The origin of fibonacci phyllotaxis—an analysis of Adler's contact pressure model and Mitchison's expanding apex model

J. theor. Biol. (1978) 74, 217-233

The Origin of Fibonacci Phyllotaxis-an Analysis of Adler’s Contact Pressure Model and Mitchison’s Expanding Apex Model DAVID W. ROBERTS Unilever Research Laboratory, Port Sunlight, Wirral, Merseyside, L62 4XN, England (Received 25 July 1977, and in revisedform

31 March 1978)

Adler’s contact pressure model for Fibonacci phyllotaxis is examined theoretically. It is shown that the model, as it stands, does not account for Fibonacci phyllotaxis, since it requires, but does not provide, a mechanism for initiating new primordia with increasingly greater precision as phyllotaxis rises. Modifications are suggested which remedy this deficiency in the model; one of these modifications involves a combination of Adler’s model with Mitchison’s model. From a comparison of the ranges of divergence angles permitted by Adler’s model against Fujita’s measurements of divergence angles in plants with low phyllotaxis, it is shown that the modified contact pressure model, if based on the concept of mechanical pressures between primordia in contact, cannot account for the divergence angles found in low phyllotaxis systems. However it is shown that this deficiency can be overcome if the contact pressure effect is regarded as a chemical phenomenon, mediated by a growth inhibitor produced by the primordia and moving more readily in vertical directions than in other directions. Mitchison’s model, which is based on the concepts of an expanding apex and primordium initiation by existing primordia, is shown to account for Fibonacci phyllotaxis only if phyllotaxis rises sufliciently slowly; to guarantee that an F,+F,+ 1 system can develop there must already be at least F, + 1 primordia present in an Fe - 1+ F, system, at .lea.st F,, primordia in an F,- 2+Fn- 1 system, and so on down to at least three primordia in a 1+ 2 system, making a total of at least F,, s- 5 primordia (where F,=nth term of the Fibonacci series with Fl = Fz = 1). Adler’s model, modified, requires only that F,+, primordia be present with divergence angles in the range 120-180” to guarantee that an F,+F,+, system can develop. 1. Introduction Fibonacci phyllotaxis, in which the contact parastichies (lines connecting leaf primordia or analogous organs in contact with one another) occur in two oppositely curving sets, in numbers which are consecutive terms of the 217

0022-5193/78/180217+-17$02.00/O

0 1978 Academic Press Inc. (London) Ltd.

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Fibonacci series (1, 1, 2, 3, 5, 8, 13, . . .), is well known in botany. Many theories have been put forward to account for this phenomenon, but none has so far been found to be completely satisfactory (Adler, 1974, reviews earlier theories and analyses their defects). Recently Adler (1974, 1977) and Mitchison (1977) have developed different models which are intended to account for Fibonacci phyllotaxis. The object of this paper is to provide a theoretical investigation of the validity of these models, and to consider their applicability. 2. Adler’s Contact Pressure Model Adler (1974,1977) accounts for Fibonacci phyllotaxis by a contact pressure model which may be summarized as follows: (1) Primordia are assumed to be initiated along an ascending spiral, with a divergence angle between 120” and 180“. A Richards-type (1948) field theory or a Snow and Snow-type (1962) space-filling theory can be shown (Adler, 1975) to account for the divergence angle falling in this range (but see next section). (2) When the expanding primordia experience contact pressure, they are assumed to move to new positions, such that each primordium is equidistant from its nearest neighbours. Adler refers to this assumption as the “MAXIMIN PRINCIPLE". The distance between two primordia has, in Adler’s representation of the apex as a cylinder, a vertical component and an angular component. An assumption which Adler does not state as such, but which is implicit in his model, is that for an F, + F,, 1 contact parastichy system to be generated at least F,,+ 1 primordia must be simultaneously mobile. (3) The ratio of the vertical separation between consecutive primordia to the girth of the cylinder (Adler refers to this parameter as the RISE; it is analogous to the plastochrone ratio in a disc representation) is assumed to be a monotonic decreasing function of time. (4) The first primordium P(0) is assumed to experience contact pressure either before the lifth primordium P(5) emerges or when the rise, r, is large enough for P(O)% two nearest neighbours to be P(1) and P(2) or P(2) and P(3). If this condition is not met anomalous phyllotaxis results unless, when contact pressure begins, the divergence angle happens to be such that P(O)% nearest neighbours are P(F,) and P(F,,, 1), F, and F,, 1 being consecutive Fibonacci numbers. (5) The condition that the divergence angle d be in the range 120-180” ensures that, when r becomes low enough for P(2) and P(3) to be the two nearest neighbours of P(O), P(2) and P(3) are on opposite sides of P(0).

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Contact pressure then causes a’ to take a value such that P(0) is equidistant from P(2) and P(3), and, for subsequent primordia, P(n) is equidistant from P(n+ 2) and P(n+ 3). The contact phyllotaxis is now 2+ 3. Since P(2) is vertically nearer than P(3) to P(O), P(3) must be angularly nearer than P(2) to P(O), i.e. 3d-360” < 360”-2d, d < 144, so that the divergence angle is now within the range 120-144”. The permitted range of d is in fact narrower (128*57-142.11”) since r has a finite value (in the range 0.0456O-1237 for 2+ 3 phyllotaxis) which contributes to the distance between primordia. (6) Restriction of d within the range 120-144” ensures that P(5) and P(3) are on opposite sides of P(O), and that P(5) is the first primordium to arise in the angle between P(O) and P(2). Consequently, when r becomes low enough, P(5) displaces P(2) as a nearest neighbour of P(0). Contact pressure then causes d to take a value such that P(0) is equidistant from P(3) and P(5), which means that P(5) is angularly closer than P(3) to P(O), i.e. 720” - 5d < 3d- 360”, d > 135”. Thus the contact phyllotaxis is 3 + 5 and d is within the range 135-144” [as in (5), the permitted range is somewhat narrower; 135.92-142.11” in this case]. (7) Restriction of d within the range 135-144” means that P(8) will be in a position to replace P(3) as nearest neighbour of P(O), thereby changing the contact phyllotaxis to 5 + 8, when r becomes low enough. (8) In general, if the divergence angle is such that primordia P(F,-,) and P(F,) are the nearest neighbours of P(0) and are equidistant from and on opposite sides of P(O), then P(F,,+l) will be the next primordium to arise in the angle between P(F,,-I) and P(O), so that if Y falls sufficiently, the contact phyllotaxis will change from F,,- 1 + F, to Fn + F,,+ I and d will become confined to a narrower range? within the range 360” Fn- ,/F,,, 1 360” F,,;,IF,,+2’ 3. What the Contact Pressure Model Lacks A weakness of this model as it stands is that it requires, but does not provide, a mechanism whereby new primordia are initiated with progressively greater precision as phyllotaxis rises. Consider firstly the case where the rise, Y, reaches a value such that primordium P(F,+,), if present, could replace P(F,- 1) as nearest neighbour of P(O), but P(F,+ I> has not yet been initiated. (Case II of Adler, 1977, p. 41.) The condition that each primordium be initiated in the larger gap between its two immediate predecessors, and tangentially nearer to the older, is asserted by Adler (1977, p. 48, 1975, p. 444) to be sufficient, when com7 The narrower range has the limits (Adler, 1974): 4F:-t(-l)“+z 4F,2-,+(-1)“+1 and 2W~s-(--l)“) 2(2F,2+,+(--1)“*‘). 8

220

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bined with the contact pressure model, to provide a complete model of Fibonacci phyllotaxis. In the case under consideration, this condition means that the lower limit for the initial divergence angle, di, of P(F,+,), is given by (360” -d)/2, where d is the divergence angle for the existing primordia P(1) to P(F,, I- 1). The upper limit for di is taken as 180” rather than 144” as given by Adler (1975), for the following reason. The upper limit of 144” results from Adler’s assumption that a, < 2/3 in the equation d,, = a,, (360”-d,-,). He asserts that this is the weakest assumption needed to ensure that the genetic spiral does not change direction (i.e. no divergence angle is greater than ISO’), apparently reasoning that if d,,-2 has the largest possible value (180°) and a,-, has its smallest possible value (l/2), then d,-, has the lowest possible value (go”), and for d,, to be less than 180” requires the assumption that a, be less than 2/3. He then shows that, if all a,,‘s have the equal value a, successive values of d,, converge to a limiting value d = 360 a/(a+ l), from which, by substitution of a = 2/3, he obtains the upper limit of 144” for d. But the condition a < 2/3 remains only a mathematical assumption, and Adler suggests no biological reason why a should not exceed 2/3. The justification for taking 180” as the upper limit for d,, is that the parameter a, may be regarded as quantifying the resultant of two opposing biological effects: the tendency of P(n-- 1) to influence (by chemical inhibition or by spacefilling) initiation of P(n) so that it arises as far as possible, i.e. opposite, from P(n - 1), and the weaker (by virtue of greater distance from the initiation zone and/or attenuation of effect with time) tendency of P(n - 2) to influence the initiation of P(n) so that it arises opposite P(n-2). Hence the maximum value for a,, corresponds to P(n - 2) having no effect, such that the effect of P(n- 1) is unopposed and d, = 180”. Tlhus the total permissible range for di is (360”-d)/2 to 180”. But for phy lotaxis to be able to change from F,- 1+ F, to F,+ F,, 1, P(Fn+ 1) must arise in the smaller gap between P(0) and P(F,- 1), so that there is what may be termed an “enabling range” for di. This enabling range for di may be calculated, assuming that d has a value close to that at which the transition to higher phyllotaxis is made. These values can be obtained, for different values of F,,, from Adler, 1974 (Table 1, p. 57). In Table 1 here, the total permissible range for d is compared against the enabling range which allows a normal change in phyllotaxis. In Fig. 1 the two ranges are shown diagrammatically for F,, 1 = 8. It is clear from Table 1 that the enabling range for di constitutes a progressively smaller proportion of the permissible range as phyllotaxis rises. Exceptionally in the case of the 2 + 3 to 3 + 5 transition, the enabling range extends outside the permissible range. It may be noted that the total per-

TABLE

1

108~95-180” 112W-180” 110*93-180’ 111.36-180” 111*2&l 80” 111~26-180” 111.24-l 80” 111.25-l 80”

Total range of permissible values for d, [=(360-d)/2 to 180°] 108.95+151.58” 128.57-176.33” 113.03-142.33” 134.57-152.74” 128.02-139.23” 136.45-143.38” 133.92-138.24’ 137.21-139.82’

60.0% 70.3 % 42.4% 26.5 % 16.3 % 10.1% 6.3 % 3.8%

Range of permissible values of di enabling normal change in phyllotaxis as % of total range in degrees

would also allow normal change in phyllotaxis, but are not permissible.

142.11” 135.92” 138.14” 137.27” 137*60” 137.47” 137.52” 137.50”

Divergence angle d for existing primordia (taken from Adler, 1974)

t Values in the range 7579-108.95

2+3 + 3+5 3+5 -+ 5+8 5+8 --f 8+13 8+13-+13+21 13+21+21+34 21 i-34 --+ 34+55 34+55-+55+89 55+89+89+144

Change of phyllotaxis, Fn-1-4-Fn +F2-Fncz

Total permissibleranges and enabling rangesfor the initial divergence angle di of primordium P(F,+ 1) when contact phyllotaxis changes

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D. W. ROBERTS

FIG. 1. Transition from 3 +5 to 5-j-8 contact phyllotaxis. Total permissible range (I for initiation of (p)8 and enabling range (II) within which (p)8 must be initiated for normal rise in phyllotaxis.

missible range always extends below 120”, although Adler (1975) has shown that his assumption (I, section l), for primordium initiation gives convergence of the divergence angle to a value above 120”. This discrepancy arises because Adler’s, 1975 analysis does not apply to individual divergence angles when the determining primordia do not remain in their original positions. For the second case, where P(F,,,) is already present when r reaches a value such that P(F,+ I) can replace P(Fn-,) as nearest neighbour of P(0) (Case I of Adler, 1977, p. 41), the above analysis still holds. For phyllotaxis to rise normally, or to continue normally without rising, P(F,+,) must have an initial divergence angle in the appropriate enabling range. The limits for this enabling range and the lower limit for the total permissible range depend on the value of the divergence angle for the existing primordia when P(F,+ 1) is initiated, and are intermediate between the values shown in Table 1 for the transitions F,,-2SF,,-1 to Fndli-F,, and F,-,-i-F, to F,+F,,,. For example, if P(8) is initiated when d and r are in the ranges appropriate to 3 f 5 phyllotaxis (O-0177 2 r I 0.0456; 135.92 I d I 142*11”, from Adler, 1974, p. 57), the lower limit for d*i is between 108.95” (enabling range 108.95 to 151*58”) and 112GY’ (enabling range 128*57-176*33”), depending on the value of d when P(8) is initiated. Thus the contact pressure model as it stands does not provide, in combination with the Richards/Snow & Snow initiation rule, a complete model of Fibonacci contact phyllotaxis. 4. Modification

of the Contact Pressure Model

The weakness demonstrated in section 3 would not arise if each new primordium P(x) first experienced contact pressure from the older primordia P(x - 2) and P(x - 3) [or, before that, from P(x - 1) and P(x - 2)] and then passed through all the intermediate contact systems before becoming part of

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contact phyllotaxis system. Two variants of Adler’s model an 4+Fn+l would allow this: (a) All the primordia are initiated before contact pressure begins. Then contact pressure begins, when r has a value such that the nearest neighbours of P(x) are either P(x+ 2), P(x+ 3), P(x- 2), P(x - 3) or P(x+- I), P(x + 2), P(x- I), P(x-2). As r decreases (this could be envisaged in terms of an increase in the girth of the apex, as in the development of a composite flower bud), phyllotaxis is compelled to rise normally. (b) Near the initiation zone of the apex, r has a high value, so that a new primordium P(X) first experiences contact pressure from P(x- 2) and P(x- 3) [or, before that, from P(x- 2) and P(x- l)]. As the initiation zone grows away from P(x), the value of r in the region of P(x) falls, so that P&--5) displaces P(x-2) as a nearest neighbour of P(x). The further the initiation zone grows away from P(x), the lower the value of r in the vicinity of P(x). Thus P(X) progresses through higher and higher contact phyllotaxis systems, up to the point in distance or time at which either internodal extension begins or the primordia lose their mobility. This variant of Adler’s model may be obtained by replacing Adler’s assumption: r is a monotonic decreasing function of time; by the assumption: ri, the normalized vertical separation between two consecutive primordia P(i) and P(i- l), is a monotonic decreasing function of the number of primordia between P(i) and the initiation zone. This variant of the contact pressure model is consistent with the observation that in composite capitula the contact phyllotaxis is often lower near the centre than it is near the circumference (Church, 1904, gives a detailed description of this phenomenon in Helianthus annus). For variant (a), the angular movement required for a given primordium could be substantial. If the average divergence angle is initially L&, then for contact pressure induced transformation to a system with divergence angle d,, the nth primordium moves through an angle n(df -dJ. Thus for the development of a sunflower capitulum (d, 137.5”) containing about 250 florets from a system with an initial divergence angle of 135”, variant (a) would require the youngest tloret primordia to make about 1.7 turns about the axis of the apex. For the development of a vegetative bud, with df 137.5 and about 40 leaf primordia, in the most extreme case, with di almost 180”, the youngest primordia would make about five turns around the axis of the apex. For variant (b), the change in angular position for a given primordium is more dependent on circumstances. If the initial position of each new primordium P(x) is assumed to be determined by its two immediate pre-

224

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decessors P(x - 1) and P(x- 2) (cf. Adler, 1975), the total movement required of P(X) depends on the extent to which P(x- 1) and P(x- 2) have already moved towards their final angular positions when P(X) is initiated. If P(x-- 1) and P(x- 2) are still in their original angular positions when P(x) is initiated, the total angular movement required for each primordium is the same as in variant (a), i.e. n(df-d,) for the nth primodium. However, if P(x- 1) and P(x- 2) have already become part of a 2 + 3 contact phyllotaxis system, the divergence angle diC,-lj between them is likely to be different from di, the initial divergence angle between P(x) and P(x- 1). The total angular movement that will be required of P(X) is given by [die,_ 1j- di] + n[d,- die,- 1J. The angle dj,,- 1j will be restricted to the range 128-57-142.11, i.e. no more than 9” below the Fibonacci angle and no more than 5” above it. The angle di may have any value in the range 109-180”. Thus for the development of a vegetative bud, with df 137.5 and about 40 leaf primordia, in the most extreme case the youngest primordia would make about one turn about the apical axis. 5. Mitchison’s Model

At this point it is appropriate to consider an alternative explanation, offered by Mitchison (1977), for the phenomenon of Fibonacci phyllotaxis. Mitchison’s model is based on an inhibitor field initiation mechanism (cf. Richards, 1948 ; Thornley, 1975) whereby : (1) Existing primordia and the apical tip produce an inhibitor which prevents the initiation of new primordia. (2) As the apical tip grows away from the existing primordia, a point arises where the inhibitor concentration is low enough to allow initiation of a new primordium. (3) The position of the new primordium is determined mainly by the two primordia which make the largest contribution to the inhibitor field at the point where the inhibitor concentration reaches a value low enough to allow initiation. Since the older of these existing primordia is vertically further away from the initiation zone, the new primordium arises in a position angularly nearer to the older than to the younger of the two determining primordia. Mitchison claims that Fibonacci phyllotaxis follows as a mathematical necessity from the combination of an expanding apex (this concept is mathematically equivalent to Adler’s assumption of decreasing r) and the above initiation mechanism, or a mathematically equivalent mechanism for positioning new primordia (Mitchison suggests as possible alternatives depletion of a compound or competition for it by developing primordia; a space-filling

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theory is a further alternative), without the need to invoke contact pressure effects. He bases this claim on his demonstration that, if P(x-F,1) and P(x-F,) are the determining primordia for initiation of a primordium P(X), apical expansion causes P(x - F,, + 1) to displace P(x- F,, _ I) as a determining primordium. However, although this demonstration is based on the assumption of a regular lattice, with equal divergence angles between each successive pair of leaves, no mechanism is provided for readjusting the gaps between primordia after the primordia are formed (cf. the criticism in Adler, 1974 of the initiation theory given in Richards, 1948). Without such a mechanism, Mitchison’s model requires an ever increasing number of leaves between rises in phyllotaxis. Thus, using the expression “m+n initiation phyllotaxis” to refer to the situation where P(x-m) and P(x- n) are the determining primordia for initiation of P(X) : With 1 + 2 initiation phyllotaxis, successive divergence angles converge to a value within the range 120-180”. Three successive divergences in this range are required to guarantee that P(x- 3) is angularly between P(x - 1) and P(x), so that initiation phyllotaxis can change to 2+3 on expansion of the apex. With 2+3 initiation phyllotaxis, successive divergence angles converge to a value within the range 120-144”. Five successive divergence angles in this range are required to guarantee that P(x- 5) is angularly between P(x-2) and P(X), so that initiation phyllotaxis can change to 3+5 on further expansion of the apex. With 3$5 phyllotaxis, successive divergence angles converge to a value within the range 135-144”. Eight successive divergence angles in this range are required to guarantee that P(x--8) is angularly between P(x-3) and P(x) so that initiation phyllotaxis can change to 5 + 8 on further expansion of the apex. The ranges within which the divergence angles can converge for the various initiation phyllotaxis systems are somewhat narrower than those given above; they are the same as those given in section 2 for contact phyllotaxis according to Adler’s model. However, the wider ranges are sufficient to enable the changes in phyllotaxis to occur. In general, F,,, 1 successive divergence angles in the range appropriate to F,- 1 + F, initiation phyllotaxis are required to guarantee that phyllotaxis can change from F,,- 1+ 17, to F, + Fn + 1 when the apex expands sufficiently. This signifies that Mitchison’s model, if it is to account for Fibonacci phyllotaxis, requires a restriction on the way r decreases. The nature of this restriction may be derived as follows: For the most rapid rise in Fibonacci phyllotaxis which can be guaranteed by Mitchison’s model, 3 primordia are initiated under conditions of 1+2

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initiation phyllotaxis, 5 under conditions of 2 + 3 phyllotaxis . . . F, + 1 under conditions of F,- 1 + F,, phyllotaxis. The total number of primordia N which must be present when r reaches the value appropriate for the transition from F,-, + F, to F,,+ FH+, phyllotaxis is therefore given by: N = 3+5+

. . . +l$+Fn+l

= Fn+3-5.

For higher values of IZ, 4s3Fn [where qf~is the golden ratio, (1 +J5)/2] is a good approximation for Fn+3 (f or n = 5, $3Fw = 54” gives 21.18, which is less than 1% out from the correct value of 21 for F& so that, approximately F, = d-3(N+5).

From Adler (1974, p. 23) the value of r when F,,- 1 + F,, phyllotaxis to F,, + F, +.1 phyllotaxis is given by : /^ ?“=

d/”

changes

--

2[2F,2+(-I)“]

’

which may with little error (for IZ = 5 the error is 2 %) be approximated

Substitution

of 4-3(N+

5) for F, gives the approximate

to :

expression:

r = 7.77(N+5)-‘,

which gives the minimum permissible value for r when N primordia (excluding any early primordia initiated before successive divergence angles have converged to within the range 120-180”) are present. Thus the conditions under which Mitchison’s model gives Fibonacci phyllotaxis are narrower than those applying to Adler’s model modified as in section 4 above. Also, Mitchison’s model would not be able to explain a rapid rise in phyllotaxis. For example, the divergence angle should not regularly fall within about 1.5” of the Fibonacci angle (this implies at least 5,8 initiation phyllotaxis) until at least the 16th primordium. These restrictions on the applicability of Mitchison’s model result from the lack of a mechanism for altering the positions of the primordia after they are formed. However, Mitchison’s model can be combined with Adler’s to provide a third variant, which overcomes the deficiencies of Adler’s original contact pressure model. Thus if contact phyllotaxis is F,,-l + F”, and all existing primordia have, as a result of contact pressure induced movement, taken up positions appropriate for Fn-l +F, contact phyllotaxis, then initiation according to Mitchison’s model ensures that each new primordium P(x) is intiated in the angle between P(x- F,-,) and P(x-F,), i.e. with a divergence angle already appropriate to F,-, + F, phyllotaxis. Thus the Adler-Mitchison model, like the other two variants of Adler’s model,

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requires the presence of only F,,, 1 p rimordia for the development of F,, + F,, + I phyllotaxis. The change in angular position for a given primordium may be estimated for the Adler-Mitchison variant. The total angular movement required by the uppermost primordium in the transition from F,-, +F,, to F,,+ F,, 1 contact phyllotaxis is given by n(d, - dJ where 12 2 Fn+ 1 and df and di have values appropriate to Fn + F, + 1 and F,.- r + F, contact phyllotaxis respectively. Thus for development of a vegetative bud, with df in the range 137.3-1376” (i.e. 13+21 phyllotaxis), and about 40 leaf primordia, if 13+21 phyllotaxis is not attained until after the 40th primordium has been initiated, the 40th primordium would move through, at most (i.e. assuming40 primordia move) : 35” if it was initiated when phyllotaxis was changing from 5+ 8 to 8+ 13, 83” if it was initiated when phyllotaxis was changing from 3 + 5 to 5 + 8, 194” if it was initiated when phyllotaxis was changing from 2+ 3 to 3 + 5. These maximum figures may be compared with those calculated for development of a similar hypothetical bud according to variants (a) and (b) of section 4: Variant (a) about 5 turns around the apical axis. Variant (b) about 1 turn around the apical axis.

6. The Nature of Contact Pressure-Induced Movement As shown in sections 4 and 5, the angular movement required of primordia by Adler’s model can be substantial. This contact pressure-induced movement could be envisaged in one of two versions, assuming that the primordia are not free to move about the surface of the apex, but are firmly attached to it: (1) When a primordium moves under contact pressure, that part of the apex to which it is attached moves also. This implies twisting of the apex, so that the tip turns through a considerable angle, as calculated for hypothetical examples in sections 4 and 5, during a rise in phyllotaxis. (2) The primordia do not move under contact pressure, but grow preferentially in directions away from other primordia with which they are in contact. Version 1 is compatible with any of the three proposed variants of Adler’s model. Version 2 is only plausible if the change of position required is small enough not to require actual movement of primordia; it is therefore most compatible with the Adler-Mitchison variant, since that requires the smallest changes of position.

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7. What the Modified Contact Pressure Model Can and Cannot Explain With any of the three modifications suggested in sections 4 and 5, Adler’s contact pressure model provides an explanation which is both conceptually simple and biologically feasible for the generation of phyllotaxis systems where the two sets of contact parastichies occur in numbers which are consecutive terms of the Fibonacci series. The contact pressure model also provides a simple explanation for the common observation that the more numerous set of contact parastichies is the less curved; this follows inevitably from Adler’s MAXIMIN PRINCIPLE [(2), section 21 and the fact that the rise r is finite and positive. However, a complete theory of phyllotaxis must explain not only the prevalence of Fibonacci numbers in systems of high contact phyllotaxis, but also the facts of low contact phyllotaxis. Adler’s model, suitably modified, can account for the former; whether or not it can account for the latter may be judged from a comparison against published observations for systems of low contact phyllotaxis. Fujita (1939) measured divergence angles, to an accuracy of about 1”, at the apices of 30 species, including several with l-i-2 or 2+ 3 contact phyllotaxis. In every case, except for apices with non-Fibonacci contact phyllotaxis, the divergence angles most frequently observed were 137” and 138”. In Table 2 Fujita’s findings for 1 + 2 and 2+ 3 contact systems are compared against the ranges permitted by Adler’s model. It is clear from Table 2 that Adler’s model does not account for Fujita’s findings, which are supported by those of Church (1920), Snow & Snow (193 1) and others (see Snow, 1955 for further references), that angles close to the Fibonacci angle (ca. 137-5”) predominate even in systems of low contact phyllotaxis. However, Mitchison’s model, within its limitations as discussed in section 5, can explain close approximation to the Fibonacci angle for systems of low contact phyllotaxis, if it is assumed that the inhibitor concentration contour for each primordium is axially elongated (Mitchison, 1977, pers. comm.). This would mean that the position of a new primordium could be determined by primordia older than its topographical nearest neighbours. The precision associated with high initiation phyllotaxis could thereby occur in a system of low contact phyllotaxis (it being assumed that nearest neighbour phyllotaxis and contact phyllotaxis would be identical). 8. A Chemical Contact Pressure Model So far the contact pressure effect has been treated as involving mechanical forces between primordia which are in physical contact with each other.

table

t Fujita’s

Maximum for species one 322

species: 82

fdrdb~m

71t

73

121

measured

72 as the number

var.

var.

gives

13 species

piha

Agrimonia Eupatoria

campantda

punctata typica

Lysimachia clethroides

Species

2+3

1+2

contact Phyllotaxis 2

129

5

137” and

8

138”

6

11

11

5

6

18

9

but a summation

14

4

5

10

7

7

7

3

5

5

of observations

in all cases.

8

16

of the number

predominating

3

6

5

3

4

1

1

3

144

for each

5

1

4

132 Number 133 134 of135divergence 136 137 angles 138 139 having140 the 141 value: 142 143

mean 135.8 (median 136.0) mean 139.3 (median 137.4) medipn 136.0 (mean 135.8) medran 137.9 (mean 138.6)

5

131

to above,

3

130

of amgles measured,

Lowest Highest Lowest Highest

Similar

00311301

2224633

1

128

3

0

3

146

individual

1

2

3

145

2

137.7

136.6

136.9

ofd

Mv;eten

gives 71.

138.7

136.6

137.4

k&e;: ofd

angle

0

3

147

1286-142.1

1286-180

Range permitted for d b;ofA~er’

Low contact phyllotaxis-comparison of Fujita’s measurementswith rangespermitted for the divergence angled by Adler’s contact pressure model

TABLE 2

230

D. W. ROBERTS

However, in another context it has been suggested (Roberts, 1977) that a contact pressure effect could alternatively be considered as a chemical effect whereby a growth inhibitor released from primordium P(A) retards growth of P(B), the effect being greatest on parts of P(B) closest to P(A). An effect of this nature would seem to be a necessary assumption if any variant of Adler’s model is to account for Fibonacci phyllotaxis in ferns, where the primordia are not in contact with one another, e.g. Dryopteris dilatata, stated in Mitchison, 1977 to show 5+ 8 nearest neighbour phyllotaxis. For this type of contact pressure, which will be referred to as chemical contact pressure, it is not necessary to assume that the effect would be equal in all directions. It could be postulated that vertical translocation of the inhibitor is more facile than angular translocation.? This postulate might enable Adler’s model, if based on the concept of chemical contact pressure, to account for low phyllotaxis in the same way that Mitchison’s model can. For this purpose a further postulate is necessary; that the inhibitor affects growth in the tangential direction more than it affects growth in the vertical direction. Otherwise, the primordia would take the shape of their inhibitor concentration contours, and chemical contact phyllotaxis would be identical to true contact phyllotaxis. There are three possibilities to consider:

(b)

(a)

Cc) 2. Chemical Contact Pressure. Dotted lines indicate the positions of the primordia. Full lines indicate concentration contours. Nearest neighbour phyllotaxis 2,3. (a) Growth inhibitor moves preferentially downwards. P(5) exerts chemical contact pressure on P(O), but P(0) does not influence P(5). (b) Growth inhibitor moves preferentially upwards. P(0) exerts chemical contact pressure on P(5), but P(5) does not influence P(O). (c) Growth inhibitor moves preferentially in either vertical direction. P(5) and P(0) exert chemical contact pressure on each other. Chemical contact phyllotaxis is 3,5. t I am indebted to the referee, Dr Tim Poston of the Battelle Advanced Studies Center in Geneva, for drawing my attention to this point. FIG.

ORIGIN

OF

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PHYLLOTAXIS

231

(a) The growth inhibitor moves downwards from its source more readily than it moves in other directions. The concentration contours produced by the primordia have the shape indicated in Fig. 2(a). With such concentration contours, the chemical contact pressure effect is not a reciprocal one; younger primordia can exert chemical contact pressure on primordia which are older than their topographical nearest neighbours, but older primordia cannot influence younger primordia which are younger than their topographical nearest neighbours. It follows that, for a given primordium P(x) to become part of an F,+F,,+, chemical contact phyllotaxis system, there must be at least Fn+* primordia above it, with divergence angles in the range appropriate to F,-, + F, phyllotaxis. To guarantee this, there must in turn be at least Fn primordia above P(x+ Fn+ &, with divergence angles in the range appropriate to F,-, + F,-, phyllotaxis. Hence, by an argument similar to that used in section 5 for Mitchison’s model, to guarantee that a given primordium P(x) will become part of a chemical contact system which restricts the divergence angle to the range appropriate for F,,+ F,,, 1 phyllotaxis, there must be at least F, + 3 - 5 primordia younger than P(x) already present. This version of chemical contact pressure therefore gives rise to a model which is no more general than Mitchison’s model alone. (b) The growth inhibitor moves upwards from its source more readily than it moves in other directions. The concentration contours have the shape indicated in Fig. 2(b). As for (a) above, the chemical contact pressure effect is not a reciprocal one; in this case older primordia can influence younger primordia more distant than their topographical nearest neighbours, but not vice versa. By similar arguments to that used in section 5, there must be at least F, + 3 - 5 primordia present before F, + F, + 1 chemical contact phyllotaxis can be guaranteed to rise. Like version (a) above, this version of chemical contact pressure is no more general than Mitchison’s model alone. (c) The growth inhibitor moves equally readily in both vertical directions and less readily in other directions. In this case the concentration contours have the shape indicated in Fig. 2(c), and the chemical contact pressure effect is a reciprocal one; if P(A) exerts chemical contact pressure on P(B), then P(B) exerts chemical contact pressure on P(A). Hence for this version of chemical contact pressure the same arguments apply as for the mechanical contact pressure model discussed in sections 2 to 6. The minimum number of primordia which must be present before the divergence angle can be guaranteed to be in the range appropriate for F,, + F,,, 1 phyllotaxis is I;,+ 1. Thus a chemical contact pressure model, coupled with any of the three modifications to Adler’s model suggested in sections 4 and 5, can account for close approximation to the Fibonacci angle for systems of low contact phyllotaxis, provided it is assumed that the growth inhibitor moves equally

232

D.

W.

ROBERTS

readily in both vertical directions and less readily in other directions. For the Mitchison-Adler variant of the chemical contact model, it is only necessary to postulate one inhibitor, since the same substance could function as both an initiation inhibitor and a growth inhibitor. The chemical contact pressure concept also makes possible a complete model for semi-decussate and related phyllotaxis in terms of contact pressures. It has previously been shown (Roberts, 1977) that contact pressure effects of a more limited kind than in Adler’s model, acting on a system where the divergence angles are already close to the Fibonacci angle, can account for semi-decussate and related phyllotaxis. A complete model is obtained if it is postulated that chemical contact pressure effects give rise to an initial Fibonacci system, which is then converted by mechanical contact pressure effects into a semi-decussate or related system. 9. Conclusions Adler’s contact pressure model, suitably modified and coupled with the concept of chemical contact pressure, can account for the Fibonacci angle in both high and low contact phyllotaxis systems. Any of the variants could require a substantial rotation of the tip of the apex (sections 4 and 5) and experimental observation of such rotation during formation of a bud (not during growth of a stem, where secondary effects might occur in the course of intermodal extension) would provide a useful test of the model. Even when modified, Adler’s model is not completely general. It could not account for the regular occurrence of divergence angles in the range appropriate to F,, F,,, phyllotaxis before F,,,, p rimordia were already present. A statistically based study of phyllotaxis in seedlings, to establish how rapidly the Fibonacci angle is approximated, would be useful in this context: such a study could determine whether Mitchison’s model alone, a contact pressure model, or neither model can account for the development of Fibonacci phyllotaxis. This paper has been concerned entirely with phyllotaxis theories whose fundamental premise is that the divergence angle is determined by the existing primordia. Other theories have been put forward according to which the positions of primordia are determined by other factors, such as the intersections of the two oppositely curving “lines of equipotential” (Church, 1904) or the action of an “apical organiser” on “foliar helices” (Plantefol, 1948), and Larson (1977) has suggested, on the basis of anatomical observations and [14C] translocation experiments in Popuhs, that the developing vascular system plays a role in the development of the phyllotaxis system. A weakness of such theories has so far been that they do not explain the

ORIGIN

OF

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PHYLLOTAXIS

233

prevalence of Fibonacci phyllotaxis over other types of phyllotaxis. In a subsequent publication a new theory of this type will be developed and evaluated along with Adler’s and Mitchison’s models against published experimental and observational data. REFERENCES ADLER, I. ADLER, I. ADLER, I. CHURCH,

(1974). J. theor. Biol. 45, 1. (1975). J. theor. Biol. 53, 435. (1977). J. thew. Biol. 65, 29. A. H. (1904). On the Relation of Phyllotaxis to Mechanical Laws. London: Williams & Norgate. CHURCH, A. H. (1920). On the Interpretation of Phenomena of Phyllotaxis. Oxford: Oxford University Press. FUJITA, T. (1939). Bot. Mag. Tokyo 53, 194. LARSON, P. R. (1977). Planta 134,241. MITCHISON, G. J. (1977). Science 196, 270. PL,ANTEFOL, L. (1948). La Theorie des Helices Foliaires Multiples. Paris: Masson et Cie. RKHARDS, F. J. (1948). Symp. Sot. exp. Biol. 2,217. ROBERTS, D. W. (1977). J. theor. Biol. 68, 583. SNOW, M. & SNOW, R. (1962). Phil. Trans. R. Sot. B244,483. SNOW, M. & SNOW, R. (1931). Phil. Trans. R. Sot. 221, 1. SNOW, R. (1955). Endeavour 14,190. TWORNLEY,

J. H. M.

(1975).

Ann. Bat. 39,491.