Plant leaves are arranged around the stem in a beautiful geometry that is called phyllotaxis. In the majority of plants, phyllotaxis exhibits a distichous, Fibonacci spiral, decussate, or tricussate pattern. To explain the regularity and limited variety of phyllotactic patterns, many theoretical models have been proposed, mostly based on the notion that a repulsive interaction between leaf primordia determines the position of primordium initiation. Among them, particularly notable are the two models of Douady and Couder (alternate-specific form, DC1; more generalized form, DC2), the key assumptions of which are that each leaf primordium emits a constant power that inhibits new primordium formation and that this inhibitory effect decreases with distance. It was previously demonstrated by computer simulations that any major type of phyllotaxis can occur as a self-organizing stable pattern in the framework of DC models. However, several phyllotactic types remain unaddressed. An interesting example is orixate phyllotaxis, which has a tetrastichous alternate pattern with periodic repetition of a sequence of different divergence angles: 180°, 90°, −180°, and −90°. Although the term orixate phyllotaxis was derived fromOrixa japonica, this type is observed in several distant taxa, suggesting that it may reflect some aspects of a common mechanism of phyllotactic patterning. Here we examined DC models regarding the ability to produce orixate phyllotaxis and found that model expansion via the introduction of primordial age-dependent changes of the inhibitory power is absolutely necessary for the establishment of orixate phyllotaxis. The orixate patterns generated by the expanded version of DC2 (EDC2) were shown to share morphological details with real orixate phyllotaxis. Furthermore, the simulation results obtained using EDC2 fitted better the natural distribution of phyllotactic patterns than did those obtained using the previous models. Our findings imply that changing the inhibitory power is generally an important component of the phyllotactic patterning mechanism.

Phyllotaxis, the beautiful geometry of plant-leaf arrangement around the stem, has long attracted the attention of researchers of biological-pattern formation. Many mathematical models, as typified by those of Douady and Couder (alternate-specific form, DC1; more generalized form, DC2), have been proposed for phyllotactic patterning, mostly based on the notion that a repulsive interaction between leaf primordia spatially regulates primordium initiation. In the framework of DC models, which assume that each primordium emits a constant power that inhibits new primordium formation and that this inhibitory effect decreases with distance, the major types (but not all types) of phyllotaxis can occur as stable patterns. Orixate phyllotaxis, which has a tetrastichous alternate pattern with a four-cycle sequence of the divergence angle, is an interesting example of an unaddressed phyllotaxis type. Here, we examined DC models regarding the ability to produce orixate phyllotaxis and found that model expansion by introducing primordial age-dependent changes of the inhibitory power is absolutely necessary for the establishment of orixate phyllotaxis. The simulation results obtained using the expanded version of DC2 (EDC2) fitted well the natural distribution of phyllotactic patterns. Our findings imply that changing the inhibitory power is generally an important component of the phyllotactic patterning mechanism.

Plants with orixate phyllotaxis and their positions in the order-level phylogenetic tree of angiosperms based on Angiosperm Phylogeny Poster [33].

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Plant material

Terminal winter buds ofO.japonica that had been collected in July from nine plants growing at the Koishikawa Botanical Gardens, Graduate School of Science, The University of Tokyo were used for morphological analyses.

Microscopic observation of winter buds

The winter buds were fixed with 5% v/v formalin, 5% v/v acetic acid, 50% v/v ethanol (FAA), dehydrated in an ethanol series, and finally infiltrated in 100% ethanol. For light microscopic observation, the dehydrated samples were embedded in Technovit 7100, cut into 5-μm-thick sections using a rotary microtome, and stained with 0.5% w/v toluidine blue. The center of gravity was determined for each leaf primordium on the section with ImageJ (https://imagej.nih.gov) and was used as its position when measuring morphometric data.

For scanning electron microscopy (SEM), the dehydrated samples were infiltrated once with a 1:1 v/v mixture of ethanol and isoamyl acetate and twice with isoamyl acetate. Subsequently, the samples were critical point dried, sputter coated with gold–palladium, and observed using SEM (Hitachi S-3400N).

DC1 model

The essential points of the DC1 model are as follows [15,16].

1. The shoot apex is considered as a plane.
2. Each leaf primordiumL emits a constant level of an inhibitory power, which generates an inhibitory field around it.
3. The inhibitory field strength decreases as a function of the distance,d.
4. Formation of new primordia is restricted to the SAM periphery represented by the circleM with a radiusR₀ at the shoot apex (Fig 2A).
5. New primordia are formed one by one at a regular time interval,T.
6. The point onM at which the inhibitory field strength is smallest gives the radial position of the formation of a new primordium.
7. Primordia move away from the center of the shoot apex with a radial velocity ofV(r) that is proportional to the radial distancer because of the exponential growth of the shoot apex.

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Fig 2. Schematic views of the shoot apex with coordinates in DC models.

The shoot apex is considered as a plane in DC1 (A) and as a cone in DC2 (B).

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At the time when then^th primordiumL_n is arising, for a position (R₀ cosθ,R₀ sinθ) on the circleM, the inhibitory field strengthI(θ) is calculated by summing the inhibitory effects from all preceding primordia,L₁ toL_n−1, as follows:(1)whered_m is the distance between the position (R₀ cosθ,R₀ sinθ) and them^th primordium (r_m cosθ_m,r_m sinθ_m) andk is a proportional coefficient (Fig 2A). In this equation, the inhibitory field strength is assumed to be inversely proportional to theη^th power of the distance from the point emitting the inhibitory power.

Considering assumptions 5 and 7, the distance from the center of the shoot apex to them^th primordium (r_m) is expressed with the initial radial velocityV₀ as:(2)

The total inhibitory field strengthI is expressed as:(3)whereG is defined asG≡V₀T/R₀ = ln(r_m/r_m+1). Morphometrically,r_m/r_m+1 is identical to the “plastochron ratio” introduced by Richards [34].

The point (R₀ cosθ,R₀ sinθ) whereI(θ) is smallest is chosen for the position of a new primordium. Note thatη andG are the only relevant parameters that influence the behavior ofI(θ) in DC1.

DC2 model

The essential points of the DC2 model are as follows [17].

1. The shoot apex is considered as a cone with an apical angle ofψ.
2. Each leaf primordiumL emits a constant level of an inhibitory power, which generates an inhibitory field around it.
3. The inhibitory field strength decreases as a function of the distance,d.
4. The formation of new primordia is restricted to the SAM periphery represented by the circleM with a distance ofR₀ from the conical vertex.
5. When the inhibitory field strength falls below a given thresholdE_s somewhere onM, a new primordium is formed immediately at that point.
6. Primordia move away from the center of the shoot apex with a radial velocity ofV(r) that is proportional to the radial distancer because of the exponential growth of the shoot apex.

Positions on the conical surface are expressed in spherical coordinates (Fig 2B). The inhibitory field strengthI(θ) at the position onM is calculated by summing the inhibitory effects from all preceding primordia,L₁ toL_n−1, as follows:(4)whered_m is the distance between them^th primordium and the positiond₀ is the maximum distance within which an existing primordium excludes a new primordium, andE is the inhibitory effect from the preceding primordium, which is defined as a monotonically decreasing, downward-convex function:(5)where, ifI(θ)<E_s, a new primordium is placed at the position. Throughout this study,E_s = 1.

Because of assumption 6, the distance from the center of the shoot apex to them^th primordium on the conical surface (r_m) is expressed with the time after its emergenceT_m and the initial radial velocityV₀ as:(6)By usingt_m≡T_mV₀/R₀, a standardized age of them^th primordium defined as the product ofT_m and the relative SAM growth rateV₀/R₀,r_m is more simply expressed as:(7)

The DC2 model is characterized by three parameters:α,, and. These parameters represent the steepness of the decline of the inhibitory effect around the threshold, the flatness of the shoot apex, and the ratio of the inhibition range to the SAM size, respectively.

In DC2, as a distance between points and on the conical surface, instead of the true Euclidian distance, its slightly modified version (as defined in the following equation) was used to avoid the discontinuity problem [17]:(8)

Computer simulation

Model simulations were implemented in C++ with Visual C++ in Microsoft Visual Studio 2015 as an integrated development environment. Contour mapping was performed using OpenCV ver. 3.3.1 (https://opencv.org/).

Computer simulations using DC2 and DC2-derived models were initiated by placing a single primordium or two primordia at a central angle of 120° on the SAM periphery. In the former initial condition, the second primordium arises at a certain time or immediately after the first primordium, in dependence on parameter settings, at the opposite position, and in some cases, more primordia are immediately inserted at middle positions. Thus computer simulations with this condition substantially cover situations starting with 1×2^x primordia (x = 0,1,2⋯) evenly distributed on the SAM periphery. Similarly, simulations with the latter condition substantially cover situations starting with 3×2^x primordia (x = 0,1,2⋯). We also tested simulations with another initial condition, in which two primordia were placed at opposite positions with a central angle of 180°, but they returned completely same results as simulations initiated by placing a single primordium did and are therefore omitted.

Computer simulations were performed with an angle resolution of 0.1°. DC2 and DC2-derived models were simulated with a time step of Δt_m = 0.001.

In all model simulations, calculation was iterated until the total number of primordia reached 100. For alternate patterns generated by simulation, the last nine primordia were used to judge the stability and regularity of divergence angles. For the other patterns, the last two nodes were used to judge the stability of the number of primordia per node. Then the patterns were categorized and displayed as shown inFig 3.

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Fig 3. Color legend for the phyllotactic patterns generated in computer simulations.

The phyllotactic patterns generated in computer simulations were classified into an alternate pattern with a constant divergence angle or a two-cycle change in the divergence angle; a tetrastichous alternate pattern with a four-cycle change in the divergence angle; a whorled pattern; and other patterns. Whorled patterns were further classified into decussate (“opposite phyllotaxis” typified by true decussate), tricussate, and other whorled patterns. These patterns were distinguished using different colors. For regular alternate patterns with a constant divergence angle, the divergence angle was indicated by a color hue from cyan (0°) to red (180°). In the case of alternate patterns with a two-cycle divergence angle change, the color hue was assigned for the mean value of the successive divergence angles. In these two-cycle alternate patterns, small-to-large ratios of two successive plastochron times and two successive divergence angles were represented by lightness (full lightness for 0) and saturation (full saturation for 1), respectively. Tetrastichous alternate patterns with a four-cycle divergence angle change were similarly expressed by color brightness and saturation based on their ratios of plastochron times and divergence angles; however, instead of the divergence angles themselves, the absolute values of divergence angles were used to calculate the ratio of divergence angles. As the divergence angle of this type of alternate pattern changes in the sequence ofp,q, −p, and −q (−180°<p,q≤180°), |q|/|p| gives the ratio of the absolute values of divergence angles if |p|>|q|. Typical examples of phyllotactic patterns are marked with circled numbers in the color legend and their schematic diagrams are shown at the bottom.

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Morphological characterization of phyllotaxis inOrixa japonica

First, we performed an anatomical analysis of the apical winter buds ofO.japonica, to characterize morphologically its phyllotaxis. In the transverse sections of the winter buds, there was a very obvious tetrastichous pattern of leaf primordia, which were arranged in opposite pairs on either of two orthogonal lines (Fig 4A). This pattern looked similar to decussate phyllotaxis; however, unlike decussate phyllotaxis, it was not symmetric. Opposite pairs of primordia varied in size and radial distance and, in each pair, a smaller primordium was positioned closer to the center of the shoot apex. Such asymmetry was also clearly recognized in the longitudinal sections and by observations performed using SEM (Fig 4B and 4C). Importantly, SEM observations detected incipient primordia that were not paired (Fig 4C). Therefore, the asymmetric arrangement of leaves was attributed to the alternate initiation of leaf primordia instead of the secondary displacement of originally decussate leaf primordia. The divergence angle between successive primordia changed in the sequence of approximately 180°, 90°, −180° (180°), and −90° (270°), and this cycle was repeated a few times in the winter bud (Fig 4D). These results confirmed that the phyllotaxis ofO.japonica is genuinely an “orixate phyllotaxis”.

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Fig 4. Orixate phyllotaxis in the apical winter buds ofOrixa japonica.

(A) Transverse section.O points to the summit of the SAM, and leaf primordia are designated asP₁,P₂,P₃, etc., withP₁ being the youngest visible primordium. Black lines represent orthostichies drawn by joining the gravity centers of leaf primorida andO. The four orthostichy lines can be roughly approximated by two orthogonal lines (pale gray broad lines). (B) Longitudinal section.I₁ indicates the incipient primordium. (C) Scanning electron microscopic image. (D) Divergence angles measured using the transverse sections. Divergence angles close to 180° show opposite positioning of the successive primordia (blue), while angles near 90° or 270° show adjacent positioning (yellow). (E) The natural logs of plastochron ratiosOP₂/OP₁ andOP₃/OP₂ are plotted based on whether the two primordia are located in an adjacent or opposite position. In (D) and (E), points linked by a line represent data from the same sample, and red points indicate data obtained from the section of (A).

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Richards’ plastochron ratio was found to oscillate in relation to the divergence angle. Plastochron ratios measured from the adjacent pairs of primordia with a divergence angle of approximately ±90° were significantly larger than those measured from the opposite pairs with a divergence angle of approximately ±180° (Fig 4E). A similar relationship between divergence angles and plastochron ratios had been, albeit fragmentarily, described for the orixate phyllotaxis ofK.uvaria [32]; thus, it is likely to be a common feature of orixate phyllotaxis.

Computer simulation assessment of DC1 regarding the ability to produce orixate phyllotactic patterns

DC1 is an inhibitory field model specialized for alternate phyllotactic patterning. DC1 assumes one-by-one formation of leaf primordia at a constant time interval, which strongly limits the model flexibility [16]. Nevertheless, as this constraint makes the patterning process simple and possible to be dealt with theoretically, it is worth investigating DC1 as a primary model for generation of any types of alternate phyllotaxis.

To test whether DC1 can produce orixate phyllotaxis, we re-examined this established model via detailed computer simulation analysis using exhaustive combinations of the determinant parameters,η andG. As reported previously [15,16], distichous and relatively major spiral phyllotactic patterns, i.e., alternate patterns with a regular divergence angle near 180°, a Fibonacci angle (137.5°), or a Lucas angle (99.5°), were generated as stable patterns over broad ranges ofη andG in these simulations (Fig 5). Of note, whenη andG were set to 1–3 and about 0.2, respectively, tetrastichous patterns were formed that resembled orixate phyllotaxis, as they showed a four-cycle periodic change of the divergence angle in the order ofp,q, −p, and −q (−180°≤p≤180°,|p|>|q|) (Fig 5A). In these patterns, however, the larger absolute value of the divergence angle was considerably deviated from 180°, whereas this should be very close to 180° in orixate phyllotaxis (Fig 5B). These patterns showed nonorthogonal tetrastichy, which is distinct in appearance from the orthogonal tetrastichy of orixate phyllotaxis (Fig 5C). Therefore, we concluded that the tetrastichous patterns found in simulations with DC1 are not orixate and that DC1 does not generate the orixate phyllotactic pattern at any parameter setting. The absence of the occurrence of normal orixate phyllotaxis, the divergence angles of which are exactly ±180° and ±90°, in the context of DC1 can be explained analytically (S1 Text).

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Fig 5. Phyllotactic patterns generated in computer simulations using DC1.

(A) Computer simulations using DC1 were performed under various settings of parametersG andη (101×101 conditions), and the patterns obtained are displayed according to the color legend shown inFig 3. (B) The black and red dots indicate the absolute values of divergence angles of the tetrastichous alternate patterns generated in (A) and real orixate phyllotaxis observed for winter buds ofO.japonica (data ofP₁~P₂ andP₂~P₃ inFig 4D), respectively. The blue dots show the averages determined from the real data ofP₁~P₂ toP₆~P₇ (Fig 4D) for each winter bud ofO.japonica. In this panel, alternate patterns with a four-cycle change in divergence angles in the sequence ofp,q, −p, and −q (|p|>|q|) were plotted at the point (|p|,|q|). (C) An example of the tetrastichous alternate patterns, which was produced by computer simulation atG = 0.3 andη = 1.5. This pattern has a divergence angle change in the sequence of 165°, −91°, −165°, and 91° and, unlike orixate phyllotaxis, exhibits a distorted tetrastichy, rather than an orthogonal tetrastichy.

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Expansion of DC1 by introducing age-dependent changes in the inhibitory power

Next, we examined whether modification of DC1 could enable it to produce orixate phyllotaxis. In an attempt to modify DC1, we focused on the inhibitory power of each leaf primordium against new primordium formation—which is assumed to be constant in DC models but may possibly change during leaf development—and expanded DC1 by introducing age-dependent, sigmoidal changes in the inhibitory power. In this expanded version of DC1 (EDC1), the inhibitory field strengthI(θ) was redefined as the summation of the products of the age-dependent change in the inhibitory power and the distance-dependent decline of its effect:(9)

F is defined as:(10)where parametersa andb are constants that represent the rate and timing of the age-dependent changes in the inhibitory power, respectively. Under this equation, in an age-dependent manner, the inhibitory power increases ata>0 and decreases ata<0. In the present study,η was fixed at 2 for EDC1.

Prior to computer simulation analysis with EDC1, we searched for parameters of EDC1 that can fit the requirements of normal orixate phyllotaxis. When the normal pattern of orixate phyllotaxis is stably maintained, a rectangular coordinate system with the origin at the center of the shoot apex can be set such that all primordia lie on the coordinate axes, and every fourth primordium is located on the same axis in the same direction, i.e., the position of any primordium (m^th primordium) can be expressed as (r_m cosθ_m−4i,r_m sinθ_m−4i) for integersi. Under this condition, we considered whether a new primordium (n^th primordium) is produced at the position (R₀ cosθ_n−4i,R₀ sinθ_n−4i), to keep the normal orixate phyllotactic pattern. In EDC1, as in DC1, new primordium formation at (R₀ cosθ_n−4i,R₀ sinθ_n−4i) implies that the inhibitory field strengthI(θ) on the circleM has a minimum atθ_n−4i. For this reason, we first attempted to solve the following equation:(11)

This equation was numerically solved under two geometrical situations of primordia: the divergence angle between the newly arising primordium and the last primordium is ±90° (situation 1) or ±180° (situation 2) (S1A Fig). The solutions obtained identified parameter sets that satisfied the above equation under both these two situations (Fig 6A,S1B Fig). The calculation ofI(θ) using the identified parameter sets showed thatI(θ) has a local and global minimum aroundθ_n−4i with large values ofG, such as 0.5 or 1, while it has a local maximum instead of a minimum aroundθ_n−4i with smallG values, such as 0.1 (S1C Fig). This result indicates the possibility that EDC1 can form orixate phyllotaxis as a stable pattern under a particular parameter setting with largeG values.

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Fig 6. Mathematical and computer simulation analysis of EDC1.

(A) Numerical solutions of parameters that fit the mathematical requirements for normal orixate phyllotaxis in EDC1. The two curves show the solutions obtained using variousG values. The closed circles indicate the solutions obtained withG set at 0.1 intervals between 0.1 and 1.0. (B) Stable patterns generated in computer simulations using EDC1 under various parameter settings (201 settings fora, 101 settings forb, 3 settings forG, and thus 201×101×3 = 60,903 simulations in total). The patterns obtained are displayed according to the color legend shown inFig 3. The white crosses (+) indicate the parameter conditions obtained as numerical solutions of, giving a minimum ofI(θ) aroundθ_n−4i for normal orixate phyllotaxis, whereas white saltires (×) indicate the parameter conditions obtained as numerical solutions of giving a maximum ofI(θ). (C) Schematic diagrams of typical examples of the phyllotactic patterns generated in the computer simulations. The circled numbers relate the diagrams to the parameter conditions shown in (B).

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Generation of orixate phyllotactic patterns in a computer simulation using EDC1

We conducted computer simulations using EDC1 over broad ranges of parameters and found that EDC1 could generate tetrastichous alternate patterns in addition to distichous and spiral patterns (Fig 6B). The tetrastichous patterns included orthogonal tetrastichous ones with a four-cycle divergence angle change of approximately 180°, 90°, −180°, and −90°, which can be regarded as orixate phyllotaxis (Fig 6C,S2 Fig). Under the conditions of assuming an age-dependent increase in the inhibitory power (a>0), these orixate patterns were formed within a rather narrow parameter range ofG = 0.5~1,a = 1~2, andb = 4~9 around the parameter settings that were determined by numerical solution, to fit the requirements for the stable maintenance of normal orixate phyllotaxis (Fig 6B and 6C). When assuming an age-dependent decrease in the inhibitory power (a<0), orixate phyllotaxis appeared at a point ofG = 0.1,a≈−10, andb≈3.5 (Fig 6B and 6C). These values ofa andb represent a very sharp drop in the inhibitory power at the primordial age corresponding to approximately three plastochron units. Around this parameter condition, there were no numerical solutions for normal orixate phyllotaxis; however, patterns that were substantially orixate, although they were not completely normal, could be established. The orixate patterns that were generated under the conditions in which the inhibitory power increased and decreased were visually characterized by sparse primordia around the small meristem and dense primordia around the large meristem, respectively (Fig 6C).

In the results of computer simulations with EDC1, besides the orixate patterns, we also found peculiar patterns with anx-cycle change in the divergence angle consisting of 180° followed by an (x−1)-times repeat of 0° (S3 Fig). Such patterns were generated when all the parametersa,b, andG were set to relatively large values and are displayed as periodic distribution of black regions in the upper right area of the middle and right panels ofFig 6B. In these patterns, asb is increased, the number of repetition times of 0° is increased, resulting in the shift fromx-cycle to (x+1)-cycle. This shift is mediated by the occurrence of spiral patterns with a small divergence angle, and the transitions fromx-cycle to spiral and from spiral to (x+1)-cycle takes place suddenly in response to a slight change ofb (S3 Fig).

Computer simulation assessment of DC2 regarding the ability of producing orixate phyllotactic patterns

DC2, as DC1, is an inhibitory field model but is more generalized than DC1 [17]. Unlike DC1, DC2 does not assume one-by-one formation of primordia at a constant time interval and thus does not exclude whorled phyllotactic patterning. Indeed, DC2 was shown to produce all major patterns of either alternate or whorled phyllotaxis depending on parameter conditions [17]. To test whether DC2 can generate orixate phyllotactic patterns, we carried out extensive computer simulation analyses using this model. Our computer simulations confirmed that major phyllotactic patterns, such as distichous, Fibonacci spiral, Lucas spiral, decussate, and tricussate patterns, are formed as stable patterns in wide ranges of parameters, and also showed formation of tetrastichous alternate patterns with a four-cycle change of the divergence angle atN = 1 andΓ≈1.8 when initiated by placing a single primordium at the SAM periphery (Fig 7A). The possible inclusion of orixate phyllotaxis in these tetrastichous four-cycle patterns was carefully examined based on the ratio of plastochron times and the ratio of absolute values of divergence angles, which should be much larger than 0 and close to 0.5, respectively, in orixate phyllotaxis. Although all the tetrastichous four-cycle patterns detected here had a divergence angle ratio near 0.5, their ratios of plastochron times were too small to be regarded as orixate phyllotaxis, and the overall characters indicated that they are rather similar to decussate phyllotaxis (Fig 7B and 7C). These results led to the conclusion that the DC2 system does not generate orixate phyllotaxis under any parameter conditions.

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Fig 7. Phyllotactic patterns generated in computer simulations using DC2.

(A) Computer simulations using DC2 were performed under various settings of parametersα andΓ (101 settings forα and 101 settings forΓ) withN = 1,1/3, or 1/5, and the resultant patterns are displayed for the cases ofN = 1 and 1/3 according to the color legend shown inFig 3. Simulations were started by placing a single primordium or two primordia at a central angle of 120° on the SAM periphery. (B) The regular alternate, two-cycle alternate, and tetrastichous four-cycle alternate patterns generated in computer simulations using DC2 in (A), including simulations withN = 1/5 as well asN = 1 and 1/3, were plotted using the ratio of absolute values of two successive divergence angles as the abscissa and the ratio of two successive plastochron times as the ordinate. The red dots indicate tetrastichous four-cycle patterns, while the black dots indicate regular alternate and two-cycle patterns. The blue dots show the data of real orixate phyllotaxis observed for winter buds ofO.japonica (calculated from the data ofP₁~P₂ andP₂~P₃ inFig 4D). (C) Magnification of the lower-left corner of (B).

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Expansion of DC2 by introducing age-dependent changes in the inhibitory power

Similar to the approach used for DC1, we expanded DC2 by introducing primordial age-dependent changes in the inhibitory power. In this expanded version of DC2 (EDC2), the inhibitory field strengthI(θ) was redefined as the summation of the products of the age-dependent change in the inhibitory power and the distance-dependent decrease of its effect:(12)whereF is a function expressing a temporal change in the inhibitory power, defined as:(13)

Generation of orixate phyllotactic patterns in a computer simulation using EDC2

Computer simulations using EDC2 were first conducted under a wide range of combinations ofA andB at three different settings ofΓ (Γ = 1, 2, or 3) and fixed conditions forα andN (α = 1,N = 1/3) (S4 Fig). In this analysis, tetrastichous four-cycle patterns were formed within the parameter window whereA was 3–7 andB was 0.4–1, which represents a late and slow increase in the inhibitory power during primordium development (Fig 8A). Further analysis performed by changingΓ,α, andN showed that small values ofα, which indicate that the distance-dependent decrease in the inhibitory effect is gradual, and large values ofΓ, which indicate that the maximum inhibition range of a primordium is large, are also important for the formation of tetrastichous four-cycle patterns (Fig 9,S5 Fig). All of these four-cycle patterns were found to be almost orthogonal and to have a sufficiently large ratio of successive plastochron times, thus fitting the criterion of orixate phyllotaxis (Fig 8B,S7 Fig). Furthermore, the plots of these patterns lied within the cloud of the data points of real orixate phyllotaxis, and therefore we concluded that they are orixate. A typical example of such orixate patterns was obtained by simulation using the parameters,A = 4.8,B = 0.72,Γ = 2.8,N = 1/3, andα = 1, and is presented as a contour map of the inhibitory field strength inFig 10A, which clearly depicts orixate phyllotactic patterning. Under this parameter condition, the inhibitory field strength on the SAM periphery was calculated to have a minimum close to the threshold at 0° at the time of new primordium formation when the preceding primordia were placed at 0°, 180°, and ±90° (S8 Fig). This landscape of the inhibitory field stabilizes the orixate arrangement of primordia. In summary, our analysis demonstrated that orixate phyllotaxis comes into existence in the EDC2 system when the inhibitory power of each primordium increases at a late stage and slowly to a large maximum and when its effect decreases gradually with distance.

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Fig 8. Phyllotactic patterns generated in computer simulations using EDC2.

(A) Computer simulations using EDC2 were performed under various parameter settings (201 settings for −20≤A≤20, 101 settings for 0≤B≤1, andΓ = 1, 2, or 3) with fixed parametersα = 1 andN = 1/3, and the patterns obtained are displayed according to the color legend shown inFig 3. Simulations were started by placing a single primordium on the SAM periphery. (B) Computer simulations using EDC2 were performed under various settings of parameters (101 settings for 0≤A≤20, 101 settings for 0≤B≤1, andΓ = 2, 2.5, or 3) with fixed parametersα = 1 andN = 1/3. The graph shows a scatter plot of alternate patterns with a constant divergence angle or a two-cycle change in the divergence angle (black), and tetrastichous alternate patterns with a four-cycle change in the divergence angle (red) generated in the computer simulations. In this graph, each pattern was plotted based on the ratio of absolute values of two successive divergence angles (abscissa) and the ratio of plastochron times (ordinate). The black dots surrounded by an orange circle represent semi-decussate-like patterns that occurred in the vicinities of orixate phyllotaxis in the parameter space, which are indicated by blue asterisks inS6A Fig. The blue dots indicate the data of real orixate phyllotaxis observed for winter buds ofO.japonica (calculated from the data ofP₁~P₂ andP₂~P₃ inFig 4D).

https://doi.org/10.1371/journal.pcbi.1007044.g008

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Fig 9. Effects of the inhibition range and increase in inhibitory power on phyllotactic patterns in EDC2.

Computer simulations were performed using EDC2 withα = 1, 2, or 4 under various settings ofΓ andA (101×101 conditions), which reflect the maximum inhibition range of a primordium and the primordial age-dependent increase in the inhibitory power, respectively. The initial value of the inhibitory power was fixed to 0.047, i.e.,A×B was fixed at 3.N was fixed at 1/3. The simulation was started by placing a single primordium on the SAM periphery. The patterns obtained are displayed according to the color legend shown inFig 3.

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Fig 10. Characteristics of orixate patterns generated in computer simulations using EDC2.

(A) Contour map of the natural log of the inhibitory field strengthI within the shoot apical region that generated orixate phyllotaxis in the computer simulation using EDC2. A value of 0 implies that the inhibitory field strength is equal to the threshold for primordium formation. (B) Relationship between plastochrons and divergence angles in orixate patterns generated in computer simulations using EDC2. For a pair of successive primordia,L_m andL_m+1, a standardized plastochron was calculated ast_m+1−t_m = ln(r_m/r_m+1). Orixate patterns were plotted based on their two standardized plastochrons: one for the pair of opposite primordia with a divergence angle of approximately 180°, and the other for the pair of adjacent primordia with a divergence angle of approximately ±90°.

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In the orixate phyllotactic patterns generated by EDC2, the plastochron time oscillated between two values together with a cyclic change in the divergence angle: the longer plastochron was observed for the adjacent pairs of primordia with a divergence angle of ±90° and the shorter plastochron was recorded for the opposite pairs with a divergence angle of ±180° (Fig 10B,S1 Movie). This relationship between the plastochron and the divergence angle agreed with the real linkage observed for the plastochron ratios and divergence angles in the winter buds ofO.japonica (Fig 4E).

Distribution of phyllotactic patterns in the parameter space of EDC2

Based on a comprehensive survey of the results of the computer simulations performed using EDC2, we examined the distribution of various phyllotactic patterns and the possible relationships between them in the parameter space of EDC2 (Figs8A and9,S4,S5,S9 andS10 Figs). Major phyllotactic patterns, such as the distichous, Fibonacci spiral, and decussate patterns, occupied large areas in the parameter space, and the Lucas spiral pattern occupied some areas. Depending on the initial condition, the tricussate pattern also took a considerable fraction of the space. In the parameter space, the distichous pattern adjoined the Fibonacci spiral pattern, while the Fibonacci spiral adjoined the distichous, Lucas spiral, decussate, and tricussate patterns. The regions where the orixate pattern was generated were located next to the regions of the decussate, Fibonacci spiral, Lucas spiral, and/or two-cycle alternate patterns. This positional relationship suggests that orixate phyllotaxis is more closely related to the decussate and spiral patterns than it is to the distichous pattern. The two-cycle patterns formed in a narrow parameter space next to the region of orixate phyllotaxis and had a divergence angle ratio of approximately 0.55 and a plastochron time ratio of approximately 0.2 (Fig 8B,S6A Fig); thus, they are similar to semi-decussate phyllotaxis, which is an alternate arrangement characterized by the oscillation of the divergence angle between 180° and 90° (S6B Fig). These semi-decussate-like patterns were not observed in the computer simulations performed using DC2 (Fig 7B and 7C); rather, they were produced only after its expansion into EDC2.

The overall distributions of major phyllotactic patterns in the parameter space were compared between DC2 and EDC2 using color plots drawn from the results of simulations conducted for EDC2 with various settings of the inhibition range parameterΓ and the inhibitory power change parameterA (Fig 9). In these simulations, largeA values accelerated the age-dependent increase in the inhibitory power of each primordium; ifA is sufficiently large, the inhibitory power is almost constant during primordium development and the EDC2 system is almost the same as DC2. Therefore, the colors along the top side of each panel ofFig 9, whereA was set to 20, which is a high value, show the phyllotactic pattern distribution againstΓ in DC2, while the colors over the two-dimensional panel show the phyllotactic pattern distribution againstΓ andA in EDC2. The order of distribution of the distichous, Fibonacci spiral, and decussate patterns was unaffected by decreasingA and, thus, did not differ between DC2 and EDC2. As reported in the previous study of DC2 [17], on the top side ofFig 9, the stable pattern changed from distichous to Fibonacci spiral, and then turned into decussate asΓ decreased. In the parameter space of EDC2, this order of distribution of major phyllotactic patterns was not affected much by decreasingA to moderate values; however, whenA was further decreased, the orixate pattern appeared in the region of the Fibonacci spiral (Fig 9,S10 Fig). AsA decreased, the range ofΓ that produced a Fibonacci spiral became wider and the transition zone between the distichous and Fibonacci spiral patterns, where the divergence angle gradually changed from 180° to 137.5°, became narrower (Fig 9). This result indicated that Fibonacci spiral phyllotaxis is more dominant when assuming a delay in the primordial age-dependent increase in the inhibitory power.

(A) Gradual increase of the inhibitory power with a relatively small size of SAM. (B) Sudden decrease of the inhibitory power with a relatively large size of SAM. EDC1 can establish orixate phyllotaxis under either of these conditions while EDC2 can only under the former condition.

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S1 Text.Mathematical analysis of the stability of the normal orixate phyllotactic pattern in DC1.

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S1 Fig.Mathematical analysis of the conditions required for normal orixate phyllotaxis in EDC1.

(A) Two different situations of the arrangement of the four preceding primordia,L_n−4,L_n−3,L_n−2, andL_n−1, relative to the incipient primordiumL_n in normal orixate phyllotaxis. (B) The blue and red curves show numerical solutions of in situations 1 and 2, respectively. Their intersection points are expected to give the parameter conditions of EDC1 that are required for stabilizing the normal orixate phyllotaxis. (C) Inhibitory field strength on the periphery of SAM in situation 1 (blue) and situation 2 (red) at the parameter settings determined as solutions of that are common to both of these situations. Graphs were drawn withθ_n−4i as 0°.

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S2 Fig.Divergence angles of the tetrastichous alternate patterns generated in computer simulations using EDC1.

For tetrastichous alternate patterns with a four-cycle change in the divergence angle generated in computer simulations using EDC1 under the conditions ofG = 0.1 anda<0 (A),G = 0.5 anda>0 (B), andG = 1 anda>0 (C), the absolute values of divergence angles were plotted using the larger value as the abscissa and the smaller value as the ordinate, such that a pattern with a divergence angle change in the sequence ofp,q, −p, and –q (|p|>|q|) was represented by a black dot at the position (|p|,|q|). The blue dots show the averages determined from the real data ofP₁~P₂ toP₆~P₇ (Fig 4D) for each winter bud ofO.japonica.

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S3 Fig.Transitions of phyllotactic patterns by a slight increase ofb in the computer simulation with EDC1.

Computer simulations with EDC1 were performed under the parameter condition ofG = 0.5,a = 10, andb = 5, 5.1, or 5.2. Changes in the divergence angle fromL₇₀~L₇₁ toL₉₉~L₁₀₀ are shown for the resultant patterns. A small-angle spiral was obtained atb = 5.1 (blue circle), while five-cycle and six-cycle alternate patterns were produced atb = 5 and atb = 5.2, respectively.

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S4 Fig.Phyllotactic patterns generated in computer simulations using EDC2 with a broad range of settings ofA.

Computer simulations using EDC2 were performed under various parameter settings (201 settings for −100≤A≤100, 101 settings forB, and 3 settings forΓ), and the patterns obtained are displayed according to the color legend shown inFig 3. Simulations were started by placing a single primordium on the SAM periphery.N was fixed at 1/3.

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S5 Fig.Computer simulations using EDC2 over a wide range of combinations of five parameters.

A computer simulation was performed using EDC2 under various settings of five parameters, 101 settings forA (0≤A≤20), 101 settings forB (0≤B≤1), 3 settings forα (α = 1, 2, or 4), 9 settings forΓ (1≤Γ≤3), and 2 settings forN (N = 1/3 or 1). The patterns obtained are displayed in theAB space according to the color legend shown inFig 3. Simulations were started by placing a single primordium or two primordia at the central angle of 120° on the SAM periphery.

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S6 Fig.Semi-decussate-like patterns generated in computer simulations using EDC2.

(A) Computer simulations using EDC2 were performed under various parameter settings (101 settings for 0≤A≤20, 101 settings for 0≤B≤1, andΓ = 2, 2.5, or 3) with fixed parametersα = 1 andN = 1/3. The patterns obtained were converted into colors according to the color legend (Fig 3), and the areas containing semi-decussate-like patterns (blue asterisks) were cut out from the color diagrams. (B) Contour map of the natural log of the inhibitory field strengthI within the shoot apical region generating semi-decussate-like phyllotaxis with divergence angles of 171° and 89° in the computer simulation using EDC2 under the indicated parameter condition.

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S7 Fig.Divergence angles of the tetrastichous alternate patterns generated in computer simulations using EDC2.

For tetrastichous alternate patterns with a four-cycle change in the divergence angle generated in computer simulations using EDC2 atΓ = 2 (A),Γ = 2.5 (B), andΓ = 3 (C) under the condition ofα = 1,N = 1/3, andA>0, absolute values of divergence angles are plotted using the larger value as the abscissa and the smaller value as the ordinate, such that a pattern with a divergence angle change in the sequence ofp,q, −p, and −q (|p|>|q|) is represented by a dot at the position (|p|,|q|). The blue dots show the averages determined from the real data ofP₁~P₂ toP₆~P₇ (Fig 4D) for each winter bud ofO.japonica.

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S8 Fig.Analysis of the stability of the normal orixate phyllotaxis in EDC2.

To analyze the stability of the normal orixate phyllotaxis in EDC2, we arranged primordia artificially in the normal orixate pattern with a four-cycle divergence angle change in the sequence of exactly 180°, 90°, −180°, and −90° and with a standardized plastochron that oscillated between 0.1 and 0.325. We then tested whether the inhibitory field strength could assign the position of a new primordium to maintain the normal orixate pattern in the EDC2 system at the parameter condition (A = 4.8,B = 0.72,Γ = 2.8,N = 1/3,α = 1), with which EDC2 generated a realistic orixate pattern in computer simulation (Fig 10). (A) Contour maps of the natural log of the inhibitory field strengthI in the shoot apical region, at which the preceding primordia were artificially arranged in two situations of the normal orixate pattern. (B) The inhibitory field strength on the SAM periphery at the time of formation of then^th primordiumL_n was calculated for situation 1 (blue) and situation 2 (red). The inhibitory field strength had a minimum close to the threshold at position 0° in both situations, which allows the positioning of a new primordium to maintain the normal orixate pattern.

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S9 Fig.Characteristics of phyllotactic patterns generated in computer simulations with EDC2 as influenced by the parameterB.

Computer simulations using EDC2 were performed under 101 settings ofB (0≤B≤1) atΓ = 1 or 3,A = 4 or 10, andα = 1. Divergence angles and plastochron times determined from the last nine leaf primordia (L₉₂ toL₁₀₀) are shown for the patterns obtained with variousB settings, which represent characteristics of phyllotactic patterns as influenced by the timing of the increase of the inhibitory power.

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S10 Fig.Characteristics of phyllotactic patterns generated in computer simulations with EDC2 as influenced by the parameterΓ.

Computer simulations using EDC2 were performed under 101 settings ofΓ (1≤Γ≤3) atα = 1 or 4,A = 4 or 10, andA×B = 3. Divergence angles and plastochron times determined from the last nine leaf primordia (L₉₂ toL₁₀₀) are shown for the patterns obtained with variousΓ settings, which represent characteristics of phyllotactic patterns as influenced by the ratio of the inhibition range to the SAM size.

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S1 Movie.Orixate pattern generation in the computer simulation using EDC2.

Contour map of the natural log of the inhibitory field strengthI within the shoot apical region showing orixate pattern generation in the computer simulation using EDC2 under the condition ofA = 5,B = 0.74,α = 1, andN = 1/3 (for the color legend, seeFig 10A). In this simulation, to facilitate stable pattern formation,Γ was expediently set to depend on the time after the start of simulation,t_sim [18]:whereΓ_i = 50,Γ_f = 2.8,t_i = 0.2, andτ = 0.3.

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S1 Code.Source codes for DC1 and EDC1 simulations as a compressed archive.

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S2 Code.Source codes for DC2 and EDC2 simulations as a compressed archive.

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S1 Dataset.Numerical data of the results of computer simulations with DC1.

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S2 Dataset.Numerical data of the results of computer simulations with EDC1.

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S3 Dataset.Numerical data of the results of computer simulations with DC2.

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S4 Dataset.Numerical data of the results of computer simulations with EDC2.

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