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Evolution of Land Plant Architecture: Beyond the Telome Theory

Posted on: Wednesday, 20 September 2006, 06:00 CDT

By Stein, William E; Boyer, James S

Abstract.-For well over 50 years, the telome theory of Walter Zimmermann has been extremely influential in interpreting the evolutionary history of land plant architecture. Using the "telome/ mesome" distinction, and the concept of universal "elementary processes" underlying the change in form in all plants, the theory was an ambitious synthesis based on the proposition that evolutionary change might be understood by a simple set of developmental or evolutionary rules. However, a major problem resides in deciding exactly how assertions of change are to span both developmental and evolutionary domains simultaneously, and, we argue, the theory critically fails testability as a scientific theory. Thus, despite continued popularity for the descriptive terms derived from the theory in evolutionary studies of early land plants, time has come to replace it with a more explicit, testable approach. Presented here is an attempt to clarify perhaps the most important issue raised by the telome theory-whether simple changes in basic developmental processes suffice to describe much of early land plant evolution. Considering the morphology of Silurian- Devonian fossil members, it is hypothesized that early land plants possessed a common set of developmental processes centered on primary growth of shoot apical meristems. Among these were (1) the capacity to monitor and act upon internal physiological status here modeled as "apex strength," (2) a mechanism for allocation of apex strength in a context-dependent way at each point of branching, (3) a rule for context-dependent apex angle for branches, (4) a largely invariant phyllotaxis unrelated to physiological status, and (5) a simple switch for terminating primary growth, based in part on genetics. Implemented as a set of developmental rules within a simple L-system model, these aspects of primary development in plants determine a sizable range of resultant morphologies, some of which are highly reminiscent of the early fossils. Thus, some support is found, perhaps, for Zimmermann's intuition. However, traditional concepts of growth patterns in plants, including the contrast between epidogenesis and apoxogenesis, require updating. In our reformulation, developmental processes, stated as rules of developmental dynamics, together constitute what we term the plant's developmental state. Using a hypothetico-deductive format, one may hypothesize intrinsic (or genetic) developmental processes that play out as realized developmental activity in specific spatial/temporal contexts, as modified by multiple context factors. The resultant plant morphology is highly dependent on multiple and simultaneous pathway ontogenetic trajectories. Within a likely set of developmental rules reasonably inferred from plant development, some of Zimmermann's elementary processes are perhaps recognizable whereas others are not. Progressively "overtopped" morphologies are easily produced by modifying intrinsic branch allocation. However, even so, the other developmental rules have a profound effect on final architectures. Planate architectures and circination vernation, often treated as special cases by plant morphologists, are perhaps better understood in terms of recurrent or iterative developmental relationships. Much analytic work remains before a completely specified system of rules will emerge. A well- articulated relationship between ontogeny and phylogeny remains fundamentally important in assessing evolutionary change. Fossil and living plants make it abundantly clear that current evolutionary concepts involving modification of a single ontogenetic trajectory from ancestor to descendant need to be greatly expanded into consideration of the entire logical geometry of causation in development. A mechanism for testing is also required that need not wait for complete elucidation at the molecular level. The relative simplicity of plant development, combined with an outstanding fossil record of early members, offers unique opportunities along these lines.

Introduction

In the study of the evolutionary origins of plant form, probably no idea has exerted greater influence than Zimmermann's telome theory (Zimmermann 1930, 1935, 1952, 1959, 1965). Departing from previous idealisms that considered organs such as shoot, leaf, and root as "fundamental" units of form (Goethe 1790 in Arber 1946), Zimmermann proposed a broadly comparative concept focused on elements he termed telomes-"undifferentiated elements in the organization of the oldest land plants" (Zimmermann 1952: p. 456; 1965: p. 1). Based on prior theoretical work (Bower 1908; Lignier 1908; Potoni 1912) and exquisitely preserved fossil shoots of Aglaophyton (formerly Rhynia) major and other plants from the Lower Devonian Rhynie Chert (Kidston and Lang 1917, 1920a,b,c), telomes were conceived to be a simple, dichotomously branched system (syntelome) of leafless stems terminating in vegetative tips (phylloids) or solitary sporangia. From this, Zimmermann (1952: p. 456; 1965: p. 1) offered narrower topographic definitions "in a restricted sense": telome-"single-nerved terminal branches from the last ramification to the tip of the plant," and mesome- "corresponding parts (of a branch system) between two ramifications."

TABLE 1. Summary of the telome theory of Zimmermann (1952).

In the layout of the telome theory, at least two phases critical to the evolution of plant form were envisioned (Zimmermann 1952). The first, mostly forgotten today, centered on the origin of ancestral growth patterns within primitive telomes (Table 1A). A second phase, still familiar to students of plant morphology, involved hypothesized changes in plant form after origin of basic telome architecture (Table 1B). Comparison of observed variations across both fossil and living forms was considered important in these proposals, especially as differences relate to the origin of major groups of vascular land plants at high taxonomic level.

The heart of the comparative approach advocated by the telome theory after the origin of telomes was the idea of modification of plant form by means of elementary processes. This concept has a developmental flavor, but when examined closely its meaning is problematic. According to Zimmermann (1952: p. 547, 1965) "the term 'elementary processes' implies that these elements of evolution proceed in ontogeny and phylogeny independently of each other." In the bulk of his writings on telomes, Zimmermann repeatedly pointed to examples of increasing complexity in both living and fossil plants that may be interpreted as involving assembly of different combinations of the same elementary processes. From this, it is reasonable to infer that both independence and universality of five or six elementary processes modifying telomes was asserted (Fig. 1). In addition, he hinted that the list of elementary processes was not fixed, but might instead be augmented by other elementary processes for certain comparisons, including (his examples) "anatomical transformations" involving leaf vascular differentiation, transformations in stelar structure, and changes in differentiation of cell type (Zimmermann 1965).

Although the basic structure of the telome theory is quite simple, the objective of the approach is astonishingly comprehensive- nothing less than a unified theory relating basic architectural, anatomical, and developmental/ontogenetic features of vascular plants to their evolutionary radiation assessed by comparing living forms and the fossil record. According to Zimmermann (1952: pp. 469- 470): "The main value of the telome theory and, besides fossil findings, at the same time its most important argument of proof, is its general validity and uniformity." If true, then the telome theory must certainly be considered a major achievement (Kenrick 2002). Even if considered today to be overly optimistic, one must nevertheless acknowledge that Zimmermann's framework has been and continues to be the ruling paradigm for a very wide array of evolutionary problems in plants (e.g., Wilson 1942, 1953; Bailey and Swamy 1951; Florin 1951; Canright 1952; Zimmermann 1969; Jennings 1979, Stewart 1983; Herr 1995, 1999) and commands at least some mention in nearly every textbook on plant morphology or evolution written over the past 50 years. Indeed, given this history it is very difficult, if not impossible, to describe evolutionary patterns exhibited by the fossil record of early vascular plants without invoking the concepts (Wilson 1953; Stidd 1987) and descriptive lexicon (Niklas 1997a) provided by the telome theory-whether one views the theory to be currently viable or not. With increased interest in the role of developmental genetics in evolutionary history and process (e.g., Gerhart and Kirschner 1997; Davidson 2001; Wilkins 2002; Cronk et al. 2002; Mller and Newman 2003; Friedman et al. 2004), the telome theory, and elementary processes, potentially takes on new importance. Because of notable advance since Zimmermann's day, there now exists the exciting possibility of identifying evolutionary changes in gene families across dramatically different morphologies that might fundamentally redefine and empower at least the developmental aspect of his comparison-based proposals of evolutionary change in plants.

FIGURE 1. Transformations by "elementary processes" of telomes in Zimmermann's telome theory-here \divided into two groups. A, Elementary processes involving single telomes. Starting from fertile telome, center, arrows indicated possible changes: c, telome "recurvation"; d, telome "reduction"; s, telome "sterilization" (Kenrick and Crane 1997). B, Elementary processes involving plant architectures larger than a single telome (with mesome). Starting from a telomic truss, center arrows indicate possible changes: a, truss "overtopping"; b, truss "planation"; C^sub A^, truss vascular syngenesis "Verwachsung im Achse," in which the entire truss is incorporated as a multifasciculate vascular system of a single axis; C^sub B^, truss webbing "Verwachsung im Blatt."

In a cogent reappraisal, Kenrick and Crane (1997) suggested that Zimmermann's system of independent and multiply combinable elementary processes, never viewed by him in explicit phylogenetic terms, should now be reinterpreted within a cladistic framework. According to them, transformational hypotheses of evolutionary change, presumably involving one or more of Zimmermann's elementary processes, should be mapped onto cladograms produced from taxic homologies and assessed thereby as potential synapomorphies. An interesting example of this approach was provided by these authors (1997: pp. 228-292) in which one or more instances of a new elementary process they termed "sporangium sterilization" (involving origin of microphylls by sterilization of lateral sporangia within the zosterophyllophyte-lycopsid clade) was considered more parsimoniously distributed over a cladogram than multiple instances of Zimmermann's theory by "overtopping"-"planation"- "reduction," or Bower's (1935) hypothesized origin by "enation."

We agree with Kenrick and Crane that offering hypotheses such as these provides an opportunity to test specific transitional events related to the telome theory within a more general phylogenetic paradigm. Moreover, if elementary processes can be interpreted as cladistically mappable evolutionary changes in developmental regulatory networks, then direct comparison is facilitated between morphology and underlying genetic structure. However, to accomplish this comparison it is essential to move beyond Zimmermann's descriptive categorization of process (see Niklas 1997a) to a more completely specified system, and thus fundamental issues raised by the telome theory, for nearly 50 years unchallenged, need to be revisited.

Does the telome/mesome distinction of the telome theory faithfully capture fundamental organization or homology in plants? The answer clearly is no. Abundant evidence suggests that all past and present plants are serially developed. New primary tissues are continuously derived by activity of the shoot/root apex and it is reasonably certain that the shoot/root apical meristem constitute the fundamental organizers of the primary body in all plants (e.g., Lyndon 1990; Tooke and Battey 2003; Friedman et al. 2004). During primary growth, each apical meristem may be viewed as a self- replicating developmental processor, autonomously responsible for the production of new tissues including new apices by bifurcation or production of lateral primordia (Stein 1998). Both meristematic activity and cell maturation occur in a spatially and physiologically context-dependent way, using information and signals sent by the apex itself, and by other parts of the plant body. Also important are more general internal or external "environmental" cues, including position in the body, physiological state of the shoot, and water relations. A new apical meristem, once established as an autonomous developmental processor, proceeds on its own, subject to its specific spatial/physiological and sometimes temporal (e.g., latent bud) context. All of this strongly indicates that there is nothing significant, either in development or in evolution, about Zimmermann's topographically defined "telome" and "mesome." As development proceeds, telomes regularly become mesomes, and there is little point-for-point correspondence between forms at any stage in development, beyond the apical meristems themselves, that might be called homologous.

Do elementary processes in the telome theory faithfully capture natural units of process in either development or evolution? This question may be directed not only toward Zimmermann's original list of elementary processes (Table 1), but also to additions/ modifications proposed by others over the years. In short, we argue for all variants of the telome theory that "elementary processes" as currently conceived fail to do so. To describe process unambiguously, analogy may be made with mathematical functions or their derivatives, including "operators,""procedures,""rules," or "logic gates" in computer modeling (Stein 1998). To be useful, all of these entities must have a well-defined scope, often expressed in terms of domain (before action) and range (after action), as well as a clearly defined mode of action, or mapping, between domain and range. In this context, it is fair to say that all elementary processes try to map some aspect of telomes from before to after application of an elementary process. However, in terms of the mappings involved, they are a mixed set. For instance, "incurvation" or "sterilization" might be viewed as a phylogenetic change in the rules of development for a single axis (or telome) from ancestor to descendant (Fig. 1A) whereas "planation" or "overtopping" involves larger portions of ancestral or descendent syntelomes (Fig. 1B). Overriding all of these concerns is the question of scope-whether all, or any, mappings should be considered developmental changes versus evolutionary changes in form (Stein 1998). Although undeniably attractive as verbal descriptions of differences between fossil or living forms in a comparative sense, Zimmermann's elementary processes are inadequately specified to describe unambiguously either development or evolution separately, let alone both together.

Presented here is an attempt to clarify perhaps the most important issue raised by the telome theory-whether evolution especially in early plants represented by Silurian-Devonian fossils can be understood in a synthetic way by recourse to only a few basic developmental changes potentially applicable in a wide array of phylogenetic settings. Using a modeling approach, the objective is to bridge the gap between Zimmermann's amorphous comparative concepts on the one hand and the logic of known or soon to be known genetic regulatory pathways on the other (Niklas 2003; Friedman et al. 2004). Our rule-based model is similar to that of Niklas (e.g., Niklas and Kerchner 1984; Niklas 1997b) in unambiguously specifying a core geometry for branching in primitive dichotomous aerial shoot systems observed in early plants. However, here we are more specifically concerned with regulation of developmental activity within autonomous shoot apical meristerns by spatial/physiological context during serial primary development. Such an approach will, we hope, permit a more natural description of the relationship between the genetically specified logic of development, and the context- dependent consequence of this logic as development proceeds observed both within and between individuals, species, or higher groups. Similar in spirit to Zimmermann, we treat each developmental rule proposed here as an independent developmental process within bounds set by a single or shared ontogenetic system. In addition, we hope to identify the same, or substantially similar, developmental context factors at work in multiple phylogenetic settings. Also, like the telome theory, the set of rules remains open-allowing for appraisal, modification, or deletion of each, as well as for the proposal of new ones. However, unlike the telome theory, the model is explicit, permitting evaluation and testing of quantitative predictions of final form using measurements from living or fossil plants.

Methods

The methodology used here, as well as the model presented below, is based on a few assumptions:

1. Tissues of each part of the primary body in early land plants, and comparable modern plants, are traceable to the developmental activity of only a single apical meristem that produces replicate daughter apices over time. Thus, homology in these plants resides in the activity of these meristems rather than in topographic relationships (such as "telome" or "mesome") based on the mature plant body. In order to address issues raised by the telome theory, modeling will be restricted to the aerial shoot system in early land plants. However, in principle this approach seems applicable to other parts of the plant body and to other organs in more advanced plant groups including angiosperms.

2. Each apical meristem is an autonomous agent of development capable of reading internal physiological condition, signals from other meristems, as well as internal or external environmental cues such as temperature or amount of light, height above ground, length of the transpiration stream, or other factors unique to position within a growing shoot system. Thus, each apex contains a full set of internal (genetic) instructions for development and must combine these with context information to produce a context-dependent subset of possible outcomes. In the model below, we examine only a very simple set of internally specified developmental processes, thus restricting outcomes even further.

3. Developmental activities of each apical meristem are fully determined by developmental rules; nothing is stochastic. Each rule is a quantitative statement, analogous to a function or logic gate (Fig. 2), in which antecedent parameter values encapsulating aspects of internal physiological state, developmental signals, or environmental factors are passed to the rule domain. Rule action is then taken with resultant values or relationships sent to the rule range. Of course, the greater the numberof rules specified with increased causal relationships between them, the greater fidelity and potential unpredictability of the result (Stein 1998). In the model outlined below, only five quantitative developmental rules somewhat suggestive of telomic "elementary processes" are proposed (Fig. 3). Although clearly insufficient for capturing all aspects of primary development or evolutionary change envisioned by Zimmermann, they are sufficiently complex in their interaction to produce highly suggestive plant architectures.

FIGURE 2. Scheme of a developmental gate (see also Stein 1998). Inputs to the domain are indicated by the "IN" arrow at left, output in the range of the gate by the "OUT" arrow at right. Each developmental gate (box) involves "developmental logic"-a function or mapping between input(s) and output(s). For instance, some process may be viewed as conforming to a logical "AND" (⋁) requiring appropriate inputs from both of two input leads to produce a single output.

FIGURE 3. Developmental rules of the model displayed in the "developmental gates" form of Figure 1. See text for formal definitions. In all, domain to the left, range to the right. R^sub 1^-determining the trajectory of apex strength. Inputs are apex strength S^sub t^ (at plastochron t), and trajectory exponent E. Output is ontogenetic trajectory F^sub t^. Logic of the developmental gate is the relation for F given in equation (3) including boundary variables S^sub min^ and S^sub max^. R^sub 2^- rule for dichotomous branch allocation of apex strength S. Inputs F^sub t^ from R^sub 1^ above, apoxogenetic-epidogenetic parameter r and intrinsic allocation parameter D. Outputs are apex strengths for daughter apices S^sup 0^^sub t+1^ and S^sup 1^^sub t+1^ (at plastochron t+1). Logic of R^sub 2^ is found in equations (5-7) utilizing realized allocation of apex strength On. R^sub 3^-rule for orientation of daughter apices relative to the parent axis. Input are apex strength S^sup x^^sub t+1^ (for each daughter apex x at plastochron t+1), angle coefficient p, angular exponent K, and Φ, original apex orientation (at plastochron t) Output are Φ^sup x^^sub t+1^ angles of apex orientation. Logic of R^sub 3^ is in equations (9, 10) involving ontogeny shape shape-position parameter G^sup x^ and boundary variables Φ^sub max^ and Φ^sub min^. R^sub 4^-rule for phyllotaxis. Input is prior phyllotactic angle Θ^sup x^^sub t^ (for apex x at plastochron t). Output is new phyllotactic angle Θ^sup x^^sub t^. Logic of R^sub 4^ involves augmentation by a constant orthostichous or irrational primary divergence angle &920; specified in each case. R^sub 5^-rule for apex termination and conversion to a sporangium. Input is apex strength S^sup x^^sub t+1^ (for daughter apex x after branching at plastochron t+1). Output is terminal state X^sup x^^sub t+1^ (see Fig. 3). Logic is that of a simple switch with values of S^sup x^^sub t+1^ falling below preset boundary S^sub min^ triggering R^sub 5^ function.

Developmental Time in Fossil and Living Plants

The first issue to be resolved in modeling developmental processes allowing comparison of living and fossil plants is an adequate but sufficiently simple description of developmental time. A very general approach might be use of differential equations involving infinitesimals of absolute time (measured in seconds). In many biological systems, however, developmental processes are highly dependent on environmental factors such as temperature, moisture availability, time of day, and physiological stress. Although it might be possible to construct partials covering most of the important factors, such equations become forbiddingly complex. Moreover, the equations would be of little use in evaluating how closely predictions of the model match living, and especially fossil, plant architectures, because after development is completed very few of these variables might be independently estimated from static specimens. However, serial development in plants provides an unreversed time-ordering of development with iterative structural markers and thus offers a reasonably natural (if somewhat imprecise) concept of developmental time. A commonly used unit is the "plastochron"-an interval of developmental time, as well as physical distance, between successive initiations of leaf primordia by the shoot apical meristem (Erickson and Michelini 1957; Esau 1965).

At present, there is very little direct evidence on how shoot apices of primitive Devonian plants with simple dichotomous branch systems were structured or developed. In particular, little is known about the developmental process resulting in axis dichotomy and the relationship, if any, to some pattern of initiation of lateral appendage primordia. For our purpose here, we simply define plastochron in early dichotomous vascular plants to be that interval of developmental time inferred by the presence of successive axis dichotomies in the mature condition. In the model (Fig. 4), plastochron time is treated as an integer with successive plastochrons labeled {1, 2, 3, . . . , t, t + 1, . . . }. For the purpose of modeling mature form, each successive plastochron interval is also visualized as a segment of axis between successive branch points. Although clearly an oversimplification, this definition nevertheless allows for direct morphological interpretation of model results.

The Model

The model described here is a quantitative, three-dimensional, rule-based system centering on developmental activity of a single shoot apical meristem, and its derivatives, through successive plastochrons, as defined above. In this paper, five interrelated rules of development (R^sub 1^-R^sub 5^) are specified (Fig. 3). Described below are the variables and a brief rationale for each rule. Specific logic and mathematical relationships appear in the Appendix.

R^sub 1^ - Apex Strength Trajectory.-Owing to a combination of factors, in part genetically based but also reflecting circumstances in which an apex finds itself, the "vigor" or "growth potential"- that is, the capacity of an apex for continued growth-appears a natural and important feature of plant development (Eggert 1961; Prusinkiewicz and Lindenmayer 1990; Tooke and Battey 2003). A way to treat this phenomenon quantitatively is by means of a developmental trajectory in which a real variable for "vigor" increases or diminishes in a specified way over plastochron time. In our model, we use a recursive concept of vigor termed apex strength (S). For non-branching shoot systems successive values of S imply distinct growth regimes for which terminology already exists (see Eggert 1961 for original morphological/anatomical definition of terms, but note our qualifications below): apoxogenetic-a diminishing regime simulated by decreasing values of S over successive plastochrons; epidogenetic-an increasing regime, simulated by increasing values of S, and menetogenetic-a steady state between the other two implied by unchanging values of S, In the model, we use a linear function for calculating values of apex strength (S) with a single apoxogenetic- epidogenetic control variable (r). This implementation may be viewed as a subset of the more general allometric approach. However, because the definition of S is recursive (values of S^sub t+1^ depend on S^sub t^, at each plastochron t), ontogenetic trajectories produced by the linear function follow geometric progressions (Fig. 5).

FIGURE 4. Geometry and notation of the rule-based model presented in this paper. A, State of the model at plastochron t = 0. Axis geometry is initialized at the base, B^sup 0^, with a single apex A^sup 0^^sub t^. This serves as the "axiom" to the L-system. Curve in B^sup 0^ is cosmetic-intended to help recognition of B^sup 0^ in later plastochrons. B, C, State of model in successive plastochrons t = 1 to t = 2. Iteratively invoked developmental rules R^sub 1^- R^sub 4^ generate internodal segments I^sub 1^, . . . I^sub 4^ corresponding to each plastochron. In addition, each apex divides at each plastochron to produce a pair of daughter apices A^sup x^^sup t^ (x = 0 or 1 to identify them). Also indicated are change in apex orientation &934; (Φ^sup x^^sub t+1^ - Φ^sup x^^sub t^) governed by R^sub 3^ and change in phyllotaxis &920; (Θ^sup x^^sub t+1^ - Θ^sup x^^sub t^) governed by R^sub 4^. D, State of the model at termination governed by R^sub 5^. When apex strength S^sup x^^sub t^ (for some apex x at plastochron t) falls below preset minimum S^sub min^, R^sub 5^ terminates the apex and draws sporangium T^sup x^^sub t^.

From observation of developing plants, as well as branch systems represented as fossils, it is clear that apex "vigor" does not simply reflect some fixed parameter such as the apoxogenetic- epidogenetic control variable (r) by itself, but instead exhibits spatial and temporal context dependence. For instance, both size and complexity of branch systems are clearly dependent on whether growth occurs near the base or tip of the plant, or lateral branch system, among other factors. To model this, we use a scaled concept of ontogenetic trajectory (F) in order to define trajectories of change in the values of S between arbitrarily defined maximum (S^sub max^) and minimum (S^sub min^) bounds in apex strength. In our model, a trajectory exponent (E) serves to identify one from an infinite family of possible context-dependent F curves (Fig. 6).

For the purpose of visualization (Fig. 7), changes in values of S after application of the ontogenetic trajectory at each plastochron are displayed as changes in subtending internode length (L) and scaled width (W). This appears reasonable, at least to some extent, because apex vigor in plants is typically assessed by increased or decreased amounts of tissues produced over axis length or specified time interval.

R^sub 2^ - Branch Al\location.-At successive plastochrons, the shoot apex in living plants divides to produce multiple daughter apices with intrinsic properties and context inherited, with modification, from the parent apex. In early Devonian plants, shoot systems are typically dichotomous, implying two daughter apices produced at each plastochron (but see other possibilities in modified rule *R^sub 2^ and Figs. 20-24 below). Because we wish to incorporate context dependence embodied by the ontogenetic trajectory (F) into allocation of apex strength values for branches, a distinction is made in the model between intrinsic properties of shoot development, presumably based in genetics, and realized properties describing how the intrinsic properties are exhibited in a given developmental context. For allocation of apex strength (S) to daughter apices, a single intrinsic allocation parameter (D) is hypothesized. At maximum, D = , dichotomies occur with equal- strength daughter apices. For values of 0 < D < , realized allocation of apex strength (D^sub n^), differing from intrinsic allocation (D), is calculated by utilizing the ontogenetic trajectory (F) introduced above (see Appendix for details). Key relationships, and how values of realized allocation play out on a dichotomous branch system for different values of trajectory exponent (E), are shown in Figure 8, and specific values of apex strength are plotted for one of many possible ontogenetic trajectories in Figure 9.

FIGURE 5. Apoxogenetic (diminishing) geometric progression of apex strength governed by equation (2) with linear coefficient r = 0.8, S^sub max^ = 10, S^sub min^ = 0; diamonds represent S^sub t+1^ apex strength at plastochron t + 1 ; triangles represent ΔS^sub t+1^ change in S from time t to t+1.

R^sub 3^ - Apex Orientation.-To determine the angle of apex orientation of daughter apices relative to the parent following branching (Φ^sup x^ for each daughter apex x), we use a similar context-dependent approach (Figs. 3, 10). In living and fossil plants this angle typically also follows an ontogenetic trajectory related to axis size, or vigor, ranging between minimum (Φ^sub min^ [asymptotically =] 0) and maximum (Φ^sub max^ [asymptotically =] 90) angular bounds. If intrinsic branching factor D < , then each daughter apex follows a unique ontogenetic trajectory after branching. These trajectories (Fig. 11) are calculated after initiation of daughter apices at each plastochron by using angle coefficient (p) analogous to r, and ontogeny shape- position parameter (G^sup x^ for each daughter apex x) analogous to F, with angular exponent (K) analogous to E (see Appendix for formulae).

FIGURE 6. Effect of trajectory exponent E on ontogenetic trajectory F defined in equation (3). F is calculated for fixed values of S^sub t^ with E = 0.5 top curve, E = 1.0 middle curve, E = 2.0 bottom curve.

FIGURE 7. Endpoint model morphologies showing effects of changing apex strength S over plastochron time in apoxogenetic (A) and epidogenetic (B) growth regimes. Internode lengths and widths are scaled to S^sub t^ at each plastochron t.

FIGURE 8. Different values of realized allocation of apex strength D^sub n^ and (1 - Dn) to daughter apices at dichotomy for different values of main axis apex strength S^sub t^. Pairs of curves, one above (1 - D^sub n^) and the other below (D^sub n^) equal allocation: D^sub n^ = (1 - D^sub n^) = 0.5 represent ontogenetic trajectories followed by an axis pathway for different values of F as defined in equation (5). Labels for each curve represent values of E (and therefore F) defined in equation (3). In these examples, intrinsic allocation of apex strength D is set at 0.35.

R^sub 4^ - Phyllotaxis.-In higher plants, "phyllotaxis" refers to the helical, whorled, or other kinds of deployment of leaves on the stem in an aerial shoot system (Erickson 1983; Jean 1994). The telome theory, supported by strong paleobotanical evidence, suggests the origin of many leaves from lateral branch systems of Devonian plants, so the concept may be reasonably applied to the ancestral condition as well. For the model to be fully determined in three dimensions, phyllotactic angle, Θ^sup x^^sub t+1^ for each daughter apex x in the transverse plane at right angles to angle of apex orientation (Φ^sup x^^sub t^), must also be specified at each plastochron t (Fig. 3). In living plants, abundant evidence suggests that angle Θ = Θ^sup x^^sub t+1^ - Θ^sup x^^sub t^, termed the (primary) divergence angle (Erickson 1983), shows relatively little context dependence based on apex vigor as introduced above for the other rules (Gregory and Romberger 1972; Niklas 1988; Green 1992; Lyndon 1994). Although sometimes variable, and in part dependent upon previous behavior of the apex especially in establishment of phyllotaxis in seedlings (Williams 1975), primary divergence angles more or less rapidly approach stable limit values, such as 90 (decussate), 120 (helical), 180 (distichous), or ~137.5 (helical-Fibonacci). The fossil record suggests that Fibonacci phyllotaxis, based on an irrational divergence angle, originated relatively late (Late Devonian), perhaps exclusively within a probable Archaeopteridalean + Seed Plant clade (see Meyer- Berthaud et al. 2000-phyllotactic pattern estimated by us from their Fig. 3). Orthostichous patterns involving rational primary divergence angles of 0, 120, 180, and especially ~90, are common earlier. In the model, different constant (therefore context independent) values are employed. Although allowing full geometric interpretation, this approach leaves unresolved the issue of underlying developmental cause for these angles (see Sachs 1991). Here we simply state that the angles exist.

FIGURE 9. One of a family of curves showing ontogenetic trajectory in apex strength S for a pathway along an axis involving unequal realized apex strength allocation, E = 4.0, D = 0.35, r = 1.5, S^sub max^ = 10, S^sub min^ = 0. Triangles represent successive maximum values of S for the larger daughter apex of each dichotomy at plastochron t+1 given value at plastochron t. Diamonds represent successive values of S for the smaller daughter apex at each dichotomy along the same pathway.

FIGURE 10. One of a family of curves showing ontogenetic trajectory in angle of apex orientation Φ for a pathway along an axis involving unequal realized apex strength allocation S. Values of Φ^sub t+1^ in plastochron t+1 are based on apex strength S^sub t^ in plastochron t as determined by equations (9, 10); E = 4.0, S^sub max^ = 10, S^sub min^ = 0, D = 0.35, r = 1.5, p = 1.0, K = 2.0, Φ^sub max^ = 90, Φ^sub min^ = 0. Triangles represent successive apex orientation values for the largest apex at each successive dichotomy. Diamonds represent the apex orientations of the smaller apex for the same pathway.

FIGURE 11. Three ontogenetic trajectories for equally dichotomous branching (D = 0.5) given different values of p and K. Resultant structures in Figure 14. Squares (top curve), p = 1.0, K = 2.0. Diamonds (center curve), p = 1.1, K = 1.0. Triangles (bottom curve), p = 4.5, K = 0.1.

R^sub 5^ - Shoot Termination.-Although individual aerial shoots in plants are conceivably indeterminate in the sense of allowing an infinite steady-state growth trajectory (e.g., menetogenesis of Eggert 1961), in reality they rarely, if ever, achieve such a condition. For a variety of reasons, probably including a combination of physical and genetic factors, individual apical meristems are typically apoxogenetic (diminishing), with shoot tips ultimately exhibiting some sort of vegetative or reproductive termination. In fossils of early vascular plants, we are commonly presented with a view of termination by formation of a sporangium (i.e., Zimmermann's fertile telome). The process is implemented here as a simple switch. Once apex strength (S) declines below a specified minimum limit (S^sub min^) in the ontogeny of the shoot or lateral branch, the apex is converted into a sporangium. For the purpose of display (Fig. 3), but perhaps also reflecting reality to some extent, the size of the sporangium is scaled to S^sub min^. It is likely that the ancient plants also terminated growth with vegetative terminal segments (i.e., Zimmermann's phylloid), and it would be possible to implement a separate rule for this situation as well. For simplicity, however, we have chosen not to do so here because this would require defining some mechanism for the choice between fertile and sterile termination based on apex strength (S) plus at least one other factor. However, it is unclear what this factor might be in general or in different phylogenetic contexts. Thus, our rule may be viewed as the simplest of a set of possible approaches to rule-based shoot termination in early plants, probably requiring elaboration in more complex later forms. As a result, displayed sporangia in the model, although highly suggestive of specific plant fossils, in fact serve only as markers of apex termination.

Implementation of the Model

The rule-based system with discrete iterations of developmental time and real variables, described above, may potentially be implemented by a variety of programming techniques (e.g., Honda et al. 1982; Niklas 1982, Niklas and Kerchner 1984; Perttunen et al. 1996). Here we use a commercially available augmented Lindenmayer system with turtle graphics called "CPFG" (Mech 1998) within "L- studio" (Prusinkiewicz et al. 2000; Karwowski 2001). Lindenmayer systems (Lindenmayer 1968a,b) are rule-based serial rewriting algorithms that are particularly useful in simulating multiple parallel processes and branched architectures during iterated discrete time intervals. They have been used extensively in modeling a variety of problems, including studies simulating plant form and development (Lindenmayer and Prusinkiewicz 1988; Pr\usinkiewicz and Lindenmayer 1990; Guzy 1995; Fournier and Andieu 1998; Jirasek et al. 2000; Kurth 2000; Salemaa and Sievaneni 2002). CPFG and L- studio were explicitly designed in part to model plant form, and include a wide range of useful programming features, as well as the capacity to produce the excellent graphics of resultant plant architectures, presented here. Compared with several published models using this remarkable software, our implementation is actually quite limited. However, the criticism has been raised that some of the more sophisticated models, involving perhaps debatable concepts or processes, lack general botanical relevance (Fisher 1992). Whether true or not, our intent here is to bridge the gap between two more or less independent disciplines-the modeler and the paleobotanist-with fidelity to the evolutionary problem at hand, involving early land plants and the telome theory, the foremost consideration. Figure 3, we believe, represents a reasonable and simple set of developmental hypotheses centered on our current understanding of early land plants represented primarily by Silurian- Devonian fossils. Implementation of the model as an L-system is arbitrary but owing to existing software offers considerable convenience. A complete listing of our (remarkably short) L-studio script, based in part on helpful examples provided with the L- studio system itself, is available from the authors upon request.

Results

The developmental rules R^sub 1^-R^sub 5^, described above and implemented as an L-system, define a multidimensional "morphospace" circumscribed by a set of tunable parameters. Not all parameters, however, are equivalent as dimensions of biological variability, nor do they have equivalent biological meaning. Moreover, parameters circumscribing the morphospace are decidedly un-orthogonal, making it difficult to use standard statistical methods to describe the effect of varying each parameter, as well as the relationships between parameters. Although several classifications are possible, we divide the parameters into four somewhat overlapping categories:

1. Constituent (or boundary condition) variables useful for defining number of time-based iterations of the model (STEPS) and basic geometry of the plants (W, L) not otherwise specified by the rules.

2. Growth/state variables (S, Φ, Θ, D, D^sub n^), some of which are recursively defined.

3. Theoretical limit values (Φ^sub min^, Φ^sub max^, S^sub max^, S^sub min^) with partial basis in biology but also for computational convenience.

4. Tunable parameters defining context dependence (r, E, p, K).

Each instance of assigning values to the above variables of the "morphospace" implies one, or more often several, pathway trajectories (F, G) determined by branching. In addition, there is a final plant architecture resulting from the context-dependent realization of developmental rules R^sub 1^-R^sub 5^ as described above. Because of interrelations of the developmental rules and number of parameters involved, the space of developmental trajectories/final architectures is very large, and rich in complexity. In what follows, we examine portions of this space by modifying different parameters, and comparing the resultant model plant architectures with well-known examples from living or fossil plants. Each topic below represents our attempt to organize the results in terms of what we believe to be important patterns in the evolution of primitive plant architectures more or less along lines sympathetic to the traditional aims of the telome theory.

FIGURE 12. Implementation axis length and width at each plastochron determined by a constant ratio. A-C, Effect of varying the ratios of constituent variables for axis width W and internode length, L for the same ontogenetic trajectory. A, W/L = 0.1. B, W/L = 0.35. C, W/L = 0.7.

General Axis Geometry.-Actual plants vary greatly in axis width versus internodal length, ranging from gracile terminal shoot segments on the one hand to extreme forms of cauliflory on the other. In the model, both axis width (W) and length (L) produced per plastochron are dependent only on apex strength (S). For the purposes of display, values of both L and W are initialized independently at maximum (S = S^sub max^) and thereafter vary in proportion to the geometric series observed in S at successive plastochrons. Different initial values of L/W produce greatly differing appearances (Fig. 12), but this is a simple matter of initial scaling. In actual plants scaling of this type is likely influenced by multiple factors probably requiring additional developmental rules. However, in our investigation of rules R^sub 1^- R^sub 5^, these factors are not pursued further.

Apex Vigor during Growth.-The aerial shoot or branch in both living and fossil plants often shows evidence of epidogenetic (increasing) development near the base, and apoxogenetic (diminishing) development elsewhere (Eggert 1961). Presumably between these broad growth regimes is a menetogenetic equilibrium, perhaps rarely achieved, during which apex vigor measured by volume of tissue produced per plastochron, or by serial xylem volumes or diameters, is theoretically self-sustaining without increase or decrease. Heretofore, all presentations of this concept, for instance in lycopsids (Eggert 1961), sphenopsids (Eggert 1962; Daviero et al. 1996), or progymnosperms (Scheckler 1976, 1978), have stopped at this point. However, treating aerial shoot systems involving contributions of multiple branches or appendages in a "syntelomic" system-by far the more common condition for all plants- the issue of diminishment and augmentation immediately becomes more complex, hence more interesting. The concepts of apoxogenesis/ epidogenesis must now be expanded to treat two distinct phenomena: (1) trajectory properties of individual apices along single defined pathways of apex derivatives, and (2) trajectory properties of the entire shoot system simultaneously taking into account the contributions of all apices of the system as development progresses. The present model demonstrates well the importance of making this distinction (Fig. 13).

FIGURE 13. Model shoot systems analyzing apoxogenetic- epidogenetic growth in self-similar (E = 0) branch systems, S^sub max^ = 10, S^sub min^ = 1.5. All shoots the same size (W = 1.0) at the base, drawn approximately to scale. A-C, Model shoots with equal allocation to daughter apices at each dichotomy (D = 0.5). A, All paths epidogenetic, r = 2.3. B, All paths at equilibrium, r = 2.0. C, All paths apoxogenetic, r = 1.7. D-F, Model shoots with unequal allocation to daughter apices at each dichotomy (D = 0.4). D, Some paths epidogenetic, others apoxogenetic, r = 1.9. E, Near equilibrium for the largest path, r = 1.6. F, All paths apoxogenetic, r = 1.3.

In isodichotomous systems where D = D^sub n^ = , equations (6, 7) in the Appendix may be solved for the equilibrium point (S^sup x^^sub t+1^ = S^sub t^) for any sequence of derivative apices, yielding r = 2. In the growth regime r > 2, the shoot system is clearly epidogenetic with each daughter apex exceeding its parent in strength (Fig. 13A). At r = 2, each daughter apex has the same strength as its parent and in the model produces internodal segments of constant length and width (Fig. 13B). Note, however, that summing apex strength for all daughters at each plastochron, ΣS^sup x^^sub t+1^ = 2ΣS^sub t^, implies a doubling of the underlying physiological processes of the shoot system, supporting values of S for the entire system at each plastochron. Such geometric increase in physiology is clearly unsustainable over the long term. At r = 1, summed apex strength is at physiological equilibrium: ΣS^sup x^^sub t+1^ = ΣS^sup x^^sub t^. However, in isodichotomous shoot systems, the apices produced at each plastochron are only half the strength of their parent, and the model shoot rapidly diminishes to termination in only a few iterations. Most "natural-looking" branching architectures (i.e., those most closely resembling the fossils in our opinion) appear in the interval 1 < r < 2, where summed apex strength at each plastochron is epidogenetic (increasing) but the apex strength of each individual apex is simultaneously apoxogenetic (diminishing), with each developmental trajectory proceeding at the same rate toward termination (Fig. 13C).

In branch systems with self-similar (E = 0) anisodichotomous apex allocation (D < ), solving equations (6, 7) in the Appendix (where D^sub n^ = D) results in corresponding equilibrium points r^sup 0^ = 1/D for the smaller, and r^sup 1^ = 1/(1 - D) for the larger, daughter axis respectively. At each equilibrium value of r^sup x^ (x = 0 or 1), ΣS^sup x^^sub t+1^ = r^sup x^S^sub t^. For dichotomous systems where E ≠ 0 and D < , apoxogenetic- epidogenetic equilibrium points for individual branch points are generalizable to r^sup 0^ = 1/ D^sub n^ and r^sup 1^ = 1/(1 - D^sub n^) where D^sub n^ = - ( D)[(S^sub t^ - S^sub min^)/(S^sub max^ - S^sub min^)]^sup E^ from equations (3) and (5) in the Appendix. In each of these cases, ratios of (ΣS^sub t+1^/S^sub t^) remain fixed at r^sup x^, as above. In epidogenetic growth regimes for individual axes, it is necessary in the model to extrapolate some or all values of S^sup x^^sub t+1^ above maximum bound S^sub max^, and because of the geometric progressions produced by recursion at each plastochron, values of S very rapidly become large (Fig. 13A,D).

For unequally dichotomous model branch systems, the most natural appearing results occur near, but somewhat below, equilibrium values for the largest daughter axis (Fig. 13E). Significantly below equilibrium (Fig. 13F) branch systems rapidly attenuate, even though summed axis strength in some cases may be epidogenetic (ΣS^sub t+1^/S^sub t^ >1). Above equilibrium (Fig. 13D), total biomass rapidly i\ncreases. Presumably in living and fossil plants, physiological capacity of the plant sustaining the shoot system severely constrains such growth. In unequally dichotomous branch systems (Fig. 13D) epidogenesis and apparent indeterminance in specific ontogenetic pathways may occur even though many, or most, pathways through successive apices are apoxogenetic (S^sup x^^sub t+1^/S^sub t^ < 1) and ultimately terminate. Although data probably exist to distinguish these different cases in fossils, none have been analyzed in this way. It is clear that in work involving branching architectures-the usual case-previous analyses have significantly underestimated the developmental significance and scope of the problem.

Apex Branching Angles.-In the above examples, the angle of apex orientation for each daughter apex (Φ^sup x^ for each daughter apex x) was constrained to a low value (Φ = 10) to permit a simple display. However, in living and fossil plants more variable angles are the norm, and species apparently differ in how these angles are primarily controlled. In some, variation in angles seems intrinsically determined, perhaps following an ontogenetic trajectory from base to tip of the shoot system. In others, it is equally clear that extrinsic factors, including gravitropisms and phototropisms, affect not only apex orientation but also, through a probable auxin intermediary, growth capacity (Silverton and Gordon 1989; Hopkins and Huner 2004). No doubt there is complex control in some taxa, involving developmental rules that are perhaps specific to major groups. As an empirical matter, it may be possible to distinguish between intrinsically controlled trajectories and directional extrinsic effects, even in fossil plants, by careful measurement of changes in angles from base to tip (i.e., angle trajectories) within and between specimens (Bateman 1992), although few data exist at this time.

Ours is an entirely intrinsic approach (Figs. 10, 11). Although there are exceptions, branch systems of living and fossil plants consisting mostly of primary tissues typically exhibit apex orientation values of 90, with larger axes usually showing lower values of Φ than smaller axes. Setting desired constraints on other parameters (including Φ^sub min^, Φ^sub max^, E, D, r) variation of angular parameters p (or Φ^sub max^), and K produces a set of plant architectures. All three plant architectures in Figure 14 involve equally dichotomous systems with all pathways through successive apices following the same ontogenetic trajectory in angle of apex orientation Φ^sup x^. Figure 14A shows results from the curvilinear trajectory of Figure 13A-in which there is rapid change in Φ^sup x^ at the base of the structure with relatively little change in Φ^sup x^ [asymptotically =] Φ^sub max^ = 90 toward the tips. The opposite situation is shown in Figure 14C where the greatest rate of change in Φ^sup x^ occurs near Φ^sub min^. Between these extremes is a linear rate of change (Figs. 13B, 14B). Although many resultant architectures produced in this manner are not easily distinguished by eye, they nevertheless invite quantitative comparisons with measurements derived from specimens (see "Discussion"). However, because no relevant quantitative data exist at this time, we can adopt values only of p and K that appear to produce natural-looking results.

FIGURE 14. Model branch systems resulting from the three ontogenetic trajectories of Figure 13, varying parameters p and K in equations (9, 10). A-C, Results from A-C respectively in Figure 13.

Differences in Apex Allocation.-One of the remarkable features observed in the fossil record of early land plants is the progression of forms through geological time starting with simple mostly isodichotomous aerial shoot systems of the Late Silurian and Early Devonian through progressively more complex anisodichotomous forms in the Middle and Upper Devonian showing varying degrees of differentiation between main axis and lateral branches or "proto- leaves" (Gensel and Andrews 1984; Knoll et al. 1984; Taylor and Taylor 1993; Kenrick and Crane 1997). Zimmermann referred to all of this change as "overtopping" and, for lack of a viable alternative, we will keep this word in our lexicon here, used in a general descriptive sense. However, from a developmental perspective centered upon growth dynamics of dichotomizing shoot apices, it is clear that multiple processes must surely have been, and continue to be, involved (see "Discussion"). A surprising result to us is the ease with which realistic plant architectures exhibiting overtopping can be produced in the model by modifying only a single parameter- the intrinsic (or genetic) parameter of apex allocation (D). When trajectory exponent E = 0 and therefore realized allocation D^sub n^ = D, context independent overtopping is simulated (Fig. 15). Changing values of D within the developmental regime 1 < (ΣS^sup x^^sub t+1^/S^sub t^) < 2 results in anisodichotomous shoot architectures that show natural-looking diminishment in degree of overtopping from base to tip. The amount of overtopping exhibited by a shoot system as a whole is directly dependent on D. As intrinsic allocation of axis strength increasingly departs from isodichotomous (D < ), a main axis progressively emerges in the modeled branch system (Fig. 15B-D,F-H). However, because changes in D also result in a shift in the equilibrium point of apoxogenetic- epidogenetic growth, modifying D also has an effect on total amount of growth exhibited by the entire shoot system before termination. This strongly suggests a functional/developmental relationship, in addition to historical or ecological correlation, between overtopped form and general size in Lower Devonian versus Middle Devonian and later fossil plants. If the allocation to the larger daughter apex is fixed for each plastochron (S^sup 1^^sub t+1^/S^sub t^ is set to a fixed value < 1), then the overall complexity of the entire branch system decreases with greater difference in allocation (decreasing D). Growth becomes progressively restricted to an increasingly prominent main axis (Fig. 15A-D). However, if apoxogenetic- epidogenetic parameter r remains fixed (Fig. 15E-H), then (S^sup 1^^sub t+1^/S^sub t^) increases as D decreases, resulting in elaboration of the shoot system before final termination (shown by sporangia) again associated by an increasingly prominent main axis. Thus, developmentally it is clear that overtopping in our model can be accomplished in a variety of ways with the following extremes: (1) by progressive restriction in capacity for growth, or (2) by augmentation of growth in some parts of the shoot system over a roughly steady state in others. In evolutionary terms, absolute and relative sizes in shoot systems in hypothesized ancestors versus descendants emerge as important features to measure.

FIGURE 15. Model results derived from varying intrinsic apex allocation D. A-D, Varying D and apoxogenetic-epidogenetic coefficient r to maintain a constant ratio of S^sup 1^^sub t+1^/ S^sub t^ = 0.9 for pathways involving the larger derivative apex at each plastochron; STEPS = 15, W = 1.5, S^sub max^ = 10, S^sub min^ = 1.0, E = 0, p = 0.15, K = 1.5. A, D = 0.5, r = 1.8; B, D = 0.4, r = 1.5; C, D = 0.35, r = 1.384615; D, D = 0.3, r = 1.285714. E-H, Varying D with constant apoxogenetic-epidogenetic coefficient r = 1.4; same as above for other variables except STEPS = 20. E, D = 0.5; F, D = 0.45; G, D = 0.4; H, D = 0.35.

FIGURE 16. Model results from varying trajectory exponent E in a branch system with constant but unequal intrinsic allocation of branch strength (D = 0.35) and holding all other parameters constant; STEPS = 20, W = 1.5, S^sub max^ = 10, S^sub min^ = 1.0, Φ^sub max^ = 90, Φ^sub min^ = 0, r = 1.5, p = 0.3, K = 1.5. A, E = 0.2; B, E = 0.5; C, E = 1.0; D, E = 2.0; E, E = 8.0.

Differences in Allocation Trajectory.-The self-similar branch systems in Figure 15 are certainly suggestive. However, allocation of vigor in a shoot system, estimated in specimens by widths or lengths of branch segments or total amount of tissue present in a shoot system above a certain level, is typically context dependent in both fossil and living plants. Specifically, most axes show progressively less difference in allocation between daughter branches as development proceeds acropetally. As described above, context dependency is handled in the model by developmental trajectory F (Figs. 6, 8) driving apex strength S. When intrinsic apex allocation is made unequal (D < ), specific ontogenetic trajectories controlled by E also influence overtopping. The resultant plant architectures are also highly suggestive (Fig. 16). Specifically, when other parameters are held constant, low values of trajectory exponent E (Fig. 16A,B) produce architectures with a prominent main axis. Lateral branches are also strongly overtopped with prominent secondary, tertiary, and higher-order axes, each producing much smaller laterals. At higher levels of E (Fig. 16C- E), progressively more compact branch systems are produced overall. Although each model system has the same value for intrinsic allocation of apex strength D, trajectory exponent E profoundly affects realized allocation of apex strength D^sub n^. At high values of E (16E) the main axis bears lateral branch systems that show more or less isodichotomous branching with very little internal overtopping. Following the paleobotanical literature, one might be tempted to interpret these patterns as comprising a main axis bearing incipient leaves or leaf precursors. However, it is important to note that the model is continuous, and has built into it no such distinction.

FIGURE 17. Model results from varying S^sub min^-threshold value for invoking termination rule R^sub 5^; STEPS = 20, W = 1-5, S^sub max^ = 10, Φ\;^sub max^ = 90, Φ^sub min^ = 0, r = 1.4, E = 0.1, D = 0.4, = 0.15, K = 1.5. A, S^sub min^ = 0.5; B, S^sub min^ = 1.0; C, S^sub min^ = 2.0.

Axis Termination.-Both minimum (S^sub min^) and maximum (S^sub max^) values for apex strength in the model act together as boundaries for triggering application of developmental rule R^sub 5^ in specific realized contexts. Holding other parameters constant, different S^sub max^/S^sub min^ ratios have profound consequences on overall model plant architectures (Fig. 17). Given an overall architectural plan defined by the other variables, a relatively high value of S^sub max^/S^sub min^ has the effect of producing greater shoot complexity. Thus, timing of this switch is very important to realized plant architectures, above and beyond intrinsic settings in the other variables. Relative size of reproductive versus vegetative portions in plants, and by inference relative commitment of resources by different plant architectures, is an area requiring morphometric attention, especially in the fossils. Except for absolute scale, the same geometry can be produced by modifying either S^sub max^ or S^sub min^ alone, and holding the other parameter constant.

Phyllotaxis.-In the model, primary divergence angle in phyllotaxis (Θ) is also an intrinsic boundary variable set at a constant value for each daughter apex upon branching. In all of the above examples, Θ has been set to 90 to simulate early Devonian plants, and perhaps similar modern plants such as Psilotum, with phyllotaxis roughly near this angle (there are currently few morphometric data). Middle Devonian and later overtopped fossil shoot systems increasingly exhibit distichous and helical patterns as time progresses. Changing the value of Θ to 120, commonly encountered in Middle Devonian aneurophytalean progymnosperms, produces model patterns highly reminiscent of the fossils. Figure 18 displays a range of possibilities generated by different values of primary divergence angle Θ in the model for a significantly overtopped (D < ) and determinate shoot system. Because orientation of branches at each plastochron involves a developmental consequence of combining orthogonal factors Θ^sup x^ and Φ^sup x^, the results in three dimensions are complex. Low values of Θ and Φ generate corkscrew-like patterns (Fig. 18A). Increasing values of Θ (Fig. 18B-K) serve to progressively define orientation of the main axis. Resonant patterns are generated when Θ divides evenly into 180 (Fig. 18B-D7K) and in each of these patterns, prominent orthostichies can be recognized (Fig. 18B with six orthostichies, C with four orthostichies, D and E with three orthostichies; K with two orthostichies). The fewer the orthostichies, the greater the overlap of lateral branches in axial view. Such a situation may be viewed as disadvantageous because of structural/developmental conflict such as overlapping developmental fields, or for ecological reasons including shading (Honda and Fisher 1978). For values of Θ between resonant orthostichy numbers 2 and 3, a series of patterns show progressive "planation" and overlap in axial view as Θ nears 180 (Fig. 18I-K). There is also a range of Θ values (Fig. 18F-H) including so-called Fibonacci phyllotaxis (Θ = Limit(F^sub n^/F^sub n+2^).360 [asymptotically =] 137.5 where F^sub n^ are integers belonging to the Fibonacci series F^sub n^ = 1,1,2,3,5,8,13. . . with index n), exhibiting a broad minimum of overlap. It has been argued that the irrational Fibonacci primary divergence angle (Fig. 18G) represents a theoretical overlap minimum (Niklas 1988). However, for determinate shoots systems such as modeled here, overlap produced by nearby angles (Fig. 18F,H) is not much different. This argument invites collection of morphometric data combined with some independent estimate of performance in specific shoot systems in order to determine whether such a theoretical limit value has biological meaning.

FIGURE 18. Model results A-D and F-K in axial viewpoint below base looking acropetally; E in lateral view. A-K, Different values of primary divergence angle Θ^sup x^. A, 30; B, 60; C, 90; D, E, 120; F, 130; G, 137.5; H, 150; I, 160; J, 170; K, 180. STEPS = 20, W = 1.2, S^sub max^ = 10, S^sub min^ = 3, Φ^sub max^ = 90, Φ^sub min^ = 0, r = 1.5, E = 1.0, D = 0.35, p = 0.6, K = 1.5. Here, base of axis (B in Fig. 4) is straight.

FIGURE 19. Four planate cases produced by modifying primary divergence angle Θ^sup x^ in the model; x = 1 is the larger and x = 0 is the smaller apex at each plastochron. STEPS = 20, W = 1.2, S^sub max^ = 10, S^sub min^ = 2, Φ^sub max^ = 90, Φ^sub min^ = 0, r = 1.35, E = 0.33, D = 0.36, p = 0.65, K = 1.5. A, (Θ^sup 0^ = 0, Θ^sup 1^ = 180); B, (Θ^sup 0^ = 180, Θ^sup 1^ = 180); C, (Θ^sup 0^ = 180, Θ^sup 1^ = 0); D, (Θ^sup 0^ = 0, Θ^sup 1^ = 0).

Model branch systems that may be considered "planate" can be obtained by setting primary divergence angle Θ to 0 or 180. It is interesting to note that when branch allocation is unequal (D < ) and depending on whether the larger or smaller apex of each plastochron receives the above values (and handedness of the underlying coordinate system), four planate cases are distinguishable (Fig. 19). Thus, a simple change in primary divergence angle is sufficient to simulate Zimmermann's elementary process of "planation," but with multiple developmental interpretations. In two instances (Fig. 19A,B) the primary divergence angle of the larger apex (Θ^sup 1^) at each plastochron is set to 180, creating a plane of mirror symmetry along the main axis and regular zigzags based on the angle of apex orientation Φ^sup 1^ for the same apex. In the other two instances (Fig. 19C,D), Θ^sup 1^ = 0 and the main axis deflects in the same direction again based on Φ^sup 1^. Within each pair of cases, setting primary divergence angle for the smaller apex (Θ

Source: Paleobiology

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