THE INTERPRETATION OF PHYLLOTAXIS

BY MARY SNOW, M.A., AND R. SNOW, M.A. (Fellow of Magdalen College, Oxford).

(Received September 20, 1933.)

WE are glad that Priestley and Scott (1933), in a recent article in this periodical, have ranged themselves with those who consider that the positions in which new leaves arise are determined by the positions of those already present; for we claim to have produced direct evidence (1931, 1933) showing that the truth lies with the theories of this kind. But as to the exact manner in which the positions of the new leaves are determined by those of the older ones, several points in the article by Priestley and Scott seem to us to need comment.

Some of these points concern Hofmeister’s rule. The authors write as if Hofmeister’s rule were that the next centre of vigorous growth must be established as far as possible from the previous one (p. 243). Again they write (p. 25 I ) that, for Fibonacci phyllotaxis, a simple extension of Hofmeister’s rule would be that “each successive primordium tends to be as nearly opposite its predecessor as possible, allowing for the fact that the primordium two before it in origin. . .is also as nearly as possible opposite to this same predecessor and is still growing.” If we rightly understand this statement, it means that (in Fibonacci phyllotaxis) each new leaf tends to arise as far as possible from both the two previous leaves.

But there is no need to “extend” Hofmeister’s rule in order to make it apply to all the systems of Fibonacci phyllotaxis: for Hofmeister himself considered the higher systems as well as the lowest system, and formulated his rule so as to make it apply to them all (1868, pp. 488 seq.). Moreover the “extension” offered by Priestley and Scott is untenable, for reasons given below.

Hofmeister began his famous chapter on phyllotaxis with the statement, “ It is a fundamental experience that new leaves (or side-shoots) arise in those positions on the stem apex (or axis) which are furthest from the side edges of the bases of the most closely neighbouring leaves that are already present” (1868, p. 482). By the expression “ the most closely neighbouring leaves ” he meant the leaves that form the uppermost cycle or ring round the apex, as is clear from his subsequent dis- cussion. He went on to illustrate this rule with examples of various phyllotaxis systems ; and he made it clear that it is only in “ distichous ” or two-rowed phyllo- taxis that the position of the new leaf depends on that of the immediately previous one alone (p. 485). For, as he pointed out, in this system each leaf covers an arc of more than 18oO, so that the new leaf arises above the gap between the two edges of the previous one, on the opposite side of the apex (p. 485). He further pointed out that in higher systems of phyllotaxis, in which each young leaf covers an arc

The interpretation of phyllotaxis I33 of less than ISOO, there are at any time two or more gaps between the leaves of the uppermost cycle round the apex. In order, therefore, to include these systems also under his rule, he formulated it as follows : “ The new leaf arises above the one gap, or, if several gaps are present, above the widest gap1” (p. 488).

Now when there are several gaps between the leaves of the uppermost cycle, or (as comes to the same thiig) when there are several leaves in that cycle, it may be expected that the position of the next leaf will depend most directly on the positions of those two older leaves which lie on each side of the gap above which it will arise; and evidence will be given below to show that this expectation is correct. But these two older leaves are by no means always the two immediately previous leaves, upon which Priestley and Scott (pp. 251, 256-7) state that the position of the next leaf mainly depends. It is true that in the system with contact parastichies I + 22 the next leaf will arise above the larger of the two gaps between the two immediately previous leaves: for in this system there are only two leaves in the uppermost cycle. But in the system with parastichies 2 -k 3, which is very common, there are at any time three leaves in the uppermost cycle, and the next leaf, which may be called x, will arise not above the gap between the two previous leaves x- I

and x - 2, but above the gap between the leaves x - 2 and x- 3, with which it will make contact. Similarly in the system 3 + 5, there are five leaves in the uppermost cycle, and the next leaf “ x ” will arise above the gap between the leaves x - 3 and x-5 . In order to see that this is so, one need only look at bud sections of plants with these systems (e.g. van Iterson, 1907, Figs. ++, 47) or at theoretical diagrams (ibid., Plate 10).

There is therefore no reason to expect that the two immediately previous leaves will be of special importance in determining the position of the next leaf except in the system I + 2 (and also the “ decussate ” system 2 + 2), in which there are only two leaves in the uppermost cycle. Indeed in Lupinus albus, which has a 2 + 3 system3, our experiments (1931, pp. 17, 21 ; 1933) show that the leaf x- I is of much less import+nce (if any) in determining the position of leaf x than are the leaves x- 2, x- 3, between which the leaf x will arise, and with which it will make contact. For by operating in certain ways on the part of the apex from which the next leaf (which we called 11) was due to arise, we caused very little change, if any, in the position

“Oberhalb der einzigen, oder wenn mehrere vorhanden der breitesten Liicke tritt das neu entstehende Blatt.. .hervor.” See also p. 508.

By the statement that a phyllotavis system possesses m+n contact parastichies, it is meant simply that near the apex one can see m curved paths, along which the leaves are in contact, winding obliquely in one direction, and n curved paths crossing these and winding in the other direction. These two sets of contact parastichies are the only ones, if each leaf touches only two older leaves below it. But in some plants each leaf touches three older leaves below, and there are then three sets of contact parastichies, two sets winding one way (with different steepness) and one set the other way. The system then possesses I+m+n contact parastichies. We give reasons later for considering that the Schimper-Braun “fractional” classification cannot be applied to phyllotaxis systems near the apex.

* To speak s&ctly, the system is I + z + 3, since the edges of the stipules of successive leaves just touch along the genetic spiral. But the central part of each leaf x arises in the gap between the leaves x-2 and x-3, and makes contact with these leaves only. Consequently it can be understood how it was that in the experiments the stipular contact with leaf I- I was of much less importance in determining the position of the central part of leaf x.

‘34 MARY SNOW A N D R. SNOW of 1, (the next leaf after 11), but big changes in the positions of I3 and subsequent leaves. Also similar operations on Pl (the youngest of the leaves already visible) caused little change, if any, in the position of 11, but considerable change in the positions of I, and I,.

Here we may remark incidentally that at one point Priestley and Scott (p. 244) advance the theory that the position of a new leaf depends not only on the positions of the older leaves, but also on those of the leaves younger than it (which do not yet exist when its own position is determined). This theory would necessitate the very difficult assumption that a leaf already determined can be somehow displaced by those that are determined after it. The theory is inconsistent with the authors’ own remarks on pp. 256-7, where they state that the position of each new leaf is deter- mined mainly by the two most recent of the previous leaves, and there is absolutely no evidence that supports it.

Since the work of Hofmeister (1868) it has been generally admitted, as a fact of observation, that each new leaf arises in the largest gap between the previous ones, in most species at least. But the question remains, how is this fact to be explained? Priestley and Scott consider that each new leaf tends to arise as far as possible from the previous leaf (p. 243), or (in Fibonacci phyllotaxis) from the two previous leaves (p. 251), with which it is in some sense “competing.” It is indeed very natural to assume some such tendency (except that in 2 + 3 and higher systems, as already pointed out, it is not only the two previous leaves which must be considered), and a similar assumption has been made by Schmucker (1933). But is such an assump- tion really necessary?

With regard to this question, Priestley and Scott give no account of the very valuable and important work of van Iterson (1907), who succeeded in explaining most of the main facts of phyllotaxis without postulating any tendency for the next leaf to arise as far as possible from previous leaves. One of the simple facts of observation on which his theory is based, is that the new leaves arise in the largest (or as he prefers to say “ larger ”) gaps between the previous ones ; but, strictly speaking, he is bound to assume further that the positions of the new leaves are actually determined by the positions of these gaps. We claim now to have produced direct experimental evidence showing that they are indeed so determined (1931, 1933). Now in order to explain how it is that each successive new leaf is determined in the largest gap between the previous ones, one need only suppose that, before a leaf can be deter- mined, there must be available on the surface of the apex a space of some minimum size at some minimum distance below the extreme summit or “ growing-point.” For the largest gap between the previous leaves will, as a general rule, be the one in which, through the growth of the apex, the minimum space necessary for leaf- formation will first become available. A conception of this kindseems to us necessary for the further working out of van Iterson’s theory, and we have tried elsewhere to develop it more fully (1931, p. 16; 1933, pp. 360, 396 seq.).

There is therefore no need to assume that the leaf that is arising tends in any way to be repelled away from the older leaves, and, further, the available evidence is against this assumption. For, on this assumption, it could not be understood how

The interpretation of phyllotaxis I35 it is that in Lupinus albus, as already pointed out, the position in which leaf x will arise depends fiuch more on the positions of leaves x - 2 and x - 3 than on that of leaf x- I. We have also found that in our experiments the outlines of the wounds which we made on the apex acted in practically the same way, in determining the positions of subsequent leaves, as did the young leaves round the apex. Now if the young leaves acted by means of any tendency to repel the subsequent leaves, this would be dif€icult to understand.

At various points in their article (e.g. pp. 256, 262), Priestley and Scott briefly discuss questions which van Iterson discussed very thoroughly. These are con- cerned with the conditions on which depend the different systems of phyllotaxis found in different plants. It is indeed one of the main conclusions of van Iterson’s book (1907) that the phyllotaxis of any plant depends in the main on two things, firstly the relative sizes of the young leaves and the apex, and secondly the way in which the system started. In the mathematical first part of his book he investigated the geometry of the various possible “similar systems of touching circles” on the surfaces of cylinder, cone and plane, and showed that when these circles cover certain fractions of the circumference, only certain contact systems are possible. These results are summarised graphically in P1. 2, fig. 2. In the botanical second part he made the hypothesis that the insertions of the young leaves form a “ similar system of touching circles’’ on a cone surface in the sense previously defined. He tested this hypothesis by examining bud sections of many species, and observing whether the appearances of the sections and the relations between the contact systems and the ratios of size of leaf to size of apex are those which, if the hypothesis is correct, are to be expected on the basis of the geometrical investigation. He concluded that, for the species investigated, the hypothesis is correct, but pointed out that, in various other species, the young leaves are probably not circular in outline (p. 295).

In order to explain the causes through which a phyllotaxis system, once estab- lished, is continued, he postulated, as already stated, that the new leaves arise in the larger gaps between the previous ones. But in order to understand how the various phyllotaxis systems are first established, and how it comes about that the Fibonacci systems are so common, it is necessary to consider how these systems originate in seedlings and axillary shoots. Concerning these questions, reference may be made to van Iterson’s observations and theoretical considerations (1907, pp. 273-89, 253-73). His treatment of all these questions is to some extent based on that of Schwendener (1878) and other earlier workers.

Priestley and Scott make some remarks which imply that the ratio between size of young leaf and size of apex is important as a factor on which the phyllotaxis depends (p. 256), but in this connection they do not refer to van Iterson (1907), nor to Hofmeister (1868) or various subsequent workers, who all recognised the importance of this ratio.

Here we may remark also that the conclusion of Priestley and Scott (pp. q6-7), that the genetic spiral is only a secondary phenomenon, is implicit in the work of all those who have followed Hofmeister’s lead, for instance Schwendener (1878),

136 van Iterson (1907)~ Schoute (1913). It is also pointed out by Church (1902,

Some comment seems to be needed also on the Schimper-Braun ‘ I fractional ” classification of phyllotaxis systems which Priestley and Scott employ. For it is disappointing to find that an attempt is still made to use this classification, even after the very thorough manner in which Church (1901) criticised it and showed the difficulty of applying it. Priestley and Scott write (p. 243) : ‘ I The usual method for describing the leaf arrangement. . . is to express it in terms of the fraction of the circumference of the axis between one leaf and the one immediately succeeding. Thus the simplest types of phyllotaxis are I /Z and 113, and other more complex types are 215, 3/8, 5/13, etc.” They then proceed to discuss phyllotaxis systems at or near the apex in terms of this classification, But clearly this classification could not usefully be applied to the phyllotaxis systems that are actually found near the apex, unless it were found on investigation that near the apex the divergence angles between successive leaves in various species did really fall into separate groups, having values of approximately 1/2 of 360’, 1/3 of 360”~ I/S of 360’, etc.

Now concerning the first of these groups, of angle 180’~ there is no difficulty: for this angle is found in the familiar ‘ I distichous ” phyllotaxis. Also some plants show divergence angles of 120°, or only a little more, e.g. various Cyperaceae and mosses. But as to the other groups, 2/5 or I++’, 3/8 or 135’, etc., there is no evidence that the divergence angles found near the apex in various species tend to fall into separate groups with approximately these values, and it is unlikely that they do so. Such measurements as are available do not indicate that in the neighbourhood of the Fibonacci angle (137.5’ approx.) there is anything but a continuous distribution of the mean divergence angles of different species, with a maximum frequency not far from the Fibonacci angle itself1. Near the apex, therefore, so far as is known, phyllotaxis systems of 215, 318, 5/13, etc., as separate classes, do not exist: and for this reason discussions of such supposed phyllotaxis systems and constructions to illustrate a transition from 215 to 3/8 phyllotaxis, such as Priestley and Scott give (p. 255), have no application to anything that has been found in shoot apices.

In order to classify phyllotaxis systems, one must use classes into which these systems are actually found by observation to fall. The classification by the numbers of the parastichy curves seen near the apex, which was used by Church (1901) and was explained above, is useful and easy to apply. Van Iterson classifies by means of the numbers of steps or ‘ I plastochrons ” intervening between the times of origin of the leaves that make contact. This classification gives numerically the same result as that of Church, except when m and n have a common factor greater than I . It has the advantages that it is based on the contacts of the young leaves close to the apex only, and that one can often see with which leaves the youngest visible leaf makes contact, even when the parastichies cannot easily be counted.

It should also be noted that though the “fractional” classification cannot be

MARY S N O W -4ND R. S N O W

PP- 106-7)-

There are, however, some indications that angles a few degrees above the Fibonacci angle are probably more common than angles a few degrees below it (see Church 1904, p. 430, van Iterson 1907, p. 208 and p. 212).

The interpretation of phyllotaxis I37 applied to phyllotaxis systems near the apex, yet it often can be applied to the systems lower down the stem. For lower down the stem, the divergence angles, as is com- monly agreed, tend to approximate to the angles corresponding to the Schimper- Braun fractions, as a result of slight secondary torsions: and it is probably for this reason that the fractional classification has remained in use for so long. It is indeed clear that Priestley and Scott (p. 253) contemplate applying the fractional classifica- tion to phyllotaxis systems as seen well below the apex, though they are discussing the origin of these systems at the apex.

An explanation of these secondary torsions was advanced by Schwendener (1883) and confirmed by Teitz (1888) with various observations. It is accepted by van Iterson (1997, pp. 227 sep.) and Schoute (1913, p. 287). It is briefly that the leaf traces usually connect up so as to form strands which, near the apex, wind obliquely round the stem. These strands sometimes form along the so-called “orthostichies,” which near the apex are slightly curved. But lower down, the elongation of the stem sets up tensions in these strands, which tend to pull them straight. As the strands straighten, they force the whole stem to twist slightly, until the leaves which are situated along the strands come to be situated in vertical rows. And when this happens, the divergence angles between these old leaves reach values corresponding to some one of the Schimper-Braun fractions.

It seems to us that discussions of the factors determining the positions of leaves are not likely to help matters much except in so far as they are based on observations or experiments on shoot apices. But there is a great deal of relevant observational work that needs to be done, even apart from the method of direct experiment which is now open.

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CHURCH, A. H. (1901, 1902, 1904). On the Relation of PhyUotaJcis to MechanicalLmus, Parts I, 2

HOFMEISTER, W. (1868). Al&emeine Morphobgik der Gewc2hse. Leipzig. PRIBSTLBY, J. H. and Scorn, L. (1933). “Phyllotaxis in the dicotyledon from the standpoint of

SCHMUCKER, TH. (1933). “ Zur Entwicklungsphysiologie der schraubigen Blattstellung.” Plan&,

SCHOUTR, J. C. (1913). ‘‘Ekitriige zur Blattstellungslehre.” Rec. T Y ~ . Bot. Nderhd. 10, 153. SCHWENDENER, S. (1878). Mechunkhe T h e 6 der BlattsteUungm. Leipzig. - (1883). “Zur Theorie der Blattstellungen.” Sitaungsber. d. kgl. Preucs. A M . d. Wiss. Berlin,

SNOW, M. and SNOW, R. (1931,1933). “Experiments on phyllotaxis,” Parts I and 2. Phil. Trcms.

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