A contact pressure model for semi-decussate and related phyllotaxis

a/. tlwo,.. Bid. (1977) 68, 583-597

A Contact Pressure Model for Semi-decussate and Related Phyllotaxis D. W. ROBERTS Unilever Research Port Sunlight Laboratory, Unilever Ltd. Port Srmligkt, Wirral, MerseJlside L62 4XN, England (Received 23 February

1977, and in revised,form

23 May 1977)

Semi-decussate phyllotaxis, in which leaves arise singly and the divergence angles between successive pairs of leaves alternate between approximately 90” and approximately 1SO”, is accounted for by a contact pressure model. It is assumed that leaf primordia are initiated at a divergence angle close to the Fibonacci angle of 137.5”, that the primordia move under contact pressure, and that when a primordium first experiences contact pressure all other primordia are fixed. Extensions of the model account for: psuedodecussate phyllotaxis, where the leaves appear to arise in pairs; semitricussate and pseudo-tricussate phyllotaxis, where the leaves are arranged in, respectively, dissolved or apparent trimerous whorls; and phyllotaxis of the 1,3 series, where the divergence angle is about 100”. The compatibility of the model with current theories of Fibonacci phyllotaxis is discussed.

1. Introduction Spiral phyllotaxis, where leaves arise singly on a stem at a more or less constant divergence angle one from the next, has been the subject of many theoretical papers (see Adler, 1974, for a historical review). Whorled or multijugate phyllotaxis, where more than one leaf arises at each node, is usually covered in these papers as either a distorted form (e.g. Thornley, 1975) or as a multiple version (e.g. Adler, 1974) of spiral phyllotaxis. A third type of phyllotaxis is also common, whereby leaves arise singly, but the divergence angle is alternately close to 180” and close to 90”. This type of phyllotaxis will be referred to here as semi-decussate phyllotaxis. It is often observed in the early stages of growth, later developing into spiral phyllotaxis, as successive divergence angles deviate increasingly from 90” and 180”. However, in some plants, e.g. Lycopersicon esculentum and some varieties of Helianthus annus, pseudo-decussate phyllotaxis can persist 583

584

I).

W.

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throughout the entire vegetative growth phase. In spite of its widespread occurrence, this type of phyllotaxis seems to have attracted little theoretical consideration. A mechanism is proposed here whereby semi-decussate phyllotaxis arises by distortion of a system which would otherwise show spiral phyllotaxis.

2. Basic Assumptions It is assumed that: (1) The pattern of leaf initiation is spiral, such that without other effects the divergence angle would be roughly equal to the Fibonacci angle of 137.5’ (the degree of latitude allowable is discussed later) which is usually observed in apices of plants with spiral phyllotaxis (Fujita, 1939). (2) The position of a leaf primordium may be altered, after initiation, as a result of contact pressures from older primordia. (3) By the time a primordium begins to experience contact pressure, the positions of all older primordia are fixed. These assumptions may be compared with those made by Adler to explain the origin of Fibonacci phyllotaxis in terms of a contact pressure model ( 1977). AdIer assumes that : (a) The initial value of the divergence angle is in the range 120”-180”. (b) The leaf primordia move, under the influence of contact pressure from one another, to positions such that each primordium is as far away as possible from its nearest neighbours. This is a specific case of the more generalized assumption (2) above. (c) The normalized vertical separation (in a cylindrical representation) between successive primordia is a decreasing monotonic function of time, so that the role of nearest neighbour to a given primordium is assumed by primordia progressively further separated chronologically from it. (d) When contact pressure begins, a primordium’s two nearest neighbours are separated from it by F, and F,,, plastochrones (plastothrone = time interval between initiation of consecutive leaves), where 4 and K+, are consecutive terms of the Fibonacci series (1, 2, 3. 5, 8 . . . 1. (e) An assumption which Adler does not state as such, but which is implicit in his model, is that for an Fn + F,, r contact parastichy system to be generated according to his model, at least F,, r primordia must be simultaneously mobile whilst experiencing contact pressure. This may be contrasted with assumption (3) above.

A

CONTACT

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581

3. The Development of Semi-decussate Phyllotaxis Given the basic assumptions of section 2, the development of semidecussate phyllotaxis may be envisaged as follows: (a) Primordia Pl and P2 are initiated at positions PI i and P2i; P2 moves under contact pressure from PI to a final position P2, opposite PI i [Fig. 1(a)]. This change of position need not involve physical movement of the entire primordium: it could be visualized in terms of preferential growth of P2 away from PI. The contact pressure effect could alternatively be considered as a chemical inhibition effect whereby a growth inhibitor from Pi retard\ growth of P2, the effect being greatest on parts of P2 closest to PI.

PI

FI

i

1~‘~. 1. Development of semi-decussate phyllotaxis-tangential

shifts.

(b) P3 is initiated at position P3,. As it develops it encounters contact pressuresfrom PI and P2, as a result of which it moves to position P?, [Fig. I(b)], mid-way between PI, and P2,. This implies that PI and P2 exert equal and opposite contact pressure effects which in turn implies one of two situations: either Pl’s lead in development is counterbalanced by P2’s greater radial and vertical proximity to P3 (Fig. 2) or, by the time P3 begins to develop, PI and P2 are not much separated vertically or radially and P:! has caught up in development with PI.

586

D. W. ROBERSS

Development of semi-decussate phyllotaxis-vertical

shifts.

(c) P4 is initiated in position P4, and moves under contact pressurefrom P3 and/or from PI and P2 to position P4J [Fig. I(c)] opposite P3 and midway between Pl and P2. (d) P5 is initiated, and moves from position P5, to P5,. [Fig. I(d)], ita position being determined by P3 and P4 in the sameway as P3’s position was determined by PI and P2 [cf. (b) above]. If P4 is assumedto be influenced by Pi [cf. (c) above], the effect of P2 on P5 should also be considered. Since P5 is initiated almost directly above P2. the most likely effect is that P5 would be displaced vertically. This would lead to a long internode between P4 and P5. (e) P6 is initiated, and moves from position P6i to P6, [Fig. I(e)] [cf. (c) above]. Positions of further primordia are fixed similarly. It may be noted that every odd numbered primordium P,, arises between primordia one and two plastochrones older (P,- , and P,,-?) and separated by IgO”, and with a primordium three plastochrones older (P,,-,) mid-way between P,,- , and P,-, and directly below the final position of P,,. The effect of P,,-3 on P,, would probably be as described under (d) above for P2 on P5. The overall result would be an alternation in internode length: long internodes between leaves separated by 90” and short internodes between leaves separated by 180”. The above mechanism is somewhat idealized. Small deviations from the ideal will produce imperfect versions of semi-decussatephyllotaxis. For example, P3 might become fixed slightly nearer (in angular terms) to PI than to P2, P4 then becoming fixed opposite P3, i.e. PI and P2 have much lesseffect than P3 on the position of P4. Continuation of this processwould lead to phyllotaxis as shown in Fig. 3(a), with the divergence angle alternating; IgO”, 90”+, 180”, 90”+, etc. In the extreme, where P3 becomesfixed directly above PI, i.e. rather than PI and P2 having equal contact pressureeffects, only the effect of P2 is significant, the result will be alternate phyllotaxis with a constant divergence angle of 180”. Another deviation from ideal semi-decussatephyllotaxis may arise if, as above, P3 tends to become fixed closer to PI than to P2, but P4 tends also

A CDNTAC-I’

PRESSURE

(b )

(a) FIG.

3. Semi-decussate

587

MODtl

phyllotaxis-deviations

from

ideality.

lo become fixed closer to PI than to P2, i.e. Pl and P2 have a large influence on the position of P4. Continuation of this process, where the younger of a leaf’s two neighbours has the larger contact pressure effect, leads to phyllotaxis as shown in Fig. 3(b). The divergence angle fluctuates, after P2, between 90“ + . 180 -, 90”+, 180” -, etc., and the phyllotaxis may be regarded as intermediate between semi-decussate and spiral phyllotaxis. 4. Pseudo-decussate

Phyllotaxis

In section 3 it was shown how an alternation in internode length may arise in semi-decussate systems, as a result of vertical movement of odd-numbered primordia P, under the influence of primordia P,-,. This effect may be reinforced by vertical displacements caused by other contact pressures, as follows : If P3 becomes fixed at about the same time as contact pressure from PI pushes it into contact with P2 (see Fig. 2) or vice-versa (depending on whethei the initial divergence angle is greater than or less than 135”), there will be no vertical displacement. If, on the other hand, P3 is still mobile whilst experiencing contact pressure simultaneously from PI and P2, it will be displaced upwards, resulting in a long internode between P2 and P3 (Fig. 2). Similar vertical displacement of P4 by PI and P2 would lead to a normal internode between P3 and P4 (there is no reason to suppose that P4 would be displaced to a greater extent than P3). Continuation along this pattern, with P5 and P6 being vertically displaced by P3 and P4, and so on, would lead to a semi-decussate system in which long and short internodes alternate; long internodes between leaves separated by 90” and short internodes between leaves separated by 180”. Thus there are two effects tending to give rise to an alternation of internodes in semi-decussate phyllotaxis. In one extreme, this alternation may be w

slight I .H.

as to be imperceptible.

so that

an even

vertical

spacing

of the

leaves 39

S88

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ROBLRTS

is observed. In the other extreme, where the short internodes do not dc~~lop perceptibly, the leaves will appear to be arranged in opposite paira with successive pairs at right angles to each other. This case will be referred to as pseudo-decussate phyllotaxis. It is possible that many cases of decussatc phyllotaxis are in fact pseudo-decussate rather than true decussate (where primordia are initiated simultaneously, as a result of twofold symmetry in the initiation system). It is not necessary to postulate large vertical displacements of the primordia to account for the large differences in internode length in pseudo-decussate stems. It is possible that leaf bases in contact (PI with P2. P3 with P4, but not P2 with P3 since P3 is displaced vertically from P2) become fused to each other. This would prevent internodal extension between Pl and P2 and between P3 and P4, whilst allowing extension between P2 and P3. Schoute (1925, 1938) proposed a similar model, whereby he suggested that many case:, of multijugate phyllotaxis arise from spiral phyllotaxis: a number of suecessive leaf bases bind to each other, so that when the bud develops they arise as a whorl (referred to by Schoute as a binding whorl). In Shoute’s terminology, pseudo-decussate phyllotaxib would be described as bi.jugate binding whorls. In sonic cases. it is possible to distinguish pseudo-dccussate from true decussate phyilotaxis. The former may be assumed to occur in species where any one of the following is regularly observed: (a) The apparently decussate system changes to a semi-decussate system further up the stem. This change may be caused by an increase in growth rate. such that internodal extension begins before the leaf bases have time to bind to each other. Alternatively the transition could be attributed to an increase in girth of the apex, so that the bases of successive primordia no longer conic into contact. (b) The system changes from apparently decussate to apparently tricussatc (whorls of three). this change being in reality a transition from pseudodecussate to pseudo-tricussate (tlkfe i~fia) phyllotaxis. (c) The reverse of (a) or (b) above occurs. 5. Semi-tricussate

and Pseudo-tricussate

Phyllotaxis

Occasionally a type of phyllotaxis is found in which the sequence ot divergence angles is 120”, 120”, 180”. 120”, 120”, 180”, etc. (angles approximate). This type of phyllotaxis, here referred to as semi-tricussate, may be rationalized in terms of the following assumptions: (1) Basic assumptions (1) and (2) of section 2, as for semi-decussate phyllotaxis, still apply.

A CONTACT

PRESSURL

MOrILI.

589

(2) When a primordium begins to experience contact pressure, its immcdiate predecessor is still able to move. (3) Contact pressure effects may occur between primordia separated by up to three plastochrones, but not between primordia separated by five or mot-c plastochrones. Given these assumptions, the development of semi-tricussate phyllotaui> may be envisaged as follows: (a) PI and P2 are initiated. Possibly, but not necessarily, P2 tno\ch to it position opposite PI before experiencing contact pressure from P3 [Fig. 4(a)].

FIG. 4. Development

of semi-tricussate

phyllotaxis.

(b) P3 is initiated at position P3,. As it experiences contact pressure from PI and P?. it moves to position P3,, whilst P2 moves to P2,, so that the three primordia PI. P2 and P3 are spaced at 120” to one another [Fig. 4(a) and (b)]. (c) P4 is initiated at P4i and moves under contact pressure from PI and P2 to P4,, mid-way between PI and P3,. and opposite P3, [Fig. 4(b) and (c)l. P3, being constrained by PI and P2, does not move. (d) PS is initiated at P.5, and P6 at P6i. They move under contact pressure from each other and from P4 to P5, and P6, [Fig. 4(c) and (d)]. P4 does not move under pressure from P5 since it is constrained by PI and P2. During this m-positioning, P5 also experiences contact pressure from P2 and P3. whilst P6 experiences contact pressure from P3. (e) 1’7 is initiated at P7,, and moves under contact pressure from P4 and I’5 to position P7,, mid-way between P4 and P5,. opposite P6,. and abovf ?‘4’ I./L.

590

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ROBERIS

P2 [Fig. 4(d)]. P6 does not move under contact pressure from 1’7, since it I\ constrained by P3. (f) By continuation of this development pattern, semi-tricussatc phyllr)taxis develops. Pseudo-tricussate phyllotaxis, where the leaves appear to arise in symmetrical whorls of three, may be derived from semi-tricussatephyllotaxis in the same way as pseudo-decussatewas derived from semi-decussatephyllotaxis. 6. Phyllotaxis of the 1,3 Series as a Modification of Pseudo-tricussatePhyllotaxis Schoute (1936, 1938) describes how a phyllotaxis system of the type ~1, m+ 1 can arise as a modification of a system of binding whorls with 111 members per whorl. Schoute’s model may be combined with the model presented here for semi-tricussate phyllotaxis, to show how an initialI> Fibonacci system can be transformed to a system of the I ,3 serifs, as follows: (I) Primordia are initiated with an initial divergence angle close to the Fibonacci angle. (2) Contact pressure effects give rise to a semi-tricussate system, a:, described in the previous section, with some vertical displacement. as shown in Fig. 5(a). Since the first three primordia experience no contact pressures from below, it is reasonable to suppose that the vertical spacings between primordia of the first cycle may be different from those in subsequentcycles. as in Fig. 5(a). (3) Leaf basesbind to each other: PI to P2 and P3 to P3. If P3 binds to PI, the first three primordia develop into a whorl, and further whorls develop similarly, giving pseudo-tricussate phyllotaxis [Fig. 5(b) and (c)l. If, however. P3 binds instead to P6, its nearest neighbour tangentially on its unbound side, the first cycle of primordia becomesconnected to the second cycle [Fig. 5(d)]. In the second cycle, P6 binds to P4f and P4 to P5, but P5 is unable to bind to P6, which is already bound to P3. Instead, P5 binds to its nearest neighbour tangentially, P8 in the third cycle. Continuation of this process gives a continuous binding spiral of primordia. in which the binding sequenceand the divergence angles between subsequentprimordia of the binding spiral arc: --P5-Pl---P2---P3--P6--P4 120” 120” 60” 120” 120” 60” P&---P9 -P7--PIO-120” 120” 60”

120

PI1 --PI2 120”

t It is reasonable to postulate P6 binding to P4 even though P3 does not bind to PI. because the vertical spacings between primordia in the first cycle PI-P3 need not be the same as in the second cycle P4-P6 and subsequent cycles.

?) I PI (f _

-

- -c. I’?, -\

i

P-l

2 P

P? ps P?

P3 0

P5

PI

!d)

PI P4 P9 P6 Pi

P?

Pa -2 @

Cc)

bpI

(ei

FK. 5. Derivation of pseudo-tricussate phyllotaxis and phyllotaxis of the I ,3 series from semi-tricussate systems. (a) Semi-tricussate before binding. Genetic spiral shown as a solid line. (b, Pseudo-tricussate before elongation. Binding whorls indicated by dotted linea. (c) Pseudo-tricussate. (d) Binding spiral (dotted line) formed. (e) Binding spiral become\ apparent genetic spiral (solid line). Apparent divergence angle 100.

(4) As the apex develops into a mature stem by elongation, the binding spiral is stretched, so that the vertical sequenceof the mature organs become\ the sameas the sequenceof primordia in the binding spiral and the divergence angles along the binding spiral become averaged OLIN [Fig. 5(d) and (r)] al 100’ ( = [ 120+ 120+60]/3). This angle is close to that characteristic of the I.3 series (99.5”). Hence the stem could show I + 3. 3 +4, 4+ 7, 7 + I I 01. I 1+ 18 phyllotaxis, depending on which parastichies are conspicuous. It must be pointed out that this mechanismcannot be the only one whereby phyllotaxis of the I ,3 seriescan arise. Parastichy numbers as high as 76+ I??

7. The Transition

from Semi-decussate

to Fibonacci Phyllotaxis

At least two mechanisms can be postulated whereby a semi-decussatc system changes to a Fibonacci system: (1) The apex becomes larger, so that primordia are more fully developed before they experience contact pressure, and become fixed before completion of the shifts necessary to produce semi-decussate phyllotaxis. As the apex expands, the contact pressure-induced movements of primordia become more limited, and the semi-decussate system changes, as successive divergence angles deviate increasingly upwards from 90” and downwards from 180”, to a spiral system where the primordia remain in their original positions with a divergence angle close to the Fibonacci angle. (2) There is an increase in the rate at which primordia are initiated and/o] the rate at which they enlarge, so that the primordia experience contact pressure earlier in their development. The stage will be reached where, when a primordium P, first experiences contact pressure, its predecessor P,-, is still mobile. If this situation persistsover scverai plastochrones, the effect will be a transition from semi-decussateto semi-tricussatephyilotaxis. With furthel increase in primordium initiation and/or growth rate, progressively more and more primordia become simultaneously mobile, sothat a stageis reached where the conditions of Adler’s theorems apply. Contact pressure-induced movcment of the primordia then gives rise to Fibonacci phyllotaxis ( Adler, 1977). It is likely that both mechanismsoperate in plants, although clearly both cannot operate simultaneously. The well documented (Schoute, 1938) transition from decussate to Fibonacci phyliotaxis r~ia “dissolving whorls” (in the terminology of this paper, pseudo-decussateto Fibonacci viu semidecussate phyliotaxis) is more simply explained in terms of mechanism I, since that mechanism can also account for the pseudo-dccussateto semidecussatetransition. 8. Initial Divergence Angles Different from the Fibonacci Angle So far it has been assumedthat the initial divergence angle is close to the Fibonacci angle. The model postulated here for the development of semidecussate and semi-tricussate phyilotaxis is incompatible with an initial divergence angle greatly different from the Fibonacci angle, as illustrated by the following examples.

(.A\

51 MI-f)l’fLMA-l’l~

PHYLI.OTAXIS. WITH INI U/\I. I)IVER(;I-I\( (CHARA~TERIS.~I(' 01: 'IHF I.? xfws)

I .\s;(,I 1 ')'),<

PI. 1’2. I’3 arc initiated and move as :,h~~n ilt t:ig. h(a). So iiu, WI~~IJccussate phyllntaxis is developed. P4 is now initiated. at position P4, (3 x 99 = 297’ clockwise from PI). In order to reach position P4,., :LX required for the continuance of semi-decussatephyllotaxis, it would ha\,e to pass to the other side of PI [Fig. 6(b)]. If it is assumed that this is possible (be:11ing in mind that P4 is supposed to arrive at P4f as LLresult of contacf

pressure from PI and P2) the implication is that contact pressure from P3 i\ strong enough to force P4 past PI against contact pressure from the latter. If P4 does reach P4,, then P5 must be considered. P.5arises at P5;, and, ti)r semi-decussatephyllotaxis, should move to a position directly above P2. But since it was assumedthat P3 forces P4 past PI, it would be inconsistent to assumethat PI can force P5 past P4. It is more likely that P5 would becomc tixed at P5,, above PI, and that P6 would become fixed at P6,. [Fig. 6(c)]. The result would be a deviation from semi-decussatephyllotaxis, whereh!; P5 and P6 are 180” out of position. (B) SEMI-DECUSSATE PHYLLOTAXIS, WITH INITI/\L DIVERGEUC‘t ANGLE 151 (CHARACTERISTIC OF THE 2, 5 SERIES)

Primordia PI to P6 are initiated and move to the final positions shown in Fig. 7. So far. semi-decussatephyilotaxis is developed. P7 is now initiated at position P7; (6 x 151 = 906” clockwise from PI). For semi-decussntephylfotaxis. P7 should move, under contact pressurefrom P5 and P6, to a position above P4. But the path from P7, to P4/ is blocked by P5, and P7 must move instead to P7,-, above P3, (Fig. 7). Thus the regular semi-decussatesequence persists only as far as P6. It is interesting to speculate on the fate of P8.

594

FIG. 7. Breakdown is 151”.

of model for semi-decussate phyllotaxis when initial divergence angle

0PC PfJr P7,

/ 0

PI P4

p3

P8f

(a)

Fz F%p;

/ PB;

69 P5

P4 PBPIZ ”

‘, PII P7 P3

Cl

(bl

P2 P6 PI0

FIG. 8. Breakdown of model for semi-decussate phyllotaxis when initial divergence angle is slightly less than 180”.

A

CONTACT

PRESSURE

595

MODEL

arising in the 90’ angle between P6 and P7 (Fig. 7). It seems unlikely that it would be able to pass to the other side of P6: more probably it would become fused to either P6 or P7 or would bc separated from them by an unusually long internode. ((‘)

StMI-DLCUSSATE

PHYLLOTAXIS,

WITII LESS

TK4N

INITIAL

DIVERtiLNCL

ANGLE

SLIGII

IL \

180

Primordia PI to P4 are initiated and move to the final positions shown III Fig. 8(a), as for semi-decussate phyllotaxis. P5 and P6 are now initiated at positions P5, and P6, and move (little movement is required) to positions P5, and P6,, above PI and P2 respectively. P7 and PS are initiated and move to positions above P3 and P4 respectively [Fig. 8(a)]. Continuation of thi:, process gives a four-ranked type of phyllotaxis in which successive divergence angles (mcasurcd in the same circular direction each time) arc 180‘. 90

I’crn~i.wibIe

ranges qf’ initial divergerm angle semi-tricussate phyllotasis

Number

of leaves

8 9 10 II I’ 13

14 15

I6 17 18 19 20 21 22

90.0-l 80.01 120.0-18O.OiI 12.5-I 57.5.b 13hGl57~5 I ‘6G 150.0 I28~6--1504 1286 146.3 I30~~146~3 I3OG144.0 130.9-144.0 130.9-142.5 131.5-142.5 131.5-141.4 132~0-141~4 132G140~6 132.4-140.6 132+140~0 132.6-140.0 132.6-l 39.5 132.9-139.5

+ Rangeswhichcanbeaccountedfor by Adler’s1975 theories

alone

are marked.

.vctlli-~l~~ll.psati~ a/d

Range(in degreesjof initial divergence angleallowing phyllotaxisto continue: Semi-decussate

5 6

for

Semi-tricussatc .~ .~-. 0~0-180~0+ I70.0--160@ I 20G15001’0~0~-150~0’ 130~0-150~0 130+145.7 130~0~145~7 133.3-145.7 133.3-144.0 133.3-144-o I35Gl44.0 135~0-14.3~1 135~0-143~1 136~0-143~1 136.0 -142.5 136.0 -142.5 136.7-142.5 136.7-142.1 136.7-142.1 137.1-142.1 vercion

of the
or tield

596

Il.

W.

ROHtR

I-S

180’. 270 . 180’, 90’. 180”, 270”, etc. This type of phyllotaxis is co~nparcd with semi-decussatephyllotaxis in Fig. 8(b) and (c). It has been observed by R. Snow ( 1958)to occur regularly in buds of Kniplwfia uvariu and K. pwnib, whilst semi-decussatephyllotaxis was found in K. trrhergekna. Presumably leaf initiation is alternate in some speciesof Kniphofin and spiral in others. The ideal initial divergence angles, which would allow semi-decussateot semi-tricussate phyllotaxis to continue over an infinite number of leaves. are respectively 135’ [ = (90+ 180)/2] and 140” [ = ( I20 + I20 + I X0)/3] (derived by Schoute, for pseudo-decussateand pseudo-tricussatephyllotaxis). The range of possible initial divergence angles allowing semi-decussateor semi-tricussate phyllotaxis narrows as the number of leaves in the semidecussateor semi-tricussatesystem increases,as shown in Table 1. The ranges may be calculated as exemplified for semi-tricussate phyllotaxis, referring to Fig. 4: Let the initial divergence angle be D (assumedIO be constant). For semi-tricussate phyllotaxis (STP) to continue as far as P4, P4 must arise between PI and PZ,, i.e. 360 < 3 D < 480”, 120 < D -c 160 For STP to continue to PS,P5 must arise between P2 and P3, i.e. 480 c 4D < 600”, 120’ < D < 150”. For STP to continue to P6, P6 must arise between P3 and P4, i.e. 600” < 5D < 780“, I20 < D < 156”. But since STP must continue to P5 if it is to continue to P6, the range must be narrower: 120’ < D < 150”. For STP to continue to P7, P7 must arise between P4 and P5,. i.c. 7X0- < 6D < 900”, 130’ < D < 150”: etc. 9. Compatibility of the Proposed Model with Theories on the Origin of Fibonacci Phyllotaxis The Richards (1948. 1951) field theory and the Snow & Snow ( 1962) space-filling theory both involve the assumption that a new primordium arises in the larger gap between its two predecessors,and somewhat nearer to the older of the two. This assumption, coupled with the further assumption that the new primordium divides the larger angle between its predecessors in a ratio no more than 2: I in favour of the younger, accounts for a di\ergence angle between 120’ and 144” (Adler, 1975), but neither theory can account for the prevalence of divergence angles closer than this to the Fibonacci angle. Thesetheories alone cannot account for the initial divergence angle falling in a range which allows a semi-decussatesystem of 6 or more leaves, or a semi-tricussate system of 7 or more leaves (seeTable 1). Adler (1974, 1977) proposesa contact pressuremodel which, combined with either the field theory or the space-filling theory (or with any other theory which provides a regular distribution of primordia at the apex) accounts for the Fibonacci angle. However. although Adler’s contact pressure model iq

A CONTACT

PRESSURE

MODEL,

T-47

probably relevant in many cases of normal Fibonacci phyllotaxis, it cannot be invoked to account for the initial divergence angle in semi-decussate and related phyllotaxis. To do so would be to suggest that Adler’s conditions simultaneously apply and do not apply. It follows that if the present model is to account for semi-decussate, scmit ricussate, pseudo-decussate and pseudo-tricussate systems with large numbers of leaves, it is necessary to account for close approximation to the Fibonacci angle without invoking contact pressures. Thornley’s morphogen field model (1975), whose parameters describe the production, diffusion and degradation of a morphogen, goes some way to meeting this requirement; for those values of the parameters which give convergence of successive divergence angles, the divergence angles converge to values in a number of well-defined bands, one of which is the range 136143 (the others are 78-85”, 97-lOI”, 105-I IO”. 132-134’and 146-180.). However, Thornley’s model at present lacks a biological mechanism for ensuring that the parameters have those values which give the divergence angle in the range 136-143’. An alternative theory, which is intended to account for close approximation to the Fibonacci angle without invoking contact pressure, is at present being prepared for publication. The author is indebted to Dr Tim Poston, Batelle Advanced Studies Center, Geneva, for drawing his attention to the potential of the theoretical approach used here. and to Dr H. A. McAllister. Ness Botanic Gardens, Liverpool University. fm helpful discussions. REFERENCES ADLER, I. (1974). J.

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