Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis
Plants bear leaves around the stem in a regular arrangement; this is termed phyllotaxis ., Across diverse plant species , phyllotaxis has common characteristics , which are often described mathematically and are reflected in a limited variety of phyllotactic patterns , including the distichous , decussate , tricussate , and Fibonacci spiral ( spiral with a divergence angle close to the golden angle of 137 . 5° ) patterns 1 ., The origin of the regularity of , and the few particular patterns that are allowed in , phyllotaxis have long been fascinating questions for botanists ., In the early days , morphological studies attributed phyllotactic patterning to Hofmeister’s axiom , which claims that , on the periphery of the shoot apical meristem ( SAM ) , a new leaf primordium is formed in the largest gap between existing primordia and as far away as possible from them 2 ., Following this axiom , many theoretical models have been proposed to explain the generation of phyllotactic patterns 3–21 ., Such theoretical models are based on a common concept: the existence of an inhibitory field created by a repulsive , either physical or chemical , interaction between leaf primordia , which conforms to Hofmeister’s axiom ., Among them , the two mathematical models proposed by Douady and Couder 15–18 are particularly notable ( they will be referred to as DC1 and DC2 hereafter ) ., The key assumptions shared by DC models are that each individual leaf primordium emits a constant power that inhibits the production of a new primordium near it and that the inhibitory effect of this power decreases as the distance from the emission point increases ., In DC1 , it is additionally assumed that leaf primordia are formed one by one at a constant time interval , i . e . , plastochron; thus , DC1 deals only with alternate phyllotaxis 15 , 16 ., In contrast , DC2 does not deny the simultaneous formation of leaf primordia or temporal changes of the plastochron and can deal with both alternate and whorled phyllotaxis 17 ., Computer simulations using DC models demonstrated that they can generate various major standard phyllotactic patterns as stable patterns that depend on parameter settings 15–17 ., In the early 2000s , experimental studies showed that auxin determines the initiation of shoot lateral organs and that its polar transport serves as a driving force of phyllotactic patterning 22–24 ., Briefly , the auxin efflux carrier PIN1 , which is localized asymmetrically in epidermal cells of the shoot apex , polarly transports auxin to create auxin convergence , thus directing the position of lateral organ initiation ., Subsequently , assuming the existence of a positive feedback regulatory loop between the auxin concentration gradient and PIN1 localization , a novel mathematical model was developed to explain the spontaneous formation of the auxin convergence ., It was further shown by computer simulation analysis that these models can produce several typical patterns of standard phyllotaxis 25 , 26 ., In the auxin-transport-based models , auxin polar transport toward the auxin convergence removes auxin from its surroundings , which prevents the formation of a new , vicinal auxin convergence ., This effect is considered to correspond to the repulsive interaction between primordia described in the previous models ., The parameters of the auxin-transport-based model were mapped on the parameters of DC2 27 , 28 , which shows that DC2 can be treated as an abstract model of the auxin-transport-based models ., DC models and the auxin-transport-based models , DC2 in particular , have been studied extensively regarding the ability to produce the various phyllotactic patterns that are observed in nature 15–17 , 25–26; however , several types were never addressed in the studies that used these models ., An interesting example is orixate phyllotaxis , which is named after Orixa japonica ( Rutaceae , Sapindales ) 29 ., Orixate phyllotaxis is a tetrastichous alternate phyllotaxis that is characterized by the periodic repetition of a sequence of different divergence angles: 180° , 90° , −180° ( 180° ) , and −90° ( 270° ) ., Although plant species that show orixate phyllotaxis are uncommon , they are found in several distant taxa ( Fig 1 ) ., Many species of Kniphofia ( Asphodelaceae , Asparagales ) display a tetrastichous arrangement of leaves 30 , and K . uvaria , K . pumila , and K . tysonii exhibit orixate phyllotaxis 31 , 32 ., Lagestroemia indica ( Lythraceae , Myrtales ) and Berchemiella berchemiaefolia ( Rhamnaceae , Rosales ) are also known as species with orixate phyllotaxis 29 ., The rare and sporadic distribution of orixate phyllotaxis among plants suggests that this peculiar phyllotaxis occurred independently a few times during plant evolution ., Therefore , it is likely that orixate phyllotaxis is generated by a common regulatory mechanism of leaf-primordium formation under some particular condition rather than by an orixate-unique mechanism ., If this is true , mathematical models that account fully for the spatial regulation of leaf-primordium formation should be able to produce not only major phyllotactic patterns , but also orixate phyllotaxis ., In this study , we re-examined the original DC models exhaustively under various parameter conditions , to test whether they can produce orixate phyllotaxis ., We then expanded DC models by introducing primordial age-dependent changes in the inhibitory power ., Our results indicate that a late and slow increase in the inhibitory power is critical for the establishment of orixate phyllotaxis and imply that changing the inhibitory power is generally an important component of the mechanism of phyllotactic patterning ., Terminal winter buds of O . 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 ., 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 ) ., The essential points of the DC1 model are as follows 15 , 16 ., At the time when the nth primordium Ln is arising , for a position ( R0 cos θ , R0 sin θ ) on the circle M , the inhibitory field strength I ( θ ) is calculated by summing the inhibitory effects from all preceding primordia , L1 to Ln−1 , as follows:, I ( θ ) ≡∑m=1n−1k ( dm ( θ ) ) −η=k∑m=1n−1 ( R02+rm2−2R0rmcos ( θ−θm ) ) −η2 ,, ( 1 ), where dm is the distance between the position ( R0 cos θ , R0 sin θ ) and the mth primordium ( rm cos θm , rm sin θm ) and k is a proportional coefficient ( Fig 2A ) ., In this equation , the inhibitory field strength is assumed to be inversely proportional to the ηth 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 the mth primordium ( rm ) is expressed with the initial radial velocity V0 as:, rm=R0eV0R0 ( n−m ) T ., ( 2 ), The total inhibitory field strength I is expressed as:, I ( θ ) =kR0η∑m=1n−1{1+e2 ( n−m ) G−2e ( n−m ) Gcos ( θ−θm ) }−η2 ,, ( 3 ), where G is defined as G≡V0T/R0 = ln ( rm/rm+1 ) ., Morphometrically , rm/rm+1 is identical to the “plastochron ratio” introduced by Richards 34 ., The point ( R0 cos θ , R0 sin θ ) where I ( θ ) is smallest is chosen for the position of a new primordium ., Note that η and G are the only relevant parameters that influence the behavior of I ( θ ) in DC1 ., The essential points of the DC2 model are as follows 17 ., Positions on the conical surface are expressed in spherical coordinates ( r , ψ2 , θ ) ( Fig 2B ) ., The inhibitory field strength I ( θ ) at the position ( R0 , ψ2 , θ ) on M is calculated by summing the inhibitory effects from all preceding primordia , L1 to Ln−1 , as follows:, I ( θ ) ≡∑m=1n−1E ( dm ( θ ) d0 ) ,, ( 4 ), where dm is the distance between the mth primordium and the position ( R0 , ψ2 , θ ) , d0 is the maximum distance within which an existing primordium excludes a new primordium , and E is the inhibitory effect from the preceding primordium , which is defined as a monotonically decreasing , downward-convex function:, E ( x ) ≡Es−1+ ( tanhαx ) −1−1+ ( tanhα ) −1 ,, ( 5 ), where , if I ( θ ) <Es , a new primordium is placed at the position ( R0 , ψ2 , θ ) ., Throughout this study , Es = 1 ., Because of assumption 6 , the distance from the center of the shoot apex to the mth primordium on the conical surface ( rm ) is expressed with the time after its emergence Tm and the initial radial velocity V0 as:, rm=R0eV0R0Tm ., ( 6 ), By using tm≡TmV0/R0 , a standardized age of the mth primordium defined as the product of Tm and the relative SAM growth rate V0/R0 , rm is more simply expressed as:, rm=R0etm ., ( 7 ), The DC2 model is characterized by three parameters: α , N≡sinψ2 , and Γ≡d0R0N ., 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 ( r ( 1 ) , ψ2 , θ ( 1 ) ) and ( r ( 2 ) , ψ2 , θ ( 2 ) ) 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:, d≡ ( r ( 1 ) −r ( 2 ) ) 2N+2Nr ( 1 ) r ( 2 ) {1−cos ( θ ( 1 ) −θ ( 2 ) ) } ., ( 8 ) 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 1×2x primordia ( x = 0 , 1 , 2⋯ ) evenly distributed on the SAM periphery ., Similarly , simulations with the latter condition substantially cover situations starting with 3×2x 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 Δtm = 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 in Fig 3 ., First , we performed an anatomical analysis of the apical winter buds of O . 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 of O . japonica is genuinely an “orixate phyllotaxis” ., 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 of K . uvaria 32; thus , it is likely to be a common feature of orixate phyllotaxis ., 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 and G . 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 of η and G in these simulations ( Fig 5 ) ., Of note , when η and G 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 of p , 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 ) ., 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 strength I ( 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:, I ( θ ) ≡∑m=1n−1{k ( dm ( θ ) ) −ηF ( n−m ) } ., ( 9 ), F is defined as:, F ( Δt ) ≡11+e−a ( Δt−b ) ,, ( 10 ), where parameters a and b 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 at a>0 and decreases at a<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 ( mth primordium ) can be expressed as ( rm cos θm−4i , rm sin θm−4i ) for integers i ., Under this condition , we considered whether a new primordium ( nth primordium ) is produced at the position ( R0 cos θn−4i , R0 sin θn−4i ) , to keep the normal orixate phyllotactic pattern ., In EDC1 , as in DC1 , new primordium formation at ( R0 cos θn−4i , R0 sin θn−4i ) implies that the inhibitory field strength I ( θ ) on the circle M has a minimum at θn−4i ., For this reason , we first attempted to solve the following equation:, dI ( θ ) dθ|θ−θn−4i=0=0 ., ( 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 of I ( θ ) using the identified parameter sets showed that I ( θ ) has a local and global minimum around θn−4i with large values of G , such as 0 . 5 or 1 , while it has a local maximum instead of a minimum around θn−4i with small G 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 large G values ., 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 of G = 0 . 5~1 , a = 1~2 , and b = 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 of G = 0 . 1 , a≈−10 , and b≈3 . 5 ( Fig 6B and 6C ) ., These values of a and b 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 an x-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 parameters a , b , and G 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 of Fig 6B ., In these patterns , as b is increased , the number of repetition times of 0° is increased , resulting in the shift from x-cycle to ( x+1 ) -cycle ., This shift is mediated by the occurrence of spiral patterns with a small divergence angle , and the transitions from x-cycle to spiral and from spiral to ( x+1 ) -cycle takes place suddenly in response to a slight change of b ( S3 Fig ) ., 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 at N = 1 and Γ≈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 ., 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 strength I ( 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:, I ( θ ) ≡∑m=1n−1{E ( dm ( θ ) d0 ) F ( tm ) } ,, ( 12 ), where F is a function expressing a temporal change in the inhibitory power , defined as:, F ( t ) ≡11+e−A ( t−B ) ., ( 13 ), Computer simulations using EDC2 were first conducted under a wide range of combinations of A and B at three different settings of Γ ( Γ = 1 , 2 , or 3 ) and fixed conditions for α and N ( α = 1 , N = 1/3 ) ( S4 Fig ) ., In this analysis , tetrastichous four-cycle patterns were formed within the parameter window where A was 3–7 and B was 0 . 4–1 , which represents a late and slow increase in the inhibitory power during primordium development ( Fig 8A ) ., Further analysis performed by changing Γ , α , and N showed that small values of α , which indicate that the distance-dependent decrease in the inhibitory effect is gradual , and large values of Γ , 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 , and α = 1 , and is presented as a contour map of the inhibitory field strength in Fig 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 ., 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 of O . japonica ( Fig 4E ) ., 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 ( Figs 8A and 9 , S4 , S5 , S9 and S10 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 parameter Γ and the inhibitory power change parameter A ( Fig 9 ) ., In these simulations , large A values accelerated the age-dependent increase in the inhibitory power of each primordium; if A 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 of Fig 9 , where A was set to 20 , which is a high value , show the phyllotactic pattern distribution against Γ in DC2 , while the colors over the two-dimensional panel show the phyllotactic pattern distribution against Γ and A in EDC2 ., The order of distribution of the distichous , Fibonacci spiral , and decussate patterns was unaffected by decreasing A and , thus , did not differ between DC2 and EDC2 ., As reported in the previous study of DC2 17 , on the top side of Fig 9 , the stable pattern changed from distichous to Fibonacci spiral , and then turned into decussate as Γ decreased ., In the parameter space of EDC2 , this order of distribution of major phyllotactic patterns was not affected much by decreasing A to moderate values; however , when A was further decreased , the orixate pattern appeared in the region of the Fibonacci spiral ( Fig 9 , S10 Fig ) ., As A decreased , the range of Γ 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 ., Orixate phyllotaxis is a special kind of alternate phyllotaxis with orthogonal tetrastichy resulting from a four-cycle change in the divergence angle in the order of approximately 180° , 90° , −180° ( 180° ) , and −90° ( 270° ) ; this phyllotaxis occurs in a few plant species across distant taxa 29–32 ., In the present study , we investigated a possible theoretical framework behind this minor but interesting phyllotaxis on the basis of the inhibitory field models proposed by Douady and Couder 16 , 17 , which were shown to give a simple and robust explanation for the self-organization process of major phyllotactic patterns by assuming that each existing leaf primordium emits a constant level of inhibitory power against the formation of a new primordium and that its effect decreases with distance from the primordium ., Re-examination of the original versions of Douady and Couder’s models ( DC1 and DC2 ) via exhaustive computer simulations revealed that they do not generate the orixate pattern at any parameter condition ., The inability of DC models to produce orixate phyllotaxis prompted us to expand them to account for a more comprehensive generation of phyllotactic patterns ., In an attempt to modify DC models , we introduced a temporal change in the inhibitory power during primordium development , instead of using a constant inhibitory power ., Such changes of the inhibitory power were partly considered in several previous studies ., Douady and Couder assessed the effects of “the growth of the element’s size” , which is equivalent to the primordial age-dependent increase in the inhibitory power and found that it stabilizes whorled phyllotactic patterns 17 ., Smith et al . assumed in their mathematical model that the inhibitory power of each primordium decays exponentially with age and stated that this decay promoted phyllotactic pattern formation de novo , as well as pattern transition , and allowed the maintenance of patterns for wider ranges of parameters 9 ., A DC1-based model equipped with a primordial age-dependent change in the inhibitory power was also used to investigate floral organ arrangement 35 , 36 ., In these studies , however , temporal changes in the inhibitory power were examined under limited ranges of parameters focusing on particular aspects of phyllotactic patterning , and the possibility of the gener
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 from Orixa 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 .
plant anatomy, buds, computerized simulations, mathematical models, hormones, plant science, plant hormones, microscopy, plants, flowering plants, research and analysis methods, computer and information sciences, mathematical and statistical techniques, scanning electron microscopy, leaves, biochemistry, plant biochemistry, computer modeling, eukaryota, electron microscopy, biology and life sciences, auxins, organisms