11 Phyllotaxis: a Model
F. W. Cummings*University of California Riverside (emeritus)
Abstract
A model is proposed to account for the positioning of leaf outgrowths froma plant stem. The specified interaction of two signaling pathways providestripartite patterning. The known phyllotactic patterns are given byintersections of two ‘lines’ which are at the borders of two determinedregions. The Fibonacci spirals, the decussate, distichous and whorl patternsare reproduced by the same simple model of three parameters. *present address: 136 Calumet Ave., San Anselmo, Ca. 94960; [email protected]
1. Introduction
Phyllotaxis is the regular arrangement of leaves or flowers around a plant stem, or on astructure such as a pine cone or sunflower head. There have been many models ofphyllotaxis advanced, too numerous to review here, but a recent review does anadmirable job (Kuhlemeier, 2009). The lateral organs are positioned in distinct patternsaround the cylindrical stem, and this alone is often referred to phyllotaxis. Also, often thepatterns described by the intersections of two spirals describing the positions of floretssuch as (e.g.) on a sunflower head, are included as phyllotactic patterns. The focus atpresent will be on the patterns of lateral outgrowths on a stem. The patterns ofintersecting spirals on more flattened geometries will be seen as transformations of thepatterns on a stem. The most common patterns on a stem are the spiral, distichous,decussate and whorled. The spirals must include especially the common and well knownFibonacci patterns. Transition between patterns, for instance from decussate to spiral, iscommon as the plant grows. . The Pattern Model
The pattern model will be assumed to be based on the simplest version of the interactionof two signaling pathways. An activated signaling pathway is called an “Asp”, and thesmaller region separating two activated signaling pathways will be called the “Margin”region. In each growth cycle, the pattern will be then tripartite, consisting of two to-bedetermined Asp regions and a Margin region.A tripartite pattern spontaneously emerges at each growth cycle due to theassumed action of the cell upon being activated by its specific ligand. This is simplydescribed as two subsequent actions of each Asp:1) Upon activation by its specific ligand, the Asp will lead to emission offurther ligands of the same type (e.g., and R , while the corresponding ligand densities in the same small regionwill be denoted as L and L . Then we can write , RRLDtL
RRLDtL α and β are similar. The equations with α and β set to zero are the familiardiffusion equations. The “Laplace” operator, roughly speaking, denotes the differencebetween the in-flux of ligand from an average of its neighbors into a given small region,minus the out-flux to the same neighbors. The diffusion rates D , D are generally aboutthe same magnitude, perhaps even equal. Thus this is not a “Turing” reaction-diffusionmodel. A reasonable (and simplest) assumption is that R and R are each proportional totheir respective ligand densities in any small region. The receptors and their ligands havehigh affinity for each other, so that the number of active receptors is reasonablyproportional to the ligand in the same small space region at a given time. Then γ R = L , and γ R = L . 2.3)Clearly more complicated equation can easily be written (Cummings, 2006, 2009), butour purpose here is to emphasize concepts. Equations (2.1) and (2.2) can then be writtensolely in terms of the activated receptor densities R and R . Nonlinearities mustultimately enter. In this simple model, they are assumed to to provide an upper cutoff forthe R’s (and L’s). A particular form will not be of interest here; the shape of the Aspsmay be affected but the positions of the Margin regions separating the two different Aspregions will be largely unaffected by the constraining nonlinearity. Our main interest hereis to illustrate the concept of how patterns of gene activity may form due to two simpleactions of cells. Such actions are as yet to be found experimentally, however.The question of stability can be examined in the usual way, by introducing anexponential space and time behavior (Murray, 1990). Then we take both activatedreceptor densities as R proportional to exp(st+κ· x), and inquire under what conditionsor parameter values the eqs. (2.1), (2.2) and (2.3) give rise to exponentially growingsolutions in time, i.e., where “s” is positive. This leads, after solution of a quadraticequation, to the condition κ < k + k ≡ k . (2.4)Spontaneous activation of the pattern from zero density occurs when the space isavailable, and a pattern does not form when the tissue is below a minimal size. Theassumption of a lower density threshold for each pattern determines the width of theMargin region, providing a line-like region, and defining a line (the bisector) dividing theMargin region into two parts.Using the definitions k = α/(γ D ), k = β / (γ D ) and f = β / α allows he steadystate version of eqs. (2.1), (2.2) and (2.3) to be written in the form,0)( fRRkR (2.5)and .0))(/( fRRfkR (2.6)The simplest example of a tripartite pattern is the axially symmetric pattern on asphere. In this case the pattern may emerge only after a critical radius R is reached bygrowth, and is given by k R =2. Observation of R then allows determination of k . Thesphere solutions can be easily verified to be, in the case that k = k ,R = C·Cos ( θ /2), and f·R = C·Sin ( θ /2). (2.7)Here C is ~ a constant maximum amplitude, which can be normalized away, and C=1. The angle θ is the usual polar angle, zero at the ‘north pole’ and π at the south pole.Figure 1 shows the two separate Asp regions as red and blue, each with a maximumamplitude at opposite poles, a minimum at the Margin region, and with the Margin region shown as a white band; a Margin ‘line’ will circle the equator, with θ = π /2. Figure 1
The tripartite spherically symmetric solution
The two regions, red and blue, each correspond to a particular genetic networkactivation. The Median region separating the two distinct regions correspond to a “toti-potent” region, which in the case of animals may be termed a “stem cell” region. If wemay view the spherical epidermal cells of Figure 1 as a many-celled plant embryo(Lyndon, 1990), Figure 1 may model the earliest but multi-cellular separation of the plantcells into meristem (red) and root (blue) regions. Polarity is then provided, and arisesspontaneously. Given the fact that the model always subdivides each original area againinto binary parts upon further growth, one expects the spherical symmetry to be brokenwith the appearance of a Cos( φ ) dependence (and multiplying a complex theta functionaldependence, dependent on the geometrical shape, where the angle ‘ φ ’ circles the polaraxis. Thus, intersections of Margin regions must occur, and these intersections willprovide the positions (‘points’) of the first leaf primordia.The next section examines the patterns of intersecting Margin regions that areallowed on a cylinder.
3. Phyllotaxis on a stem
The patterns occurring on the stems of plants (Mitchison, 1972; Douady and Couder,1991; Green, 1996; Cummings and Strickland, 1998; Kuhlemeier, 2007) is presented inlight of the present model. The suggestion is that, even though the biochemicals involvedin the two kingdoms, animal and plant, are expected to be quite different, the simpletripartite patterning model (two Asps separated by Margin cells) of the previous section isassumed to have applicability to plants as well as animals ( Cummings (2009)). It isshown that while the present patterning model gives the known arrangements of plantpatterns (“phyllotaxis”), it nevertheless explicitly forbids the number ‘four’ in a spiralarrangement with one ‘leaf’ on each level. This means that, counting up or down the stemfrom some initial leaf, (taken as at the origin) one leaf per level, the total number ofleaves until a repeat (a leaf directly above the initial leaf) occurs, ‘four ‘ is forbidden bythe present pattern algorithm. In fact, what is observed in nature is that the number four isstrikingly less frequent in this case than the numbers 2, 3, 5 or 8, which are Fibonaccinumbers. This argues the case for bias in development, albeit now for plant rather thananimal development. One is challenged to find a plausible argument from naturalselection or adaptation for such paucity of ‘four’ as is observed to occur. There is clearlyno structural reason for the absence of spiral ‘four’.A review of phyllotaxis has been given recently (Kuhlemeier, 2007), providingmuch insight into the molecular basis of this subject. Auxin plays a particularly key role(Reinhardt et al., 2003). A complex quantitative analysis, taking into account the knownmolecular components, has been given by Smith et al (Smith et al., 2006). The presentmodel hopes to augment these contributions although from a quite different (and simpler)point of view.Outgrowth of leaves from a common stem will occur from points of intersectionof two Margin regions. A prediction (perhaps better: a ‘suggestion’) is that a “masterregulatory” gene, will be found at these stem positions; in the animal case, outgrowthsfrom the main body are accompanied by activation of the distal-less gene. Leaf locationsare shown in Figure 2 as Margin intersections for the spiral case of N= 3, 5 and 8. InFigure 2, the cylinder has coordinates x/x y/y with circumference x , and stem repeatlength y The simple patterning mechanism of the present paper in fact reproduces allFibonacci spiral patterns (Mitchison, 1972; Douady and Couder, 1991; Cummings andStrickland, 1998), as well as the common decussate and distichous patterns, andnumerous others commonly seen, such as whorls. Decussate patterns are very common,and consist of two leaves at the same level, placed 180° apart on the stem, with the nextpair up (or down) along the stem rotated related to this first pair by 90°. Decussatepatterns are the simplest example of alternating ‘whorl’ patterns, which may have three,four, etc., leaves at the same level. Distichous phyllotaxis is also very common, wheresuccessive higher (or lower) single leaves are 180º from the preceding one, examplesbeing corn, ginger and ferns. Superposed whorls also commonly occur, where, in themost common case, a pattern of two leaves situated 180º at the same level, followed atanother level by two leaves directly above the original pair. This latter pattern isparticularly common in compound leaves. All of these patterns are easily reproduced bythe model of Section 2.Solutions to the model of Section 2 as applied to plant patterns on a stem aregiven. A new pair of integers is introduced by the model, the pair (p, q) designating agiven pattern. The (p, q) pair underlies and predicts the more usual ‘parastichy” integerpair (m, n) (e.g., Mitchison, 1972).The unit square is bounded by the coordinates x/x o and y/y o in Figure 2. Apositive integer pair (p, q), p ≥ q, designates a particular pattern. The axialcoordinate around the stem is x/x o , with x being the stem circumference, so thatall points at x = 0 and x = x o are the same. The normalized length up (or down)the stem is taken to be y/y o , and the horizontal line y = 0 has the same values as y= y o . Interest is focused on a repeating pattern in the coordinates, with theboundary conditions that y=0 occurs at the point x/x o = 0 or 1. The solutions ofeqs. (2.4), (2.5) and (2.6) have been constructed so that there is always anintersection of Margin (R = R ) at the origin x/x o = 0 and y/y o = 0, (i.e., the point(0,0)), as well as at the other three corners (1,0), (1,1), and (0,1). The repeatingleaf at (0, 1) is not counted in the total number ‘N’ of leaves in a pattern.The solutions to eqs. (2.5) and (2.6) can be written in terms of the variables θ (x,y) and θ (x,y) asR = (2+Sin( θ )+ Sin( θ )), (3.1a)and fR = (2 ─ Sin( θ ) ─ Sin( θ )). (3.1b)Here θ and θ are given by θ = 2 π (px/x + qy/y ), and θ = 2 π (qx/x ± py/y ). (3.2)with k A = (2 π ) (p +q )(y /x +x /y ), (3.3)The area of a single repeat along the stem is A = x y . Equation (3.3) shows that if thestem radius is kept constant as growth occurs and A increases, then there need be nochange in pattern (p, q). On the other hand, if x and y increase proportionally withgrowth, then the phyllotactic pattern (p, q) must change. Both are observed.Intersecting Margin lines (defined as R = R ) denote the positions ofleaves or florets, and are located by requiring that the argument in each Sinefunction of eqs. (3.1a, b) be 2 π times an integer ‘i’ or ‘j’ in each case. This givesat once the two equations, from eqs. (3.1a, b),y/y o = (p/q)x/x o + i/q, (3.4)y/y o = ±(q/p)x/x o + j/p. (3.5)A given leaf is designated by a particular integer pair (i, j) designating anintersection of the two straight lines within the rectangle of Figure 2. Aparticular pattern is specified by an integer pair ‘p, q’ . Equations (3.4) and(3.5) are in a standard form. The two straight lines have slopes S = (p/q) and S = ±(q/p), and the terms i/q and j/p are intercepts of the straight lines with the y/y axis. The y intercepts do not all lie within the unit square. Pattern constructionfollows at once from eqs. (3.4) and (3.5).The intersections of the lines shown inFigure 2 gives the positions of the leaves in a given pattern specified by the pair(p, q). The number of leaves in a given pattern, in the case that there is only oneleaf per level of either x or y, and (p, q) are relatively primed, is given by theexpression N(p, q) = p ± q . (3.6)This can be seen by eliminating x (or y) in eqs. (3.4) and (3.5) and observing that0 < y/y o ≤
1, which implies that the maximum number of leaves in a pattern isgiven by eqs. (3.4) and (3.5). In the case of the plus sign in eqs. (3.4) and (3.5) thetwo sets of straight lines, corresponding to a set of integers ‘i’ in the one case and‘j’ in the other, have opposite slopes, while for the negative sign in ‘N’ the twosets of straight lines both have the same slope. This is shown in Figure 2 for thecases of N = 3, N = 5 and N = 8.When there are ‘J’ leaves on the same level, the expression for the numberof leaves in a pattern becomes simply N(p, q, J) = (p ± q )/J. A ‘whorl’ patternhas p = q, the most common example being the decussate pattern, when p = q = 2,J=2 and N = 4. Figure 2
The Margin lines are shown, where R = R of eqs. (3.1a) and (3.1b) are equal. Theslopes are given by S = p/q and by S = ±q/p. The horizontal coordinates are x/ andvertical y/y Left to right, N = 5, N = 3, and N = 8.In the case of the spiral with one leaf per level, growth considered as transition from one Fibonacci pattern to the next occurs simply by adding one leaf to each existing row.
The plus/minus sign then alternates in N of eq. (3.6),and p or q is alternately increased to the next value, as shown in Table 1. Thereare R + = p – (p–q) pattern rows in the case of the plus sign, and R - = q + (p–q) rows in the case of the minus sign in the expression for N in eq.(3.6). The m’scorrespond to the usual designation of parastichies, where N = m + n. Figure 5illustrates three Fibonacci pattern spirals, N = 3 = 2 – 1 = 2 + 1, and N = 5 = 2 + 1 = 3 + 2, and N = 8 = 3 − = 5 + 3.Whorl patterns have p = q, so that N = 2p, and J = p, in the formula N= (p ± q )/J. Superposed whorl patterns have p > q = 0, when one set of lines has slopezero, and the other set has infinite slope. The common superposed whorl with p =2, q = 0, and N = p, has two leaves on the same level (J = p = 2) displaced by180°, with a superposed pair directly above. Compound leaves usually displaysuch a pattern.There is a simple analytical transformation taking the unit square into anannulus, while preserving the form of the model equations. The coordinate ‘y’ ismapped into the polar coordinate ‘r’ while ‘x’ maps into the polar angle ‘θ’. The lines y = 0 and y = y o are then mapped into two concentric circles, while the linesx = 0 and x = x o are mapped into the straight lines representing the angles 0 and2 π in the plane. In the case that x maps into an angle less than 2 π , the square mapsinto a conical figure rather than an annulus. The two sets of intersecting straightlines of eqs. (3.1) and (3.2) are in either case mapped into two sets of intersectinglogarithmic spirals.What is clear from eq. (3.6) is that the number ‘ four’ is not includedamong spiral patterns, those with a single leaf on a level, according to the model.Such is also very rare in nature. No adaptive or selective reason for this isforthcoming. Rather, it is a result of the particular pattern formation algorithm ofthe present paper.The Table 1 shows increasing Fibonacci pattern numbers determined as analternating increase in the two basic determining parameters ‘p’ and ‘q’. The more usualparastichy numbers ‘n’ and ‘m’ are shown on the right of the Table 1. Table 1: The Fibonacci Pattern
The table shows that as growth occurs, progression from one Fibonaccipattern to the next comes about by addition of one leaf to each row, as spaceallows. There are R – = q +(p-q) rows in the case of the negative sign in N= p ±q , when the two sets of straight lines have the same slopes, and R + = p – (p − q) rows in the case that the two sets have opposite slopes, and illustrated in Figure 2.The relatively primed integers p and q increase alternately with increasing area.The pair (p, q) underlie the more usual and larger ‘parastichy’ patterndesignations (m, n) shown in the right-most column.It may be briefly noted that it is to be expected that, starting from aradially symmetric embryo such as imagined in Figure 1, opposite leaf primordiawill emerge when a critical area is reached, and the pattern will subsequentlysettle into a (e.g.) spiral pattern with stem growth. For example, consider ahemisphere or a cone; given that the present pattern algorithm of Section 2, thepattern must always give rise to two ‘determined’ Asp regions separated by aMargin region. Then the two sides of the cone or hemisphere will be designatedas two different Asp regions upon further growth, outlined by two Marginregions: one Margin region will be a circle at the cone or hemisphere base, whilethe second will divide the surface into two (equal) parts, and (e.g.) go through thenorth pole. The two Margin line intersections then occur at the base, or periphery,and will determine the positions of two initial primordia.
5. Summary
The present argues for pattern formation agents different from the usual. Theusual concept of ‘morphogen’ requires a long-range diffusing substance (Kerzsberg andWolpert, 2007). The term ‘
Asp ’ is used here to refer to the density of a particularactivated signalling pathway, and thus to the density of its associated factor that activatesits specific transcription factor. The term ‘Asp’ denotes a spatial region in which specific0transcription factors have been activated in the nucleus, and where particular selectorgenes are activated (Gerhart and Kirschner, 1997). As such, no discussion of whetherligand diffuses around cells or through them is required, nor is there discussion of themethod of achieving long range diffusion. Binary patterns are given at each growth cycle,providing successive overlapping patterns, and a possible unique genetic specificationafter each growth cycle. Pattern complexity increases with each cycle, a cycle specifiedby the growth and decay of a given signaling pair.The Asps may be assumed to interact with morphogenetic movement; one Aspincreases the apical/basal ratio, while the other decreases the difference (e.g., Cummings,2006). Specific gene activation not specified here will give rise to specific cell shapes.Such epidermal sheet movements will necessarily act back to affect gene activation, butsuch feedback from gene to cell shape change has not been considered here. Nor havemany other complications been included, in particular the effect on pattern of the plantcell wall.The model hopes to introduce new possibilities for patterning of stem cells anddesignation of ‘points’ of plant stem outgrowth. As a realistic model it is clearly far toosimple. There are many omissions that could be mentioned. The assumption that the innertissues can be neglected in patterning is certainly one such omission; the relevantassumption has been made that all patterning takes place in the outer plant cell layer ofthe stem, and interaction with the inner tissues can be neglected. Relevant molecules suchas PIN1 are located in the internal cells; the present model does not integrate the eventsof cells at the surface of the stem with those of the inner tissues. Initiation of pattern inthe meristem has been mostly neglected here (Smith et al. 2006), but rather attention hasbeen focused on the pattern of lateral outgrowth from the stem. No attempt has beenmade here to make contact with the bio-molecules of the suggested coupled signalingpathways that are the proposed source of the pattern. Many perhaps most of these areprobably as yet unknown. It may be that the hormone auxin acts as a ligand, and the PINproteins catalyzes the required export of the auxin from the cell (Petrasek et al., 2006).Auxin is required for proper positioning of organs, and auxin can induce lateral organoutgrowth (Reinhardt et al., 2000) by local application. In the present model two keyligands are required, and such have not been described as yet.
References
Cummings, F. W., and J. C. Strickland, “A model of phyllotaxis”, J. Theor. Biol. , 531-544 (1998). This is a related model to the present, but the pattern has avery different steady-state energy-minimization origin.Cummings, F. (2006) “On the origin of pattern and form in early metazoans”, Int.J. Dev. Biol., , 269-281.Kuhlemeier, C. (2007) “Phyllotaxis”, Trends in Plant Sciences”, , 143-150. Thisreview includes much of the relevant biochemistry involved, a subject not covered in thepresent work.Kerzberg, M and Wolpert, L., “Specifying positional information in the embryo:looking beyond gradients”, Cell (Essay), , 205-209 (2007).Lyndon, R.F. (1990) Plant Development: the Cellular Basis, Unwin Hyman, Ltd.(London, Mass., U.S., and Sydney)Mitchison, G. (1972), “Phyllotaxis and the Fibonacci series”, Science , 270-275.Murray, J. D. (1990) Mathematical Biology, (Springer-Verlag, Berlin).Petrasek, J. et al. (2006) “PIN proteins perform a rate-limiting function in cellularauxin efflux”, Science , 914-918.Reinhardt, D. et al. (2000) “Auxin regulates the initiation and radial position of plantlateral organs”, Plant Cell, , 507-518.Smith, R., Guyomarc’h, S., Mandel, T., Reinhardt, D., Kuhlemeier, C., andPrusinkiewicz, P., (2006) “A plausible model of phyllotaxis” Proc. Natl. Sci. USA , 13012-1306. Acknowledgement
I wish to express appreciation for encouragement and inspiration provided through manyyears by my friend and outstanding scientist Brian Goodwin (1931-2009)(1931-2009)