A Model of Phyllotaxis

J. theor. Biol. (1998) 192, 531–544

A Model of Phyllotaxis F. W. C*  J. C. S Physics Department, University of California, Riverside, CA 92521, U.S.A. (Received on 10 July 1997, Accepted in revised form on 6 February 1998)

Phyllotaxis is the study of the symmetrical arrangements of plant organs, and most often associated with the Fibonacci series of numbers. The present work points out that the well known Helmholtz equation of mathematical physics correlates all of the well known patterns in one simple algorithm, involving two integers p and q: p q q q 0 accounts for spiral patterns, including ‘‘jugate’’ patterns, p = q gives the alternating whorl patterns, while p q 0, q = 0 gives superposed whorls. In spiral patterns, the integers p, q underlie the larger and more usual integers m, n. The integer number of leaves N in a pattern is given in all cases by the expression N = (p 2 2 q 2 )/J, where J is an integer giving the number of leaves on a single (e.g. stem) level. A biochemical origin of the algorithm is suggested. 7 1998 Academic Press

1. Introduction A novel approach to phyllotaxis is presented. Phyllotaxis is the study of symmetrical patterns determined by plant organs such as leaves, bracts, florets, scales and so on. A simple generative algorithm not involving numerical calculation is given; one is easily able to draw the resultant patterns free hand. The algorithm arises from a cell surface molecule model with a degree of biochemical plausibility, and is inclusive of the various phyllotactic patterns. In spite of the overwhelming diversity of plant architecture, there are common patterns that link a wide range of species. Particularly eyecatching examples include the spiral arrangement of florets in the capitula of sunflowers and daisies, and the spirals of pine cones and pineapples. The subject has a long and rich history, and both the observational and theoretical has recently been impressively and usefully reviewed by Jean (1994). Phyllotaxis may be classified into three main types * Emeritus, present address: 2365 Virginia Street, (4, Berkeley, CA 94709, U.S.A., ‘‘Cummings.nature.berkeley.edu’’, and to whom all correspondence should be addressed. 0022–5193/98/120531 + 14 $25.00/0/jt980682

seen in nature. These are (1) spiral, as seen in the vast majority of the 250 000 or so species of higher plants, and most frequently associated with the Fibonacci numbers or series (Mitchison, 1972; Douady & Couder, 1992). The spiral patterns of leaves, bracts or florets of plants (and hereafter all termed ‘‘leafs’’ for brevity) are a familiar curiosity of nature, and spiral patterns typically have one leaf per level on a stem; (2) distichous, actually the simplest spiral, as in maize or ginger, in which the leafs at successive levels along the stalk appear displaced by 180°; (3) whorled, one of the simplest and most frequent examples being decussate, for example maple or mint, in which two leaves appear at the same level on the stalk and displaced from one another by 180°, followed at the next level along the stalk by another identical pair of leaves displaced by 90° from the first pair, and so on. Higher order alternating whorled patterns also occur with some frequency, in which more than two leaves appear at the same level, followed by a similar but rotated set at the next level in such a way as to place the leafs of a higher (lower) level above the space left by the lower (higher); ‘‘parrot’s feathers’’ is an example of six on one level, when the next level is rotated by 30° relative to the last. Superposed whorls also occur reasonably frequently, in which there is no 7 1998 Academic Press

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rotation from one level to the next, the leafs forming a lattice all running parallel to the stem axis; the often seen cases of wisteria, and honeysuckle are examples of superposed whorls with only two leafs per level. The simplest case of superposed whorl is the interesting case of a single superposed row (monostichy). The successive patterns arise as a consequence of the way the plant grows, and as the area increases. The first true foliage leaves of a plant often initiate a 180° opposite (distichous) or a decussate pattern, the distichous most often giving way to a spiral. New leaves are always added in succession only in a region close to the apical tip. During the growth of a plant from embryo to maturity, its apical diameter increases. In terms of the present picture, this may be represented, following Mitchison (1972), as a gradual increase in the stem circumference, or as an increase in area for a given stem height moving down the stem. Which pattern eventually emerges, spiral or whorl, is related to the initial pattern of the leaf primordia at the meristem, as well as the rate of growth. The meristem consists of a group of rapidly dividing cells, and these may be geometrically classified into apical, lateral or intercalary meristem cells; apical meristem cells occur near the tip of a stem, lateral lie parallel to the axis of the stem, while intercalary is located somewhere along the length of a stem. It may be helpful in visualizing the process of the origin of pattern formation to consider a nested sequence of annuli of increasing radii, seen as if one were viewing a conical structure from above (see Fig. 5 and Section 10). The first leaf primordia always arise in the annulus nearest the tip of the cone, one or several leafs at a time; if two, then they are displaced by 180°. This annulus then grows out to become the next largest annulus surrounding a (new) smaller one of the original size. The next emerging leaf pattern on this new inner annulus, given the geometry of the meristem, is affected by the pattern of the second largest annulus (as well as possibly even larger annuli); the boundary conditions for the emerging pattern of primordia in the labile meristem is determined partly by the existing leafs in larger annuli, as well as by the rate of growth of the plant. The present work shows how all three main types, spiral, distichous and decussate, as well as the higher order whorled and superposed whorls, are all encompassed by the same simple algorithm. It is a somewhat separate but important aspect that the algorithm springs from a model which readily lends itself to possible biochemical interpretation.

2. The Pattern Algorithm The model from which the plant pattern algorithm springs will be briefly reviewed here (Cummings, 1994, 1996). The resultant equations of the model are the familiar Helmholtz and Laplace equations of mathematical physics. This section may be viewed as standing separate from the phyllotactic algorithm of the next section, in the sense that the model from which the algorithm arises does not claim uniqueness. Cells of an epithelial sheet are envisioned as joined at their lateral surfaces by adhesive connections, biochemicals which come in two forms. These two forms may be, for example, two conformational states of the same three domain membrane spanning molecule. The two states may represent the active and inactive states of a single biomolecule. The resultant patterning mechanism is expected to be robust since patterning is a process so crucial to successful development, a point most recently emphasized by Barkai & Leibler (1997); the present model essentially depends on a single parameter (a length, k−1 ). Each of the two states has an average density in the (middle) surface bisecting the cell heights, so that the density of each varies in space along this middle surface. The first assumption of the model is that there is an energy cost to the formation of gradients of each state. The second assumption is that there is also an (autocatalytic) tendency for one state to foster the local growth or increase of that same state; for example, a given conformational state will encourage an increase of the same conformational state in its immediate vicinity. These ideas are easily translated into mathematical form. Again, the particular formulation is of lesser import than the ideas briefly sketched above, since various mathematical formulations can lead to essentially the same patterns, the simplest being further inclusion of nonlinear terms. In a later section (Section 11), speculations will be made as to a plausible biochemical realization, or at least to animal homologues. The same biochemical mechanism for patterning in animals has been proposed earlier (Cummings, 1994, 1996, 1997). Dynamical behavior in the model will arise from the growth in time, and will be imposed as a change in the parameter A(t), the total area of the middle surface of the epithelial sheet, with A(t) assumed externally given, as an uptake of nutrients from the environment occurs, and as cell division and the accompanying growth occurs. Then the patterning changes which come about by biochemical rearrangements are supposed to take place on a shorter time scale than growth as parameterized by a change in the total area A.

    Consider the energy E of the cell surface molecules given by the expression E = ef(D1 (9f1 )2 + D2 (9f2 )2 − a 2F(f1 , f2 )) dA. (1) Here e is of dimensions energy, and the integral extends over the entire (middle) surface (Cummings, 1994, 1996, 1997) of total area A; f1 and f2 are the dimensionless ‘‘densities’’ of the two morphogens, being taken as the density divided by the maximum density of each, so that 0 E f1,2 E 1. The first two terms of eqn (1) for the energy describe the fact that there is an energy ‘‘cost’’ to formation of a gradient of either state; these first two terms alone in the absence of interaction (a = 0), would give rise to a tendency of each of f1 or f2 alone, upon minimization of the energy E to spread themselves as evenly as possible over the surface subject to any constraints (such as boundary conditions, i.e. fixed values of either in a specified region). The minimum energy configuration in this case of no interaction (a = 0) would be for each molecular type to independently satisfy the well known Laplace equation. The dimensionless constants D1 and D2 give the relative energy cost of the two (squared) gradients in eqn (1), so that we may take D1 + D2 = 1. Here, in the present example of plant phyllotaxis we also take D1 = D2 for simplicity. The third term under the integral represents the interaction and tends to a minimum energy when the two types avoid each other, that is, the energy is increasingly lowered to the extent that a single type or state dominates in a given local region. A plausible possibility is that a molecule in one of two possible conformational states tends to cause an adjacent molecule to adopt the same conformational state. Then, due to this third term in the integral alone, energy decreases locally as (f1 − f2 )2 increases, an expression of the autocatalytic local self enhancement mentioned above. A relatively high local value of f1 will cause f2 to convert to f1 , and vice versa, so that the third term under the integral will be taken as an even function of the difference between the two densities. The function F(f1 , f2 ) thus increases monotonically [note the minus sign before the term a 2F in eqn (1)] as the absolute difference between f1 and f2 increases; the energy is thus lowered by this term as (f1 − f2 )2 increases, to reflect the interconversion of f1 and f2 . The simplest model of this interconversion effect, including nonlinear effects is b a 2F(f1 , f2 ) = a 2[(f1 − f2 )2 − (f1 − f2 )4 + · · ·] (2) 2 where a 2 and b are constants. Only the first two terms

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have been kept in a Taylor expansion of F. In the model of the text, for simplicity the term proportional to ‘‘b’’ is treated as a perturbation, and the nonlinear term 0b, while very important, will only be invoked when making energy arguments based on b q 0. The constant a 2 has dimensions of (area)−1, and supplies the crucial length of the model. Minimization of the energy E of eqn (1) with eqn (2), by use of the Euler–Lagrange equation leads to two uncoupled equations (Cummings, 1994, 1996) 92c + k 2(c − 2bc 3 ) = 0,

(3)

92h = 0

(4)

Here we have defined the new variables c and as c = (f1 − f2 ),

(5)

h = D1 f1 + D2 f2 .

(6)

and

Also, we have defined the constant k 2 (dimensions of area−1 ) by k2 =

a2 . D1 D2

(7)

Equation (3) has often been invoked in the past in the context of developmental patterns (Kauffman, 1993; Kauffman et al., 1978); the linear Helmholtz equation [eqn (3)] with b = 0) is the time independent description of vibrational modes on thin plates, and for a given shape, results in the well known allowed ‘‘eigenmode’’ solutions and eigenvalues. It has been noticed in the past that there is often a striking similarity of these to animal patterns. The present model envisions that a ‘‘pattern’’ equilibrium is established on a time scale which is short compared to the growth rate; biochemical reaction times are assumed short compared to this growth rate. Time dependence in the present view has its origin in the (assumed determined externally) growth embodied in the total area A = A(t), which has its origin in the Laplacian operator 92. The system is thus an open system, with a given energy uptake (determined by conditions of available nutrients) which is exhibited as cell division and the subsequent growth and which leads to increase in the total area A. No commitment is made as to local growth rates. The energy of the configuration enters the model in an important way via the nonlinear term in the interaction energy when b q 0. The energy expression of eqn (1) is always identically zero for b = 0 at its

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minimum. This can be seen by use of the identity 9·(f9f)=(9f)2 + f92f, and use of eqn (3) (with b = 0) and eqn (4), which expresses the minimum energy condition. However, when we include the nonlinear term, b $ 0, we find by use of a vector identity and the minimum conditions of eqns (3) and (4) that the minimum energy is given by ek 2b 4 fc dA. 2

Emin = −

(8)

If b Q 0, then the minimum energy solution is c = 0, whereas if b q 0, minimum energy is achieved when c reaches its maximum or minimum allowed value (21) when this is also an allowed solution, thus giving the lowest possible energy pattern. Then it is apparent from eqn (8) that it is energetically favorable to generate as many maxima and minima values of c as is consistent with allowed solutions of the conditions of eqns (3) and (4). Further, it is also apparent that growth, i.e. increase in total area A further reduces energy, since the energy integral is proportional to area, and the greater the energy reduction to the extent that this area increase is accompanied by increasingly large numbers of maxima and minima of c. As will be seen, eqn (8) has relevance for the present phyllotaxis model in indicating some patterns which are rarely seen, having relatively high energy cost. It is easy to show by integration of the Helmholtz equation, and use of the fact that the normal to the surface is perpendicular to the ‘‘morphogen’’ gradients, that the integrals of the two individual densities over the entire area are equal. Then because of eqn (3) an integral over each morphogen f1,2 will equal A/2. In what follows, the nonlinear parameter b will be treated as a (first order) perturbation, b1, in the sense that only the linear Helmholtz equation is used in the subsequent analysis, but at the same time the nonlinearity is a key fact, since we retain and refer to the minimum energy expression of eqn (8) when appropriate. When the nonlinear Helmholtz equation is considered, b q 0, one effect is that the two regions become of unequal size, an effect not considered in the present work.

10) gives a conformal (analytic) map which has the property of retaining the form of the model Helmholtz and Laplace equations, eqns (3) and (4), and which maps the cylinder onto a (truncated) right cone, when the straight lines of the cylindrical representation map onto intersecting log spirals. The cylindrical representation envisions a right cylinder of height y0 and circumference x0 cut along its axis and folded out into a rectangle; the coordinates of the corners are then taken as (x = 0, y = 0), (x0 , 0), (0, y0 ), and (x0 , y0 ). The line (x0 , 0) to (x0 , y0 ) is the same as (0, 0) to (0, y0 ). Since interest here is focussed on allowed repetitive patterns, we require also that there be periodicity in the y direction by requiring that the line (0, 0) to (x0 , 0) have the same pattern as the line (0, y0 ) to (x0 , y0 ). With these boundary conditions, the solutions to eqns (3), the Helmholtz equation [and now taking b = 0 in eqn (3), and also D1 = D2 = 1/2] are readily verified to be 1 c = cos(2p[px/x0 − qy/y0 ]) 2 1 + cos(2p[px/x0 + py/y0 ]), 2

0 1 k 2p

2

= (p 2 + q 2 )·

0

1

1 1 + . x02 y02

(9)

(10)

The factor 1/2 multiplying each cosine term arises by requiring that the expression give the lowest minimum energy in eqn (8). The solution for h of the Laplace eqn (4) is simply h = 1. The condition of eqn (10) can be recast into the useful form in terms of the area A = x0 y0 and the rectangle ratio y0 /x0 as A=

0 1

2

2p (p 2 + q 2 )(y0 /x0 + x0 /y0 ). k

(11)

The area for a given p, q pattern will be a minimum for y0 /x0 = 1. The integers p and q are always taken so that 0 E q E p. The ‘‘leafs’’ appear at points of maximum c (e.g.); then from eqn (9) the arguments of the cosines must be 2pi and 2pj, (i and j = integers) so that p

x y −q =i x0 y0

(i = integer),

(12)

q

x y 2p =j x0 y0

(j = integer).

(13)

3. The Phyllotactic Algorithm Following the important work of Adler (1974) and others before (see Jean, 1994), the cylindrical representation, for which pattern visualization is simplest, will be used. A short later section (Section

All phyllotactic patterns of the model arise simply from the simultaneous solution for the leaf positions

    (x, y) of the two straight lines of eqns (12) and (13). Pattern construction is facilitated by putting each of the two straight lines into the standard forms y/y0 = s ·(x/x0 )+(y/y0 intercept), where s is the line slope, so that the two equations read y/y0 = (p/q)(x/x0 ) + i/q; y/y0 = (3q/p)(x/x0 ) + j/p.

(14)

All patterns, spiral, disticious, decussate, whorled, superposed whorl, are obtained from eqns (14). The running integers i, j for a given pattern (p, q) each take on a total of p + q − 1 values in order to fill the unit square in the x/x0 , y/y0 plane. The presence of the nonlinear term for b q 0, and the expression for the minimum energy of eqn (8) favors those patterns which have the greatest number of maxima and minima per unit area, that is, a high ‘‘space filling’’ ratio. As the area increases, those patterns are most energetically favored which most fully fill up the area given the size allotted each leaf. Equation (8) says that, due to the fact that the main contribution to the integral comes from the regions in the vicinity of c = 21, the minimum energy per unit area is Emin /Am 1 − (ek 2b/2)·N ·Da/Am ,

(8')

where N is the number of leaves in area Am , and Da is the ‘‘size’’ of each leaf; the minimum area Am = (2p/k)2(p 2 + q 2 ) of the pattern is given by eqn (11) with xo = yo . Equation (8') is an analogue in the present work of the well known Hofmeister’s axiom (Jean, 1994); the present criterion differs in that it is not an axiom. Rather it follows from the basic assumptions of Section 3, and it says that minimum energy per unit area configurations are favored, while ones of relatively higher energy/area are less favored, but not completely disallowed. The content of eqn (8') is that patterns (p, q) with larger ratios of leaf area to stem area are preferred, or in other words, ‘‘close packing’’ of leafs is a preferred configuration over less close packed. The maximum value of N/Am , called the ‘‘filling factor’’ hereafter, is unity; the energy/area is proportional to the negative of this factor N/Am , which expresses, (keeping leaf area Da constant) the relative space filling of a given (p, q) configuration. 4. Construction of the Patterns Construction of each pattern proceeds simply as follows. Only the two equations for straight lines of eqn (14) are needed to encompass all patterns. Each of the four sides of the square in the y/y0 , x/x0 plane are divided into p and q equal parts, the p equal parts

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marked by (say) ticks, and the q equal parts by dots. Two sets of parallel straight lines are drawn to connect these points as per the instructions of eqns (14): two sets of parallel lines having the same sign slopes s1 = +p/q and s2 = 1/s1 = +q/p in the case of one pattern, and opposite sign slopes +p'/q' and −q'/p' in the case of another pattern. The y/y0 intercepts [which lie on a line through (0, 0) and (0, 1)] are given various values by ranging the integers i and j over p + q − 1 total values for each of i and j. The y/y0 intercepts do not always lie on the square side. The intersections of these two sets of lines within the square gives the positions of the ‘‘leafs’’ or florets, etc., representing the maxima of c. A number N of leafs is counted to the first repeat of the pattern, always counting the leaf at the point x = 0, y = 0, while excluding ones at the other three corners. All allowed combinations of the integers p, q lead to observed phyllotactic patterns. 5. Fibonacci Spiral Phyllotaxis: p q q q 0, p, q Relatively Primed The first case considered in this section is p q q q 0, with p and q relatively primed, leading to spiral phyllotaxis and the well known Fibonacci series. There is in this case one leaf at each level of the stem. The distichous and decussate patterns, as well as higher order whorled, are examples of the situation when p = q, or of p q 0, q = 0 and are discussed in a later section, as are the bijugate, trigugate, . . . patterns, when p q q, with p, q not relatively primed. Figure 1 shows the simple case p = 2 and q = 1. In this figure, two situations are shown, corresponding to the two choices of sign in eqns (12), (13) or eqn (14). In one case, Fig. 1(a), the slopes s1 and s2 have the same signs, namely s1 = +2 and s2 = 1/2, and the two sets of two parallel lines are shown, along with the y/y0 intercept values of zero (always present) and 1/2. In this case i = 0, 1, while j = 0, −1. As Fig. 1(a) shows by the filled circles, this case corresponds to a total number of leaves N = 3, which occur in n = 1 rotation in one direction (shown along the dashed line), or in m = 2 rotations in the opposite sense. The number N for the case of Fig. 1(a) is given in terms of the pair (p, q) by N = 3 = 22 − 12, and also by the pair of integers m, n, when N = 3 = 2 + 1. Figure 1(b) shows the situation for p = 2 and q = 1, but for the slopes of the two sets of parallel lines with opposite signs, with s1 = +2, and s2 = −1/2. Here the number of leafs is seen to be N = 5, in three rotations (dashed lines, m = 3) in one sense, and two in the opposite sense. In this case, N = 5 = 3 + 2 = 22 + 12. The intercepts now occur for i = 1, 2, and j = 0, −1.

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Notice particularly that N = 4 is not allowed in spiral phyllotaxis, when p q q, since N = 4 is not the sum or difference of the square of two integers; it does not occur since only one ‘‘wavelength’’ is involved in the present model. Also of most interest is the fact that the case N = 6 does not occur when p q q, with p, q relatively primed, for the same reason. The case of N = 7 = 42 − 32 is allowed, but very much less likely than, say, N = 8; the N = 7 pattern takes a single turn to repeat (an observed case of ‘‘spiromonostichy’’; Jean, 1994, p. 41), as does N = 9 and N = 11, when a diagonal line of the pattern square goes through all leaves. Of note is the fact that the area [eqn (11)] in the case of N = 7 is A A (42 + 32 ) = 25 and is large compared to that of both N = 8 (A A 10) and N = 13 (A A 13). The

y/y0 s 1 =2 s 2= 1 / 2

N=3=2+1=2 2 –1 2

x/x0

y/y0 s 1=2

s 2= 1 / 2

N=5=3+2=2 2 +1 2

x/x0

F. 1. Two spiral patterns with N = 3 = 22 − 12, and N = 5 = 22 + 12. These both correspond to the integers p = 2 and q = 1. In the top figure the slopes given from eqn (14) for the two sets (—) of parallel lines are s1 = p/q = +2/1 and s2 = q/p = +1/2. (W) represent the positions of the N = 3 = 2 + 1 = 22 − 12 ‘‘leafs’’ (or leaf primordia), where only the one at the lower left hand corner has been counted in each case. The lower pattern shows the two sets of allowed parallel lines with slopes of opposite signs, with s1 = +2/1, and s2 = −1/2, and now N = 5 = 3 + 2 = 22 + 12. The case of N = 4 and N = 6 are not allowed by the algorithm in spiral phyllotaxis. (– – –) show (for each of the two patterns) one set of the ‘‘opposing visible parastichy’’ pair, when now N = m + n, while the opposing set is given by (—). These latter opposing visible parastichies always have opposite sign slopes.

s 2= 1 / 3 s 1 =3

N=5+3=3 2 –1 2=8 N=m+n=p 2 ±q 2 F. 2. N = 8 = 5 + 3 = 32 − 12. The same sign slopes are s1 = p/q = +3, and s2 = q/p = +1/3. The m = 5 parastichy lines are shown dashed, and the n = 3 parastichies are solid, and coincide with the 1/3 slope lines.

special cases when (q = p − 1), and also the cases of (q = 1, p q 3) will be discussed further shortly. Figure 2 shows the case for p = 3, q = 1, and corresponding to a total number of leafs given by N = 8 = 32 − 12 = 3 + 5. The slopes here have the same sign, s1 = +3 and s2 = +1/3; there are five visible rotations (m = 5 ‘‘parastichies’’ shown dashed) in one direction and three (n = 3) in the opposite. As becomes quickly clear by construction, in the spiral pattern case p q q (p, q relatively primed) the total number of leaves N in a repeating pattern has the general value N = p 2 − q 2, in the case that the straight line slopes s1,2 have the same sign, or N = p 2 + q 2, when slopes are of opposite sign. As we ask for increasing number N of leaves which can be expressed in this way, corresponding to increasing areas, we find that the Fibonacci series 1, 1, 2, 3, 5, 8, 13, . . . is predominant. Thus, only integers N = (total number in a pattern) which can be expressed as the sum or the difference of the squares of integers form spiral patterns. Further, those patterns are preferred energetically which have the highest N/A ratio. Table 1 exhibits the fascinating numerical patterns resulting from constructions in the case that p q q q 0, when p, q are relatively primed and Nk are Fibonacci numbers. Formulas for the k th integer in Table 1 are given by F2k = Fk2 + Fk2 − 1 , when slopes have opposite signs, and N = p 2 + q 2, and F2k − 1 = Fk2 − Fk2 − 2 , when pattern slopes have the same sign, and N = p 2 − q 2. Here, Fk = 1, 2, 3, 5, 8, 13, 21, . . . for k = 1, 2, 3, . . . , the Fibonacci series. Mitchison (1972) has emphasized that the successive Fibonacci patterns arise as a consequence of the

    way the plant grows, and as the area increases. The first true foliage leafs of a plant often initiate a 180° opposite (distichous), with one of these leafs displaced along the axis from the next. The subsequent primordia at the apical meristem then appear as if positioning themselves in a region not forbidden by the previous two leafs, or as if a forbidding diffusing substance emanated from the previous leafs. At any rate, such a situation eventually gives way to a spiral (Mitchison, 1972), as new leaves are added in succession in a region close to the apical tip as its apical diameter increases. The present picture also envisions a gradual increase in the stem circumference, and an increase in area at a given stem height, moving down the stem (Fig. 5). As the apical diameter increases, so also will the divisions into p or q equal segments increase, one at a time. As the total area k 2A A p 2 + q 2 increases as we proceed down the stalk axis, so also does the number of divisions of the stalk axis and circumference, p and q, as progressively higher order patterns are allowed to appear (Table 1). As Table 1 shows, going to higher values of p, q, and progressively higher values of N (leaf number) in the Fibonacci pattern, each successive number N contains only one of the numbers p, q from the previous number N so that one set of divisions into p or q equal parts of the rectangle remains the same as we go from one N to the next allowed; only one of the p, q is being increased from one pattern to the next, with neither p nor q decreasing, as area increases. T 1 The Fibonacci series N(p, q) = p2 2 q2 k 1 2 3 4 5 6 7 8 9 10 11

Nk 2 3 5 8 13 21 34 55 89 144 233

=(p 2 2 q 2 ) 2

2

1 +1 22 − 12 22 + 12 32 − 12 32 + 22 52 − 22 52 + 32 82 − 32 82 + 52 132 − 52 132 + 82

=m + n

k 2Am /2 = (p 2 + q 2 )

1+1 1+2 2+3 3+5 5+8 8 + 13 13 + 21 21 + 34 34 + 55 55 + 89 89 + 144

2 5 5 10 13 29 34 73 89 194 233

N2k + 1 = Nk2 + Nk2 − 1 , N2k = Nk2 − Nk2 − 2 . Fibonacci numbers N are shown in the second column, as well as in columns headed by both p and q. The number N is the total number of leafs (florets, bracts, . . .) of an allowed pattern. The model requires that in the spiral case when p q q an allowed N must equal p 2 2 q 2, where p and q are integers. As a result, as p and q increase the integer N is always also the sum of the two previous integers above it in the table, N = m + n. The fourth column shows the number of the visible opposed parastichies, m and n. The minimum (dimensionless) area Am of a pattern (p, q) from eqn (11) when the ratio y0 /x0 = 1. As the area of a pattern increases, so also do N, p and q.

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All Fibonacci patterns given in Table 1 display the property that a pair of families of spirals are visible and opposed (Adler, 1974; Jean, 1994), with all leafs lying on m turns in one sense, and also n turns in the opposite, and where N = m + n, m q n, (with three of the visible parastiches shown by the dashed lines in Figs 1 and 2). The present model thus uncovers a ‘‘sublevel’’ of Fibonacci numbers (p, q) underlying the usual integers (m, n), where now N = m + n = p 2 2 q 2. The pair (p, q) is associated with the fundamental generative algorithm. The usual phyllotactic nomenclature (Adler, 1974; Jean, 1994) is to describe a pattern by the integers (m, n) rather than by the smaller integers (p, q) which have been introduced here. In contrast to the pair of families of spirals which are visible and opposed, and described by the two usual integers m, n, so that N = m + n, the present construction introduces pairs of parastichies which are less visible and only opposed in half of the patterns of Table 1, and as shown by the solid lines of Figs 1 and 2. In Jean’s model (Jean, 1994, p. 139) Fibonacci patterns arise for N = 3, 5, 8 and 13 only; the other Fibonacci numbers given in Table 1 correspond in Jean’s model to rare non-Fibonacci spiral patterns. The divergence angle of any pattern is the shortest distance on the x/x0 axis between two successive leafs, along the n opposing parastichies (see Figs 1, 2); this is 1/3 for N = 3, corresponding to an angle of 120°; it is 2/5 for N = 5, with angle of 144°; 3/8 for N = 8, and 135°; and so on, giving an angle which oscillates about while quickly converging to the limit of 137.5° as N increases. The longer distance between leafs, from the opposing parastichy, will quickly converge to the ‘‘golden ratio’’ = 0.618 . . . as N increases: 2/3, 3/5, 5/8, 8/13, 13/21, . . . Jean (1994, p. 37) refers to this as Adler’s theorem. By construction of the patterns of Figs 1 and 2, the interesting fact emerges that each pattern of a given N = m + n, when superposed on a pattern with N = m on the same unit square shows all leafs of both patterns lying on just one set of n parallel lines. The new leafs (or primordia) of an emerging N = m + n pattern appear on (when y0 /x0 = 1) or near one set of n spirals of the N ' = m preceding pattern. 6. Non-Fibonacci Spirals with p q q q 0, and p, q Relatively Primed Other spirals are allowed beyond those whose total number N is a Fibonacci number. For example, any odd integer 5, 7, 9 , 11, . . . can be written as the difference of squares of two adjacent integers, which becomes clear upon writing p + q = p 2 − q 2, when

. .   . . 

538

T 2 A short summary of allowed patterns in present model N

Type 137)

p, q

(p 2 2 q 2 )/J;

n, m

sw fs;aw sw fs aw fs ss bs aw ss fs ts ss s bs ss aw fs aw s

1 2 2 3 4 5 5 6 6 7 8 9 9 10 10 11 12 13 14 15

1, 0 1, 1 2, 0 2, 1 2, 2 2, 1 3, 2 4, 2 3, 3 4, 3 3, 1 6, 3 5, 4 3, 1 4, 2 6, 5 6, 6 3, 2 7, 7 4, 1

(Jean’s model does not give whorl patterns) 1, 1 (n, m only for spirals) 2, 1 (N = 4 not allowed in spiral phyllotaxis) 2, 3 1, 4 21, 2 (all alternating whorls (‘‘aw’’) have N = 2p) 1, 6 3, 5 31, 2 1, 8 3, 7 22, 3 1, 10

ts bs s

15 16 17

6, 3 6, 2 4, 1

N/Am ;

minEb (Jean, pp. 135,

1 2 + 12

1.0

(decussate)

22 − 1 2

0.60

1.08

22 + 1 2 32 − 2 2 (42 − 22 )/2

1.0 0.38 0.6

2.03 (fs preferred) 1.98

4 2 − 32 32 − 1 2 (62 − 32 )/3 5 2 − 42 32 + 1 2 (42 + 22 )/2 6 2 − 52

0.28 2.83 0.80 2.79 0.6 2.89 0.22 (very rare) 1.0 (mistake for N = 7) 1.0 3.53 (very rare; Jean, p. 174)

5, 8

32 + 2 2

1.0

4.35

4, 11

42 − 1 2

0.88

(mistaken for N = 4, 11)

32, 3 23, 5 4, 13

(62 + 32 )/3 (62 − 22 )/2 4 2 + 12

1.0 (preferred) 0.8 NA (mistaken for N = 4 or 13)

Note that every even N can possibly be an alternating whorl, p = q, with N = (p 2 + q 2 )/p = 2p. For all odd N, N can be written as p 2 − (p − 1)2. The first column shows the type of pattern, where the following abbreviations have been used: fs = Fibonacci spiral; s = non-Fibonacci spiral; aw = alternating whorl; sw = superposed whorl; bs = bijugate spiral; ts = trijugate spiral; ss = single spiral (spiromonostichy). The second column shows the total number N in a (non-repeating) pattern, including bijugate and trijugate. The third column gives the corresponding integers p and q of the present model, while the fourth column gives the values of the visible opposed parastichies, where N = m + n, which is Pr of Jean’s model. The fifth column gives the values of N(p, q) only in the spiral case, by the criterion of the present model. In all cases, both spirals and whorls, the value of N is given by N = (p 2 2 q 2 )/J, where J is the number of leafs on a single level; J = p in the case of whorls, and is the common integer factor of p, q in the spiral case. The sixth column shows the factor N/Am , the (number of leafs)/(area of pattern) at fixed leaf area which is proportional to the negative of the energy/area of the model. For a given area Am , larger values of N/Am are favored. In particular it appears that a pattern with a value of N/Am Q 0.28 occurs very rarely, while Fibonacci patterns generally have N/Am e 0.8. The last column shows the values of Jean’s ‘entropy’’ function Eb , from Jean’s interpretative model (1994, Chap. 6, p. 135). There are more allowed spiral patterns in Jean’s model than are allowed by the present model, and Jean’s model gives no whorl patterns. There can be two different classifications for a given pattern; for example, Jeans gives the pattern with N = 7 = 1 + 6 as a member of the accessory series 1, 6, 7, 13 . . ., while in the present model it is a (spiromonostichy) case of 7 = 42 − 32.

q = p − 1, e.g. 9 = (5 + 4)(5 − 4). By construction it is then seen that spirals of any odd number N which repeat in one turn may occur, and are called ‘‘spiromonostichy’’ patterns. Jean (1994, p. 41) observes that spiromonostichy is observed in the shoots of plants such as Costus, and he calls this ‘‘a genuine puzzle’’, in that it violates Hofmeister’s axiom; they are however allowed by our eqn (8'), although predicted to be rare since they have a low N/Am ratio. While spiromonostichies of N = 7, 9, . . . are allowed, they should appear infrequently, as they are far from space filling; while N = 7 is rare, an N = 9 simple spiral does not occur. The ‘‘filling factor’’ N/Am is proportional to the negative of the energy/area, [from eqns (8), (8'), and (11)] and in the case of q = p − 1 A (p 2 − q 2 )/ (p 2 + q 2 ) = [p2 − (p − 1)2 ]/[p 2 + (p − 1)2 ]

= (2p + 1)/(2p 2 − 2p + 1) and becomes 1 − 1/p as p gets large. Then for example, the case of N = 9 = 52 − 42 has an N/Am ratio of 9/41 = 0.22; for N = 11, N/Am = 11/61 = 0.18; such cases have not been seen, whereas the case of N = 7 with a N/Am ratio of 0.28 is very rare, suggesting that a N/Am ratio of about 0.3 is the lower limit of occurrence. This is to be compared to, say, the Fibonacci spiral case of N = 5 = 22 + 12, when the filling factor N/Am = 1. The case of N = 32 − 22 = 5 = 4 + 1, a single spiral with N/ Am = 9/13 = 0.69, is classified by Jean (1994) as a member of the accessory series 1, 4, 5, 9 . . . rather than a spiromonostichy, and such patterns do occur. Also by way of comparison are the Fibonacci spiral cases of N = 8, when N/Am = 0.8, or N = 13, when N/Am = 1. There appear to be no known spiral

    patterns with a filling ratio N/Am less than 0.28 and N q 7 (see Table 2). Other non-Fibonacci spirals are allowed. It is of interest to note that the case of the integers N = 5 (in the case when N = 32 − 22, Am = 13, with filling factor N/Am = 5/13 = 0.38), N = 10 and N = 15, with N = 42 − 12 gives members of the Lucas series 1, 3, 4, 7, 11, . . . for the m and n in N = m + n, not for N. The Lucas series does not otherwise occur in the model as a simple spiral. These Lucas numbers m, n are N = 5 = 4 + 1 = 32 − 22; N = 10 = 3 + 7 = 32 + 12; and N = 15 = 4 + 11 = 42 − 12. The Lucas numbers appear as understandably mistaken readings for N in the cases of the spirals N = 10, 15 and 17, as shown in Table 2. Six references are known which mention the series 3, 7, 10, 17, . . . (Jean, 1994, p. 154), of which N = 10 is a member. The type of pattern represented by this sequence is not allowed by Jean’s model. It is to be noted that for a plant progressing through the Fibonacci sequence N = 1, 2, 3, 5, 8 . . . as growth occurs and area increases, all adjacent Fibonacci patterns N and m share a single set of n parastichies; in this sense, N = 10 is not the pattern expected to follow N = 8, except possibly as a transitional state, since the leafs of both N = 8 = 3 + 5, and N = 10 do not all share a single set of five parallel lines, as do the patterns N = 13 = 5 + 8 and N = 8 = 3 + 5. There is a more general and important point to be made regarding those patterns with N = p 2 2 12, for p e 3. Since the slopes of the two sets of parallel lines are p and 2(1/p), and the slopes become increasingly steep as p increases, it is easy to make an observational mistake regarding the actual N, and mistakes are expected to be of increasing frequency as p increases, being exaggerated also as leaf size increases, and also as the ratio y0 /x0 becomes large. Figure 3 illustrates the point by consideration of the case N = 10. Here p = 3, and q = 1. The dashed line of Figure 3 shows a possible ‘‘mistake’’, in that the leaf indicated by the arrow is close to the (x0 , 0) to (x0 , y0 ) axis and may be mistaken for the repeat of the leaf at (x0 , 0). In this case, the pattern could mistakenly be denoted as a case of N = 7, belonging to the series 2, 5, 7, 12, . . .. The case of N = 42 − 12 = 15, with four parastichies in one direction and 11 in another illustrates the point more clearly. This could easily be mistaken for either a case of N = 4, or N = 11. The case of N = 17 = 42 + 12 can easily be mistaken for a case of N = 4 or N = 13, etc. This is indicated in Table 2. It is to be noticed that the patterns mentioned in Table 2 for N = 10 (3, 7) and N = 15 (4, 11) are not

539

N=7+3=3 2 +1 2 =10 F. 3. N = 10 = 7 + 3 = 32 + 12. The figure illustrates that cases where N = p 2 2 12, and p q 3 are easily mistaken for lower order patterns. (– – –) indicates that the leaf indicated by the small arrow can be mistaken for a ‘‘repeat’’ of the leaf at the point (1, 0), especially as the leaf size increases, and also as the ratio y0 /x0 increases, and as p increases. The case of N = 7 shown by the dashed curve is then a mistaken observation. The case of N = 10, while allowed, is relatively rare, and does not follow the case of N = 8 as growth occurs, since a relatively large rearrangement of leafs is required compared to the next Fibonacci pattern of N = 13, since this latter pattern shares a single set of n = 5 parallel lines with N = 8.0

allowed to exist according to Jean’s model. In the present model rare patterns are obtained for low values of N/A, while in Jean’s model rare patterns are given by high costs corresponding to high values of Eb . Patterns not allowed in the present model, but allowed in Jean’s model are shown in Table 3. 7. Bi, Tri, . . . Jugate Spirals; p q q q 0, p, q Not Relatively Primed The bijugate, trijugate, . . . spiral phyllotaxis arises when p q q are not relatively primed, and when there are 2, 3, . . . leafs on the same level. Examples are the cases p = 4 and q = 2, with the common integer factor of 2, which is either the bijugate spiral case of N = (42 + 22 )/2 = 10 = 2(22 + 12 ), or the bijugate spiral N = 6 = 2·3=(42 − 22 )/2 = 2(22 − 12 ). Construction gives, in the first case of N = 10, two T 3 Some patterns not allowed in present model, but occuring in Jean’s model, (1994, p. 135, prop 3c) N(0Pr )

(m, n)

7 11 12 14 15

3, 4 4, 7 4(1, 2) 5, 9 5(1, 2)

(min. entropy, Eb , Jean, pp. 135,

137) s s s s s

2.83 3.63 3.79 4.5 (not provided)

This table shows spiral cases given by Jean’s interpretative model (Jean, 1994, pp. 135, 137) but not allowed in the present model.

540

. .   . . 

vertically (i.e. one above the other) repeated patterns each of N = 10, where each pattern of N = 10 consists of two superposed Fibonacci spiral patterns of N = 5, as in Fig. 1(b), but now having two leafs on each level instead of one, each leaf displaced from the other by 180°. The bijugate pattern of N = 6 = (42 − 22 )/ 2 = 2(22 − 12 ) also shows two interlaced Fibonacci spiral patterns of N = 3 [see Fig. 1(a)], with two leafs on each level, again displaced by 180°. Another example of a bijugate spiral is N = 16 = (62 − 22 )/2. Generally one has a number in a spiral pattern given by N = (p 2 2 q 2 )/J, where J = 1, 2, 3, . . . is the uni, bi, and trijugate, . . . number. The value N = 9 in Jean’s model also gives trijugacy, but the value N = 15 gives penta-jugacy 5(1, 2) instead of the trijugacy of the present model, when N = 15 = 3(22 + 12 ). Examples of trijugate spirals occur with p = 6, q = 3, when N = 15 = (62 + 32 )/3 = 3(22 + 12 ), and also the case of N = 9 = (62 − 32 )/3 = 3(22 − 12 ). In all trijugate cases by the usual construction one obtains three vertical repeats of the trijugate spiral, each of the three repeats corresponding to a basic trijugate pattern. In each basic trijugate pattern, with 15 leafs (N = 3·5) in the first example, and N = 9 = 3·3 in the second, there are three superposed Fibonacci N = 5 patterns in the first case and N = 3 in the second [see Fig 1(a, b)], but now with three leafs on each level displaced from each other by 120°, and J = 3 (Table 2). Very generally in the present model, all allowed patterns have a pattern number N given by N = (p 2 2 q 2 )/J. J is the number of leafs on a single level; when p q q q 0, J is the common integer factor of p and q (the ‘‘jugacy’’); when p = q, (alternating whorl) or q = 0, (superposed whorl), then J = p.

8. Distichous Phyllotaxis; p = q = 1 The common case of N = 2 = 12 + 12 does not satisfy the p q q criterion of spiral systems. This special case of p = q = 1 corresponds however to distichous phyllotaxis, in which successive leafs are at 180° from the preceding one on a higher or lower level. Examples are the corn or ginger plant, and most ferns. Since this pattern has opposite sign slopes, s = 21, the picture is particularly simple: the two straight lines of eqn 14. forming a figure ‘‘X’’ within the (y/y0 ), (x/x0 ) square. This is the simplest spiral, and at the same time the simplest whorled pattern, since p = q. We notice that the energy/area of eqn (8') attains its minimum value of −1, or a filling factor of N/Am = 1 for this often seen pattern.

Decussate N=4=p+q=2p F. 4. The Decussate pattern. This is a case of the simplest alternating whorled pattern, and this ‘‘decussate’’ pattern occurs for p = q = 2. Higher order alternating whorled patterns occur when p = q, and the number of leafs in a repeating pattern is N = p + q = 2p. The ‘‘superposed whorls’’ occur when p q 0, q = 0, a fascinating example of which is the case of ‘‘monostichy’’ corresponding to p = 1, q = 0, in which a single superposed column runs parallel to the stem axis; the case of p = 2, q = 0 is not rare (e.g. roses, wisteria).

9. Whorled Phyllotaxis; p = q q 0, and p q q = 0 The case of alternating whorled patterns corresponds to p = q. When p = q = 2, this common pattern is known as decussate, and is shown in Fig. 4. The number of leaves is now N = 4 = 2p. The decussate pattern then corresponds to the simplest alternating whorled configuration (except for the distichous pattern of the previous section): two leaves at the same level but 180° apart, followed by two more along the stem axis, these two also separated from each other by 180°, but rotated relative to the first pair by 90°. Higher order whorled patterns also occur, in which p leafs occupy the same stem level at equal angular intervals, followed by p leafs at a lower (higher) stem level, etc., but rotated by p/p from the previous level, and correspond to p = q q 2. When p = q all intersecting straight lines of eqn (14) have slope of either 1 or −1. In the case of alternating whorls, the total number N of leafs per pattern is always given by N = 2p. The delicate ‘‘Parrot’s feathers’’, corresponding to p = q = 6 is a striking example. From consideration of the energy/area of eqn (8') alone it is expected that decussate patterns (p = q = 2) are somewhat frequent, with a filling factor N/Am = 0.5, while alternating whorled patterns are expected on this basis to show decreasing frequency of occurrence as the value of p = q increases; since from eqn (8') we have that energy/area A − N/Am = −2p/(p 2 + q 2 ) = −2p/ (2p 2 ) = −1/p.

    Although alternating whorls are found in the majority of symmetrical (non-spiral) plant constructions, superposed whorls, at least p = 2, q = 0 whorls, are also not infrequent. In the present model, superposed whorls occur when p q 0 and q = 0. In this case the two slopes of eqn (14) are either zero or infinite, and the two sets of parallel lines are parallel to either the x axis or the y axis. The superposed whorl configuration with p = 2 and q = 0 is seen (e.g.) quite often as wisteria, mahonia and honeysuckle; this is the pattern of two leafs on the same level displaced from each other by 180°, followed at the next level with a repeating patten whose two leafs are directly above (below) those of the previous level. The leaflets of compound leaves very often show this p = 2, q = 0 pattern. A very interesting case is that of ‘‘monostichy’’, consisting of a single column of superposed leafs parallel to the stem axis, with p = 1, q = 0. Such a plant actually exists (Jean, 1994, p. 175). Of course, the energy/area expression of eqn (8), or eqn (8') indicates that such a plant should be rare, as observed (but not forbidden by the present model as it is by Hofmeister’s axiom) given the low space filling of the available area. Certain cacti show superposed whorl patterns with p q 2, q = 0, (as do some insects). In Jean’s model (Jean, 1994), whorls appear as extreme cases of spirals. There appears to be no statistical information regarding whorled patterns (both cases, p = q q 0, and p q 0, q = 0) but a casual perusal of gardens tells one that both alternating whorls and superposed whorls are not uncommon. The cases p = q = 2 (decussate, e.g. fuchsia) and p = 2, q = 0 (e.g. rose or wisteria) are often seen. While mindful of the pitfalls inherent in the collection of accurate data, Jean (1994) has given a compilation of many observations collected over 160 years. For the general information of the reader, it may be helpful to give a rough idea of the relative frequencies of the various patterns, although as pointed out previously, there is no simple single criterion for predicting frequency. Data is given for almost 13000 observations on more than 650 species. Excluding pine cones and pineapples (where virtually all reports synchronize the Fibonacci patterns), Fibonacci spirals are about 92% of all spiral observations, while ‘‘Lucas’’ patterns 1, 3, 4, 7, 11, 18 . . . and other ‘‘anomalous’’ spirals occur in less than 2% of the observations. The Lucas series cases arise in about 1.5% of all cases, as compared to 6% for the bijugate Fibonacci spirals, 21, 2, 3, 5, 8, . . ..

541

The present model does not allow Lucas phyllotaxis. In Section 6 it was pointed out that there are allowed patterns in the present model which can be easily mistaken for Lucas patterns, and a slight twist of the stem can be the cause of such a misreading. Both Jean’s model and the present one do not allow N = 4, an early Lucas number in the series 1, 3, 4, 7,. Inclusion of the presently neglected nonlinear term in the Helmholtz equation, b q 0, may be expected to lead to some Lucas patterns, since such are often so close to allowed patterns. Confirmation of this speculation requires further study. 10. Transformation from Cylinder to Cone A simple analytical transformation takes all results for the rectangle or cylindrical geometry so far discussed to a conical geometry, as shown in Fig. 5. The important point of an analytic transformation is that it leaves the form of both the Helmholtz and the

r2

R1

0

R2

r1

=sin -1 (

0/ 2

);

2 r 1,2=R 1,2

0

F. 5. Analytic map from rectangle to cone. The analytic function Reiu = R1 eiz/l maps the rectangle in the (x, y) plane into the truncated cone in the R, u plane. The analytic transformation preserves the form of the Helmholtz and Laplace equation of the model. The m and n intersecting opposed visible parastichies given by N = m + n in the (x, y) plane then go over into oppositely oriented intersecting log spirals in the cone plane. The bottom part of the figure illustrates how cones are joined, with patterns of increasing areas and decreasing cone angles as they recede from the apical meristem.

. .   . . 

542

Laplace equation invariant, so that we are truly finding the solution on the cone via the transformed plane or cylinder. The new coordinates defined by the analytic function Reiu = R1 ·eiz/l takes the unit square in the x, y plane into the (truncated) conical region shown in Fig. 5. Here z = x + iy, (i = imaginary unit, l is a length) and the two planes are related by R = R1 exp(−y/l),

and

u = x/l.

Thus the lines of the rectangle at y = 0 and y = y0 are mapped into the circle segments at R = R1 and R = R2 , respectively. The lines x = 0 and x = x0 are mapped into two straight lines through the origin in the R, u plane, x = 0 mapping into u = 0, and x = x0 going into u = u0 . The half cone angle is given by a = sin−1(u0 /2p); u0 = 2p corresponds to a flat disc, when the half cone angle a = p/2; a zero cone angle a = 0 corresponds to a cylinder, when r1 = r2 . The area of a single conical region shown in the figure is A = (R12 − R22 )u0 /2 = 2p(R1 − R2 )(r1 + r2 )/2, where the relationship 2pr1,2 = u0 R1,2 has been used. Also, the ratio y0 /x0 = ln(R1 /R2 )/(2p sin(a)). The two sets of parallel straight lines in the x, y plane (e.g. Figs 1, 2) determined by integers p q q then go over into two sets of intersecting log spirals in the R, u cone plane. The leaf patterns also fall on two other sets of m and n straight parallel lines always with opposite slopes in the x, y rectangle plane, where N = m + n, and these go under the mapping into two other distinct oppositely directed intersecting log spirals in the R, u plane, the opposed visible parastichies (Jean, 1994; Adler, 1972). The patterns on cylinders of changing areas can now be viewed as joined smoothly to those on cones by the joining of truncated cones of changing cone angles and R1 /R2 ratios; pictured in Fig. 5(b) is a succession of joined cones, each consisting of one periodicity and each joined smoothly to the next with a progressively smaller cone angle a, as we proceed from the conical meristem to the cylindrical shape. Then the cone comprising the apex meristem will have the smallest inner radius, and thus be able to accommodate a given number (presumably most often one or two) of leaf primordia. 11. Possible Biochemical Origin of the Algorithm A speculation regarding possible biological realizations of the model set forth now follows. The two kinds of pattern forming entities or biomolecules with the requisite properties as envisioned in Section 2 are unknown at present, in both animals and plants. In spite of the many differences between plants and animals, there may nevertheless be homologous

molecules or mechanisms for certain key developmental processes in both kingdoms. In animals, two different kinds, or two different states of the same cell surface adhesion molecule (CAM) on the lateral surfaces of cells comprising an epithelial cell sheet is a promising possibility for patterning (Cummings, 1994, 1996, 1997). Such adhesion molecules are known in animals, the so called RPTP adhesion molecules, which make homotypic adhesive connections with neighboring cells (Brady-Kalnay & Tonks, 1995; Tonks & Neel, 1996). In conjunction with the receptor tyrosine kinases, these three domain membrane spanning adhesion molecules, phosphatases (in that they detach phosphate groups from tyrosine residues) are known to be in the signalling pathway to the gene, and intimately involved in the crucial processes of development: cell shape changes, control of cell division, and transcription. Further, this cell-membrane-spanning (three domain) tyrosine phosphatase adhesive molecule (RPTP) comes in two forms (Fischer et al., 1991; Gebbink et al., 1993; Eijgenraam, 1993). Although these remarks are explicitly relevant only to animal signalling pathways, the assumption here is that homologous mechanisms operate in plants. In plants, presumably there is also some way in which information is transmitted from cell to cell. Since plant cells are immobile, the contact from cell to cell is either via the plasmodesmata and thus through the symplast, or by the secretion and recognition of molecules across the plasma membranes and cell walls (Lyndon, 1990). However, it is unknown whether such plant homologues to the animal systems mentioned above even exist, or if so, how such animal cell–cell information transfer homologues might operate in plants. In the face of this ignorance, the speculation is nevertheless that in a general manner of speaking, patterning mechanisms in multicellulars, both plants and animals, are most closely linked to the signalling pathway to the gene. The linkage may be so intimate that the two previously disparate concepts of ‘‘pattern formation’’ and ‘‘signalling pathway’’ are effectively merged; patterning is part of the signalling pathway rather than some very separate mechanism. The above speculation has interesting implications for the possible connection between the present model and other models of plant patterning. Numerous experiments (Green, 1992; Green & Baxter, 1987) have repeatedly emphasized a primary role for geometrical changes in the pattern formation itself, and not merely a consequence of an underlying ‘‘prepattern’’. While in the present model as in most others the leafs are treated as either points or circles, in Green’s approach the leaf is dealt with as a

    hoop-reinforced appendage. Using polarized light he described what happens in the microstructure of the apex epidermis as leaf primordia are initiated (Green, 1992; Green & Baxter, 1987) and concludes that as an organ is initiated, a region of the surface (tunica) layer of the apical dome bulges under the pressure of the shoot apex. On the other hand, Hernandez and Palmer (Jean, 1994; Hernandez & Palmer, 1988) showed the subsurface origin of the disc florets at early stages of sunflower capatuli development with the scanning electron microscope. They found that the first cell divisions creating the initials occur well before the raised pattern can be seen. Further, the pattern is seen to be determined before the florets come into contact, so that contact (Adler, 1974) appears to be a subsequent phenomenon and does not induce the pattern (Jean, 1994, p. 254). The present model envisions biomolecules which are closely coupled to the cellular elements determining cell shape changes and cell division, so in that sense the present view is not far from Green’s (1992). However, some greater or lesser time delay between patterning onset and actual large scale geometrical changes is to be expected in the present patterning mechanism; thus in the present view neither contact pressure nor geometrical changes initiate the pattern, but rather are seen as consequences of it. There is also no long range diffusion involved in the patterning in the present model. It is a striking fact that the linear Helmholtz equation is able to correlate so many of the phyllotactic patterns seen in nature. In summary, the integers p and q give rise via eqn (11) to (1) simple Spirals: p q q q 0, and p, q are relatively primed (2) jugate spirals (bijugate, trijugate, . . .): p q q q 0, where p, q are not relatively primed (e.g. p, q = 4, 2); (3) alternate whorled patterns, with p = q q 0; and (4) superposed whorled patterns, when p q 0 and q = 0. While these patterns arise from the Helmholtz equation quite independently of its origin or derivation, an underlying biochemical origin is strongly suggested. Future work would envision a better understanding of the effect of the changing geometry on the pattern as the primordia become leafs, as emphasized by Green, since the present work has only considered the pattern as arising from a prescribed geometry. This extension entails the much more complicated problem of the coupling of pattern to geometry (Cummings, 1994, 1996, 1997). Further, the effect of the nonlinearity b q 0 of eqn (11) as other than a first order perturbation is another area of future interest; the regions marked out by the two morphogens are more generally of different size.

543 12. Conclusion

A simple algorithm, given in eqn (14), encompasses the principal phyllotactic patterns, plus other less frequently seen patterns seen as well. The dominant spiral patterns, described by the Fibonacci series, are obtained principally as a result of conditions dictated by the nonlinear algorithm. In the spiral case, p q q q 0, the number of leaves N must be the sum or difference of the squares of two non-equal and relatively primed integers p and q, as seen in Table 1. In every pattern, the number of leafs in a pattern is given by the simple expression N = (p 2 2 q 2 )/J. In the case that p q q q 0, the integer J is the common factor of p and q. In the whorl cases, both alternate (p = q) and superposed (q = 0), J = p. In any case, J is the number of leafs on a single level of the stem. An expression which is reminiscent of Hofmeister’s axiom (Jean, 1994) results from the basic assumptions of the algorithm. However, it is stressed that the implications of eqn (8'), a consequence of the model, and Hofmeister’s axiom are not identical: eqn (8') is not an axiom, and it gives an expression for the relative energy/area of a configuration, favoring close packing over less close packed. Thus, configurations not allowed by Hofmeister’s axiom are allowed by the present model, but may be so energetically unfavorable as to be rarely (N = 7) or never seen (N = 9 = 52 − 42, and N = 11 = (62 − 52 )). It is not known how to relate the energy/area of the present model to frequency of occurrence in a quantitative way, and the energy expression given in the present work can at best serve as a rough guide; this is partly because the uncertain leaf areas Da, remembering that the filling factor N/Am must be multiplied by Da to give the negative of the energy/area. As was pointed out in the introduction, patterns are expected to be determined by several factors, importantly including how the plant grows, especially in the vicinity of the meristem; the energy cost is no doubt but one factor among others. Relative energy costs (all assuming the same Da) have nevertheless been indicated in the text in the case of the spiral systems on the basis of the present model; Table 2 shows a summary of a number of patterns, both spiral and whorled, with the filling factor N/Am given in the second to last column on the basis of the present model. The last column gives the entropy Eb on the basis of the interpretative model of Jean (1994, Chap. 6). According to his principle of minimum entropy production, at plastochrone m + n the spiral pattern which is selected is the one which minimizes the function Eb , and such values are shown in Table 2 in the last column. Future work will hopefully clarify the relationship between the

544

. .   . . 

interpretative model of Jean and the present model. Although the frameworks in which the two models are formulated are quite different, there are nevertheless some striking coincidences in results. As a result of the present model, the total number of leafs in a spiral pattern is given, as is more usual, in terms of two families of opposing visible parastichies (Adler, 1974), where N = m + n, and n and m are the number of turns in each of two opposite directions. In the spiral case, growth accompanies progressively larger values of p and q by eqn (11), and the Fibonacci series is unique in that only one of the p, q increases as we go to progressively higher N or larger area, as growth proceeds and the apical radius increases and also the circumference of the stem gradually increases away from the apex. Bijugate, trijugate, . . . spirals arise from cases with p q q where p and q have a common integer divisor. The integers p = 1, q = 1 correspond to the distichous pattern with two leafs occurring in one rotation (e.g. corn, ginger). This is the simplest spiral pattern, and at the same time the simplest whorled pattern. Non-spiral patterns occur for p = q and p q 0, q = 0. A number of leaves N = 4 = p + q corresponds to two equal integers p = 2, q = 2, and gives the simplest whorled, or decussate pattern (e.g. iris, fuchsia). Higher order alternating whorls correspond to values of p = q q 2. Superposed whorls correspond to p e 1, q = 0, with slopes of the two sets of parallel straight lines in the x, y plane given by the algorithm of eqn (14) as zero and infinity (‘‘orthostichies‘‘), when columns of superposed leafs run parallel to the stem axis. The superposed whorl pattern of p = 2, q = 0 appears quite commonly (e.g. rose, wisteria, etc.). The model is easily falsifiable in that it predicts, among many other things, that simple spiral patterns (one leaf per level) of N = 4 or N = 6 are not allowed, or that certain simple spiral patterns (e.g. N = 9 = 5, N = 11), although allowed, do not appear because they have such small space filling. Informative conversations with Paul Green are gratefully acknowledged. We also thank an anonymous referee for many helpful and insightful comments. Consultation on botanical matters with Lynn Winter-Duggan is also appreciated.

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