Ontogenetic constructions of some fossil plants

Review o f Palaeobotany and Palynology, 23 (1977): 337--357 ~)Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

O N T O G E N E T I C C O N S T R U C T I O N S O F SOME F O S S I L P L A N T S

KARL J. NIKLAS The New York Botanical Garden, Bronx, N.Y. 10458 (U.S.A.)

(Accepted August 18, 1976)

ABSTRACT Niklas, K.J., 1977. Ontogenetic constructions of some fossil plants. Rev. Palaeobot. Palynol., 23: 337--357. This article presents a survey of ontogenetic studies in paleobotany, and of biologically relevant mathematical results available from such techniques as finite element analyses, algorithmic systems, and computer simulation. Dynamic representations of growth are possible when the observed cellular arrangements in fossils are mathematically described. Successive computer solutions of parameterizing equations allow for the extrapolation of ontogenetic trends forwards and backwards in time (i.e., more and less mature stages, respectively), as well as the interpolation of missing stages or portions of an organism's development. The hypothetical constructions derived from these techniques may be tested against direct comparisons with the fossil being simulated and/or proposed modern analogues. Similarly, multicellular organisms or portions of organisms (e.g., leaves, sporangia) may be constructed as arrays of symbols -- each symbol representing a cell or group of cells. Development in such models is simulated by providing instructions for cell division, cell death, or alteration in cellular states, e.g., vegetative to reproductive. Illustrative simulations of Parka, Mastopora, Rhynia and Calamites are presented and paleobotanical conclusions concerning their respective growth patterns are drawn. INTRODUCTION This article has as its objective the d e m o n s t r a t i o n o f relevant m a t h e m a t i c a l modelling s y s t e m s w h e r e b y the g r o w t h p a t t e r n s o f fossil plants m a y be d e t e r m i n e d . M a t h e m a t i c a l c o n s t r u c t i o n s w h i c h p u r p o r t to describe biological s y s t e m s are at best a p p r o x i m a t e s w h i c h classify p l a n t m o r p h o l o g y as either integral o r iterative in nature, i.e., sequences w h i c h c o n t a i n no e l e m e n t s possessing the same p a t t e r n as the entire s t r u c t u r e in c o n t r a d i s t i n c t i o n to t h o s e sequences w h i c h are dimensionless and m a n i f e s t a r r a n g e m e n t s o f integral units. Similarly, m a n y s y s t e m s a d m i t o f alternate c o n t i n u o u s and discrete descriptions w h i c h ignore the principal o f g r o w t h c o n t i n u i t y in time. These disadvantages are balanced, however, by the explicit m a t h e m a t i c a l descriptions o f g r o w t h and cell g e o m e t r i e s w h i c h allow for the c o n s t r u c t i o n s o f o n t o g e n y . O n t o g e n e t i c simulations based on c o m p u t e r or algorithmic t e c h n i q u e s o f t e n include the p a r a m e t e r o f time t h e r e b y e x t e n d i n g h y p o thetical g r o w t h sequences t o m o r e m a t u r e and juvenile stages, as well as interp o l a t e d stages where empirical data are lacking. This " t i m e p a r a m e t e r i z a t i o n " p r o p e r t y is o f p a r t i c u l a r significance t o p a l e o b o t a n i c a l studies w h e r e preservation m a y be p o o r , i n c o m p l e t e or a m b i g u o u s . A c o m p l e t e d e s c r i p t i o n o f

338 a developing organism would have to specify all the essential processes and structures in the proper time sequence at all levels of organization; such a description, while perhaps attainable, has a level of information redundancy useless to the paleobotanist. For this reason, the models discussed in this article will de-emphasize biochemical and physiochemical factors and stress cell geometries and division patterns. Since cells appear to be the only functionally a u t o n o m o u s units (with genetic continuity) in higher plants, they are assumed to be the basic units in developmental studies and will be used as the fundamental "mathematical" unit. Previous ontogenetic studies of fossil plants have followed two general schemata: (1) the reconstruction of growth sequences based on relationships seen in large fossil suites; and (2) ontogenies based on filial cell relationships in individual specimens. These approaches may be termed allometrie and positional reconstruction, respectively, since the former designates ontogeny by the differences in proportions correlated with changes in absolute magnitude of the total organism or specific parts while the latter studies emphasize the apparent developmental history of juxtaposed cells. Valuable contributions to paleobotany have been made by allometric reconstructions (Walton, 1935; Morgan, 1959; Eggert, 1961, 1962; Stidd and Phillips, 1968; Phillips et al., 1972; Basinger et al., 1974) and positional reconstructions (Delevoryas, 1955, 1956, 1958; Melchior and Hall, 1961; Crepet and Delevoryas, 1972; Good and Taylor, 1972), while excellent reviews of previous studies are available (Delevoryas, 1964, 1967). STRATEGY OF MODEL CONSTRUCTION A number of assumptions must be made concerning the nature of plant growth and the parameter of time before simulations of ontogenies are attempted; the selection of an appropriate mathematical technique is also dependent upon the particular paleobotanical problem: (1) The cell is assumed to be the basic morphological and mathematical unit. Reasons given to support this preliminary assumption are: (a) cells are metabolically a u t o n o m o u s units in all higher organisms; (b) synthetic activity is mediated at the cellular level; (e) all cells in an organism are descendents from a primary cell -- initially the egg, and subsequently, meristematic progenitors; (d) the property of genetic continuity among cells and their descendents allows for the application of "programs" or "algorithms". An apical cell or zygote carry developmental programs with "instructions" for the timing and geometries of major cell events. Cell division, enlargement, death and differentiation may be envisioned as linear continua or discontinuous processes which may be defined as symbolic sequences (Lindenmayer, 1975), map sets (Marshall, 1971), or finite elements (Niklas and Chaloner, 1976; Niklas and Phillips, 1976; Niklas, 1977). (2) All ontogenetic patterns are integral and/or iterative. Patterns in plant morphology may be broadly defined as either integral or iterative. Integral patterns are those with elements having different patterns from that of the

339 entire structure such as leaf shape, and fructification/sporangial shape. In contradistinction, iterative patterns are dimensionless and describe arrangements of integral units such as phyllotaxi, root hair and stomatal distributions. (3) All ontogenetic sequences are either continuous or discrete. Ontogenetic processes which relate the manner in which a system's observables change in time (as a function of their instantaneous values) are continuous, and take the form of first-order differential equations (Ritger and Rose, 1968). That branch of mathematics which most closely relates to these biological processes is control theory (el. Arbib, 1973; Milsum and Roberge, 1973, p.20) -- such phenomena are steady-state systems with physiochemical feedback loops dictating morphogenesis. Discrete descriptions are defined, on the other hand, by finite sets or stages in development which may be broadly defined as catastrophic. The study of such processes lies within the province of a u t o m a t a theory (Harrison, 1965; Spira and Arbib, 1967). Normally, the properties of these two descriptive techniques and the interpretation of results obtained through their application are different in emphasis and format, but by no means are they mutually exclusive. Studies stressing ontogenetic sequences of entire plants incorporate both continuous and discrete patterns (Niklas et al., 1976; Niklas, 1976b--c). In vascular plant growth, apical cell division patterns may be continuous, while vascularization, d i c h o t o m y and reproduction may be non-continuous (discrete) phenomena. Many attempts have been made to develop relationships between these two descriptive modes via reduction (i.e., the paraphrasing of one system into another) and correspondence (i.e., both descriptive techniques are describing the same p h e n o m e n o n and, therefore, correspondence between their respective languages must exist). The tool of finite elements, which describes nonlinear continua, appears to have the greatest application to in to to simulations. (4) Cell states are parameterized by discrete time descriptions. The computation of cell states occurs at specified intervals, and cell division, death, and transitions are not continuous functions but rather occur at certain time intervals based upon previous states and inputs. Clearly, this is an operational decision as to what cell p h e n o m e n a are termed "discrete" or " c o n t i n u o u s " . In most simulation studies a compromise is reached between the high stability of cellular states and those occasional shifts into other states. The parameter of time is "dissected" into periods or cell cycles which seem appropriate. The selection of any particular modelling system from the available techniques is perhaps most rigorously controlled by the state of preservation seen in the fossil or fossils being studied, as well as the primary objectives of the researcher. Where large numbers of specimens are available, and allometric constructions are sought, finite element simulations are the optimal approaches. Similarly, this technique offers the most precise and detailed ontogenetic constructions currently available. Where a more restricted number of fossil specimens are available and positional constructions are attempted, algorithmic systems afford practical and useful methods for simulations. The theory, use, and application of finite element techniques

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have been reviewed (cf. Niklas and Phillips, 1976; Niklas et al., 1976; Niklas, 1977), while c o m p u t e r programs are available for two-dimensional ontogenies and related problems (cf. Ede and Law, 1969; Korn, 1969a, b, 1974; Niklas, 1976d). An increasing application of finite automata theory to biological problems has generated extremely elegant techniques and much useful data (see for example Baker and Herman, 1972a, b; Frijters and Lindenmayer, 1974). Development of multicellular organisms is mathematically defined in such techniques as growing arrays of finite automata; each morphogenetic state in such an array is an instantaneous description of the organism, and the entire development is the cumulative sequence of such arrays. Transitions in cell states (e.g., vegetative to reproductive, living to dead, non-specialized to specialized) consists, in general, of a state symbol into a new state symbol, or the substitution of more than one symbol. Propagative deterministic context-free Lindenmayer systems (PDOL-systems) (Lindenmayer, 1968a, b, 1971) correspond to these outlines, and provide a valuable tool. The approach of automata arrays in modelling development can be applied to any multicellular structure, whether filamentous, sheet-like or threedimensional; such sequences may be finite (analogous to determinate development) or infinitely long (indeterminant growth). Since this method also considers growth within and without bounds, we may properly speak of terminating growth (where the sequence is finite) or of limited growth (where it is periodic) or of proper growth (where size increases without bound). For purposes of illustration, selected fossil plants representing diverse growth modes, taxonomic affinity, and morphogenetic complexity, will be discussed within the context of techniques applied to simulate their respectire ontogenies. While emphasis will be placed on the plants selected, passing reference will be made to other potential applications, e.g., fossil angiosperm leaf simulations, phylogenetic implications, information storage and evolution. ILLUSTRATIONS

Mastopora pyroformis Bassler 1909 (p.59, pl.7) The genus Mastopora Eichwald (1840), though not figured until 1890,.is recognized as a representative of the Dasycladales (Johnson and HCeg, 1961) and is placed in the sub-tribe Mastoporinae along with Apidium and Epimastopora by Pia (1927). The genus has a stratigraphic range from the Ordovician to Silurian and has been described from Esthonia (Stolley, 1896), India (Salter, 1851), England (Nicholson and Etheridge, 1878), and Maryland--Virginia, U.S.A. (cf. Bassler, 1909). The thallus is ovoid to spherical or club-shaped to cylindrical. Cortical cells are gently concave, prismatic to funnel-shaped and are the tubular ends of lateral branches; calcification occurs almost exclusively around cortical cells (cf. Johnson and H~eg, 1961, p.46, pl. 18). In surface view the lateral branches appear as densely packed hexagonal facets. Owing to the simple morphology and presumed growth

341 pattern represented in Mastopora, the genus was selected for mathematical simulation in an a t t e m p t to determine the nature in which facets were accreted over the plant's surface. Based on the observations of ten specimens (NYBG 1484A-1488A), the construction of the hypothetical ontogeny of Mastopora pyroformis requires two constraints: (1) sets of radiating spirals, which represent the surface patterns of hexagonal facets, must be accreted so that all surfaces are filled; and (2) the individual facets (tips of lateral branches) must be limited in their final size (diameter). The geometry of M. pyroformis requires that new spirals are intercalated as determined by their shape, the size of their units, and the surface over which they grow. (1) Spiral shape: Spirals may be designated as logarithmic, and have the basic form D I n R = 0, where D is the distance measured along the spiral, R the length of the radius vector, and 0 the angle swept by the spiral. At 180 °, 0 = ~ and D = n/ln R, i.e., D becomes a measure of the spiral's tightness. (2) Facet size: This morphological feature may be empirically determined by plotting the size of the facet (x-axis) versus the distance from the tip of the plant (y-axis); the resulting allometric regression (y = ax b) may be approximated, giving the rate of facet size increase (b) and the m a x i m u m value (y = total length of plant). (3) Thallial shape: For purposes of simplicity, the shape of Mastopora is assumed to be spherical. This approximation is valid, since a decrease or cessation of spiral intercalation must occur when the surface area reaches a m a x i m u m value (i.e., when the thallus becomes cylindrical or club-shaped). The critical phase in the development of lateral branches is, therefore, at the tips of the plant. Despite numerous simplifications in the manner in which the morphological features of Mastopora have been described, computer-simulated shapes (Fig. 1A--D) correspond in many critical respects to the apical configuration of the thallus. The display shown in Fig.lA demonstrates the need for intercalated facets as determined by the numerous empty areas produced when programming of this feature is deleted. This display shows eight spirals generated by the rate of facet size increase determined from actual specimens. Fig.lB demonstrates the simulated morphology of Mastopora with intercalated facets -- the spirals of the original simulation (Fig.lA) are represented by e m p t y circles, while first- and second-degree intercalated facets are stippled and blackened circles, respectively. The distributional patterns shown in this latter figure are most congruous with actual specimens. By varying the tightness of the simulated spiral (Fig.IC--D, tightness equal to 5 and 70, respectively) and correlating the appearance of simulated and actual shapes, it is possible to determine the precise pattern in which lateral branches were formed. The optimal simulation (i.e., the one most in agreement with the specimens examined) is shown in Fig.lC, and indicates that lateral branches are accreted in an almost whorl-like pattern. Tight spirals add relatively fewer facets, while loose spirals result in greater frequency of facet intercalation than that which was observed. Assuming an apical growth typical of

342

A

i Fig. 1. Four computer displays showing the simulated morphologies of Mastopora, an Ordovician--Silurian alga. All simulations depict the polar view of the growing apex, and circles represent the distal tips of lateral branches (facets). The computer displays represent the eight cycles of growth (as determined by the number of facets produced per spiral). Simulations in which intercalated growth was not programmed result in spaces between branches and an incomplete surface (A). Intercalated branches (B) produce simulations similar to the general appearance of fossils (stippled and blackened circles represent the seventh and eighth cycle of intercalation, respectively). Figures C--D show the effect of spiral "tightness" (parameter D) on the appearance of simulations. For further details see text. m a n y algal lines, t h e " i n t e r c a l a t i o n " o f l a t e r a l b r a n c h e s n e c e s s i t a t e d b y t h e i n c r e a s i n g s u r f a c e a r e a o f t h e t h a l l u s t i p suggests a c o a l e s c e n c e o r i n t e r d i g i t a t i o n o f l a t e r a l b r a n c h e s . T h e c u m u l a t i v e p a t t e r n s o b s e r v e d in t h e s e fossils m a y , t h e r e f o r e , b e c o n s i d e r e d t h e r e s u l t o f p a c k i n g , as well as g r o w t h . The s i m u l a t i o n o f Mastopora has a g r e a t e r g e n e r a l i t y , since it p r o v i d e s t h e d e s c r i p t i o n o f a basic a n d c o m m o n f o r m in b o t a n y , e.g., p h y l l o t a x i c p a t t e r n s . T h e s i m u l a t e d g r o w t h p a t t e r n o f this alga is b a s e d on a v e r y s i m p l e g e o m e t r i c r e l a t i o n s h i p b e t w e e n u n i t s h a v i n g a f i n i t e size a n d t h e s u r f a c e a r e a in w h i c h t h e y are p a c k e d . G e o m e t r i c c o n f i g u r a t i o n s o f this s o r t have b r o a d a p p l i c a t i o n b u t are o f a s u p e r f i c i a l n a t u r e since r e l a t e d s i m u l a t i o n s are n o t boua fide o n t o g e n i e s b u t r a t h e r r e p r e s e n t t h e s t a t i c g e o m e t r i e s o f o b s e r v e d

343 specimens. Modifications of this approach by means of finite elements can provide a " g r o w t h " sequence; however, little additional information concerning Mastopora can be derived. Simulations of Mastopora demonstrate that an apparently complex morphology may be reduced to a few describing parameters. The complexity of morphology is the result of parametric interactions rather than their number (Raup, 1968).

Parka decipiens Fleming 1831 Parka, a thalloid plant represented by impression and compression fossils from the Lower Devonian of North America and Great Britain is one of a few Devonian fossils having similar morphology (Spongiophyton, Orestovia, Protosalvinia) which due to poor preservation makes taxonomic considerations highly speculative (Don and Hickling, 1917; Johnson and Konishi, 1958). Chemical analyses of the coalified "cuticles" of Parka suggests a green algal affinity (defined in avery broad sense) (Niklas, 1976a), while SEM and mathematical considerations indicate a pseudoparenchymatous construction (Niklas, 1976d). Owing to the two-dimensional growth habit of this genus the application of finite a u t o m a t a series (L-systems) may be used to construct hypothetical growth sequences. The reader is referred to Lindenmayer (1968, 1971 ), and Herman and Rozenberg (1975) for detailed treatments of finite a u t o m a t a series theory and application; the simulation language (CELIA, i.e. cellular linear iterative array generator) allows for L-system analyses of growth functions (cf. Baker and Herman, 1972). L-systems are of particular value when dealing with simple and/or branched filamentous growth, as it is hoped Parka will illustrate. L-systems represent a Chopskian grammar in which symbols stand for cell states and strings of symbols stand for cell ontogenies. By analogy, strings of symbols (finite automata arrays) are similar to frames on a film; the running backward or forward of the film loop corresponds to more juvenile and more mature aspects of development. The most distinctive characteristic of L-systems is a certain level of repetitiveness in substrings which they produce. In these cases, formulae are determined which obtain all the strings produced and which specify each string as a related series ( " c o n c a t e n a t i o n " ) of some previous string in the sequence. This interdependence has been termed the "recurrence p r o p e r t y " ; in the specific case of Parka the repetitive property will be shown to be a function of filament dichotomy. On the bases of SEM observations made on latex replicas (Niklas, 1976d), the outlines of cells and filament d i c h o t o m y were determined. Since Parka is basically circular in outline, concentric regions characterized by greater and lesser frequency of filament dichotomy could be determined, and a simple geometric relationship drawn between distance from the thallus center and frequency of cellular division. With a considerable degree of reliability, the overall growth of a filament can be described; the develop-

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m e n t a l states o f each cell can be q u a n t i f i e d with t h e help o f t h e a t t r i b u t e " p r o b a b i l i t y o f cell division in the n e x t c o n c e n t r i c region'.' F i l a m e n t s have a basal and distal pole (thallial c e n t e r and margin, respectively) and collectively radiate o u t w a r d s during g r o w t h to p r o d u c e , as a result o f f i l a m e n t d i c h o t o m y , a p e l t a t e thallus with intact margin. If u is the life span o f t h e apical cell, and v is t h a t o f t h e basal cell, t h e n the m e a n values ( u , v ) describe the w h o l e e v o l u t i o n of an observed filament. Cell families (i.e., sets o f coexisting cells which are the total derivates o f a single cell) exist and r e p r e s e n t a genealogical relationship b e t w e e n all cells p r o d u c e d by a f i l a m e n t ' s apical cell b e f o r e d i c h o t o m y occurs; each t i m e an apical cell divides anticlinally to p r o d u c e a d i c h o t o m y , t w o cell families are f o r m e d . Let: Si = Si

u + Siv

(1)

refer to the r e c u r r e n c e p r o p e r t y o f P a r k a ' s g r o w t h , w h e r e i is a p o w e r function o f cellular t r a n s f o r m a t i o n and u , v r e f e r t o the m a x i m u m n u m b e r o f d e v e l o p m e n t a l states seen b e t w e e n the apical and basal cells o f a s u b f a m i l y (S). F i l a m e n t g r o w t h t h e n b e c o m e s diagrammatically: a0~al-~a2~

"''-~a,

,,~''"

~au

1

(2)

with a0, a~, a2 . . . . . a . . . . . . au being the d e v e l o p m e n t a l states o f a cell; the t w o t e r m s S i , and S i ~ (Eq.1) c o r r e s p o n d to t w o celt subfamilies issuing f r o m a simple apical cell, and t h e l o o p e d arrows indicate anticlinal apical cell division and s u b s e q u e n t d i c h o t o m y . T h e p a r a m e t e r s u , v in e f f e c t r e p r e s e n t a f e e d b a c k s y s t e m which d e t e r m i n e s f i l a m e n t d i c h o t o m y ( D i S t e f a n o et al.,

1967). At any given time, the n u m b e r o f cells N i in a developing f i l a m e n t (simulated by eq.2) will be 2 ~', w h e r e ~ is the n u m b e r o f cellular divisions. F o r the s e c o n d d i c h o t o m y o f a filament, the n u m b e r o f cells p r o d u c e d will be: Ni = N

2,, + N,-u-~, + N

v ,, + Ni-2v

(3)

In this p a r t i c u l a r case, t h r e e subfamilies a k are necessary t o describe t h e w h o l e f a m i l y a ° [here the u p p e r index c o r r e s p o n d s to t h e n u m b e r o f d i c h o t o m y , ~; t h e lower index refers to a set o f ~ w h e r e k = (0,1 . . . . , ~)]. T h e frequencies o f a~ are given b y b i n o m i a l coefficients: ( ~ ) - k~(1 -- k),

(4)

the n u m b e r s o f cells in each a~ class for a given bipartitioning degree is: NI~ = N i --- (~ - - k ) u - - k v

(5)

and a c c o r d i n g t o eqs.3--5, t h e cell n u m b e r o f t h e w h o l e f i l a m e n t will be d e f i n e d by: N i= E t~ -

N i 0

(~--k)u--kv

(6)

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The mathematical display of the cellular relationships described b y eqs. 1--5 is shown in Fig.2. In this graphic representation of Parka, the first cell (Fig.2A) is shown to give rise to repeatingly dichotomized filaments. (Fig. 2B--F) {cells are replicated as circles with straight lines representing genealogical relationships) radiating in 360 ° thereby producing circular thalli with intact margins (Fig.2G). Owing to the iterative nature of cell subfamilies (Fig.2F) which are by analogy the "unit crystal" of the cellular "lattice", an intercalated pattern of cells (Fig.2G, d o t t e d lines) must be incorporated in the growth simulation to allow for an increase in thallus diameter without marginal fraying. Simulations of growth based on this technique allow for a precise description of hypothetical cell division patterns which in turn may be used to describe relationships with modern analogues, e.g., comparisons between Parka and Coleochaete (Niklas, 1976d). Using simulations, apical and intercalary growth seen in Parka have been used to depict the distribution of

B

c

A F

% Fig.2. C o m p u t e r displays of the h y p o t h e t i c a l growth of Partza. Sequence A - - F displays the d e v e l o p m e n t of a p s e u d o p a r e n c h y m a (cells are represented by circles) where genealogical relationships are s h o w n by means of solid lines. The unit of growth (F) may then be reiterated to f o r m the characteristic shield-shaped thallus (G), and with the aid of a n o t h e r unit of cells (shown by d o t t e d lines) successively larger thalli may be simulated. For further e x p l a n a t i o n see text.

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sporangia on the thallus dorsal surface, and the relative rates of filament elongation. On the basis of finite automata studies of Parka and other organisms, it can be shown that the growth function given in: h

f(n) = Y~pi(n)c 7

(7)

i=l

describes all systems where cellular interactions are absent, thus even physiological conclusions may be drawn from fossil data. Finite automata arrays may be applied to a broad spectrum of biological phenomena: (1) the vascular arrangements observed in fossil and living plants (see Calam ites section); (2) the simulation of c o m p o u n d structures, e.g., leaves, inflorescences, flowers; (3) distributional patterns of roots; (4) the appearance of structures at discrete times in ontogeny, e.g., heterocysts in the cyanophytes (Baker and Herman, 1972); and (5) algal growth (Liick, 1975).

Rhynia Kidston and Lang 1917 The Silurian--Lower Devonian flora has figured prominently in consideration of primitive land plants evolution (Zimmerman, 1949; Banks, 1968), with Cooksonia and Rhynia suggested as the prototypic levels of organization for the earliest tracheophytes. Our understanding of Lower Devonian plants is based upon four major lines of inquiry: (1) the evaluation of previously described plants has led to significant reinterpretations, e.g., Psilophyton princeps (sensu Dawson) has been shown to be a chimera (Hueber and Banks, 1967; Hueber, 1967, 1971); (2) numerous new plants have been described anatomically as well as morphologically, e.g. Kaulangiophyton (Gensel et al., 1969); Krithodeophyton and Steganotheca (Edwards, 1968, 1970); (3) the stratigraphic and therefore chronological arrangements of major land flora, and of botanical structures have been reassessed thereby making evolutionary patterns more solidly based on the geological data, e.g., the Rhynie Chert (Chaloner, 1970); and (4) broad taxonomic arrangements of plant fossils have suggested evolutionary trends among early tracheophytes (Cronquist et al., 1966; Banks, 1968). More recently, studies have provided insights into the transitional plant organizations between the rhyniophytes and zosterophylls, e.g., Renalia (Gensel, 1976), the trimerophytes and progymnosperms, e.g., Ooearnpsa (Andrews et al., 1975), and other groups, e.g., Pertica (Kasper and Andrews, 1972). While the possible significance of many early land plants is clear, morphological (Merker, 1958, 1959; Pant, 1962), anatomical (Satterthwait and Schopf, 1972), and reproductive features (Lemoigne, 1969) of even so well known a taxon as Rhynia remain in doubt. The application of simulation techniques may provide a valuable adjunctive approach in understanding ontogenetic and phylogenetic trends. For purposes of illustration, the anatomical features of Rhynia were recast into finite elements (for a detailed explanation of this technique see Niklas, 1977), and c o m p u t e r simulations attempted. Two growth aspects of this genus were examined: (1) apical meristem configuration, and (2)

347 p a t t e r n s seen in b r a n c h i n g . T h e i n t e r d e p e n d e n c e o f (1) a n d (2) led to t h e m o r e general c o n s i d e r a t i o n o f t h e b r a n c h i n g p a t t e r n s seen in t h e e v o l u t i o n a r y series f r o m t h e r h y n i o p h y t e s to t h e t r i m e r o p h y t e s . This s e c t i o n will theref o r e t r y t o bridge t h e u n c o m f o r t a b l e aspects o f o n t o g e n e t i c and p h y l o g e n e t i c simulations. T h e a n a t o m y o f Rhynia reveals a m o r e or less t e r e t e x y l e m s t r a n d w i t h c e n t r a l l y l o c a t e d p r o t o x y l e m indicating a c e n t r a r c h m a t u r a t i o n ; cortical p a r e n c h y m a , w h e r e k n o w n , is relatively simple. S h o o t apices w i t h cell detail are n o t u n c o m m o n , b u t t h e n u m b e r o f initial cells c o u l d n o t be d e t e r m i n e d in t h e sections m a d e ( N Y B G 1 4 9 1 - 1 5 2 3 ) . T h e s i m u l a t i o n s o f t h e h y p o t h e t i c a l apical m e r i s t e m are b a s e d solely o n e x t r a p o l a t i o n s o f m o r e p r o x i m a l cell detail (i.e., p o s i t i o n a l c o n s t r u c t i o n ) ; cell lineages associated w i t h apical d i c h o t o m y are b a s e d u p o n (1) t h o s e o b s e r v e d in m a t u r e d i c h o t o m i e s , and (2) p u r e l y m a t h e m a t i c a l c o n s i d e r a t i o n s w h i c h p r o v i d e t h e m o s t simple " i n s t r u c t i o n s " t o a single cell t o divide and g e n e r a t e a d i c h o t o m y . T h e finite e l e m e n t display o f t h e h y p o t h e t i c a l a p e x o f Rhynia is s h o w n in Fig.3; such s i m u l a t i o n s are e x t r e m e l y crude, b u t p r o v i d e a w o r k i n g m o d e l and generative source for simulating branch patterns.

E

F

@@ J°l°l~ • o

.~

~ l , • .

w°, o ° °

,e° ,

•

• • °oe._

°°o°

°

G"

H"

,

"':

I°~

I."

A@ t

B

•

C

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D

Fig.3. Finite element simulations of iso- (A--F) and anisodichotomy (A--F') are used to construct the dichotomous (G,G') to pseudomonopodium (I,I') growth habit transition. Hypothetical regions of cells (shown by lined boundaries) in polar views of the apices are shown to dichotomize in iso- (sequence A--F) and anisodichotomous (sequence A--F' ) patterns. Mathematical relationships between the crotch of branches and their respective tips (Cr--I) are shown by means of dots, while line drawings (G'--I') connecting these dots show the general appearance. For further details see text.

348 Finite element displays (polar views) of the hypothetical apices of R h y n i a (Fig.3A--F) depict a sequence of d i c h o t o m y ; lined boundaries represent groups of cells which have similar genealogies and behave ontogenetically as a group. By manipulating the relative size of apical cell derivatives or their numbers (dictated by the relative frequencies of cell divisions) both iso- and anisodichotomous growth sequences may be simulated (Fig.3A--F', respectively). The configuration of hypothetical apices is similar in both " t y p e s " of dichotomy, and isodichotomy simulations require the least number of information units to describe " g r o w t h " , i.e., a less cumbersome algorithmic description is necessary. A mathematical description of the growth patterns seen in R h y n i a and Cooksonia (Fig.3G, G') is less complex than anisodichotomous, e.g., Hicklingia (Fig.3H, H') or pseudomonopodial growth habits (Fig.3I, I'). Based on relatively simple geometric relationships, a simulated phylogeny from isodichotomous to pseudomonopodial growth may be described (Fig.3, G'--I'). A small bias in the frequency of cell division between the two apices of a dichotomy results in progressively more pronounced anisodichotomy, and eventually a p s e u d o m o n o p o d i u m is produced. In living analogues, such a bias may be the result of auxin gradients and/or the division pattern of the apical cell(s). A relationship on purely mathematical grounds may also be provided to explain the branching pattern transition seen between the rhyniophytes and trimerophytes. This formulation requires an understanding of the optimal design of plants, i.e., the design of an organism which allows the performance of necessary functions with a minimum expenditure of energy and material in both the performance of its functions and construction (Rashevsky, 1943; Cohn, 1954, 1955). All organisms must perform a number of functions -- some of which are biochemical in n a t u r e , w h i l e others are purely mechanical. A tree trunk must be thick enough to sustain its own weight, while photosynthates must be produced at a greater or equal rate to their consumption for a photoautotroph to survive. The absence of locomotion in plants requires a much larger area for food production; two optimal designs are available: (1) dorsiventrality, as in some of the algae and bryophytes, and (2) the general "branching" character of vascular plants. The branching optimal design may be thought of, and constructed mathematically with, only a few approximations. While plant structures are very heterogeneous and each part has its own density, an average density 5 may be considered. The main axis of a plant is also irregular in nature and, therefore, an average length lo and radius r0 must be considered. Let the total number of branches be n, their average length ~, and average radius be r. The mass M of the whole plant is then: M =~

(~or~, + nqr 2)

(8)

In the rhyniophytes and trimerophytes the principal metabolic processes must be proportional to ~he total area of all n branches since they possess no leaves. If M is the total mass, q is the average rate of metabolism pe r unit

349 mass, and k is a constant which includes 2n, then the proportionality of metabolic rate is given by q M ~- kng.r

(9)

From mechanical laws, the length of a branch cannot exceed a value determined by its radius; the strength of a branch is dependent upon its average density 5 and its radius r. Therefore, if f means a "mathematical function": =f(r,5)

(10)

The flow of metabolites through the axis occurs in the vascular tissues. The larger the radius of the main axis r0, the larger the total cross-sectional area and surface area of the vascular tissues available for flow (which is a function of the metabolic rate, q M , the density of the vascular tissues 5v, and the area of the stele). In plants with large protosteles, 5v ~ 5. Metabolic rates, differential tissue densities, and plant size and degree of branching dictate plant optimal design and growth habit. The r h y n i o p h y t e to trimerophyte transition (Fig.3G'--I') may crudely be described by these mechanical and metabolic parameters in an evolutionary reciprocity (see Niklas, 1976b--c), where: 4,, = M / n S r ~ - - n~(r2/r~)

(11)

and: ~1~o = qal3M~13 /( C - - g s13M~13)

(12)

The rigorous proof of eqs.11--12 may be found in Niklas (1976b). For a constant metabolic rate (an apparently valid assumption, since the mechanism of photosynthesis is t h o u g h t not to have changed significantly in evolutionary time) and a low proportion of vascular to non-vascular tissues, the simple dichotomous growth habit is optimal, with increasing crosssectional areas in the main axis the p s e u d o m o n o p o d i u m is optimal*. The " m a t h e m a t i c a l p r o o f " presented may be in all fairness criticized as an exercise in abstraction based on very crude approximations. However, considerations of surface and cross-sectional areas, metabolic rates, and their precise relationships are the foundation of modern physiology and afford the paleobotanist trends from which the physiological/biochemical evolution of plants may be deduced. Morphological data provide the structural expressions rather than the biochemical impetus of botanical evolution. C a l a m i t e s Schlotheim 1820 (p.398, pl.20)

The calamites were in appearance much like the extant equiseta but had an arborescent habit. The stems show conspicuous nodes and internodal ridges that alternate across nodes in many species. Leaves were in alternate *For branches of unequal lengths, a distributional function n(q ) must be introduced such that fo'n d~ = n.

350 whorls and inserted in one of two patterns; branching was nodal or interfoliar and not necessarily occupying all available positions. Most had a pith cavity (the result of early growth cessation of a complete pith and axial lateral expansion). The equisetoid nature of these plants is also seen in the conspicuous p r o t o x y l e m lacunal of young stems. Primary xylem maturation has been described as endarch, but caution must be exercized since the apparent endarchy of E q u i s e t u m is ontogenetically mesarchy. Eggert (1962) has shown epido-, meneto- and apoxogenetic phases in development similar in effect to those reported in some arborescent lycopsides. The genus C a l a m i t e s was selected for purposes of illustration since (1) its ontogeny parallels in many respects a modern analogue, i.e., E q u i s e t u m (Golub and Wetmore, 1948a, b); (2) the epido-, meneto-, and apoxogenesis of the genus parallels the growth patterns reported in many Carboniferous plants; and (3) the reported ontogeny of C a l a m i t e s is well known and may conveniently be used to illustrate L-system simulations. An idealized diagrammatic reconstruction of the xylem (parallel lines represent primary xylem strands) of an arborescent sphenopsid is shown in Fig.4B, while a simulated menetogenetic xylary pattern (Fig.4A) has been constructed from parallel arrays of L-systems which "cross-hatch" each other in a b o t t o m right to upper left (solid lines) and b o t t o m left to upper right (dotted lines) pattern. Simulations of apoxo- and epidogenesis employed the concatenative series shown in Fig.4C and D, respectively. The simulations of growth patterns seen in C a l a m i t e s employing arrays of finite automata are extremely simple to formulate, and yet manifest both continuous (menetogenetic) and discontinuous growth aspects (epido-; apoxogenesis). The model here proposed for C a l a m i t e s requires a system of cellular interactions (IL-systems), and is based on essentially " f i l a m e n t o u s " growth behaviour. Changes of cell states (Fig.4C--D) are programmed by the following rules (where p and q stand for any pair of symbols from the set (ao, al, • • • ,an ; bo, bl . . . . . b~ }, p:/=q and x > O, y >~ 0):

(q0, qx) -~ qx+l

(13)

(P0, P~, qx) -+ q0; (P_~0,P0) -' P0

-

The change of state is given in each rule for a segment which is in the underlined state as influenced by either the immediate left or right neighbor symbol in the array. The developmental sequence (hypothetical) thereby generated from these rules is shown in Fig.4B. The mathematical " s h o r t h a n d " of L-system simulations may be related via reductionism (see Introduction) to other ontogenetic patterns for purposes of taxonomic and/or phylogenetic comparison, e.g. E q u i s e t u r n with C a l a r n i t e s spp. Relationships may also be drawn between the points of leaf and branch attachments, increase in leaves per node, and dimensions of the primary body at different soma levels all within the c o n t e x t of symbol arrays. -

351

i

i°2

ooOO°

B

...°

..L ..% "Y'" "hr" " ( ~ .,-

,-'

,

[.-"

'a2 a1

°6

a2

Fig.4. L-system s i m u l a t i o n s of t h e vascular p a t t e r n s seen in s o m e a r b o r e s c e n t s p h e n o p s i d s (Calamites). S i m u l a t i o n s of t h e m e n e t o - (A), epido- (C), a n d a p o x o g e n e t i c (D) s e q u e n c e s m a y be used to r e c o n s t r u c t t h e p a t t e r n s o b s e r v e d o n fossil s t e m s (B) b y m e a n s of finite a u t o m a t a series ( a , , a ~ , . . . , a n ; b o , b l , • . • , b n ) . F o r f u r t h e r details see text. DISCUSSION

Ontogenetic studies have revealed a number of growth patterns which have parallel expressions in both fossil and extant plants, e.g., arborescent Carboniferous Lycopsida and Sphenopsida (Eggert, 1959), many coenopterid ferns (Phillips, 1974). Numerous other studies suggest a unified theory of apical and secondary tissue development where conceptual relationships of apparently diverse phenomena may be expressed geometrically. Unless directly observed growth sequences are determined by inferential rules governing the geometric relationships among cells within a time sequence; spacial configurations and the chronology of cells, tissues, and organs based on modern analogues are used to reconstruct patterns seen in fossil plants. The logic of these rules of inference may be mathematically formalized in biological models and information obtained concerning " h y p o t h e t i c a l " growth patterns. The use of models in understanding the ontogeny of shape is therefore a valuable adjunctive approach to paleobotany in testing (but not formalizing) ideas and diagrammatically expressing morphological relationships. In a preliminary a t t e m p t to illustrate various techniques available to paleobotanists (e.g., finite elements, L-systems, algorithms, simulations, optimal design), aspects of growth seen in Parka, Mastopora, Rhynia and Calamites

352

were discussed. It should be emphasized that while specific ontogenetic (and in some cases, phylogenetic) conclusions were drawn from these examples, the primary objective was to present (within a familiar context) applications of modelling techniques rather than a detailed exposition of specific ontogenies. The motivation for the attempts at modelling presented in this article is the conviction that the complexity of completed form can be generated by the interaction of only a few parameters. The abstract models determined for Parka, Mastopora, Rhynia and Calamites show varying degrees of relationship to actual modes of growth, yet correspondence of abstracted simplicity with actual development is by no means excluded. Patterns observed in the growth of Parka and Mastopora are mathematically simulated to a high degree of resolution. Parameterization of Rhynia and Calamites describes growth relationships in a precise albeit abstracted manner. The advantages of biological modelling are many, and for the paleobotanist, important features may be: (1) the ability of some techniques to construct missing stages in development or portions of individual specimens (cf. Niklas and Chaloner, 1976; Niklas and Phillips, 1976); (2) the precision whereby morphological relationships may be expressed in modelling techniques, thereby allowing inter- and intraspecific comparisons; (3) the feasibility of relating morphological features to phylogenetic trends; and (4) the use of optimal design modelling to provide clues concerning physiological aspects of now extinct plants (see Rhynia section). To illiJstrate some of these points, a simple relationship between sporangial size and intersporangial distances on a fructification may be employed. A model was constructed in which sporangia (depicted as spheres), with a proximal to distal geometric diminution in diameter, were positioned on an axis where intersporangial distances could be varied (Fig.5). The distances between "sporangia" in the first simulation (Fig.5A) were determined such that the spherical sporangia were densely packed against one another. The axis of this fructification was then allowed to elongate exponentially for 5 and 10 iterative sequences (Fig.5B--C). The simulated " m o r p h o l o g y " of these fructifications may be interpreted as ontogenetic, in which case variations in appearance represent growth stages and intersporangial distances are the result of axial elongation, or phylogenetic, in which case variations are the result of genetic alteration and intersporangial distance becomes a valid taxonomic criteria. Problems involved in the distinction of various species (e.g. Zosterophyllum) often involve sporangial dimensions, arrangements and axial configuration. The determination of possible morphologies which are logical extrapolations of observed preservational patterns in a suit of fossils, may provide a valuable adjunctive consideration in questions of t a x o n o m y and evolution. The models presented in this article were restricted in the most part to specific generic applications and by de-emphasizing physico-chemical relationships. Numerous researchers, however, have provided a wider scope to modelling and its application to evolution on both a molecular and morphological

353

Js A

B

C

Fig.5. Hypothetical relationships between sporangia (circular outgrowths) having a geometric proximal to distal decrease in diameter and the axis bearing them. Successive geometric elongation of the axis (B--C) results in greater intersporangial distances from the initially densely packed simulation (A). This very crude simulation may be used to interpret ontogenetic and phylogenetic relationships if the simulation dimensions are scaled to the specific paleobotanical problem. For further details see text.

level (Gould, 1966; Eigen, 1971; Gatlin, 1972; Sugita, 1975; Lasker, 1976; Kuhn, 1976). Paleobotanical studies have provided valuable data on ontogenetic sequences (Eggert, 1961, 1962; Hall, 1961; Delevoryas, 1955, 1956, 1964; Good and Taylor, 1972). It is hoped the techniques now available may provide additional impetus to such studies. REFERENCES Andrews, H.N., Gensel, P.G. and Kasper, A.E., 1975. A new fossil plant of probable intermediate affinities (Trimerophyte--Progymnosperm). Can. J. Bot., 5 3 : 1 7 1 9 - - 1 7 2 8 . Arbib, M.A., 1973. Automata theory in the context of theoretical neurophysiology. In: R. Rosen (Editor), Foundations of Mathematical Biology, Vol. III. Academic Press, New York, N.Y., pp.193--282. Baker, R. and Herman, G.T.; 1972a. Simulation of organisms using a developmental model. Part I. Basic description. Int. J. Bio-Med. Comput., 3: 201--215. Baker, R. and Herman, G.T., 1972b. Simulation of organisms using a developmental model. Part II. Heterocyst formation problem in blue-green algae. Int. J. Bio-Med. Comput., 3: 251--267. Banks, H.P., 1968. The early history of land plants. In: E.T. Drake (Ed.), Evolution and Environment. Yale University Press, New Haven, Conn., pp.73--107.

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