator below this concentration. Computation produces a single AC (labelled ‘0’ in this and subsequent figures). Cells were numbered sequentially to facilitate computation of eqn (2) in the Appendix, and do not necessarily reflect the age of cells (or impose any constraints on transport).

 


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  		Fig. 3. Cell outlines for 2-d prothallus, adapted from Tilney et al. (1990, Fig. 4B). Computation produces a single AC. Cell ‘1’ is its most recent daughter. For shading and cell numbering, see Fig. 2 legend.

 


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  		Fig. 4. Cell outlines for 4-d prothallus, adapted from Tilney et al. (1990, Fig. 4C). Computation produces a single AC. Cell ‘1’ is its most recent daughter. For shading and cell numbering, see Fig. 2 legend.

 


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  		Fig. 5. Cell outlines for 5-d prothallus, adapted from Tilney et al. (1990, Fig. 5). Computation produces a single AC. Cell ‘1’ is its most recent daughter. For shading and cell numbering, see Fig. 2 legend.

 


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  		Fig. 6. Cell outlines for 9-d prothallus, adapted from Tilney et al. (1990, Fig. 6). Computation produces a single AC. Cell ‘3’ is its most recent daughter. For shading and cell numbering, see Fig. 2 legend.

 


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  		Fig. 7. Cell outlines for 30-d prothallus, adapted from Tilney et al. (1990, Fig. 7). Computation produces a single AC. Cell ‘1’ is its most recent daughter. For shading and cell numbering, see Fig. 2 legend.

 


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  		Fig. 8. Cell outlines as in Fig. 7. Cell shading as in Fig. 2. Morphogen concentrations were computed as in Fig. 7, then intercellular transport cut off. Results are shown at a later time, when all original non-apical cells in the prothallus have achieved sufficient activator to become AC-type. The original AC is temporarily below threshold, in response to the abrupt cessation of transport.

 
Our approach to gametophyte differentiation is similar to that taken by Meinhardt (Gierer and Meinhardt, 1972; Meinhardt, 1982, 1993) for Hydra development. The particular activator/inhibitor kinetics in this work are from Gierer and Meinhardt (1972; equations in Appendix). The same rate constants are used for all six (as in Tilney et al., 1990) developmental stages computed and it is assumed that these are time-independent as the gametophyte develops.

In our model, the activator is an activator of AC type, and the inhibitor is an inhibitor of the activator. (Though the inhibitor is not excluded from having a direct effect on AC differentiation, the important property for producing AC localization and redifferentiation is its cross-regulatory effect on the activator.) The kinetics produce highest inhibitor concentration where the activator is highest, so that the AC cell can be viewed as a source of inhibitor (Fig. 1C; also see Discussion).

Intercellular transport
In this model, intercellular communication depends on transport of the inhibitor. The activator stays fixed in cells. This provides for the extreme localization of AC type in gametophytes. With respect to transport, pattern formation in this model is more like that of a Wigglesworth (1940) inhibitory field model than the full Gierer–Meinhardt (1972) model (activator also moving intercellularly). But the model retains the Gierer–Meinhardt reaction terms and hence the regulation necessary for cell multipotency (i.e. relaxation to uniform steady-state concentration in the absence of inhibitor transport, departure from this uniform concentration in the presence of transport).

In modelling molecular transport across the gametophyte, it is assumed that cell thickness is uniform and that concentrations depend on cell area (i.e. it is assumed that the gametophyte is indeed a two-dimensional geometry). We assume that cytoplasm is well-mixed within cells (for example, by cytoplasmic streaming), and that the rate-limiting step in transport is across cell walls, through the plasmodesmata (i.e. transport time across a cell is negligible with respect to transport time between cells). We treat this step as a concentration-dependent exchange of inhibitor (Fick’s 1st Law) across the cell wall. (For the transport equation, see Appendix.) The amount of inhibitor exchanged per unit time depends on the conductance of the inhibitor (constant over the course of development, as with the reaction parameters) through plasmodesmata and the number of plasmodesmata in the wall (from Tilney et al., 1990). This type of exchange is diffusion, in which the spatial units are the cells [see Wolk and Quine (1975) for an earlier discussion of this idea].


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 THE MODEL
 RESULTS AND DISCUSSION
CONCLUSIONS
 APPENDIX
 LITERATURE CITED
 
Generation of apical dominance
The model successfully produces the normal AC/non-AC pattern in the six developmental stages studied. Figures 2–7 show the cell outlines, adapted from Tilney et al. (1990), for the protonema and 2-, 4-, 5-, 9- and 30-d prothallus stages, respectively. The cells are shaded according to the computed activator concentration: black, for activator at or above threshold for AC type; and grey, for activator below threshold for AC type. At all stages, the model succeeds in limiting the spatial extent of AC type to the single AC. (Localization to a single cell remains successful in computations with activator concentration initialized equally in the AC and its most recent daughter; see Appendix for details.) The sharp discontinuity in activator concentration between the AC and its neighbours is a result of the two-morphogen dynamics in the model.

Disruption of intercellular communication
A number of experiments have demonstrated the effects of disruption of intercellular communication on the AC/non-AC pattern. These include plasmolysis (Nagai, 1914; Schraudolf, 1966), cell ablation (Ito, 1962; Korn, 1974) and surgery (Albaum, 1938a; Korn, 1974). [Nakazawa (1963) demonstrated that plasmolysis leading to redifferentiation does proceed by breaking the plasmodesmata.] Isolation of the AC from the rest of the gametophyte by any of these methods results in redifferentiation of non-apical cells to AC type.

In our computer experiments, cell isolation is achieved by cutting off intercellular transport, so that only intracellular reactions are computed. This is done after the normal AC/non-AC pattern has been established (as in Figs 2–7). Upon cessation of transport, the spatial pattern of the activator relaxes to uniformity, such that every cell in the gametophyte is at the steady-state concentration marking AC type. Our computational ‘plasmolysis’ experiments produce redifferentiation at all six developmental stages. Figure 8 shows a typical result, for the 30-d prothallus.

Experimentally, relaxation to AC type has a time dependence. Ito (1962) conducted ablation experiments (on mature, 3-week-old prothalli of Pteris vittata), in which individual, non-apical cells were isolated by needle ablation from all neighbours. These isolated cells redifferentiated to AC type, with cells farthest from the original AC taking approx. 4 d to form new protonemata, and those closest to the AC taking approx. 20 d. Our computations show a similar trend: breaking intercellular transport (after achieving the normal pattern; Figs 2–7), we see a redifferentiation front moving from peripheral to apical positions. (Cessation of transport in the computation, while most apparently similar to a plasmolysis experiment, is practically equivalent to Ito’s isolation of individual cells.) Although Ito’s experiments did not cover a range of developmental stages, our computations predict that redifferentiation (both initiation and completion) should take longer in older gametophytes than in younger gametophytes.

Redifferentiation times were not as divergent in our computations as in Ito’s (1962) experiments. At the 9- and 30-d stages (the closest comparable with Ito’s 3-week-old prothalli), we obtain a roughly 1 : 1·2 ratio of times for first and last cells to redifferentiate, as compared with Ito’s approx. 1 : 5 ratio. This difference between experiment and computation could arise from differences in defining differentiation. Ito measured this as formation of new protonemal cells, whereas the computations reflect when a cell becomes specified for AC type. There may be a delay between specification and the manifestation of observable protonemal cells. Also, the low model ratio is at least partly due to the high plasmodesmal conductance necessary for apical dominance in the protonemal stage (Fig. 2; see discussion below).

Increase of AC plasmodesmata number
Tilney et al. (1990) suggested that the sharp increase (approx. 50-fold) in AC plasmodesmata number from protonema to 30-d prothallus is required for proper regulation of gametophyte development. Our model allows us to test more precisely what type of regulation this might be. Specifically, we can directly alter AC plasmodesmata number in the model (difficult or impossible experimentally). The only function of plasmodesmata in our model is to effect intercellular transport of the inhibitor. Therefore, if we find that the experimentally observed 50-fold increase in AC plasmodesmata number is necessary in the model to maintain apical dominance, this suggests that the in vivo increase is a result of transport constraints across an ever-growing gametophyte. If, on the other hand, we observe that the model does not require something like a 50-fold increase in AC plasmodesmata number to generate the normal pattern, this implies roles for AC plasmodesmata number other than transport of a cell differentiation factor across the AC walls.

Our results show the latter: the model requires little or no increase in AC plasmodesmata number, from protonema to 30-d prothallus, to produce apical dominance. In fact, for most prothallus stages, per wall plasmodesmata numbers of approx. 150 allow sufficient inhibitor to flow out of the AC (the 9-d prothallus required approx. 650). This is at least an order of magnitude lower than the numbers observed by Tilney et al. (1990; Fig. 9).

The plasmodesmal conductance of the inhibitor is a consideration in interpreting these results. This conductance is set as the slowest that will give suppression of AC character in all sub-apical cells, across the six developmental stages studied. The conductance is constrained by (needs to be highest in) the protonema stage (Fig. 2). [A plasmodesmal conductance four-times slower is sufficient for apical dominance in all prothallus stages. At this lower conductance, numbers of AC plasmodesmata required are still an order of magnitude lower than those observed by Tilney et al. (1990): 400 for the 2-d prothallus; 1400 for the 4-d prothallus; 650 for the 5-d prothallus; 2150 for the 9-d prothallus; and 2400 for the 30-d prothallus.] Constraint of plasmodesmal conductance by the protonema may be somewhat surprising if this is thought of as the initial stage in gametophyte growth. However, the length of the gametophyte does not change much over the stages studied; the heart-shaped prothallus is just much broader than the protonema. Therefore, the range of the AC-type inhibitor does not need to get much longer as development proceeds, and flow out of the AC does not have to increase greatly. The protonema is the limiting stage because it has low plasmodesmata number across basal walls (effective suppression of non-apical cells depends upon effective transport in the periphery, as well as effective transport out of the AC), and because of large average cell size. Intercellular transport [eqn (2), Appendix] is inversely proportional to cell area. Although the length of the gametophyte does not change greatly, there is a large cell proliferation, such that average cell size in Fig. 2 is approx. 1800 µm2, while that in Fig. 7 is approx. 600 µm2.

Our model results do not support Tilney et al.’s (1990) hypothesis that export of an inhibitor requires an increasing density of plasmodesmata in the apical region (specifically, the AC). Since plasmodesmata are created only during cell divisions, AC plasmodesmata number may rather reflect transport requirements between adjacent lineages of the AC (merophytes) in more basal regions of the gametophyte. Another suggestion, also from Tilney et al. (1990), is that high AC plasmodesmata number may be involved in the formation of the AC itself. Besides its effect on cell differentiation in the gametophyte, the AC is distinguished (in older prothalli, Figs 5–7) by its triangular geometry. Cell division in the AC occurs anticlinally, from the outside wall to one of the two inner walls. The inner wall chosen alternates from division to division. The wall chosen always has the highest plasmodesmata number. (Tilney et al. observed that plasmodesmata number increased in the target wall before completion of cell division.) Perhaps this observable distinction between the two inner walls is used by the mechanism responsible for spatial orientation of apical cell division.

AC type and meristematic activity
The AC/non-AC pattern, which is the focus of this paper, is correlated with meristematic activity: the AC is the fastest dividing cell in the gametophyte. If the inhibitor in our model is also identified as a cell division inhibitor, then the AC becomes a local source to inhibit meristematic activity. This is akin to Linsbauer’s (1926) original one-morphogen model (Fig. 1A), but based on cell types defined by chemical dynamics, rather than pre-defined, terminal cell types.

Future work
We would like to extend the current model for chemical pattern formation to incorporate cell division and growth (following Holloway and Harrison, 1999a), enabling computation of gametophyte development from protonema to mature prothallus in one run (rather than being bound to the present six stages). This would allow us to address a number of issues, including: (1) the possible effects of AC plasmodesmata number on cell division orientation [see Harrison (1993) Sec. 3·2, for initial theoretical work connecting chemical patterning with division orientation]; (2) the effects of altering AC plasmodesmata number on inter-merophyte transport; (3) the association between the AC-type inhibitor and inhibition of cell division, using cell division data; (4) the effects of cell size at intermediate stages, with a bearing on how the AC character is always retained in the original AC, and never transferred to its latest daughter (see Appendix for preliminary results); (5) more accurate cell and gametophyte sizes than those based on Tilney et al.’s (1990) partially cropped micrographs, which could have a bearing on plasmodesmal conductance and redifferentiation times in the model as a closer match with Ito’s (1962) times would aid in matching the transport properties of the model inhibitor to those of putative inhibitors such as IAA (Albaum, 1938b); and (6) meristem function in higher plants, including a switch to (roughly) hemispherical three-dimensional geometry (and multiple tissue layers), and use of existing transport data [e.g. Gunning’s (1978) plasmodesmal map in Azolla] and molecular data from such well-characterized systems as Arabidopsis, Antirrhinum or maize.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 THE MODEL
 RESULTS AND DISCUSSION
 CONCLUSIONS
APPENDIX
 LITERATURE CITED
 
We have developed a minimum (two-morphogen) kinetic model for generation and maintenance of apical dominance, in the geometrically simple case of the fern gametophyte. This model successfully generates the AC/non-AC pattern underlying normal development, and undergoes cell redifferentiation in response to disruption of intercellular transport. Our results do not support the hypothesis that increasing AC plasmodesmata number is necessary for effective export of differentiation factors across the AC walls. This phenomenon may rather be linked to inter-merophyte transport, or to control of apical cell division orientation.


   ACKNOWLEDGEMENTS
 
D.M.H thanks F. D. Fracchia and the School of Computing Science, Simon Fraser University for their hospitality, and L. G. Harrison, J. Dumais, T. C. Lacalli, and an anonymous referee for critical reading and helpful comments. This work was started with the support of the National Science Foundation under a fellowship awarded in 1996.


   APPENDIX
TOP
 ABSTRACT
 INTRODUCTION
 THE MODEL
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX
LITERATURE CITED
 
Reaction
The equations of Gierer and Meinhardt (1972) are used to compute changes in activator ({Delta}X) and inhibitor ({Delta}Y) concentrations due to intracellular reactions, in a time interval (time step) of {Delta}t. They are:



[For an enzyme mechanism that gives these rate equations, see Holloway et al. (1994).] Parameter values used in this work are: a = 0·0006; S = 0·4; c = 0·1; µ = 0·009; c' = 0·065; {nu} = 0·01; {Delta}t = 0·015625. With these parameters, AC-type is defined by having X >= 1·74 (the uniform steady-state concentration). (Though non-apical cells are defined in our model by any X < 1·74, in fact they have X < 0·1 when intercellular transport occurs. Apical cells, depending on the developmental stage, have X > 10. We do not, therefore, expect any natural perturbations in X or Y to change differentiation state.)

This model tends to form far more localized activated regions than other two-morphogen models (Holloway and Harrison, 1995). We therefore chose it as a starting point for describing the chemical mechanism underlying the extreme localization of AC type within the fern gametophyte. We do not propose that the Gierer–Meinhardt mechanism gives the exact in vivo biochemistry of AC differentiation, but that it serves as a starting point to capture the essential kinetic features of this phenomenon.

Intercellular transport
The following expression calculates the exchange of inhibitor (Y) between a cell i and its neighbour i + 1:


where {Delta}Y[i] is the change in Y due to this exchange; CY is the plasmodesmal conductance of Y (= 40, assumed not to vary with position in the gametophyte); A[i] is the area of the cell, in µm2 (from Tilney et al., 1990); and the Y terms give the concentration difference between the two cells. npl[i] is proportional to the number of plasmodesmata in the wall between the two cells: it is equal to Tilney et al.’s values for observed plasmodesmata in a sectioned wall, plus a ‘leakage’ term of 0·6 (the minimum necessary to maintain communication throughout the gametophyte). Each plasmodesma observed in a sectioned wall corresponds to roughly 250 plasmodesmata in the entire wall (Tilney et al., 1990; Fig. 9), so the above leakage term represents approx. 150 plasmodesmata per wall.

General
Figures 2–7 represent computational results at 160 000 time steps. Figure 8 shows redifferentiation at 230 000 time steps (160 000 with transport, then 70 000 without). Concentration fluctuations were modelled as in Holloway and Harrison (1999b), assuming that a unit morphogen concentration corresponds to 10 000 molecules per cell.

Morphogen concentrations were initialized at the uniform steady-state values in all cells (corresponding to AC type). In Figs 2–8, a transient boost (in the first time step only) of X + 0·5 was given to the AC. (Such determination of initial polarity should not be necessary in future work with growth and cell division.)

To test retention of AC character in only one cell, following apical cell division, two other initialization procedures were used (on prothallus stages): (1) transient boost (first time step) of X + 0·5 in the AC and its latest daughter; and (2) transient boost (first time step) to final peak X and Y (as determined from the previous stage computation) in the AC and its latest daughter. In these cases, only one of two initially activated cells ever retained AC character. This was not, however, always the original AC, but the latest daughter in two out of five cases for procedure (1), and one out of five cases for procedure (2).

The Cell Systems Simulation Environment (Lantin, 1999) was used for graphical presentation of the results. All computations and graphics were run on Silicon Graphics O2 and Octane workstations. Programs for the numerical computation of the model equations were written in C.


   LITERATURE CITED
TOP
 ABSTRACT
 INTRODUCTION
 THE MODEL
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX
 LITERATURE CITED
 

    Albaum HG. 1938a. Normal growth regeneration, and adventitious outgrowth formation in fern prothallia. American Journal of Botany 25: 37–44.[CrossRef]

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    Cooke TJ, Lu B. 1992. The independence of cell shape and overall form in multicellular algae and land plants: cells do not act as building blocks for constructing plant organs. International Journal of Plant Sciences 153: S7–S27.[CrossRef]

    Dengler NG, Tzukuya H. 2001. Leaf morphogenesis in dicotyledons: current issues. International Journal of Plant Sciences 162: 459–464.[CrossRef]

    Ferrandiz C, Gu Q, Martienssen R, Yanofsky MF. 2000. Redundant regulation of meristem identity and plant architecture by FRUITFUL, APETALA1 and CAULIFLOWER. Development 127: 725–734.[Abstract]

    Gierer A, Meinhardt H. 1972. A theory of biological pattern formation. Kybernetik 12: 30–39.[CrossRef][Web of Science][Medline]

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D. M. Holloway and L. G. Harrison
Pattern Selection in Plants: Coupling Chemical Dynamics to Surface Growth in Three Dimensions
Ann. Bot., February 1, 2008; 101(3): 361 - 374.
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