1. Developmental Biology and Stem Cells
  2. Plant Biology
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A stochastic multicellular model identifies biological watermarks from disorders in self-organized patterns of phyllotaxis

  1. Yassin Refahi
  2. Géraldine Brunoud
  3. Etienne Farcot
  4. Alain Jean-Marie
  5. Minna Pulkkinen
  6. Teva Vernoux Is a corresponding author
  7. Christophe Godin Is a corresponding author
  1. Laboratoire de Reproduction de développement des plantes, France
  2. University of Cambridge, United Kingdom
  3. The University of Nottingham, United Kingdom
  4. INRIA Sophia-Antipolis Méditerranée Research Center, France
  5. AgroParisTech, France
  6. CIRAD, INRA and INRIA Sophia-Antipolis Méditerranée Research Center, France
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Cite as: eLife 2016;5:e14093 doi: 10.7554/eLife.14093

Abstract

Exploration of developmental mechanisms classically relies on analysis of pattern regularities. Whether disorders induced by biological noise may carry information on building principles of developmental systems is an important debated question. Here, we addressed theoretically this question using phyllotaxis, the geometric arrangement of plant aerial organs, as a model system. Phyllotaxis arises from reiterative organogenesis driven by lateral inhibitions at the shoot apex. Motivated by recurrent observations of disorders in phyllotaxis patterns, we revisited in depth the classical deterministic view of phyllotaxis. We developed a stochastic model of primordia initiation at the shoot apex, integrating locality and stochasticity in the patterning system. This stochastic model recapitulates phyllotactic patterns, both regular and irregular, and makes quantitative predictions on the nature of disorders arising from noise. We further show that disorders in phyllotaxis instruct us on the parameters governing phyllotaxis dynamics, thus that disorders can reveal biological watermarks of developmental systems.

https://doi.org/10.7554/eLife.14093.001

eLife digest

Plants grow throughout their lifetime, forming new flowers and leaves at the tips of their stems through a patterning process called phyllotaxis, which occurs in spirals for a vast number of plant species. The classical view suggests that the positioning of each new leaf or flower bud at the tip of a growing stem is based on a small set of principles. This includes the idea that buds produce inhibitory signals that prevent other buds from forming too close to each other. When computational models of phyllotaxis follow these ‘deterministic’ principles, they are able to recreate the spiral pattern the buds form on a growing stem.

In real plants, however, the spiral pattern is not always perfect. The observed disturbances in the pattern are believed to reflect the presence of random fluctuations – regarded as noise – in phyllotaxis. Here, using numerical simulations, Refahi et al. noticed that the patterns of inhibitory signals in a shoot tip pre-determine the locations of several competing sites where buds could form in a robust manner. However, random fluctuations in the way cells perceive these inhibitory signals could greatly disturb the timing of organ formation and affect phyllotaxis patterns.

Building on this, Refahi et al. created a new computational model of bud patterning that takes into account some randomness in how cells perceive the inhibitory signals released by existing buds. The model can accurately recreate the classical spiral patterns of buds and also produces occasional disrupted patterns that are similar to those seen in real plants. Unexpectedly, Refahi et al. show that these ‘errors’ reveal key information about how the signals that control phyllotaxis might work.

These findings open up new avenues of research into the role of noise in phyllotaxis. The model can be used to predict how altering the activities of genes or varying plant growth conditions might disturb this patterning process. Furthermore, the work highlights how the structure of disturbances in a biological system can shed new light on how the system works.

https://doi.org/10.7554/eLife.14093.002

Introduction

Developmental systems strikingly produce regular patterns and analysis of eukaryote development has classically been focused on regularities as the main source of information to understand these complex systems. However, it is becoming increasingly evident that intrinsic molecular noise is an inherent property of biological systems (Elowitz et al., 2002; Kupiec, 1997; Lander, 2011). This noise can be buffered, e.g. (Okabe-Oho et al., 2009), but can also theoretically propagate through scales and generate patterning disorders e.g. (Itoh et al., 2000). In this case, disorders observed during development could be informative not only on the origin of noise but also on the underlying developmental mechanisms that propagate the noise. Here we address this question theoretically using phyllotaxis, the remarkably regular geometric organization of plant aerial organs (such as leaves and flowers) along the stem, as a model system (Appendix section 1).

Phyllotaxis primarily arises at the shoot apical meristem, a specialized tissue containing a stem cell niche and located at the tip of growing shoots. Rooted in early works of pioneers such as (Bonnet, 1754; Braun, 1831; Bravais and Bravais, 1837) and after decades of research, the idea that phyllotactic patterns emerge from simple physical or bio-chemical lateral inhibitions between successive organs produced at the meristem has become largely prevalent, (Adler et al., 1997; Jean, 1995; Kuhlemeier, 2007; Pennybacker et al., 2015; Reinhardt, 2005). Microscopic observations and modeling led to propose that this self-organizing process relies on five basic principles: i) organs can form only close to the tip of growing shoots, ii) no organ can form at the very tip, iii) pre-existing organs prevent the formation of new organs in their vicinity (Hofmeister, 1868), forming altogether an inhibitory field that covers the organogenetic zone, iv) due to growth, organs are progressively moved away from the organogenetic zone, v) a new organ is formed as soon as the influence of the inhibitory field produced by the existing organs fades away at the growing tip, (Schoute, 1913; Snow and Snow, 1962; Turing, 1952). Computer simulations were used to analyze the dynamical properties of an inhibitory field model relying on these assumptions (Mitchison, 1977; Thornley, 1975; Veen and Lindenmayer, 1977; Young, 1978) and many others after them, including (Chapman and Perry, 1987; Douady and Couder, 1996a; Green et al., 1996; Meinhardt, 2003; Schwabe and Clewer, 1984; Smith et al., 2006b). In a detailed computational analysis (Douady and Couder, 1996a; 1996b; 1996c), Douady and Couder demonstrated the ability of such models to recapitulate a wide variety of phyllotactic patterns in a parsimonious way and that these patterns are under the control of a simple geometric parameter corresponding to the ratio between the radius of organ inhibitory fields and the radius of the central zone (area at the very tip where no organ can form). This modeling framework thus provides a deterministic theory of self-organizing patterns in the meristem characterized by a global geometric parameter, capturing macroscopic symmetries and orders emerging from lateral inhibitions, e.g. (Adler, 1975; Atela, 2011; Newell et al., 2008; Smith et al., 2006b). In the sequel, we will refer to this widely accepted view as the classical model of phyllotaxis.

In recent years, plausible molecular interpretations of the abstract concepts underlying the classical model have been proposed. They mainly rely on distribution in the meristem of the plant hormone auxin, a central morphogenetic regulator. Auxin is actively transported at the meristem surface, notably by both PIN-FORMED1 (PIN1) polar efflux carriers and non-polar influx carriers (AUX/LAX family). These transporters form a dynamic network that permanently reconfigures and that periodically accumulates auxin at specific locations on the meristem flanks (the organogenetic domain), initiating organ primordia (Reinhardt et al., 2003). By attracting auxin, the growing primordium depletes auxin in its vicinity, thus preventing organ formation in this region. This mechanism is now thought to be at the origin of the predicted inhibitory fields in the meristem (Barbier de Reuille et al., 2006; Brunoud et al., 2012; Jönsson et al., 2006; Smith et al., 2006a; Stoma et al., 2008). The range of this inhibition corresponds to one of the two key parameters of the classical model: as primordia get away from the tip, inhibition is relaxed and auxin can accumulate again to initiate new primordia. For the second key parameter, i.e. the size of the apical domain in which no organ can form, it has been suggested that the very tip of the meristem contains significant quantities of auxin but is actually insensitive to auxin due to a down-regulation of the effectors of transcriptional auxin signaling (Barbier de Reuille et al., 2006; Vernoux et al., 2011). A low auxin sensitivity then participates in blocking organ initiation in the central domain (where the stem cells are located) of the meristem. These molecular insights support the hypothesized structure of the classical model.

Comparatively, little attention has been paid as of today to disorders in phyllotaxis (Jean, 2009; Jeune and Barabé, 2004). However, in the recent years, the presence of irregularities in phyllotactic patterns has been repetitively observed in various genetic backgrounds (Besnard et al., 2014; Couder, 1998; Douady and Couder, 1996b; Guédon et al., 2013; Itoh et al., 2000; Landrein et al., 2015; Leyser and Furner, 1992; Mirabet et al., 2012; Peaucelle et al., 2011; Prasad et al., 2011; Refahi et al., 2011), suggesting that phyllotaxis has a non-deterministic component. In some cases, the departure from any known regular pattern is so strong that plant phyllotaxis is considered random, e.g. (Itoh et al., 2000). Recently, strong disorders have been observed and quantified in spiral patterns of Arabidopsis thaliana wild-type and arabidopsis histidine phosphotransfer protein (ahp6) mutants (Besnard et al., 2014), Figure 1. Surprisingly, a structure could be found in these disorders that corresponds to either isolated or series of permutations in the order of lateral organs along the stem when taking a perfect spiral with divergence angle 137.5° (golden angle ϕ) as a reference. Live-imaging of meristems showed that organs are sometimes co-initiated in the meristem, leading randomly to post-meristematic organ order permutations in around half of the cases. This phenomenon showed that, while the azimuthal directions of the organs are highly robust, the time between consecutive organ initiation (or plastochron) is variable, Figure 1G. Taken together, these observations called for revisiting in depth models of phyllotaxis to account for disorders in this self-organizing developmental system.

Irregularity in phyllotaxis patterns.

(A) wild type inflorescence of Arabidopsis thaliana showing regular spiral phyllotaxis. (B) aph6 mutant inflorescence showing an irregular phyllotaxis: both the azimuthal angles and the distances between consecutive organs are largely affected. (C1) Organ initiation in the wild type: the size of organs is well hierarchized, initiations spaced by regular time intervals. (C2) Organ initiation in the ahp6 mutant: several organs may have similar sizes, suggesting that they were initiated simultaneously in the meristem (co-initiations). (D) A typical sequence of divergence angles in the WT: the angle is mainly close to (≈137°) with possible exceptions (M-Shaped pattern). (E) In ahp6, a typical sequence embeds more perturbations involving typically permutations of 2 or 3 organs. (F–I) Frequency histogram of divergence angle: wild type (F); ahp6 mutant (G); WS-4, long days (H); WS-4 short days - long days (I).

https://doi.org/10.7554/eLife.14093.003

Here, we show that the same disorders, the permutations, occur in various plant species, suggesting noisy plastochrons are a characteristic of phyllotactic systems at the origin of pattern disorders. In addition, we demonstrate that inhibitory fields pre-specify a number of organogenesis sites, suggesting noise on inhibition perception as the most likely origin of disorders. Building on this observation, we developed a stochastic model of organ initiation that is fully local and relies on a stochastic modeling of cell responses to inhibitory fields. Our stochastic model fully and precisely captures the observed dynamics of organogenesis at the meristem, recapitulating both regular and irregular phyllotactic patterns. We show that the stochastic model also makes quantitative predictions on the nature of the perturbations that may arise due to different genetic and growth manipulations. Most importantly, we demonstrate that disorders in phyllotactic patterns instruct us on the parameters governing the dynamics of phyllotaxis. Disorders can thus provide access to the biological watermarks corresponding to the parameter values of this self-organizing system, providing a striking example where disorders inform on mechanisms driving the dynamics of developmental systems.

Results

The shoot architecture of a variety of plant species suggests that disorder is a common phenomenon in phyllotaxis

As permutations have been notably reported in Arabidopsis (Besnard et al., 2014; Guédon et al., 2013; Landrein et al., 2015; Refahi et al., 2011) and in sunflower (Couder, 1998), we sampled a variety of unrelated species in the wild and searched for permutations. We could easily find permutations in several other Brassicaceae showing spiral phyllotaxis as well as in either monocotyledonous or dicotyledonous species from more distant families such as Asparagaceae, Sapindaceae or Araliaceae (Figure 2) (Appendix section 1). As suggested by the results on Arabidopsis, these observations raise the possibility that these organ permutations result from a noise on the plastochron and that such perturbation could be a common feature of phyllotactic systems that occurs in meristems with different geometries. In addition this disorder is probably under complex genetic control as different unrelated genetic modifications (including the Arabidopsis ahp6 mutant) can modulate its intensity.

Permutations can be observed in various species with spiral phyllotaxis.

A schema in the bottom right corner of each image indicates the rank and azimuthal directions of the lateral branches. The first number (in yellow, also displayed on the picture) indicates the approximate azimuthal angle as a multiple of the plant’s divergence angle (most of the times close to 137° or 99°). The second number (in red) corresponds to the rank of the branch on the main stem. (A) Brassica napus (Inflorescence) (B) Muscari comosum (Inflorescence) (C) Alliara petiolata (D) Aesculus hippocastanum (Inflorescence) (E) Hedera Helix (F) Cotinus Dummeri.

https://doi.org/10.7554/eLife.14093.004

The classical deterministic model suggests inhibition perception as the most likely origin of disorders

To understand how the timing of organ initiation could be affected during meristem growth, we first analyzed the relative stability of inhibitory field minima in the classical deterministic model. For this, we implemented a computational version of the classical model based on (Douady and Couder, 1996b) (Appendix section 2). Primordia are created on the meristem surface at the periphery of the meristem central zone, at a fixed distance R from the meristem center. Once created, primordia drift away radially from the central zone at a speed proportional to their distance from the meristem center. As soon as created, a primordium q of radius r0, generates an inhibitory field, E(q) in its neighborhood such that at a point x, at a distance d(q,x)>r0, the inhibition due to q decreases with the distance to q:E(q)(x)=(r0d(q,x))s, where s is a geometric stiffness parameter (E(q) is often regarded as an inhibitory energy emitted by primordium q, e.g. Douady and Couder, 1996b). As a result, at any moment and for any point x of the meristem surface, the existing primordia create altogether a cumulated inhibition E(x) that is the sum of the individual primordium contributions: E(x)=qQE(q)(x), where Q denotes the set of all preexisting primordia. High inhibition levels on the peripheral circle prevent the initiation of new primordia at corresponding locations. However, as the preexisting primordia are moving away from the central zone during growth, the inhibitory level tends to decrease at each point of the peripheral circle. A new primordium initiates where and when the inhibitory field is under a predefined threshold on the peripheral circle.

We then performed a systematic analysis of the inhibition profiles and their dynamics along the peripheral circle and observed the following properties (Figure 3, Video 1,2):

  • Property 1: let i and ji<j, denote the numbers of contact spirals that one can observe on a stem in clockwise and counter-clockwise directions (Appendix section 1), the number of local minima l in the inhibition profile is bounded by i and j:ilj: (Figure 3A–B).

  • Property 2: The angular distance between local minima is a multiple of the divergence angle (Figure 3A–B).

  • Property 3: The difference between the inhibition values of local minima decreases monotonically with the control parameter Γ=r0/R (Figure 3C)

In addition, we noticed that depending on time, the difference in inhibition level between consecutive local minima may markedly vary. In some cases this difference is so small that a biological noise may lead the biological system to perceive the ordering between two or more local minima differently from the ordering of the actual inhibition levels. Such errors would lead to initiate several primordia together or to change the temporal order of their initiation, thus inducing perturbations in the sequence of divergence angle. Also, as suggested by property 3, the number of primordia initiation events affected by these errors would decrease with the Γ parameter and would thus depend on the geometry of the meristem.

Figure 3 with 1 supplement see all
Properties of the inhibition profiles in the classical model and effect of a forced perturbation on divergence angles and plastochrons.

(A) Inhibition variation (logarithmic scale) along the peripheral circle and its global and local minima for a control parameter Γ1 = 0.975. EkE1 is the difference in inhibition levels between the kth local minimum and the global minimum. The angular distances between the global minimum and the kth primordiu inhibition m are multiple of the canonical angle (α=137°). (B) Similar inhibition profile for a control parameter Γ2 = 0.675 < Γ1. The difference EkE1 in inhibition levels is higher than in A. (C) Variation of the distance E2E1 between the global minimum of the inhibition landscape and the local minimum with closest inhibition level (i.e. the second local minimum), as a function of Γ. (D) Inhibition profile just before an initiation at azimuth 80° and (E) just after. (F) Variation of inhibition profile in time. As the inhibition levels of local minima decrease, their angular position does not change significantly, even if new primordia are created (peak qn), color code: dark red for low inhibition and dark blue for high inhibition values. (G) Sequence divergence angles between initiations simulated with the classical model (control parameter Γ1). At some point in time (red arrow), the choice of the next initiation is forced to occur at the 2th local minimum instead of the global minimum. After the forcing, the divergence angle makes a typical M-shaped pattern and returns immediately to the α baseline. (H) Corresponding plastochrons: the forcing (red arrow) induces a longer perturbation of the time laps between consecutive organs.

https://doi.org/10.7554/eLife.14093.005
Video 1
Temporal variation of the inhibitory profile around the central zone in the classical model for a large value of the parameter Γ.

The number of inhibition mimima is stable (3) in time. When the absolute minimum reaches the initiation threshold (here E = 0), a primordium is created that instantaneously creates a strong inhibition locally, which suddenly increases the inhibition level at its location. Between initiations, local minima regularly decrease in intensity due to the fact that growth is moving existing primordia away from the center. This movement is accompanied by a slight drift in position common to all primordia (here to the right).

https://doi.org/10.7554/eLife.14093.007
Video 2
Temporal variation of the inhibitory profile around the central zone in the classical model for a small value of the parameter Γ.

The dynamics is similar to that of small Γ except that the number of local minima of inhibition is higher (here 5) and that the distance between two consecutive minima is lower.

https://doi.org/10.7554/eLife.14093.008

To investigate this possibility, we started to induce a perturbation in a stationary spiral pattern by forcing at a given time the system to initiate a primordium at the site of second local minimum instead of that of the global minimum (i.e. at an angle 2ϕ, Figure 3G, red arrow). The system was then left free to self-organize. We observed that the next primordium was always initiated at the site of the original global minimum, resulting in a divergence angle ϕ and a quasi-null plastochron (Figure 3H). The system was then able to recover from the perturbation by initiating the next primordium at the originally expected site (i.e. with a divergence angle 2ϕ), with a long plastochron, leading to a M-shaped pattern (Figure 3G, Video 3). The rest of the divergence angle sequence was then not affected and remained at a value close to ϕ while long oscillatory perturbations were observed on plastochrons (Figure 3H). Stronger perturbations induced by forcing various other local minima to initiate instead of the global minimum (Figure 3—figure supplement 1 and Appendix section 6.6 for related control simulation experiments) similarly demonstrated that the system spontaneously makes a short distorted pattern and then returns to the normal ϕ baseline in every case. We concluded that i) a noise in the perception of the local primordium order does not propagate far in the divergence angle sequence and that angle specification in the classical model patterning system is highly robust to perturbations in local minima initiation ordering and ii) however, the plastochron itself is affected during a much longer time span.

Video 3
Temporal variation of the inhibitory profile around the central zone in the stochastic model for a small value of the parameter Γ.

Here, due to stochasticity, global minimum is not always the one that triggers an initiation. The dynamics of the divergence angle and of the plastochron are shown in the bottom graphs to interpret the model's initiations based on the inhibition levels.

https://doi.org/10.7554/eLife.14093.009

Together, these results suggest that time and space in primordium initiation are largely decoupled in inhibitory field-driven self-organization as locations of primordia are strongly pre-specified and relatively stable, while plastochrons are not. This observation is in line with the observations from live-imaging of Arabidopsis shoot apical meristem that demonstrated that despite variability in the plastochron (with almost 30% of organs co-initiated) the specification of initiation sites was extremely robust (Besnard et al., 2014).

Organ initiation can be modeled as a stochastic process

Our previous results suggest that high variability in the timing of organ initiation could result from the joint effect of noise in the perception of inhibitory fields and of the decoupling between space and time in this self-organizing system. We therefore decided to revisit the inhibitory field models to integrate locality and stochasticity as central components in the patterning system. For this, we kept from the classical deterministic model the assumptions related to the movement of primordia at the meristem through growth and to the definition of inhibitory fields. However, we completely reformulated the way primordia are initiated as local stochastic processes.

At any time t, the K cells that make up the periphery of the central zone potentially may take the identity of a primordium, depending on the local value of the inhibitory field (signaling) in each cell. We assume that this switch in cell identity does not depend on a threshold effect as in the classical model but, rather, depends on the cellular perception of the inhibitory signal which, in essence, is stochastic (Eldar and Elowitz, 2010). We thus assumed that each cell k reads out the local inhibitory field value, Ek(t), and switches its state to primordium identity with a probability that (i) depends on the level of inhibition Ek(t); (ii) is proportional to the amount of time δt the cell is exposed to this level of signaling (Gillespie, 1976; 1977):

(1) P(Xk(t,δt)=1)=λ(Ek(t))δt,

where Xk(t,δt) denotes the number of primordia initiated at cell k in the time interval [t,t+δt] (Xk(t,δt) will typically have a value 0 or 1), and λ is a rate parameter that depends on the local inhibition value Ek(t) at cell k and that can be interpreted as a temporal density of initiation. To express the influence of inhibitory fields on the probability of initiation, the dependence of λ on the local inhibition Ek(t) must respect a number of general constraints: i) λ must be a decreasing function of the inhibition as the higher the inhibition level at one site, the lower the probability to observe an initiation at this site during δt; ii) for small δt, for P(Xk(t,δt)=1) to be a probability, we must have 0λ(Ek(t))δt1; iii) the ratio of the probabilities to trigger an initiation at two different sites is a function of the difference in inhibition levels between these sites. Under all these constraints, the rate parameter takes the following form (see Appendix section 3 for a detailed derivation of the model):

(2) λ(Ek(t))=eβ(Ek(t)E),

where E is a parameter controlling the sensitivity of the system to inhibition, and β is a parameter controlling the ability of the system to discriminate between inhibition levels, i.e. to respond differently to close inhibition levels (acuity). Therefore, for each cell k, the probability to initiate a primordium during a small time interval δt can be expressed as:

(3) P(Xk(t,δt)=1)=eβ(Ek(t)E)δt.

If we now assume that the probabilities to observe an initiation at a site k in disjoint time intervals are independent, the process described by Equation 3 is known as a non-homogeneous Poisson process of intensity λ(Ek(t)) (e.g. Ross, 2014). Therefore for each cell k of the periphery, our model assumes that the probability to initiate a primordium is a non-homogeneous Poisson process, whose parameter is regulated by the local level of the inhibitory field at that site.

This stochastic formulation of the model at the level of cells, called SMPmicro (Stochastic Model of Phyllotaxis at microscopic level), makes it possible to develop the calculus of different key quantities or properties of the system. For example, if we assume that recruitments of cells for organ initiation are stochastically independent from each other (the probability to draw an initiation at a site k only depends on the value of the local parameters at site k but not on what may be drawn at other places), then we can estimate the expected number of cells independently recruited for organ initiation during the timespan δt. Let us denote X(t,δt) the number of cells initiated along the peripheral circle, and for a given time t and a small time span δt, X(t,δt)=k=1KXk(t,δt). Its expectation is simply the sum of the expectations of the individual independent Poisson processes:

(4) E(X(t,δt))=k=1KE(Xk(t,δt))=k=1KP(Xk(t,δt)=1)=δtk=1Keβ(Ek(t)E),

therefore giving us an estimate of the expected number of peripheral cells initiated during time δt and for the inhibition profile E(t).

So far we have considered peripheral cells as independent sites that may independently switch to primordium identity with a probability that depends on their local level of inhibition. As a local inhibition valley may span over several cells, one might expect that the probability to trigger a primordium initiation in a valley is increased by the fact that several founder cells can potentially contribute to this initiation during time δt. To formalize this, we then upscaled our stochastic model at the level of valleys where a stochastic process is now attached to each local minimum l instead of to each cell k. This upscaled model is called SMPmacro. The idea is that the stochastic processes of all the cells k spanned by a local inhibition valley l sum up and together define a stochastic process Nl that a primordium is initiated in l at a higher level. Being the sum of independent Poisson processes, this upscaled process is also a Poisson process with intensity Λl=kKlλk(Ek(t)), where k varies over the set of cells spanned by the valley of the l th local minimum and indexed by Kl. If L denotes the number of local minima, then the expected number of primordia initiations during a small time laps δt is thus:

(5) E(Nl(t,δt))=δtl=1LΛl(t).

Therefore, at microscopic scale, SMPmicro couples i) a deterministic part inherited from the classical model and related to the geometry and the dynamics of the fields and ii) a new stochastic part related to the perception of this inhibitory field, i.e. representing signal perception capacities. It relies on the assumption that the perception sites, corresponding to the cells surrounding the central zone periphery, are stochastically independent of each other. Decisions regarding primordium initiation are taken in a cell-autonomous manner, thus reflecting more realistically the outcome of the initiation signaling pathway in each cell. At a macroscopic level, in each inhibition valley several cells may trigger primordium initiation. The probability to trigger an initiation increases in SMPmacro with the size of the valley when more than one founder cell are likely to contribute to initiation. A variant of this upscaled model consists of defining the probability for a valley to initiate a primordium by the probability of the cell with lowest level of inhibition in this valley. In this variant, called SMPmacro-max, Λl=maxkKl λk(Ek(t)).

Stochastic modeling simulates realistic phyllotaxis sequences

To study the emergent properties of this system, we implemented a computational version of SMPmacro-max and tested its sensitivity to parameter changes. In addition to the geometrical parameters of the classical model, two new parameters, β and E now reflect the ability of the system to perceive the inhibitory signal from the fields (or equivalently the initiation signal).

As expected from a phyllotaxis model, the stochastic model is able to produce both spiral and whorl modes, from either imposed initial distributions of organs (Figure 4A–C), or random starting points (Appendix section 6.1). However, the great majority of the sequences of divergence angle generated for different values of Γ=r0/R displayed divergence angle perturbations (i.e. angles different from those predicted by the classical model with the same Γ) of the type observed in Figure 4D–E. As suggested by their typical distributions (Figure 4F–H left), these perturbations correspond to permutations very similar to those observed on real plants in previous studies (Besnard et al., 2014; Landrein et al., 2015) with the appearance of secondary modes at multiples of 137, Figure 1F–I. The amplitude of these secondary modes is correlated with the amount of perturbations in the sequences.

Patterns generated by the stochastic model. 

(A) The model generates spiral patterns (in (A–C), up: sequence of simulated divergence angles, down: corresponding plastochrons). (B–C) and whorled patterns. (D) Simple M-shaped permutations simulated by the stochastic model (β = 10.0, E = 1.4, Γ = 0.625). (E) More complex simulated permutations involving 2- and 3-permutations (β = 10.0, E= 1.4, Γ = 0.9). The permutations are here: [4, 2, 3], [14, 13, 12], [16, 15], [19, 18]. (F) Typical histogram of simulated divergence angles and corresponding plastochron distribution for β = 11.0, E =1.2, Γ= 0.8. (G) Histogram of simulated divergence angles and corresponding plastochron distribution for β = 9.0, E = 1.2, Γ = 0.8 (H) Histogram of simulated divergence angles and corresponding plastochron distribution for β = 9.0, E = 1.2, Γ = 0.625.

https://doi.org/10.7554/eLife.14093.010

To complete this analysis, we looked at plastochron distributions in the simulated sequences (Figure 4F,G,H right). In all cases, distributions were displaying a single mode largely spread along the x-axis. Interestingly, the more sequences were perturbed, the more negatively skewed the distributions, showing thus a higher occurrence of short plastochrons (Figure 4H). This is reminiscent of the observation of co-initiations in growing meristems associated with perturbed phyllotaxis (Besnard et al., 2014; Landrein et al., 2015). The stochastic model is thus able to produce perturbed sequences with realistic series of divergence angles and corresponding realistic distributions of plastochrons.

The proportion of complex versus simple disorders in phyllotaxis sequences depends on the global amount of phyllotaxis disorders

We then aimed to quantitatively assess the complexity level of permutations as a proxy for plastochron noise. For this we focused on spiral phyllotaxis modes and used two measures: the density of permuted organs respectively involved in 2 and 3-permutations π2=2.σ2/Σ and π3=3.σ3/Σ where σ2 and σ3 are respectively the number of 2- and 3-permutations in the sequence and Σ the total number of organs in the sequence. The quantities σ2 and σ3 were estimated a posteriori from simulated sequences using the algorithm described in (Refahi et al., 2011). The total density of permuted organs involved either in 2- or 3 permutations is denoted by π=π2+π3.

We first explored the intensity of permutations of different natures (2- and 3-permutations) in simulated phyllotaxis sequences. For this, we carried out simulations for a range of values for each parameter Γ,β,E (Figure 5—source data 1). These values can be regarded as predictions of the model for each triplet (Γ,β,E). Remarkably, the simulations show that the values of π,π2 and π3 are not linearly related to each other. To make this relationship explicit, we plotted the proportion of organs involved in 2-permutations π2 as a function of the total proportion of perturbed organs π (Figure 5). Surprisingly, the points, when put together on a graph, were organized in a narrow crescent showing a convex curve-like relationship between π and π2, revealing a remarkable property of the stochastic model: the more perturbations there are in the simulated sequences, the higher the proportion of 3-permutations, and this independently of the model parameters. The fact that this non-linear relationship emerges from a fairly large sampling of the parameter space suggests that it can be considered as a key observable property characterizing the model’s underlying structure.

We tested this by gathering all measured values of π2 and π3 published in the literature for Arabidopsis (Besnard et al., 2014; Landrein et al., 2015) and plotted the corresponding points on the original graph showing the model’s simulations (Figure 5, red crosses). The measured points fall within the range of predicted values and show that the measured values follow the same non-linear variation as the one predicted by the model: the larger the total percentage of permutations, the larger the proportion of 3-permutations in the sequences. This confirms a first prediction from the stochastic model and indicates that disorder complexity increases non-linearly with the frequency of disorders.

Intensity of 2-permutations as a function of the total amount of perturbations.

As the perturbation intensity π increases, the percentage of 2-permutations decreases in a non-linear way to the benefit of more complex 3-permutations. The diagonal line denotes the first bisector. In red: values of 2- and 3-permutations observed in different mutants and ecotypes of Arabidopsis thaliana (Besnard et al., 2014; Landrein et al., 2015 and this study) placed on the plot of values predicted by the stochastic model.

https://doi.org/10.7554/eLife.14093.011

The amount of disorders in a sequence depends on both geometry and signal perception

Based on our simulations, we then investigated the variations of these perturbations as a function of the model parameters Γ,β.E and s. As a general trend, for a value of s fixed to 3 as in (Douady and Couder, 1996b), we noticed that for a given value of Γ, an increase in β was roughly counteracted in terms of permutation intensity by a decrease in E. Likewise, an increase of Γ was canceled by a decrease in β. We gathered all these observations in one graph and plotted the global amount of perturbation π in a sequence as a function of a combination of the three original model parameters and reflecting the observed trend: ΓP=ΓβE. In the resulting graph (Figure 6A), each point corresponds to a particular instance of the three model parameters. The points form a narrow and decreasing band associating a small set of possible perturbation intensities π with each value of ΓP. For combinations of the three model parameters leading to a small ΓP, the perturbations can affect up to 50% of the organs whereas for high values of ΓP, there may be no perturbation at all. Interestingly, ΓP being a combination of the three elementary model parameters, plants having identical geometrical parameter Γ may show substantially different intensities of perturbations if their perception for different values of parameters β,E. Consequently, since the intensity of perturbation in the system is more simply reflected by ΓP than by values of Γ,β,E taken independently, ΓP can be considered to be a control parameter for perturbations.

Figure 6 with 1 supplement see all
Key parameters controlling phyllotaxis phenotypes in the stochastic model.

Phyllotaxis sequences were simulated for a range of values of each parameter β, E, Γ. Each point in the graph corresponds to a particular triplet of parameter values and represents the average value over 60 simulated sequences for this triplet. (A) Global amount of perturbation π as a function of the new control parameter ΓP=ΓβE. (B) Divergence angle α as a function of the control parameter Γ of the classical model on the Fibonacci branch. (C) Divergence angle α as a function of the new control parameter ΓD=Γ1β1/6E1/2 on the Fibonacci branch (here, we assume s = 3, see Appendix 1—figure 6 for more details). (D) Plastochron T as a function of control parameter of the classical model Γ. (E) Plastochron T as a function of the new control parameter ΓD. (F) Parastichy modes (i,j) identified in simulated sequences as a function of ΓD. Modes (i,j) are represented by a point i+j. The main modes (1,2), (2,3) … correspond to well marked steps. (Figure 5—source data 1)

https://doi.org/10.7554/eLife.14093.013

Both divergence angles and plastochrons are controlled by a unique combination of the geometric and perception parameters

To further investigate the structure of the stochastic model, we then studied how the usual observable quantities of a phyllotaxis system, i.e. divergence angles and plastochrons, depend on the model parameters Γ,β,E and s.

In the classical deterministic model, divergence angles are a function of a unique control parameter Γ=r0/R (Douady and Couder, 1996b), meaning that the same divergence angle can be obtained in the model for different couples of r0 and R provided that their ratio is unchanged. We thus checked whether Γ could also serve as the control parameter for the divergence angles and plastochrons in the stochastic model. For this, we simulated various spiral phyllotaxis sequences (from the Fibonacci branch where divergence angles are close to 137°) by varying Γ,β,E and estimated the corresponding divergence angle α and plastochron T, Figure 6B,D. We observed that, although the point clouds evoke the corresponding curves in the standard deterministic model (Appendix 1—figure 2), significantly different values of the divergence angles (or plastochrons) can be observed for many particular values of Γ. This phenomenon suggests that Γ is not a satisfactory control parameter in the stochastic model: for a given value of Γ, varying β or E can significantly modify the divergence angle or the plastochron. We therefore tested various combinations of these parameters (Figure 6—figure supplement 1), and finally found that for ΓD=Γs/31β1/6E1/2 both cloud of points collapse into a single curves (Figure 6B–C, D–E and Appendix section 6.3). For this definition of ΓD, each value of the control parameter can be associated with quasi-unique divergence angle and plastochron, i.e. defines a precise observable state of the system.

We then checked whether parastichies were also controlled by ΓD. For this, we computed the parastichies (i,j) corresponding to each simulated sequence and plotted the sum, i+j, as a function of ΓD (Figure 6F). The resulting points were arranged on a stepwise curve, where each step corresponds to a Fibonacci mode: (1,2), (2,3), (3,5), (5,8). Each value of ΓD thus defines a precise value of the mode.

We concluded that ΓD can be considered as a second control parameter of the stochastic model, relating the system’s state to the observable variables α and T through a unique combination of the system’s parameters.

Observable variables convey key clues on the state of the phyllotaxis system

According to the stochastic model, a particular phyllotaxis system is characterized by a particular set of values of the parameters Γ,β,E. Upon growth, a specific spatio-temporal dynamics emerges that is characterized by observable variables: divergence angle, plastochron and frequency of permutations. In the current state of our knowledge and measuring means, the parameters Γ,β,E are not directly observable. Therefore, we investigated what can be learned about them from the observable variables.

For this, we can use the relationships established above between the control parameters ΓP and ΓD and the observable variables α,T,π2,π3, .... For a genotype G, we have:

(6) ΓP=ΓβEΓD=Γs/31β1/6E1/2

Using the characteristic curves of Figure 6C,E, both measurements of α and T give possible estimates for ΓD. In a consistent model, these estimates should be compatible. Similarly, an estimate of ΓP can be derived from the observation of π. In the case of the WT for example, ΓD is estimated to be in [0.425, 0.5] and ΓP in [8.30, 11.49] according to the observed value of the plastochron, divergence angle and permutation intensity. This value is itself derived from the measurement of the plastochron ratio ρ (Appendix section 4.1.3).

Based on estimated values of ΓD and ΓP, equations 6 define a system of 2 equations and 3 unknowns (s was fixed to 3 in the simulations). This system is underdetermined and does not allow us to identify exactly the values of Γ,β,E. However, being underdetermined in one dimension only, this system plays the role of a generalized control parameter: a given value of the generalized control parameter (ΓDΓP) determines the parameters Γ,β,E up to one degree of freedom. If one of the parameters is given, then the others are automatically determined according to equation 6.

Experimental observation of anti-correlated variations in disorder and plastochron is interpreted by the model as a change in inhibitory field geometry

Several recent works demonstrated that mutations or changes in growth conditions could alter phyllotaxis and disorder patterns. Our model predicts correlations between main observable phyllotaxis variables. According to the model, plastochron positively correlates with ΓD (Figure 6E). If signaling is not altered by the experimental setup (β and E are unchanged) then ΓD positively correlates with Γ=r0/R (Equation 6). Therefore, the model predicts that, like in the classical model, plastochron positively correlates with r0 (size of the primordia inhibitory fields) and negatively correlates with R (size of the central zone). However, the model makes also in this case the new prediction that plastochron negatively correlates with the frequency of observed permutations (Figure 6A).

A series of recent observations support this prediction. By changing growth conditions (plants first grown in different day-length conditions and then in identical conditions) or by using different accessions or mutants with markedly different meristem sizes from that of the wild type (Landrein et al., 2015), changes in the size of the meristem could be induced. The authors hypothesized that this change affected the size of the central zone only and not the size of primordia inhibitory fields. Corresponding changes in Γ were observed to positively correlate with the frequency of organ permutations and negatively correlate with plastochron, as predicted by our model. In addition, the stochastic model makes it possible to quantitatively estimate the changes in central zone sizes from the measured phyllotaxis disorders with an error less than 5% (Appendix section 4.3).

In a previous study on ahp6 mutants, Besnard et al. (2014) showed that the frequency of disorders could markedly augment while the size of meristems did not significantly change like in (Landrein et al., 2015). As discussed above, this change in disorder intensity could theoretically be due to an alteration of initiation perception in the mutant. However, the stochastic model suggests that it is not the case. Indeed, we re-analyzed plastochrons of the mutants and could observe that, although the change is limited, mutant plastochrons are significantly smaller than those of the wild type (Appendix section 4.2). According to Figure 6E, this means that ΓD is reduced in the mutant. If this reduction was due to an increase in either β or E, then according to the model, one should expect a corresponding increase of ΓP (Equation 6) and, thus, a decrease of disorders (Figure 6A). On the contrary a significant increase of disorders was actually observed, suggesting that perception is not altered and that, rather, a decrease in Γ=r0/R could be the source of variation. Since the size R of the meristems did not change, the model suggests that ahp6 is altered in the size r0 of the primordia inhibitory fields.

The stochastic model leads to interpret previously unexplained sequences as higher order permutations

In both the previous analysis and in related works (Besnard et al., 2014; Guédon et al., 2013; Landrein et al., 2015; Refahi et al., 2011), the permutation detection was restricted to 2- and 3-permutations. However, the stochastic model potentially predicts the existence of higher order permutations, i.e. 4- and 5-permutations, in Arabidopsis thaliana especially for small values of the control parameter ΓP. Following this prediction, we revisited the measured divergence angles in (Landrein et al., 2015) on the WS4 mutant grown in standard conditions (π2= 29.45%, π3= 14.91%) and for which 7.7% of angles were left unexplained when seeking for 2- and 3-permutations (Figure 7A). When higher order permutation are allowed in the detection algorithm (Appendix sections 4.1 and 5), most of the unexplained angles for WS4 can be interpreted as being part of 4- and even 5-permutations (Figure 7B).

Detection of higher-order permutations in WS4.

The detection algorithm (see ref [7] for details) searches plausible angle values, i.e. values within the 99% percentile given the Gaussian like distributions fitted in Figure 1, such that the overall sequence is n-admissible, i.e. composed of permuted blocks of length at most n. (A) When only 2- and 3-permutations are allowed, some angles in the sequences cannot be explained by (i.e. are not plausible assuming) permutations (the blue line of successfully interpreted angles is interrupted). (B) Allowing higher order permutations allows to interpret all the observe angles as stemming from 2-, 3- 4- and 5-permutations (the blue line covers the whole signal). Organs indexes involved in permutations: [3, 2], [5, 8, 6, 4, 7], [13, 12], [16, 14, 17, 15], [19, 18], [22, 21], [24, 25, 23], [27, 26], [30, 29], [32, 31], [35, 34], [39, 40, 38], [43, 42], [46, 45], [48, 49, 47], [52, 50, 53, 51].

https://doi.org/10.7554/eLife.14093.015

The stochastic model predicts dynamic behaviors not yet observed

Previous observations (Landrein et al., 2015) point to the existence of positive correlations between meristem size and intensities of perturbations. As discussed above, the stochastic model explains this: if a mutation, or a change in growth conditions only affects the geometry of the system Γ=r0/R, then ΓD and ΓP are both affected in the same sense (Equation 6). However, it also allows predicting new observable facts.

Assume that a mutation or a change in growth conditions affects the ability of the plant to perceive initiation signals without modifying the geometry of the system, i.e. β and/or E are modified while Γ is left unchanged. Then, according to equation 6, this induces opposite variations in ΓD and ΓP that can be detected with the observed variables. For example, assuming that a mutation decreases the sensitivity of the system to the initiation signal (decrease of E), ΓP is decreased while ΓD is increased. The decrease of ΓP induces an increase of the perturbation intensity π while the increase of ΓD induces an increase of the plastochron. The model thus predicts that, in such a case, it is possible to expect an augmentation in the disorder correlated with a decrease in organ initiation frequency. To date, such a fact has not yet been observed and constitutes a testable prediction of our model.

Discussion

We present here a multi-scale stochastic model of phyllotaxis driven by inhibitory fields and focusing on the locality of cellular decisions. A stochastic process models the perception of inhibitory fields by individual cells of the organogenetic domain and, at a higher scale, the initiation of primordia (Figure 8A). This process is continuous in essence and its results are independent of the time discretization chosen for the simulation. In contrast to previous models, the stochastic model does not use any inhibition threshold to decide either to produce an initiation or not. Instead, at each moment there is a non-zero probability to trigger initiation in any cell but this probability depends on the inhibition level in that cell, providing a realistic abstraction of the underlying signaling mechanism (Figure 8A).

Structure of the stochastic model.

(A) Inhibitory fields (red), possibly resulting from a combination of molecular processes, are generated by primordia. On the peripheral region of the central zone (CZ, green), they exert an inhibition intensity E(t) that depends on the azimuthal angle α (blue curve). At any time t, and at each intensity minimum of this curve, a primordium can be initiated during a time laps δt with a probability pk(δt) that depends on the level of the inhibition intensity at this position. (B) Relationship between the classical model parameters and its observable variables. A single parameter Γ controls both the divergence angle and the plastochron. (C) Relationship between the stochastic model parameters and its observable variables. The stochastic model of phyllotaxis is defined by 3 parameters Γ,β, E. The observable variables α, T and π,π2,π3… are controlled by two distinct combinations of these parameters: ΓD=Γ1/sβ1/6E1/2 controls the divergence angle and plastochron while ΓP=ΓβE controls the global percentage of permutated organs π, which in turns controls the distribution of permutation complexities: π2,π3….

https://doi.org/10.7554/eLife.14093.016

Noise on the timing as an intrinsic property of self-organization driven by lateral inhibitions

While the stochastic model is able to reproduce the major spiral and whorl phyllotaxis patterns, stochasticity induces alterations in the patterning process mainly affecting the plastochron i.e. the timing of organ initiation. These alterations take the form of permutations of the order of organ initiation in the meristem. If the plastochron is small as frequently observed in real sequences of permuted organs, permutations in the model can be considered equivalent to co-initiations that have been identified in the Arabidopsis meristem as the main source of permutations observed on the inflorescence stem. Less frequently, simulated permutations can have longer plastochrons. In this case, they can be interpreted as true permutations of the order of organ initiation in the meristem, consistently with the low frequency of such meristematic permutations observed also in Arabidopsis (Besnard et al., 2014). These results are in line with a previous attempt at introducing stochasticity in the classical deterministic model that could also induce permutations (Mirabet et al., 2012). However in this latter work only a limited frequency of defects could be induced (even when the noise was fixed at high levels), while requiring time discretization and post-meristematic randomization of organ order when more than one organ initiation were detected in the same simulation time step. By contrast, the capacity of the stochastic model to reproduce faithfully perturbed sequences as observed in Arabidopsis indicates that the model captures accurately the dynamics of the phyllotactic system. Taken with the fact that permutations are observed in a variety of species and genotypes from a given species, our theoretical results identify noise on the plastochron as a common characteristic of phyllotaxis systems that may generate disorders in these developmental systems.

It is important to note that our work points at stochasticity in signaling mechanisms allowing perception of inhibitory fields as the most likely origin of this developmental noise but does not entirely rule out the idea that other phenomenon might contribute (see Appendix section 6). A major contribution of stochasticity in signaling is supported by the robustness of the model to changes in different assumptions and parameters. However we have observed that spatial discretization for example can modify the frequency of permutations in the model, although this effect is limited (see Appendix section 6.2). A possible interpretation is that changes in the size of cells could also contribute to a certain point to noise on the plastochron, an idea that could be further explored.

Importantly, phyllotaxis dynamics in the stochastic model relies not only on the geometry of the inhibitory fields captured by the Γ parameter as in previous deterministic models (Douady and Couder, 1996b; Richards, 1951) (Figure 8B), but also on two new parameters E and β (Figure 8C). E and β describe respectively the sensitivity of cells to the inhibitory signal and the acuity of their perception, i.e. their capacity to differentiate close signal values. Our work thus suggests that a robust self-organization of a 3D developmental system driven by lateral inhibition depends both on the geometr of inhibitory fields in a tissue but also on the signaling capacities of cells in tissues. These theoretical observations are consistent with the key role of signaling in phyllotaxis (Vernoux et al., 2011), and with the setting of patterning dynamics in animal systems, downstream of morphogenetic signals (Kutejova et al., 2009). This pinpoints the interplay between global information provided by signal distribution and local interpretation of the information as a general principle for patterning emergence. In addition, we predict that due to pre-specification of initiation sites, noise in phyllotaxis is expected mainly on the timing of patterning. This might explain the selection through evolution of genetic mechanisms, such as the one recently described implicating the AHP6 protein (Besnard et al., 2014), to diminish noise on plastochron and disorders in phyllotaxis.

Developmental disorders reveal biological watermarks

In biological systems, disorders are frequently viewed as a result of biological or environmental noise that mainly alters systems function or development. It is in this sense for instance that noise on phyllotaxis patterns had previously been analyzed (Itoh et al., 2000; Jeune and Barabé, 2006; Peaucelle et al., 2008). Here we show that biological noise at microscopic scale may be revealed at macroscopic scale in the form of organ disorders, the permutations. Our stochastic model of phyllotaxis suggests that these disorders bear information on the more profound, hidden variables that control the phyllotaxis patterning. Much like digital watermarks that represent a copyright or any information to be hidden in images or audio signals, the actual variables (Γ,β,E) representing the state of the phyllotaxis system are not directly apparent in the plant phenotype (i.e. in the sequence of lateral organ angles and its dynamics). However, by scrutinizing carefully the image or, here, the phyllotaxis pattern and their perturbations with adequate decoding algorithms, it is possible to reveal the hidden information that was “watermarked” in the original signal. In this way, permutations together with divergence angle α and plastochron T reveal key information about the state of the system that has produced them, as their knowledge drastically reduces the set of possible Γ, β and E values. Reciprocally, any experimental alteration of these values modifies the biological watermark. Our model suggests that this change is reflected in macroscopic alterations of the phyllotaxis patterns that convey information about their possible molecular origin.

To illustrate this, we used our stochastic model to confirm that changes in permutation frequencies due to changes in growth conditions are most likely explained by a specific modulation of Γ, as previously proposed (Landrein et al., 2015) (Appendix section 4.1.5). Such a biological watermarking also suggests a different interpretation of the function of AHP6 in phyllotaxis (Besnard et al., 2014). Movement of AHP6 from organs has been proposed to generate secondary inhibitory fields that filter co-initiation at the meristem, decreasing the frequency of permutations. Based on permutation modifications, our theoretical framework suggests that AHP6 effect on the phyllotactic system does not need to be viewed as an additional specific mechanism acting on plastochron robustness but could be simply interpreted as a mechanism increasing Γ slightly. As no differences in auxin-based inhibitory fields could be detected between ahp6 mutant and wild-type plants (Besnard et al., 2014) (Appendix section 4.2), our model thus leads to a vision with composite inhibitory fields resulting at least from the combined effect of auxin-based and AHP6-based subfields. Conversely, this predicts that inhibitory fields in general cannot be explained only by auxin-based mechanisms as previously proposed.

Combined with data on divergence angles and on the plastochron, we further predict that the phyllotactic disorders could be used to identify mutants affected in biological mechanisms that controls β and/or E*. The model indicates that such mutations would have an opposite effect on frequency of permutations and on plastochron. Mutants behaving accordingly would allow not only to test this prediction of the model but also to dissect the molecular mechanisms at work. Precise and automated quantifications of the permutation, divergence angles and plastochron would allow for screening for such mutants and should become feasible with the fast development of phenotyping tools (Dhondt et al., 2013; Granier and Vile, 2014).

Using stochastic models to understand multicellular development at multiple scales

Our model only takes into account stochasticity in the perception of inhibitory fields by cells and is based on two biologically plausible assumptions: that this perception is mostly cell autonomous and that it only depends on the local level of the inhibitory signal. This provides a reasonable abstraction of local stochastic fluctuations in i) hormonal concentrations related to inhibition produced by each primordium ii) in the activity of the signal transduction pathway leading to initium creation. The detailed molecular mechanisms controlling organ initiation are for the moment only partially known. However the capacity of the stochastic model to capture accurately phyllotaxis suggests that it also captures plausible emergent properties of the underlying molecular mechanisms. This model thus not only provides a framework to understand the dynamics of patterning in the meristem but also the properties of the signaling mechanisms that process the different signals involved. Note also that the predictive capacities of our model suggest that noise on perception could be the most influential source of noise in the system. However demonstrating this would require further exploration of other potential sources of stochasticity acting at different scales, such as growth variations, spatial discretization of the peripheral zone (to account for the real size of plant cells), in order to assess their relative contribution to disorders. Moreover, similarly to the classical deterministic model of phyllotaxis, our stochastic model does not explicitly account for the cascade of molecular processes that participate to the establishment of new inhibitory fields at the location of incipient primordia. This might limit the ability of these models to fully capture the dynamics of the self-organization of the system. To do so, more mechanistic versions of this stochastic model could be developed in the future, combining more detailed cellular models of hormone-based fields, e.g. (Jönsson et al., 2006; Smith et al., 2006a; Stoma et al., 2008), and stochastic perception of these hormonal signals in 2D or 3D models with cell resolution.

Heterogeneity of biological systems at all scale has attracted an ever-growing attention in the recent years (Oates, 2011). Deterministic models do not account for the high variability that can be observed in systems behaviors, indicating that they fail to capture some key characteristics of biological systems (Wilkinson, 2009). While more demanding computationally, stochastic models are required in such cases, e.g. (Greese et al., 2014; Uyttewaal et al., 2012; Wennekamp et al., 2013), and our work illustrates how dynamic stochastic modeling can help understanding quantitatively self-organization and more broadly patterning in higher eukaryotes.

Material and methods

Stochastic model formalization

Based on the classical model of phyllotaxis (Appendix section 2), a complete and formal presentation of the stochastic model is described in the Appendix section 3. In particular, it is shown how the exponential form of the intensity law can be derived from basic model assumptions and how different observable quantities can be expressed using the model parameters.

Computational implementation of the stochastic model

A computational version of the stochastic model SMPmacro-max was implemented in Python programming language using Numpy and SciPy . Similarly to (Douady and Couder, 1996b), unless otherwise stated, the stiffness parameter was fixed in all simulations to s = 3. The non-homogeneous Poisson process was simulated using the algorithm described in (Ross, 2012). A pseudo-code version of the stochastic model algorithm is given in the Appendix section 3.2.

Estimation of phyllotaxis variables

To estimate the value of the different variables characterizing phyllotaxis α, T and π, π2, π3, etc. in either simulated or observed sequences, we used a method based on the algorithm developed in (Refahi et al., 2011) and described in the Appendix section 4. As this algorithm is central to the identification of permutations, we additionally tested its ability to detect correctly permutations on synthetic data in which known permutation patterns were introduced (Appendix section 5). Results show that the algorithm is able to detect permutations with a success rate of 98% on average.

Sensitivity analysis

The parameter space of the stochastic model was explored by varying values of Γ, β and E. 60 stochastic runs have been made for each 3-tuple of the parameter values. Each simulation run generated a sequence of 25 divergence angles and corresponding plastochrons. The different observable variables have been extracted from these simulations. Results are reported in Tables 1 and 2 of the Figure 5—source data 1.

Statistical models

The models describing the different non-linear relationships between the observable variables and the control parameters ΓD and ΓP were fitted with Gauss-Newton non-linear least-squares method (Bates and Watts, 2007). Approximate 95%-prediction bands of the response variables were computed by assuming random errors of the models independent and identically normally distributed.

Appendix 1

1 Definitions related to phyllotaxis

Phyllotactic patterns emerge from the arrangement of organs on concentric circles, each circle corresponding to a ”wave” of organ (or primordia) production, which is driven centrifugally by growth. Each circle contains the same number, called jugacy, of regularly spaced primordia, and is rotated with respect to the previous circle by a constant angle, called the divergence angle. Denoting the divergence angle by ϕ and the jugacy by j, phyllotactic patterns are traditionally classified as follows:

j=1:{distichous,  if  ϕ=πspiral,  if  ϕπ.j2:{whorled,  if  ϕ=π/j,multijugate,  if  ϕπ/j.

Note that to completely characterize a phyllotactic pattern, one also needs to specify the radii of the concentric ”co-initiation” circles mentioned above. A first parameter needed for this is the radius of the central zone, a circular region at the centre of the apical meristem where no organ is ever produced. This radius is thus a lower bound for the radii of the co-initiation circles. Some regularity has been observed (or assumed) in these radii, and they often are specified thanks to a single additional parameter, which can be either

  • the plastochron, defined as the time between two successive waves of organ formations, or

  • the plastochron ratio, defined as the ratio between the radii of two successive co-initiation circles.

The use of these quantities as characteristic parameters implies that they are constant, which is only partially corroborated by the observation. In particular, permutations of organs correspond to irregularities of the plastochron. They allow to complete the description of ”ideal” phyllotactic patterns such as those depicted in Appendix 1—figure 1.

Appendix 1—figure 1
Spiral phyllotaxis with divergence angle ϕ=137.5 and different distributions of radial positions.

(a) The first 10 primordia are depicted, with the generative spiral as a dashed line. The angle between successive organs is always equal to ϕ. The nearest neighbours of primordium i are i+3 and i+5, hence the mode of this pattern is (3,5). (b) 100 primordia are depicted, along with the 8 (resp. 13) parastichies oriented anticlockwise (resp. clockwise) indicated in cyan (resp. red), hence the mode of this pattern si (8,13).

https://doi.org/10.7554/eLife.14093.017

In the spiral case (which is the most widespread in nature), as the name indicates successive organs are arranged along a generative spiral, see Appendix 1—figure 1a. In multijugate cases, j copies of this spiral are superimposed, each rotated by an angle multiple of π/j. When a sufficient number of organs are present, other spirals naturally occur to the eye. These spirals are formed by series of nearest neighbours, or second nearest neighbours. Such spirals are called parastichies. Because they are more apparent than the generative spiral, these parastichies are often used as characteristics of a phyllotactic pattern. More precisely, the parastichies formed by sequences of nearest and second nearest neighbours appear as spiralling outwards in opposite directions. One can thus count the number of spirals oriented in each direction, which determines a pair of integers (p,q) called the mode of the phyllotactic pattern. A well-known observation is that the divergence angle is often close to the golden mean angle (137.51), which leads to phyllotactic modes (p,q) where p and q are successive numbers from the Fibonacci sequence (Fn)n1=(1,1,2,3,5,8,13,21,) is defined by F1=F2=1 and Fn+2=Fn+Fn+1. Other irrational numbers are also observed, giving rise to sequences closely related to the Fibonacci sequence. The most frequent pattern after Fibonacci is probably the so-called Lucas phyllotaxis, where ϕ99.5 and modes are successive terms in the Lucas sequence (Ln)n1=(2,1,3,4,7,11,18,29,) is defined by L1=2,L2=1, and Ln+2=Ln+Ln+1.

To generate phyllotactic patterns two main mechanisms have been proposed. In 1868, Hofmeister proposed that a new primordium in generated periodically (the period being the plastochron defined above) in the largest available space on the central zone periphery. Later in 1962, Snow & Snow suggested that a new primordium is generated on the central zone periphery where and when there is available space, given the previous history of the system. In this view, phyllotaxis (and in particular the plastochron) emerges as a consequence of a local production rule.

Before describing models in more details, let us simply incorporate the notion of permutations in the description of phyllotaxis above. The permutations observed in plants and discussed in the text concern the order of organs along the generative spiral (hence they are defined for spiral phyllotactic patterns). We call n-permutation a series of n organs whose order along the generative spiral is permuted. The order on the spiral corresponds to the order of appearance of primordia, and is materialized geometrically by the radial position which reflects the passage of time (see next section for more details). Hence, an n-permutation corresponds to a situation where n successive organs as shown on Appendix 1—figure 1a have non-increasing radial positions: for instance, there is a 3 permutation whenever the radial coordinates of organs 3, 4 and 5 verify r3>r4 and r3>r5 (with the obvious notation). A more formal definition of this notion of n-permutations was published in (Refahi et al., 2011).

2 The classical model of phyllotaxis: a brief recap

2.1 Model description

We implemented the dynamical system introduced in (Douady and Couder, 1996b) based on the rules put forward by Snow & Snow for creation of a new primordium. According to these rules, the existing primordia inhibit primordia initiation in their vicinity. A new primordium is initiated at the periphery of the central zone of the meristem where and when the sum of the inhibition generated by pre-existing organs is below a threshold. In this dynamical system the plastochron, being an emergent property of the system and not an a priori parameter, is not necessarily a constant quantity.

In the simulations, the function that models the inhibition generated by primordium q at sampling point x on the central zone periphery decreases with the Euclidean distance d(q,x) of x from the center of primordium q (by default we use polar coordinates for x and q). Unless stated otherwise, we have used the function

(1) E(q)(x)=(r0d(q,x))s,

where s denotes the stiffness of the inhibition and r0 the radius of primordium q. Following (Douady and Couder, 1996b), we used s=3 in all the simulations.

The total inhibition at a sampling point x is the cumulative inhibition of pre-existing primordia qQ:

E(x,Q)=qQE(q)(x).

In the following the set Q of pre-existing primordia will not be written explicitly when it is clear from the context or arbitrary. A new primordium appears at some point x of the central zone periphery when E(x)<ϵ, where ϵ is the inhibition threshold. In the simulations, we used ϵ=1 which guarantees that no primordium can be generated at a distance less than r0 from the center of the primordia.

Because of growth, the primordia drift away radially from the central zone at a velocity proportional to their distance from the meristem center, i.e. dr(t)dt=γr(t), (Douady and Couder, 1996b), leading to an exponential variation of the distance of primordia to meristem center:

(2) r(t)=Reγt,

where R is the radius of the central zone. The angular position of primordia, on the other hand, is supposed to remain constant in time.

In all generality, to simulate the dynamical emergence of phyllotactic patterns one needs a three-dimensional coordinate system of the form ”time, radius, angle”. However, with the choice of radial growth above, time and radial position are in fact equivalent coordinates, as detailed below, allowing us to represent the system using two coordinates only (time and angular position).

With the chosen radial growth, if there is a plastochron T the plastochron ratio is constant, equal to eγT. The radial distance between successive organs is of the form Ret(eγT-1) and is thus time dependent. The equation for r(t) provides us with a bijection between the radial position and time of initiation of primordia. At time t the radial position of a primordium that appeared at time τq is

rq(t)=Reγ(t-τq).

Reciprocally, a primordium with radial coordinate rq at time t must have appeared at time

τq=t-1γln(rqR).

The principle of our simulations is then to iteratively recalculate the positions of existing primordia for increasing time values, updating the value of inhibition functions at each time step and checking whether E(x) < ϵ for some coordinates x at the boundary of the central zone. In simulations, primordia are represented by two coordinates: their date of initiation τq and their angular position θq[0,2π). Note that by construction, at time t=τq we have rq=R, i.e. primordia appear at the boundary of the central zone.

2.2 Initial conditions

Depending on the context, different archetypal initial conditions where used to run simulations. These initial conditions are given in terms of the coordinates of pre-existing primordia, which determine the initial value of the inhibition function E.

  • Starting from cotyledons: one primordium is generated at the boundary of the central zone with a random azimuth θ0 (angular position). Then in the case of a dicotyledon plant, a second primordium is generated at the same distance of the apex center on the opposite side, i.e. at angle θ0+π.

  • Starting from a spiral pattern with divergence angle ϕ(0,2π) and a constant plastochron T > 0. The radial and angular positions (used to calculate E(x) ) of primordia are respectively of the form RenT and nϕ for successive integers n.

  • Starting from a whorled pattern, with jugacy j and constant plastochron T > 0. The radial and angular positions of primordia are respectively of the form RenT and nπj+2kπj, where 0k<j spans the j co-initiated primordia, and n are successive integers spanning a finite number of co-initiation circles, or ”whorls”.

  • Starting from a multijugate pattern of divergence angle ϕ, jugacy j and constant plastochron T > 0. The radial and angular positions of primordia are respectively of the form RenT and nϕ+2kπj, where 0k<j spans the j co-initiated primordia, and n are successive integers spanning a finite number of co-initiation circles.

2.3 Implementation

We implemented the classical model in Python programming language using NumPy and SciPy packages. A major issue in the classical model consists in deciding the moment and the place of the next initiation. In Snow and Snow simulations, time progression is continuous and the initiation time is unknown in contrast to the Hofmeister model where time periodicity is imposed. To simulate continuous time progression, small time steps dt must be used, which may lead to markedly long simulation times.

To speed up the simulation time, we implemented a dichotomic initiation time search. Based on a initially large time step Δt, we first calculate the inhibition field every Δt unit of time. Let Emin denote the minimum value of inhibition at the periphery of the central zone at time t. Once the inhibition minimum at the periphery of the central zone is below the inhibition threshold, i.e. Emin < ϵ, we take smaller time steps until the time precision dt is reached. In the following, we present the pseudo code of the algorithm to generate n primordia.

Here, the first primordium is generated at a random place on the periphery of the central zone. This can be replaced by other initial conditions as mentioned above.

Classical Model of Phyllotaxis: Pseudo code

Begint0Generatethefirstprimordiumrandomlyontheperipheryofthecentralzone.CalculatethefunctionEgiventhisprimordium.p1whilep<n:whileEmin>ϵCalculatetheprimordiapositionsatt(radialgrowth).CalculateEgivenprimordiapositions.tt+Δtt1tΔtwhile(tt1)>dt:tmid t1+(tt1)/2Calculatetheprimordiapositionsattmid(radialgrowth).CalculateEgiventheprimordiapositionsattimetmid.ifEmin>ϵ:t1 tmidelse:t tmid pp+1GenerateprimordiumattontheperipheryofthecentralzonewhereE=EminEnd

Note that not only time, but space also has to be discretized in the simulations. Formally, the boundary of the central zone is thus discretized into K creation sites, corresponding to positions

Ck=(R,θk)=(R,2kπK),0kK-1

in polar coordinates. Let Q=Q(t) denote the number of pre-existing primordia at time t, and denote by τq, the ”date of birth” of primordium q for q{1,,Q}. Primordia are supposed given in order of appearance, i.e.

τ1τ2τQ.

Given Q(t), we denote by Ek(t) the value of the inhibition function at site k, i.e. with the previous notations

Ek(t)=E(Ck,Q(t)).

Then, we denote by κq{0,,K-1} the creation site index at which primordium q has appeared. The position of primordium q at time tτq is therefore

Pq=(Reγ(t-τq),θκq).

Now, the inhibition field at position k is given more explicitly by the following expression:_

Ek(t)=q=1Q(t)(r0d(Ck,Pq))s=q=1Q(t)(r0R2(1-eγ(t-τq))2+2R2eγ(t-τq)(1-cos(θk-θκq)))s,

using the Euclidean distance in polar coordinates. The expression can be simplified by introducing the geometric scaling parameter Γ=r0/R:

(3) Ek(t)=Γsq=1Q(t)(1+e2γ(tτq)2eγ(tτq)cos(θkθκq))s2.

Because Γs appears as a global factor, the threshold parameter ϵ is defined up to a factor for any fixed choice of the stiffness s: for any ρ > 0 the exact same simulation would result from either (Γ,ϵ) or (ρΓ,ρsϵ). For this reason, we always fix ϵ=1 in the following.

2.4 Bifurcation diagram

Douady and Couder (Douady and Couder, 1996b) showed that the divergence angle and the phyllotactic mode reached by the classical model at equilibrium (if any) is controlled by the parameter Γ appearing in the inhibition function, Equation 3. Similarly to their work, we observed stable spiral patterns, whose characteristics (divergence angle and mode) varied depending on Γ and the initial condition.

To assess the consistency of our implementation with the study in (Douady and Couder, 1996b), we re-calculated the bifurcation diagram obtained by these authors. Current computational hardware allowed us to refine considerably the precision of the diagram. The variations of the divergence angles obtained by varying Γ in the classical model form a characteristic tree like bifurcation diagram of phyllotaxis, shown in Appendix 1—figure 2.

Appendix 1—figure 2
Bifurcation diagram in the Snow and Snow model.

We used a sample of the interval 0.04Γ2.9 with steps of 0.01 or less (refinements were performed in areas with higher numbers of branches). For each value of Γ, we ran simulations of the classical model with (i) spiral initial conditions for divergence angles taking integer values in [20,180], and plastochrons taking values in a sample of 128 points between 0.05 and 8.00, (ii) whorled initial conditions for the same samples of divergence angles and plastochrons and all jugacies 2j7. For each simulation, we estimated the final divergence angle ϕ and phyllotactic mode, and reported these in the graph above (abscissa: Γ, ordinate: ϕ, color code: mode).

https://doi.org/10.7554/eLife.14093.018

Note that for consistency with (Douady and Couder, 1996b), the model had to be slightly altered compared to the description given in the previous section. First, the local inhibition function took the form below instead of a power function as in (Bates and Watts, 2007):

E(q)(x)=-1+(tanhsd(x,q)r0)-1-1+(tanhs)-1,

where tanh stands for hyperbolic tangent, and s denotes the inhibition stiffness, s was set to 8 in the simulations of the bifurcation diagram.

Secondly, the meristem was not considered as a flat domain but as being dome shaped. This was performed by using of a distance function of the form below, where P0, P1 are two points of coordinates (r0,θ0) and (r1,θ1) respectively

d(P0,P1)=(r0-r1)2N+2Nr0r1(1-cos(θ0-θ1)),

where 1/N is called conicity (Douady and Couder, 1996b) and is equal to 3 in Appendix 1—figure 2. This choice was only made for this particular situation, and in the remainder of the text all simulations were made with the model strictly as defined in the previous section. The expression above does not allow to factor Γ out as in (Cinlar, 1975). Instead, following (Douady and Couder, 1996b) the new control parameter below was used:

Γ=d0N1/2R0.

The caption of Appendix 1—figure 2 describes the default sampling of parameters and initial conditions. Note for complete precision that for regions including many branches, additional sampling points were calculated a posteriori to reduce the number and width of gaps. Also, for small values of Γ were simulations are more expensive computationally, higher plastochron values (T3) were not included. Each initial condition comprised 150 pre-defined primordia and was followed by 250 primordia generated by the simulation algorithm. To assess the convergence to a stable phyllotactic patterns, the last 50 divergence angles were considered. If all these 50 divergence angle were within 5 from their average, the pattern was considered steady.

3 Stochastic model

3.1 Model description and derivation

In the stochastic model, we modify from the classical model the rule driving primordia initiation. Instead of a deterministic and global threshold value ϵ, we assume that each cell is able to decide in a cell autonomous manner to trigger initiation of not. This decision is not taken on the basis of a threshold being reached or not, but on a local probability to trigger initiation at this cell site. This probability depends naturally on the inhibition level Ek(t), Equation 3 (the lower the inhibition, the higher the probability value) and on the amount of time one waits for an initiation at this site k (the longer one waits, the higher the probability).

If Xk(t,δt) denotes the number of cells initiated at a location Ck in the time interval [t,t+δt] (Xk(t,δt) will typically take values 0 or 1, as a cell can initiate at most one primordium at a time), the probability of observing one initiation during a small time interval of length is assumed nearly proportional to δt:

(4) P(Xk(t,δt)=1)=λ(Ek(t)).δt+o(δt)=λk(t).δt+o(δt),

where λk(t)=λ(Ek(t)) is a rate parameter that can be interpreted as a temporal density of events (and o(δt)/δt0 as δt0). The function λ expresses the ability of any cell to respond to the inhibitory signal. If we assume in addition that probabilities to observe an initiation over non-overlapping time-spans are independent, the previous expression (4) defines an non-homogeneous Poisson process (Cinlar, 1975; Ross, 2014) of rate λk(t).

Derivation of the λ’s expression

As explained above, the function λ must obviously decrease with the inhibition level at site k. We assume in addition that the ratio between probabilities at any two different sites k and l only depends on the difference between their respective inhibitory energies:

(5) P(Xk(t,δt)=1)P(Xl(t,δt)=1)=λ(Ek(t))λ(El(t))=f(Ek(t)-El(t)),

where f is an unknown function to be determined. Based on the previous assumptions, it is possible to find an explicit expression for λ as a function of the inhibition. For sake of notation simplicity, let us omit the time dependence t in the following expressions.

For sites k and l with the same energies, the rates λ(Ek) and λ(El) must be equal: λ(Ek)λ(El)=1=f(0). Taking the specific value El=0, Equation 5 becomes: λ(Ek)λ(0)=f(Ek),which by denoting α=λ(0), leads to:

(6) λ(Ek)=α.f(Ek).

Therefore from Equation 5, we deduce: λ(Ek)λ(El)=f(Ek)f(El)=f(Ek-El), and by adding El to argument El in the above expression, we get:

f(Ek+El)=f(Ek).f(El),

showing that, if f is continuous, f must have the form of an exponential. Taking in addition that f(0)=1 and f is decreasing, we get:

f(x)=e-β.x

where β is a positive scalar parameter. Finally, by replacing the above expression of f in the expression of λ of Equation 6 and denoting E*=lnαβ, we get the final expression of λ:

(7) λk(t)=λ(Ek(t))=e-β(Ek(t)-E*),

where β and E* are two real parameters, with β > 0. The probability of an initiation during a small time lapse δt is thus:

(8) P(Xk(t,δt)=1)=λk(t).δt

From the theory of non-homogeneous Poisson processes (Cinlar, 1975; Ross, 2014), we have that the number of initiations is also a Poisson random variable, with rate

(9) μk(t,Δt)=tt+Δtλk(u)du,

i.e. μk(t,Δt) is the mean number of initiations between t and t+Δt at site k.

Independent Poisson processes can be superposed to yield another Poisson process, and the instantaneous rates are additive. If V{0K-1} is any set of creation sites (not necessarily contiguous; including the set {0K-1} itself), the global creation process for sites of V is a non-homogeneous Poisson process with instantaneous rate

λV(t)=kVλk(t).

For V={0K-1} in particular we denote

λ(t)=k=0K-1λk(t).

This additivity carries over to the average rate functions, giving

(10) μ(t,ΔT)=k=0K-1μk(t,ΔT),

and μV(t,ΔT) is defined similarly.

The creation process at a given initiation site is a non-homogeneous Poisson process as just described, but its rate is actually stochastic and depends on all the primordia which have been previously (randomly) created. Properly speaking, the process is conditionally (on previous creation events) a Poisson process. More precisely, from the properties of non-homogeneous Poisson processes we can derive the probability of occurrence of an initiation at site k, conditionally on the previous creation events, which with our notations are encoded by their times and sites of appearance, τq and κq respectively (for 1qQ=Q(t)). The next initiation is the (Q+1)th organ and, at time tτQ, the probability that it occurs at site k{0K-1} can be written as:

(11) P(κ(Q+1)=k|E(τQ))=0λk(t+u)eμ(τQ,u)du,

where we write the conditioning using E(t)=(Ek(t))0kK-1 for brevity.

In the classical model, the inhibition function (3) is directly compared to the threshold ϵ=1, so that all the parameters are those appearing in (3), i.e. Γ, γ and the initiation times τq, which are emerging from the model’s dynamics and are thus not parameters strictly speaking. Note also that γ is essentially a ”time unit” parameter and is thus not crucial to the geometry of phyllotactic patterns (in other words, a change of time scale with a factor γ would allow us to remove this parameter from Equation 3). In summary, Γ is truly the only control parameter in the standard, deterministic model.

In the stochastic formulation above, the Γ dependent inhibition Ek is used via the rate λ, Equation 7. This introduces the two additional parameters β and E*. As the λ is used in the integral expressions (10) and (11), there is no obvious way to reduce the dimensionality of the control space, with its coordinates (Γ,β,E*). See the main text for a heuristic approach, where some lower dimensional parametrizations are obtained empirically.

Probability of n-permutations

A number of theoretical derivations can be carried out from the model. In particular, some properties of the probability that an n-permutation occurs can be related to the model parameters. For simplicity we focus on 2-permutations. Let us assume that previous initiations have occurred close to a ”normal” spiral, with no permutations at least in the most recent organ creations. Without loss of generality, the last organ creation is supposed to have occurred at location θ0=0. According to our simulations, the minimum and second minimum of the inhibition profile {Ek(t)| 0kK-1} are then typically situated at the ”next” positions along the spiral, i.e. at sites k1 and k2 respectively very close to ϕ and 2ϕ, with ϕ the divergence angle. Then, a 2-permutation happens if the next organ creation takes place at site k2 rather than k1. Conditionally on the next initiation taking place at one of the two sites k1, k2, Equation 11 gives the probability of k2 occurring first as

Π2(t)=0λk2(t+u)exp (μk1(t,u)μk2(t,u))du.

Similarly, the (conditional) probability that the next creation occurs at site k1 is given by replacing λk2 by λk1 in the integral above. Since, as just mentioned, we observe that typically Ek1 < Ek2 and the intensity λ is a decreasing function of E, we deduce that the next initiation is more likely to occur at site k1, i.e. without a 2-permutation.

These expressions make it possible to investigate the variation of probability due to a change in model parameters. If for example the sensitivity of the system E* is increased by δE > 0, how does the 2-permutation probability vary? Remembering from (7) that we can write λk=eβE*e-βEk, we obtain

Π2(t)=eβE*0exp(-βEk(t+u)-μk1(t,u)-μk2(t,u))du.

This indicates that an increase in the sensitivity E* will tend to increase Π2. However things are more complicated, since from (9) we have μk(t,u)=eβE*tt+ue-βEk2(u)du and thus the terms μk1 and μk2 in the integral above also depend on E*, as below:

Π2(t)=eβE*0exp(-βEk2(t+u)-eβE*(tt+ue-βEk1(u)du+tt+ue-βEk2(u)du))du.

Hence, despite the first factor being an increasing function of the sentitivity E*, it may happen in general that the probability of a 2-permutation decreases as the sensitivity of the system is increased, through the non-trivial formula above. To proceed further, the relationship between E*, β, Γ and permutations is studied empirically using simulations.

3.2 Implementation

We developed two different versions of the model.

  • In the first version, called SMPmicro, every cell on the peripheral zone is considered as a potential initiation site and therefore the Poisson processes are simulated for each cell.

  • In the second version of the model, SMPmacro-max, the local minima indicate the potential initiation sites and therefore the Poisson processes are simulated at these sites assuming that a primordium is triggered if at least one of the cells spanned by the local minimum valley switches its identity to primordium.

The models are implemented using the Python programming language using the NumPy and SciPy packages. In the following, we present the pseudo code of the model to generate n primordia. We use the non-homogeneous Poisson simulation algorithm as presented in (Ross, 2012). To encompass both versions above in a single description, we use the notation S to describe either the whole set {0,,K-1} if all sites are tested, or the subset S{0,,K-1} of all local minima of the function Ek for the second version.

The variables appearing in the pseudo-code below are:

n: number of primordia generated by simulation

t: system time

p: current number of primordia

Δt: time step

Ek: inhibition level at sampling point k

U(0,1): the uniform distribution between 0 and 1

β and E*: parameters of the model

Stochastic Model of Phyllotaxis: generic SMP Pseudo code

Begint 0Generatethefirstprimordiumrandomlyonthecentralzoneboundary.CalculatethefunctionEgiventhisprimordium.p 1whilep<n:Computetheprimordiapositionsatt(radialgrowth).ComputeEk(t)fork{0,...,K1} initiation=falseforkS: simulate non-homogenous Poisson process at each cell (or each local minimum)δk0whiletrue:DrawrU(0,1)δkδkln(r)/(eβE)ifδk>Δt: breakComputeprimordiapositionsatt(radialgrowth).ComputeEk(t).DrawsU(0,1)ifseβEk:initiation=true breakifinitiation:tt+min{δ1,...,δk}i=argmin{δ1,...,δk}Generate a new primordium at cell number i (or local minimum)pp+1else:tt+ΔtEnd

In the implementation, the simulation of non homogeneous Poisson processes in cells (or in local minima) is parallelized. This speeded up the simulations and allowed us to carry out an extensive sensitivity analysis.

4 Interpreting phyllotaxis phenotypes using the stochastic model

4.1 Parameter estimation

This section details how values for the control parameters ΓD and ΓP of the stochastic model were estimated from observed data for various genotypes or growth conditions. When similar data can also be produced by simulations (e.g. divergence angle sequences, permutations) the same estimation procedures are applied to the simulated data.

4.1.1 Permutation intensities

To estimate the value of the different permutation intensities, π,π2,π3,…, in either simulated or observed sequences, we used the algorithms described in Refahi et al. (2011). The algorithms allows identification of permutation patterns of bounded length (at most n for a fixed n2) on spiral patterns with different canonical divergence angle values (e.g. φ= 137.5 for Fibonacci phyllotaxis, φ= 99.5 for Lucas phyllotaxis). The algorithm can detect truncated permutation patterns at the end of the sequences. It also allows the analysis of the sequences in the reverse direction. Most importantly in this study, it allows to characterize permutations in sequences of angles which are not strictly equal to multiples of φ, but are “close” to these, as happens in real experimental data or our stochastic simulations.

More precisely, let g(x;μq,κ) denote the probability density function of von Mises distribution (the standard circular analogue of a Gaussian distribution) with parameters μqDn (mean direction) and κ (concentration parameter), where Dn is the set of the theoretical angles (multiples of φ occurring with permutations of length at most n). We used κ=10.4 which was estimated in (Guédon et al., 2013) by fitting von Mises distributions on experimental data from A. thaliana. For a measured (simulated ) angle xi[0,360) and for a candidate theoretical angle, μqDn we calculated the posterior probability :

ω(xi,μq)=g(xi;μq,κ)ΣμrDng(xi;μr,κ)

The permutation detecting algorithm seeks candidate angles in Dn such that the sequence in composed of permutations, but only accepts them if the posterior probability is higher than a threshold whose value was set to 0.001×|Dn|. This threshold was determined empirically based on sequences whose permutation patterns were confirmed by a human expert. The factor |Dn| stands for the implicit hypothesis that the theoretical angles were a priori equally probable.

4.1.2 Average divergence angles

For a given sample of divergence angle sequences (i.e. repetitions of the same genotype in the same growing conditions, repetitions of stochastic model simulation for a given set of parameters Γ,E*,β), divergence angles were estimated as the mean value of the divergence angle over the total set of divergence angles in the whole set of sequences. The estimation takes into account potentially detected permutations (meaning that, in case of permutation detection, organs are re-ordered in the sequences to remove the permutations prior to estimating average angles).

4.1.3 Plastochrons

In some cases plastochrons were directly estimated from counting initiations of organs over different time spans. In such cases plastochrons are estimated in terms of hours (see table below). However, to compare the experimental plastochron with simulated ones, we rather estimated plastochrons (T) from experiments from plastochron ratios (ρ), themselves estimated in (Besnard et al., 2014; Landrein et al., 2015) from measurements of ratios between distances of consecutive primordia. According to section 1, the plastochron ratio ρ=eγT is constant for a constant plastochron T. Corresponding plastochrons T can thus be estimated (in units corresponding to that of the model parameter γ) from measured plastochron ratios as:

(12) T=lnργ,

where γ is a constant fixed in the model to 5s-1. For the wild type for example, the measure plastochron ratio is ρ=1.20, leading to an estimated plastochron T=0.0227 expressed in the time units defined in the simulation.

4.1.4 Control parameters of the stochastic model

Using the characteristic curves of Figure 5A,C,E (main text), it is possible to estimate values or ranges of values for the stochastic model control parameters from values of the observable phyllotaxis variables: π, α and T.

Let us illustrate the estimation method on the wild type. A value of ΓP can be estimated from a parametric regression of the could of points obtained from the stochastic model simulations Appendix 1—figure 3A. The estimated total permutation intensity is π=14.20%, which leads to a possible value of ΓP in [8.30,11.49] according to the curve of Appendix 1—figure 3A.

Appendix 1—figure 3
Estimation of control parameters from observable phyllotaxis variables.

(A) An exponential model was fitted to the simulated data of π and ΓP using the Gauss-Newton least squares method (Bates and Watts, 2007); for the fitted model, an approximate 95% prediction band was then computed by assuming the random error terms additive and i.i.d. normally distributed. The range of possible ΓP values [8.30, 11.49] that could yield the observed π value 14.2 was determined by the prediction band. (B) A Gompertz function was fitted to the simulated data of plastochron and ΓD using the Gauss-Newton least squares method (Bates and Watts, 2007); for the fitted model, an approximate 95% prediction band was then computed by assuming the random error terms additive and i.i.d. normally distributed. The range of possible ΓD values [0.427, 0.492] that could yield the observed range of plastochron values [0.023, 0.028] was determined by the prediction band. (C) A 4th degree polynomial was fitted to the simulated data of angle and ΓD using the least squares method; for the fitted model, an approximate 95% prediction band was then computed by assuming the random error terms additive and i.i.d. normally distributed. The range of possible ΓD values [0.384, 0.472] that could yield the observed angle value of 136° was determined by the prediction band. (D) Distributions of the estimated plastochrons in the groups of wild-type and mutated Arabidopsis plants in the experiments of Besnard et al., 2014. The box depicts the inter-quartile range bisected by the median, and the whiskers reach out to the extreme values in the group; the colored point denotes the arithmetic mean, and the colored dashes indicate twice the standard error of the mean; n stands for the number of plants in the group.

https://doi.org/10.7554/eLife.14093.019

We then estimate the plastochron as explained in the preceding section. For the wild type, T[0.023,0.0227] (Landrein et al., 2015). By using a parametric regression of the simulated cloud of points (ΓD,T predicted by the stochastic model, Appendix 1—figure 3, we derive an estimate value of ΓD in [0.427,0.492]. Now using the curve of Appendix 1—figure 3, and the average divergence angle of 136° measured for the WT (Besnard et al., 2014), we deduce that ΓD should be included in [0.384,0.472]. These two sets of estimates for ΓD are consistent with each other and lead to a consolidated estimate for its value of ΓD in [0.427,0.472].

Similar estimations for ΓD and ΓP of the wild-type and other plants are reported in the next section.

4.1.5 Estimated values

The following tables summarize estimated observable variables (α,T,π,π2,π3) based on data from (Besnard et al., 2014; Landrein et al., 2015) and the inferred model control parameters ΓP and ΓD. Two different day length conditions were used to affect the size of the meristem of different phenotypes and mutants (Landrein et al., 2015). In the first setting, plants were first grown in short-day conditions for one month after germination and then passed in long-day conditions, Appendix 1–table 1. In a second setting, plants were growth only in long-day conditions from germination, Appendix 1–table 2. To complete the published results, we grew ahp6 plants in long day conditions as in (Landrein et al., 2015). Results are reported in the last column of Appendix 1–table 2.

Appendix 1-table 1

Observed phyllotaxic variables on plants grown in short day and then in long day conditions. % permuted organs is shown as a number followed by 2 other numbers in parentheses, i.e. π2,3(π2,π3)

https://doi.org/10.7554/eLife.14093.020
Short day - Long dayCol0WS4clasp-1ahp6
#angles/#sequences704/291046/25619/172815/89
Estimated # 2-permutations4915428297
Estimated # 3-permutations252153
% unexplained angles%2.7%7.7%1.1%2
Estimated % permuted organs14(13.37,0.82)43.3(29.45,14.91)9.2(8.81, 0.47)25.9(20.45,5.47)
Estimated average α~136.4136138.0-
Estimated α~ standard deviation5.52.36-
# Lucas sequences0000
Average measured plastochron (h)7.55.11910.6
Plastochron ratio1.121.081.135-
Estimated plastochron (equ. 12)0.02270.01540.0253-
Estimated ΓD0.4300.3250.450-
Estimated ΓP (in [ΓPmin,ΓPmax])[8.7,11.4][5.2,5.8][9.6,12.3][6.7,8.7]
Appendix 1–table 2

Observed phyllotaxic variables on plants grown in long day conditions only. % permuted organs is shown as a number followed by 2 other numbers in parentheses, i.e. π2,3(π2,π3).

https://doi.org/10.7554/eLife.14093.021
Long day onlyCol0WS4clasp-1ahp6
#angles/#sequences1193/34487/15667/19965/27
Estimated # 2-permutations13221422
Estimated # 3-permutations0110
% unexplained angles%2.6%3.6%4.3%0.1
Estimated % permuted organs2.12(2.12,0)9.3(8.76, 0.6)4.52(4.08, 0.44)4.4(4.44,0)
Estimated average α~136.4141.4128.1-
Estimated α~ standard deviation10.76.820.46-
# Lucas sequences0060
Average plastochron (h)106.3219.5
Plastochron ratio1.151.1151.145-
Estimated plastochron (equ. 12)0.02800.02180.0271-
Estimated ΓD0.4850.4100.475-
Estimated ΓP (in [ΓPmin,ΓPmax])[12.8,15.1][10.0,12.0][11.7,14.2][11.8,14.3]

4.2 In comparison to wild-type, plastochrons are reduced in ahp6

The stochastic model suggests that a comparison between plastochrons of the wild type and the mutant plants would be informative about the potential influence of the mutation on key phyllotactic geometrical or biochemical processes (see main text). We tested this idea on ahp6 mutants. For this, we re-analysed the measurements of number of organ initiations over different time-laps carried out by (Besnard et al., 2014). These measurements were made for three experiments on organ initiation with wild-type and two mutation-types (ahp6-1) of Arabidopsis. We pooled the experiments together and combined the two mutation groups. For each plant in each group, we estimated the plastochron by dividing the total observation time of the plant with the total number of organs initiated in it within this time. The distributions of the plastochrons in the two groups (wild type, mutant) are shown in Appendix 1—figure 3. The groups differ from each other statistically significantly (in Kruskall-Wallis rank sum test, the null hypothesis ”mean ranks of plastochron values in the groups are the same” is rejected with p-value 0.01417; in one-way Welch ANOVA test, the null hypothesis ”means of the plastochrons in the groups are the same” is rejected with p-value 0.01055), and the distributions suggest that plastochrons tend to be smaller in mutant plants than in wild plants.

4.3 Interpretation of phenotypes with modified meristem size

We revisited the results from (Landrein et al., 2015) to check whether the observed correlation between meristem size and perturbation intensity could be quantitatively explained by the stochastic model.

For each genotype, (Landrein et al., 2015) assumed that the change of growth conditions induces a change in size of the central zone while the radius of the primordia inhibitory fields is mainly not modified. Therefore, for plants grown in long-day conditions, the size of the meristem decreases, and supposedly induces a decrease of the size of the central zone, thus leading to an increase of Γ. Here, as inflorescences develop in identical conditions, we assume in addition that the parameters E* and β reflecting the signal sensitivity and acuity of the biological system are not affected by the change in growth conditions. For all the plants, the different observable variables α, T, π2 and π3 were measured (Landrein et al., 2015). From these data, we can estimate for each genotype and growth condition the corresponding values of the control parameters (ΓD,ΓP) as described in the previous section. Let us denote G a particular genotype, we can derive from the model that the ratio of central zone size due to a change in growth condition is given by:

(13)R(GSL)R(GLL)=(ΓP(GLL)ΓP(GSL))17(ΓD(GLL)ΓD(GSL))67

If it is assumed that this change in central zone sizes is induced by the change in meristem size due to different growth conditions, this ratio is also the ratio of meristem sizes. From this relationship, it is possible to predict whether an increase or a decrease in the meristem size can be expected based on the measured values of the other observed variables. For the tested plants, the model predicted in all the cases a decrease of the meristem size with an error of respectively 4%, 5% and 1% for Col0, WS4 and clasp-1.

5 Assessment of pattern identification accuracy

As mentioned above our analysis of permutations and the resulting tables relied on our previously published algorithms (Refahi et al., 2011). These algorithms are based on the construction of a set of candidate labellings (organized as a tree), within which sequences containing permutations are selected provided they are close enough to the data given as input. Because these algorithms require some parametrization (to measure ”closeness to the data” in particular) and have been used extensively to generate the tables above, we performed a robustness analysis.

Namely, we randomly generated sets of sequences of divergence angles involving 2-permutations and 3-permutations with von Mises noise (circular version of the Gaussian distribution) to model uncertainty. We tested the combinatorial models pattern detection precision for different noise amplitude values as well as different permutations frequency, chosen in adequacy with the available experimental data.

Let X={X1,X2,,X60} be a set of generated sequences of multiples of ϕ=137.5 involving 2- and 3-permutations, where Xk=(x1k,,xk), =25 (a typical length in experimental data sets), and 1k60. We then introduced noise, leading to S={S1,,S60}, Sk=(s1k,,sk), such that sikP(μ,κ), where P(μ,κ) indicates the von Mises distribution with mean value μ=xik, and the concentration parameter κ. Both κ and the permutation frequencies were estimated from the measured sequences in (Refahi et al., 2011; Besnard et al., 2014; Guédon et al., 2013; Landrein et al., 2015).

We generated four sets of sequences for different values of permutation frequency and κ as described above, see Appendix 1—figure 4 for the histogram of angles. We then used the combinatorial model to analyse the sequences. By careful human inspection, we then counted the correctly labelled noisy angles and divided them by the number of all divergence angles. The resulting values are reported in Appendix 1–table 3.

Appendix 1—figure 4
Histogram of randomly generated sequences.

(a) Histogram of randomly generated sequences, κ=14.0,π2,3=24%. (b) Histogram of randomly generated sequences, κ=14.0, π2,3=43%. (c) Histogram of randomly generated sequences, κ=10.4, π2,3=24%. (d) Histogram of randomly generated sequences, κ=10.4, π2,3=43%.

https://doi.org/10.7554/eLife.14093.022
Appendix 1–table 3

Combinatorial models permutation detection precision on randomly generated noisy data.

https://doi.org/10.7554/eLife.14093.023
κ \ π2,324%43%
14100%98.5%
10.499%98%

6 Assessing the model robustness by altering implementation choices

Even though the main ideas of our model are unambiguous, a number of parameters had to be specified for implementation purposes. This raises the question of whether some of our conclusions could be dependent upon these implementation choices. To ensure that this is not the case, we considered a number of alternatives and compared simulation results with our “main choices”. The results are reported below and overall they confirm the robustness of our results, as they are not significantly changed by alternative implementation choices.

6.1 Generating spiral and whorl patterns from random initial conditions

To test the ability of the model to initiate spiral or whorl patterns de novo from unpatterned meristems, we made simulations starting from a random inhibition field Ek at each location k of the central zone periphery, k ranging over the discretized angular positions. Ek is constructed as follows: Ek=exp(15σk+10), where σk is drawn from the uniform distribution on the interval [0,1) and the overall distribution (σk)k is smoothed by applying a Gaussian filter. The value of the inhibitory field in each cell is then updated at each time step of the simulation until the first primordium initiation, by adding a random factor drawn from a uniform distribution on the interval [-0.5,+0.5]. For all the simulations, β was fixed to 11.0 and E* to 1.0.

We observed that the model is able to generate phyllotaxis patterns from random initial conditions (see movies and their legends in the main text). Depending on the model control parameter ΓD, the system was able to generate either spiral of whorl phyllotaxis. In the case of whorl patterns, the system was able to initiate the expected whorl motifs for 2 and 4 organs without any transition phase. This is due to the symmetry of the system combined with its geometry. Starting from a meristem with no preexisting organ, assume that the level of inhibition is low so that a random noise can potentially augment substantially the probability of an organ initiation at any position around the periphery. Soon, a first organ is initiated at a random location on the periphery. This breaks the circular symmetry of the system and favours slightly the site at 180 degrees. As the inhibition level is low, a second primordium is initiated immediately at this new position with a plastochron close to 0. This in turn creates a new inhibitory component to the global inhibitory field. Note that the symmetry of the system has been partially recovered as the second primordium is at the same distance of the center as the first. Therefore, if the range of the inhibition emitted by each primordium is large enough, it prevents for a while the initiation of new primordia. Atfer some growth, the two initial primordia get away from the meristem center. The inhibition decreases at the central zone periphery and after some time, new initiations will occur almost synchronously at the two possible locations perpendicular to the first organ pair direction where the inhibition field is minimal. This initiates a 2-whorled phyllotaxis.

Along similar lines, a 4-whorled pattern can be initiated if, starting from the two initial organs as above, the inhibition does not cover the entire peripheral circle. Then another pair of two primordia can immediately be generated, with plastochrons close to zero, in the direction perpendicular to the first pair and on opposite sides of the central zone. The inhibition produced by the 4 organs covers the entire meristem and another 4 organs will be generated only after some time, when the 4 initial ones have drifted away, sufficiently far from the center. In this case the phyllotaxis is 4-whorled.

6.2 Changing spatial discretization

We considered different spatial discretizations. These can be described in terms of the angle at the central zone boundary which corresponds to possible positions for organ initiations. The default discretization, used in the main document, uses a 1-degree resolution. The 2-degree and 5-degree discretizations generate roughly the same permutation rates, whereas the 10-degree discretization leads to an increase of the permutation rates, see Appendix 1–tables 47.

Appendix 1–table 4

1 degree (in the paper), permutation rates and average divergence angle.

https://doi.org/10.7554/eLife.14093.024
Γ|E* \ β1012
0.65, 130.0 (22.3, 7.7), 13618.4(15.7, 2.8), 136.8
0.75, 130.7(21.2, 9.5), 136.215.5 (11.5, 0.9), 137.3
0.85, 117.6(13.5, 4.1), 138.515.5(11.9, 3.5), 137.3
0.95, 111.7(9.5, 1.8), 139.94.8(4.4, 0.4)140.1
Appendix 1–table 5

2 degrees, permutation rates and average divergence angle.

https://doi.org/10.7554/eLife.14093.025
Γ|E* \ β1012
0.65 , 131.8(24.3,7.4), 136.419.7(18.2,1.5), 136.6
0.75, 127.1(16.9,10.2), 13611.8(10.2,1.6), 136.5
0.85, 119.8(14.7,5.2), 137.311.0(9.9,1.1), 137.7
0.95, 113.6(12.6,1.0), 138.75.3(5.0,0.3), 139.5
Appendix 1–table 6

5 degrees, permutation rates and average divergence angle.

https://doi.org/10.7554/eLife.14093.026
Γ|E* \ β1012
0.65, 132.9(25.1,7.8), 136.121.3(17.5,3.8), 136.1
0.75, 135.2(23.3,11.9), 135.718.1(16.0,2.1), 136.3
0.85, 116.6(12.9,3.7), 137.98.9(7.5,1.4), 137.9
0.95, 110.0(8.2,1.9), 139.44.3(4.3,0.0), 139.9
Appendix 1–table 7

10 degrees, permutation rates and average divergence angle.

https://doi.org/10.7554/eLife.14093.027
Γ|E* \ β1012
0.65, 136.7(24.7,12.0), 136.529.7(22.0,7.6), 136.2
0.75, 129.0(22.4,6.6), 136.925.3(20.8,4.5), 136.7
0.85, 124.2(17.4,6.8), 135.921.0(17.5,3.5), 134.8
0.95, 113.2(13.2,0.0), 139.39.4(8.8,0.6), 139.5

6.3 The effect of the inhibition function on the results and control parameters

We changed the inhibition function to the two following function, used e.g. in (Douady and Couder, 1996b)

E(q)(x)=-1+(tanhsd(x,q)r0)-1-1+(tanhs)-1,

while we repeat here our default choice (also considered in [Douady and Couder, 1996b]):

E(q)(x)=(r0d(q,x))s,

where tanh stands for hyperbolic tangent, and for both functions s denotes the inhibition stiffness. Note that though qualitatively controling the stiffness, the same quantitative value of s is expected to give different results for both functions above. In the following Appendix 1–table 8 we present permutation rates for the tanh based expression, with stiffness s=1.

Appendix 1–table 8

Permutation rates for the tanh based inhibition rate, with s=1; permutation rates and average divergence angle.

https://doi.org/10.7554/eLife.14093.028
Γ|E* \ β781012
0.45, 135.9(25.9,10.0), 138.525.1(20.9,4.2), 138.3
0.45, 1.245.9(25.6,20.3), 138.842.4(30.1,12.3), 138.526.8(21.1,5.7), 138.117.7(16.0,1.6), 137.9
0.55, 144.3(27.4,16.8), 138.036.1(26.5,9.6), 138.223.3(20.1,3.1), 137.515.9(14.5,1.3), 137.4
0.55, 1.236.7(25.8,10.9), 138.028.0(20.4,7.6), 137.917.1(15.5,1.6), 137.713.9(13.9,0.0), 137.5
0.65, 139.0(26.3,12.8), 135.730.4(23.7,6.7), 136.315.3(14.1,1.1), 136.310.3(9.7,0.6), 136.8
0.65, 1.227.7(22.2,5.6), 136.419.9(17.1,2.8), 136.710.6(10.2,0.4), 136.74.3(4.3,0.0), 136.7
0.75, 140.2(23.0,17.2), 135.232.2(20.9,11.3), 135.117.6(14.4,3.2), 135.810.9(10.3,0.5), 136.3
0.75, 1.231.1(22.6,8.5), 135.021.3(18.1,3.2), 136.011.1(10.0,1.1), 136.56.7(6.2,0.5), 137.1
0.85, 127.4(17.5,9.9), 137.619.2(14.4,4.8), 137.25.9(5.9,0.0), 138.04.4(4.2,0.2), 138.1
0.85, 1.221.5(15.8,5.8), 136.414.4(13.0,1.4), 136.95.6(5.6,0.0), 137.63.1(3.0,0.1), 138.0
0.95, 121.0(16.2,4.9), 138.914.8(10.7,4.0), 138.63.7(3.7,0.0), 139.6
0.95, 1.215.4(12.4,3.1), 138.48.5(7.3,1.2), 138.92.9(2.9,0.0), 139.4
1.05, 122.4(16.8,5.6), 139.613.1(11.0,2.1), 139.22.4(2.4,0.0), 139.6
1.05, 1.210.2(9.3,0.9), 139.55.8(5.7,0.1), 139.72.1(2.1,0.0), 139.9

The control parameters ΓP and ΓD turn out to be the control parameters of the system using this new inhibition function, as can be seen in Appendix 1—figure 5.

Appendix 1—figure 5
Role of the control parameters for the tanh based inhibition function.

(a–b) Average plastochron ratio as a function of Γ and ΓD=Γβ1/6E*1/2, respectively. (c) Number of permutations π as a function of ΓP.

https://doi.org/10.7554/eLife.14093.029

When using our default (power law) inhibition function, we have fixed s=3 in the main paper. We found the ΓD=Γβ1/6E*1/2 as the control parameter of plastochrons.

As shown in Appendix 1—figure 6, when using a ower law the parameter ΓD can in fact be related more explicitly to the stiffness s, by using

(14) ΓD=Γs/3β1/6E*1/2

instead of the previous expression.

Appendix 1—figure 6
Role of the control parameters for the power law inhibition function, with rows (a–e) corresponding respectively to a steepness s=1,2,4,5,6.

First two columns: average plastochron ratio as a function of Γ and ΓD=Γβ1/6E*1/2, respectively. Third column: average plastochron ratio as a function of ΓD=Γs/3β1/6E*1/2.

https://doi.org/10.7554/eLife.14093.030

6.4 Changing the growth rate

Because of growth, the primordia drift away radially from the central zone at a velocity proportional to their distance from the center, i.e. dr(t)dt=γr(t). In the simulations we have used γ=5, we tested γ=3, see the results in Appendix 1–table 9, which do not depart significantly from our main table, see e.g. Appendix 1–table 4 for comparison.

Appendix 1–table 9

Growth rate changed, γ=3.

https://doi.org/10.7554/eLife.14093.031
Γ|E* \ β1012
0.65, 129.0(21.0,8.0), 136.217.7(16.2,1.5), 136.6
0.75, 127.4(19.8,7.6), 135.415.8(14.6,1.2), 136.2
0.85, 114.7(11.3,3.4), 137.710.5(9.8,0.6), 138.0
0.95, 18.0(6.9,1.1), 139.34.3(4.1,0.2), 139.3

6.5 Adding noise to the primordia initiation site on the periphery of central zone

To see the effect of the noise on the sampling of the peripheral zone, we added a random noise in {-2,-1,0,1,2} degrees to the initiation site of the primordium. The permutation rates are most of the time slightly increased.

Appendix 1–table 10

Noise added to the peripheral zone.

https://doi.org/10.7554/eLife.14093.032
Γ|E* \ β1012
0.75, 129.2(19.5,9.7), 135.916.8(14.2,2.6), 136.7
0.75, 1.217.2(14.8,2.3), 136.414.6(12.5,2.1), 136.1
0.85, 123.6(16.0,7.7), 136.612.3(11.2,1.1), 137.7
0.85, 1.216.5(14.2,2.2), 135.79.2(8.3,0.9), 137.2
0.95, 112.5(11.4,1.1), 139.46.9(6.7,0.2), 139.4
0.95, 1.28.6(7.5,1.1), 138.92.4(2.4,0.0), 138.6

6.6 Introducing new primordia at arbitrary positions

As a control to the experiment reported in Figure 2G, we ran simulations where a new primordium is artificially introduced in an otherwise stable spiral pattern. While Figure 2G only considers the case where this ”forcing” takes place at the angle corresponding to the second local minimum, we report here the system’s response in cases of arbitrary angular positions.

These types of perturbations amount to exploring random initial conditions. In the bifurcation diagram of the “standard model” (cf. Appendix 1—figure 2), different initial conditions can lead to different phyllotactic patterns (the branches of the bifurcation diagram); each pattern is reached by a specific set of initial conditions called its basin of attraction. If the artificially created “initial condition” lies outside the basin of attraction of the current pattern, we expect to observe a change of phyllotactic pattern in the subsequent simulation. This intuition is confirmed by simulations: see a broad classification of the possible effects of different typical “new initial conditions” in Appendix 1—figure 7.

Appendix 1—figure 7
Effect of forcing initiation in different regions of the inhibitory field profile.

(a) As reported in Figure 2H, forcing a primordium near the first or second local minima does not affect the phyllotactic mode permanently, but simply forces a 2-permutation in the case of the second minimum. (b) When primordia are introduced near the highest or second highest local maxima, the system returns to its previous pattern. Near the third maximum or the third minimum, the system changes pattern, in the shown simulation it converges to a spiral oriented in the opposite direction to the previous spiral (i.e. φ=-137.5 instead of +137.5); a 2-permutation can be seen in the new spiral for the third maximum. New primordia introduced far from any maximum or minimum (black headed arrows) tend to disrupt the phyllotactic mode more significantly, converging to whorled patterns (primordia forced near 0 or near 290), or patterns involving high numbers of successive permutations (primordium forced near 150).

https://doi.org/10.7554/eLife.14093.033

Appendix 2

Additional videos and initial conditions for all videos

For all videos, unless otherwise stated, we used either of the following initial conditions:

Random: We assume that no organ preexists at the meristem surface and that the level of inhibition randomly fluctuates in the peripheral cells. We model this situation by defining a random inhibition field E=(Ek)k for k ranging over the discretized angular positions as follows: Ek=exp(15σk+10), where σk is drawn from the uniform distribution on the interval [0, 1) and the overall distribution (σk)k is smoothed by applying a Gaussian filter. The value of the inhibitory field in each cell is then updated at each time step of the simulation until the first primordium initiation, by adding a random factor drawn from a uniform distribution on the interval [-0.5, +0.5].

Empty: We assume that no organ preexists at the meristem surface and that the level of inhibition is zero in every cell at the beginning of the simulation (i.e. Ek=0 for all k at t=0). The first primordium is generated at a random location on the peripheral zone.

For all the simulations, β was fixed to 11.0 and E* to 1.0.

Appendix 2—video 1
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a Fibonacci spiral including two 2-permutations from a random initial inhibitory field. For more precise information about the initial condition see Supplementary information.

https://doi.org/10.7554/eLife.14093.034
Appendix 2—video 2
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a Fibonacci spiral from a random initial inhibitory field. It takes some time to converge to 137 degrees. For more precise information about the initial condition see Supplementary information.

https://doi.org/10.7554/eLife.14093.035
Appendix 2—video 3
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a Fibonacci spiral pattern including a 2-permutation pattern at the beginning before converging to 137 degrees. For more precise information about the initial condition see Supplementary information.

https://doi.org/10.7554/eLife.14093.036
Appendix 2—video 4
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone. 

The stochastic model generates a 2-whorled pattern from no pre-existing organs. For more precise information about the initial condition see Supplementary information.

https://doi.org/10.7554/eLife.14093.037
Appendix 2—video 5
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a 2-whorled pattern from a random initial inhibitory field. For more precise information about the initial condition see Supplementary information.

https://doi.org/10.7554/eLife.14093.038
Appendix 2—video 6
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a 2-whorled pattern from a random initial inhibitory field (second run). For more precise information about the initial condition see Supplementary information.

https://doi.org/10.7554/eLife.14093.039
Appendix 2—video 7
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a 2-whorled pattern from no pre-existing organs (second run). For more precise information about the initial condition see Supplementary information. 

https://doi.org/10.7554/eLife.14093.040
Appendix 2—video 8
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a 4-whorled pattern from no pre-existing organs. For more precise information about the initial condition see Supplementary information.

https://doi.org/10.7554/eLife.14093.041
Appendix 2—video 9
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a 4-whorled pattern from a random initial inhibitory field. For more precise information about the initial condition see Supplementary information. 

https://doi.org/10.7554/eLife.14093.042
Appendix 2—video 10
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a 4-whorled pattern from a random initial inhibitory field (second run). For more precise information about the initial condition see Supplementary information. 

https://doi.org/10.7554/eLife.14093.043
Appendix 2—video 11
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone.

The stochastic model generates a 4-whorled pattern from no pre-existing organs (second run). For more precise information about the initial condition see Supplementary information. 

https://doi.org/10.7554/eLife.14093.044
Appendix 2—video 12
Dynamics of the divergence angles as well as the inhibitory field at the peripheral zone showing multiple initiation sites. 

The SMPmicro variant was used instead of SMPmacro-max used in the previous simulations. With SMPmicro, a primodium may be initiated at any cell independently of the others. This makes it possible to two primordia initiation in the same valley. A single initiation site is marked in green and multiple simultaneous initiation sites are marked in red. An initial spiral pattern of 150 preexisting primordia has been used as initial condition.

https://doi.org/10.7554/eLife.14093.045
Appendix 2—video 13
Same video as Appendix 2—video 12 with lower fps.
https://doi.org/10.7554/eLife.14093.046

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Decision letter

  1. Christian S Hardtke
    Reviewing Editor; University of Lausanne, Switzerland

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for submitting your work entitled "A Stochastic Multicellular Model Identifies Disorders as Biological Watermarks in Self-organized Patterns" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Naama Barkai as the Senior Editor.

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission. All reviewers and the editors agree that your work is of potential interest, however they also identified a number of shortcomings that we would like to ask you to address. Please pay attention to the following essential revisions. The reviewers and editors agree that without them, and depending on their outcome, the paper could not be published. They are the following:

1) To convince your audience that your model is capable of the self-organization shown experimentally, you would have to demonstrate that it is able to create a whorled pattern from a random starting point. This is important because a random starting point may change the way the system responds to noise.

2) To demonstrate robustness of the model and its in vivo credibility, we believe it is important that you vary some of the assumptions and parameters in the model, such as the spatial and temporal discretization, the choice of inhibition function (which would vary the nature of the inhibition), and the presence/absence of a secondary mechanism for plastochron control.

Moreover, ideally, you could in addition demonstrate that you get similar results analyzing mistakes in an unrelated system. This would considerably strengthen the credibility of your approach.

Find more details in the reviews pasted below, and please address to your best capacity the other points raised in there.

Reviewer #1:

The authors propose that we can learn something about the mechanism that creates a pattern, not just by looking at the pattern itself, but also by looking carefully at the mistakes it makes. Motivated by recent work, the authors use phyllotactic patterns in order to explore this idea. I find the idea interesting. They suggest that the analysis of the mistakes can help us to understand the mechanism, and they give us an example, the ahp6 mutant, that their analysis shows is more likely the result of a reduction in size of the inhibitory field about primordia, rather than a change in meristem size. This would allow us to distinguish between these two choices. This is not really proof that this is the case, but rather a proof of principle that the idea has discriminating power. In this case there are other possibilities that would need to be excluded, such as increased/decreased noise in location selection due to changes in cell size, or any other input that could affect noise in the system.

I would have like to have seen more exploration of the effect of other model formulations of the model on the results they observe. For example, does the rate of permutations depend on the choice of the inhibition function? What about the spatial and temporal discretization? The growth rate? What does the shape of the curve look like in response to noise in the sampling of the peripheral circle where primordia are formed?

Specific comments:

General: I am not really fond of the comparison to watermarks, I find it a bit misleading as the perturbations are random after all.

Introduction: The presentation in the Introduction suggests that Douady and Couder pioneered the inhibition field model of phyllotaxis. Although they contributed considerably to the field, the authors should perhaps mention at least a few of the others that came before, such as Thornley (1975), Mitchison (1977), Veen and Lindenmayer (1977), Young (1978), Schwabe and Clewer (1984), Chapman and Perry (1987), etc.

Results: The authors write that it has been shown that the central zone is high in auxin. Based on the data, I would say it has been "suggested" or "proposed". Data to support this idea is only from Arabidopsis inflorescense meristem, and from the DII reporter without the recent control construct added by Weijers. The data looks completely different in tomato, where DR5 works just fine in the central zone.

Results: The authors report dislocations in other species, and then immediately assume that it is due to variation in the plastochron. One other possibility could be variations in inhibition threshold. In most inhibition models of phyllotaxis these two are tied together, but that is not necessarily the case in planta. In fact, does not their previous work (Besnard et al. 2014) claim that the timing (i.e. plastochron) is controlled somewhat independently? To report that it is a variation in plastochron, analysis of the timing of primordium initiation at the meristem is required.

Also it is not clear to me from the side views in Figure 2 if these dislocations are just random positioning, or are permutations as stated in the figure caption and text. You would need an image or diagram from the top, or the angles plotted on a graph in order to see if this is the case.

Results: The authors state that the inhibition profile can be seen as an energy. In what sense is it an energy? I think it would be less confusing for the reader, and more accurate to refer to it as the inhibition profile.

Property 2 is reported in the caption for Figure 5G in Smith et al. (2006, CJB), you should probably reference that work in this context.

In a simulation experiment, they perturb the model by initiating primordia at a local inhibition minimum that is not the global minimum and observe recovery. They need the control experiment here. What happens when they initiate a primordia at a random location? Does not the pattern also recover (as it does in ablation experiments)? How does this compare?

Also the authors argue that noise in divergence angle order does not propagate far, but that it has a long effect on the timing, but they don't really give any proof this. We would need to see the time, divergence angle pairs produced by the model, and the model would have to have a similar discretization in time as in space (in most simulation models space it much more coarsely discretized than time). Finally, I might be convinced if they could show evidence of this in planta. It seems to me that after ablation experiments the timing recovers just as fast as the divergence angle.

Stochastic model section: Again they are assuming there is no separate control of the plastochron. I wonder if their stochastic model leads to a different outcome than just adding noise to the system? An example would be the Smith et al. (2006, CJB) model, where noise is added to the peripheral circle sampling point positions, which would also have the effect of adding noise to the inhibition threshold. Don't get me wrong, I appreciate a proper stochastic model, but the worth of such models becomes apparent when they demonstrate different behavior than a simple deterministic model + noise. Is this really the case here?

Speaking of the Smith et al. model, another major source of noise in this system is the fact that the resolution of the possible locations for primordia to initiate is very low. Fate change is a cellular phenomena, and there are really not very many cells around the peripheral zone, maybe 30-40 cells. This gives a resolution of around 10 degrees, a considerable source of noise. Given this it would seem that a perturbation of the possible initiation locations would be very relevant.

Materials and methods: The common phyllotaxis angles are irrational, and have the property that new primordia will be placed far from existing ones. It is therefore not surprising that if you include higher permutations, you start to obtain a roughly uniform sampling of the entire meristem. Consideration of 5 permutations gives 6 matching points, 52.5, 105, 137.5, 190, 275, 327.5 so random angles will always be close to one of these locations, which could explain the data in Figure 7. I would need to see statistics on this to demonstrate that these angle matches are really closer than random.

Reviewer #2:

Phyllotaxis has fascinated philosophers and scientists for centuries being one of few known examples of biological patterns characterized by such regularity, and hundreds of papers were published focused on biological and mathematical interpretations of this unique trait. The submitted manuscript, however, shows that "the other side" of real phyllotactic patterns, namely their perturbations, can also be informative in terms of shoot development regulation. The authors present two models of phyllotactic pattern generation: in the first model the "classical" assumptions often used in phyllotactic pattern models are employed; in the second one stochastic factors are added. Patterns generated with the aid of two models are compared with one another as well as with the empirical data, mainly on Arabidopsis inflorescence phyllotaxis (wild type and mutants). I would like to point several advantages of the presented stochastic model: it accounts for expansion (growth) of the apex surface; primordia simulated in the model are generated by groups of "cells" which all contribute to the pattern formation; the model reproduces real phyllotactic patterns as shown by comparison of pattern parameters, in particular the disorders in the patterns; the biological meaning of the variables used in the model is thoroughly and critically discussed, and those that can have the biological meaning assigned are recognized.

I would like to point to three topics that could be addressed by the authors:

1) In the Introduction and Abstract, the authors state that developmental disorders are generally not considered as informative. I would not point it so strongly, since in fact studies on mutant phenotypes are often studies on disturbed patterns that lead us to conclusions on pattern generation mechanisms. Also some old literature was devoted to teratology, i.e. disturbed development, in order to speculate on developmental mechanisms.

2) From my knowledge on empirical and theoretical papers on phyllotaxis one of the discrepancies between the two is in that in nature, so-called fused or double primordia (or leaves) can be sometimes observed, while they are most often not generated by models. Would it be possible to generate such primordia in the stochastic model (decreasing the epsilon value)? This could provide one more example of pattern perturbation in support of the model.

3) Most reference to empirical data are done with Arabidopsis, obviously a good choice since some mechanisms on the phyllotaxis regulation are recognized based on the mutation. Nevertheless, the relative size of primordia in Arabidopsis is large, and they are not densely packed around its SAM (low numbers of contact parastichies), while in other species primordia are relatively small and densely packed. I understand that the phyllotaxis parameters used for comparison with model generated phyllotaxis, are not available for these species, but these cases could be to larger extent referred to. For example, shoots with relatively small primordia and short plastochrons (conifer twigs, Asteraceae capitulum) would support the model conclusion on relationships between plastochron duration and frequency of permutations (Results); just have a look on the already dead Christmas tree thicker twigs or trunk, and you will most likely find some permutations in support of the model.

Reviewer #3:

My ability to fully interpret this paper is very limited since it is mainly mathematical and my background is biology. However, I understand the main message to be that the authors take an inhibitory field model for phyllotaxis and integrate stochasticity into it. The stochastic model recapitulates the normal patterns of organogenesis as well as observed deviations. It also enables the authors to infer information about the sources of noise.

I think that the ability to help pin-point the root cause for why a pattern is disrupted is very useful and hence this paper is helpful to phyllotaxis researchers. However, I still have concern over the paper's general significance.

My other concern is how model-dependent the conclusions are in any case. Their model, based on abstracted inhibitory fields rather than self-organising auxin transport seems to me to lack a fundamental feature of the latter. For instance, auxin application to un-patterned meristems reveals that organ spacing is robust to random initial distributions of applied auxin (Reinhardt et al., 2003). From what I understand of the model in this paper (and I may be wrong) this fundamental property is not a feature. The ability of the system to form patterns dynamically from random starting points is however captured by less abstracted models based on polarity feed-back from auxin concentrations (Jonsson et al., 2006).

If the model used does not have this property I am worried that the conclusions are not relevant to the real plant.

[Editors' note: further revisions were requested prior to acceptance, as described below.]

Thank you for resubmitting your work entitled "A Stochastic Multicellular Model Identifies Biological Watermarks from Disorders in Self-organized Patterns" for further consideration at eLife. Your revised article has been favorably evaluated by Naama Barkai (Senior editor), a Reviewing editor, and three reviewers.

The manuscript has been improved but there are some remaining issues that need to be addressed before acceptance, as outlined below:

The reviewers and editors agree that you should change the manuscript title to reflect its more narrow proof of concept nature, for example "[…] from Disorders in the self-organizing pattern of phyllotaxis".

Please fix some of the wording, in particular the use of inhibition vs energy (see comments below), to avoid misunderstandings in the context of the wider literature.

Please address the issue of the dynamic definition of inhibitory fields. This appears to be an important limitation of your current model, and although we do not ask you to resolve it at this point, we would like to ask you to explicitly spell out this limitation If it is not captured in the model. That is, specification (or growth of the organ) and the generation of inhibition could occur simultaneously and dynamically rather than in a consecutive, step-wise fashion as is modelled here. Maybe you could also comment on how would this influence the way the system responds to noise?

Find more details in the reviewer comments below, but you only need to respond to the points raised above.

Reviewer #1:

I am mostly satisfied with the changes the authors have made to address the comments from the initial review. It is too bad that they were not able to apply their method to another patterning system, but the authors argue that it would be a complete work in itself, although I am not so sure. I think the application of the method to at least one other system would speak to the potential generality of their ideas, and without it the paper seems in a bit of a niche area (i.e. plant phyllotaxis specific).

Reviewer #2:

Judging from the answers to Editor's and Reviewers' comments provided by the authors and, most of all, from the changes introduced in the manuscript, it has been significantly improved and most of the suggested changes have been introduced: more modelling was performed widening the examined cases, and model interpretation has been improved, also the changes have been made in the main text as suggested. Nevertheless, the manuscript in the present form is for me more clear and representing a broader approach to the problem.

Reviewer #3:

The authors have now attempted to address my main point which was to test whether the model could create a whorled pattern from a random starting point. While I'm happy the authors have done this simulation I still have a remaining concern on this point. This is that in their description of this process they relate that the first primordium is specified stochastically.

The current models for phyllotaxis based on polarized auxin transport are able to self-organize auxin peaks (and hence primordia) simultaneously. Auxin is dynamically distributed over time to create spacing similar to a Turing mechanism. Again, the spacing process for all auxin peaks occurs spontaneously and dynamically. From the supplied description I am still not convinced this model captures this dynamic. In other words, in current models based on auxin transport (e.g. Johnson et al. (2006), an inhibitory field isn't specified only after a primordium is specified (as seems to be the case here), it is being generated dynamically during the process of specification i.e. auxin build-up. Hence the whole process can adjust to changing conditions as they occur – it is dynamic. In contrast, in the whorl simulations here, it seems an initial position is specified by noise and then others are specified in quick succession at a spacing that allows several to form around the apex in the same plastochron but not really in a spontaneous manner.

I guess it really comes down to the question: Can the authors map their model (with or without the stochasticity) on to the class of models represented by the current polarized transport models? For instance, as done by Newell, Shipman and Sun (2008) Journal Theoretical Biology 251, 421-439.

Overall I do think it is critical to show that the model they have chosen to use captures the same fundamental properties as exhibited by models that are more closely based on current experimental work.

https://doi.org/10.7554/eLife.14093.047

Author response

1) To convince your audience that your model is capable of the self-organization shown experimentally, you would have to demonstrate that it is able to create a whorled pattern from a random starting point. This is important because a random starting point may change the way the system responds to noise.

As suggested, we completed our analysis of the self-organization of our model by testing its ability to generate both spiral and whorl patterns from a random starting point, i.e. a naked dome, where cells are interpreting a noisy background signal. We showed that these conditions are sufficient to initiate both types of patterns (spiral and whorls) in a robust manner. Movies showing these simulations are provided as supplementary material (see Appendix, section 7 for details about each movie).

2) To demonstrate robustness of the model and its in vivo credibility, we believe it is important that you vary some of the assumptions and parameters in the model, such as the spatial and temporal discretization, the choice of inhibition function (which would vary the nature of the inhibition), and the presence/absence of a secondary mechanism for plastochron control.

We are most grateful for this comment, as it allowed us to determine a more robust formula for the control parameter ΓD that includes our previous expression of ΓD as a particular case. It turns out indeed that this parameter is also dependent on the stiffness of the inhibition function.

More specifically, we have considered variations of the discretization and inhibition function as detailed below and in our specific answers to reviewers and in all cases our main conclusions remained unchanged (except for the precise form of ΓD). These new results are reported in detail in the appendix, section 6, and correspond to several additional in silico experiments:

  • Spatial discretization: We made simulations with different angular discretizations (1°, 2°,5°, 10°; 1° being the value initially used in the original manuscript) and show that these changes affect mostly the precision of the divergence angle and only marginally the levels of permutations in the simulated sequences, as seen in our new Appendix section 6.2.

  • Inhibition function: We tested other types of inhibition function, either by varying thestiffness of the original function (values s = 1,2,3,5,6 have been tested) or by changing the form of the inhibition function itself (see Appendix section 6.3). This analysis revealed that the change in the inhibitory function per se did not modify the previous results (nature of the control parameters, permutation frequencies). However, the changes in stiffness were shown to modify the dynamics of the system, similarly to the previously defined model parameters (Γ, E*, β). This led us to modify slightly the definitions of the control parameters so that they account for the effect of s on the system. Equations in the text were changed accordingly. Importantly, this analysis led us to a more general formulation of the control parameters, including now stiffness.

Concerning temporal discretization, contrary to classical deterministic models, there is no discretization of time in our stochastic model. This is an advantage as parameters related to time discretization have been removed. The model simulates continuous time by drawing initiation events from continuous non-homogeneous Poisson process (see algorithm in Appendix section 3.2).

Concerning plastochron control, it is not necessary to integrate any secondary mechanism in the model. We did previously show that cytokinin-based secondary fields (encoded by the distribution of the AHP6 protein: Besnard et al. Nature 2014) regulate specifically the plastochron. However, reinterpretation of our previous data in the light of our new model indicates that such a field can be viewed as a component of the inhibitory fields whose effect is equivalent to an increase in Γ. Our model has the native capacity to integrate different components in the inhibition function. This aspect was discussed in the Discussion section of our original submission (see section of Developmental disorders as biological watermarks).

Moreover, ideally, you could in addition demonstrate that you get similar results analyzing mistakes in an unrelated system. This would considerably strengthen the credibility of your approach.

We agree that, ideally, similar results on other independent systems would be remarkable. We think indeed that our work lays the ground for such investigations in developmental biology. Our model is about patterning systems with lateral inhibition. We have developed it on a 2D circular system. This could similarly be developed on other developmental systems based on lateral inhibitions and that can be modelled in 2D (e.g. mouse hairs, trichomes) or 1-D (somites, anabaenae). However, such a work necessitates extensive quantitative data and analysis of the underlying structure of the data. To our knowledge, no other systems have been analysed at the moment in a way that permits such an exploration. In addition, providing a similar demonstration in any other system would be a complete work in itself (that necessitates modelling a new developmental system) and collect completely new datasets from various phenotypes and we believe this is out of the scope of the present work.

Reviewer #1:

I would have like to have seen more exploration of the effect of other model formulations of the model on the results they observe. For example, does the rate of permutations depend on the choice of the inhibition function? What about the spatial and temporal discretization? The growth rate? What does the shape of the curve look like in response to noise in the sampling of the peripheral circle where primordia are formed?

As pointed out by the reviewer, our study aims at showing the plausibility of a stochastic perception of the inhibition signal to explain observed phyllotaxis perturbation in wild-type and mutant plants. What we show here is that a noise in the perception of the signal satisfactorily explains all the data that have been recently observed about perturbations in phyllotaxis systems. However, it is true that other sources of noise could be imagined. To address this concern, we investigated various possibilities as suggested by the reviewer. We have addressed some of these in the section on essential revision and recall below our conclusions.

  • The choice of inhibition function: We tested other types of inhibition function, either byvarying the stiffness of the original function (values s = 1,2,3,5,6 have been tested, see Appendix section 6.3) or by changing the inhibition function itself. The main outcome of these new simulations is that our main results remain essentially unchanged, provided the control parameter is updated to include the stiffness parameter. This updated control parameter confirms our qualitative claims and introduce a new level of refinement, for which we are very grateful to Reviewer #1. Equations in the text were changed accordingly to include the stiffness parameter.

  • The spatial discretization of the peripheral zone: We made simulations with different angular discretizations (1°, 2°, 5°, 10°; 1° being the value initially used in the original manuscript) and show that these changes affect the precision of the divergence angle rather than the levels of permutations in the simulated sequences, as seen in our new Appendix section 6.2.

  • The temporal discretization: contrary to classical deterministic models, there is nodiscretization of time in our stochastic model. This is an advantage as parameters related to time discretization have been removed. The model simulates continuous time by drawing initiation events from continuous non-homogeneous Poisson process (see algorithm in Appendix section 3.2).

  • Growth rate: Changing the growth range does not change divergence angle, and other curves, but accelerates or decelerates plastochrons (temporal scaling). This has a low effect on the frequency of permutations (see Appendix section 6.4), suggesting that, as expected, the growth rate is not determinant for the spatio-temporal patterning of the system.

Finally, as reported in Appendix section 6.6, we have tested simulations where a random noise is introduced in the sampling of the peripheral zone. More precisely, the precise initiation position was randomly shifted by a few positions compared to the exact position induced by the inhibition field. Perhaps not surprisingly, the results in Appendix 6.6 show that this leads to a moderate increase in the number of permutations.

We have updated the Results and Discussion to highlight these new results. We believe that this strengthens significantly our conclusions.

Specific comments:

General: I am not really fond of the comparison to watermarks, I find it a bit misleading as the perturbations are random after all.

We thank the reviewer for this remark. It is true that the perturbations are random but, they are not the watermark per se. Our previous formulation was indeed ambiguous. In our analogy, the biological watermark (the hidden information) is made of the state variables Γ, E*, β. What we say is that by looking in the actual phyllotactic signal, that contains noise, it is possible to compute a perturbation intensity, i.e. this is part of the algorithm to extract the watermark. This, together with the estimation of the other observable variables, plastochron and divergence angle, provides information on the actual values of the hidden state variables, and strongly restricts their range of values. This information is our biological watermark.

To correct this ambiguity, we slightly modified the wording of the paper’s title and clarified the description of the analogy in the Discussion.

Introduction: The presentation in the Introduction suggests that Douady and Couder pioneered the inhibition field model of phyllotaxis. Although they contributed considerably to the field, the authors should perhaps mention at least a few of the others that came before, such as Thornley (1975), Mitchison (1977), Veen and Lindenmayer (1977), Young (1978), Schwabe and Clewer (1984), Chapman and Perry (1987), etc.

We agree with the reviewer and we have now changed the wording of this passage in the Introduction and added a few key references as suggested.

Results: The authors write that it has been shown that the central zone is high in auxin. Based on the data, I would say it has been "suggested" or "proposed". Data to support this idea is only from Arabidopsis inflorescense meristem, and from the DII reporter without the recent control construct added by Weijers. The data looks completely different in tomato, where DR5 works just fine in the central zone.

We agree with the reviewer and we now say that this “has been suggested”.

Results: The authors report dislocations in other species, and then immediately assume that it is due to variation in the plastochron. One other possibility could be variations in inhibition threshold. In most inhibition models of phyllotaxis these two are tied together, but that is not necessarily the case in planta. In fact, does not their previous work (Besnard et al. 2014) claim that the timing (i.e. plastochron) is controlled somewhat independently? To report that it is a variation in plastochron, analysis of the timing of primordium initiation at the meristem is required.

Modeling stochasticity of inhibition perception is actually exactly the essence of our approach. An analysis of the effect of variations in inhibition threshold was previously carried out in Mirabet et al. 2012 and concluded that it induced specifically a variation in the plastochron leading to organ co-initiations. However, in this work permutations could be induced although at a much lower frequency than what is observed in planta. Also only simple perturbations (2-permutations) could be obtained. We developed this idea further by modelling the response of the system to the inhibition signal (using a probabilistic framework rather than a deterministic threshold).

Our work also does suggest that there is a relative independence between the emergent plastochron and the spatial patterning system, as reported previously in planta in Besnard et al. 2014. Such an analysis, comparing observed wild type to mutant plastochrons, is done in the appendix (see Appendix 4—figure 1D).

In any case, we agree that an analysis of the timing of primordium initiation at the meristem in other species than Arabidopsis would be important. However, this would require performing live imaging in these various systems, which is at the moment out of reach. For this reason, we have considered permutations as a plausible signature of plastochron variation, based on our previous work in Arabidopsis. We have thus rephrased our text in the following way in order to be more careful in our conclusions and to address the reviewer’s concern:

“As suggested by the results on Arabidopsis, these observations raise the possibility that these organ permutations might result from a noise on the plastochron and that such perturbation could be a common feature of phyllotactic systems that occurs in meristems with different geometries.”

Also it is not clear to me from the side views in Figure 2 if these dislocations are just random positioning, or are permutations as stated in the figure caption and text. You would need an image or diagram from the top, or the angles plotted on a graph in order to see if this is the case.

To clarify the 3D geometric interpretation of the 2D pictures, we added as suggested at the bottom of each image a small diagram showing how the branch angles and orders should be read from the top.

Results: The authors state that the inhibition profile can be seen as an energy. In what sense is it an energy? I think it would be less confusing for the reader, and more accurate to refer to it as the inhibition profile.

It is common in the literature about phyllotaxis to refer the inhibition created by each organ in its surrounding as an abstract repulsive energy by analogy with physical systems. This term was notably used in the work of Douady and Couder (1996). We added a couple of sentences to make this definition clearer.

Property 2 is reported in the caption for Figure 5G in Smith et al. (2006, CJB), you should probably reference that work in this context.

The reviewer is right that several works before have mentioned that in a stationary regime the position of next organ to be initiated is separated from the current initium by roughly a divergence angle (Smith et al. 2006 mention this in one figure legend as well as Mirabet et al. 2011, Figure 1 for instance). However, what we say here based on our study of the dynamics of inhibitory fields is more general: at any moment, all the energy minima are separated by multiples of the divergence angle. This can be observed for example in Figure 3A,B,D,E,F and in corresponding Videos 13.

In a simulation experiment, they perturb the model by initiating primordia at a local inhibition minimum that is not the global minimum and observe recovery. They need the control experiment here. What happens when they initiate a primordia at a random location? Does not the pattern also recover (as it does in ablation experiments)? How does this compare?

We agree with the reviewer that such a control would be informative. We have performed the control experiments suggested by this comment: interrupting an on-going simulation, we have forced the initiation of a primordium at various random locations, not only local minima. These types of perturbations amount to exploring random initial conditions. In the bifurcation diagram of the standard model (cf. Appendix, Figure 2), different initial conditions can lead to different phyllotactic patterns (the branches of the bifurcation diagram), each pattern having a basin of attraction composed of those initial conditions that converge to the corresponding phyllotactic pattern. From this general principle, we expect that forcing a primordium at an arbitrary position may lead to a permanent change of phyllotaxis (if the artificially created initial condition lies outside the basin of attraction of the current pattern). This expected behavior is exactly what we observed and in fact we were able to classify the initial conditions that typically lead to a change of phyllotactic mode and those that did not (including the previously treated case of a primordium introduced at the second local minimum), as reported in Appendix, Section 6.7 and Appendix Figure 5 and we mentioned this control experiment in the main text.

Also the authors argue that noise in divergence angle order does not propagate far, but that it has a long effect on the timing, but they don't really give any proof this. We would need to see the time, divergence angle pairs produced by the model, and the model would have to have a similar discretization in time as in space (in most simulation models space it much more coarsely discretized than time). Finally, I might be convinced if they could show evidence of this in planta. It seems to me that after ablation experiments the timing recovers just as fast as the divergence angle.

We are unsure of what is the concern of the reviewer here. We have not studied the effect of noise on divergence angle but the effect of introducing artificially noise on the timing through forcing permutations in the order of organ initiation. We then observe that temporal noise does not affect significantly the angular position of the other primordia but has a long term effect on the timing in the classical model. This was observed robustly on many repetitions. In this, we are not trying to demonstrate that noise on spatial position does not propagate but only to show that spatial positioning and timing are partly uncoupled in this system and that this drove us to develop a model of signal perception to explain the observed permutations. In addition, information concerning the divergence angles and the associated plastochron produced by the stochastic model are available in Figure 4F, G, H.

Concerning the experimental evidence, we would like to point out that our previous work published in Besnard et al. 2014 (Figure 2G) shows in planta using live imaging of wild-type meristems that angle specification is extremely stable over time despite a high plastochron variability with almost 30% of organ co-initiated. These observations are indeed fully consistent with the results of our simulations, showing a partial uncoupling of organ positioning and timing of initiation. To make this point absolutely clear we have indicated this at the end of the corresponding Results section.

Stochastic model section. Again they are assuming there is no separate control of the plastochron. I wonder if their stochastic model leads to a different outcome than just adding noise to the system? An example would be the Smith et al. (2006, CJB) model, where noise is added to the peripheral circle sampling point positions, which would also have the effect of adding noise to the inhibition threshold. Don't get me wrong, I appreciate a proper stochastic model, but the worth of such models becomes apparent when they demonstrate different behavior than a simple deterministic model + noise. Is this really the case here?

We assume that when mentioning “separate control of the plastochron” the referee has in mind again the mechanisms involving AHP6 (Besnard et al., 2014). We would like to point out that we didn’t say that AHP6 controls the plastochron but that it controls the robustness of plastochron. From our work and previous theoretical/experimental work on phyllotaxis, we rather believe that plastochron is an emergent property of the system, while having different properties compared to the divergence angles. We propose that what is regulated is the different parameters of the patterning (not directly in the form of a direct regulation of the plastochron) and consequently that all the observation variables (i.e. spatial positioning and plastochron but also perturbations) are linked through the 2 control parameters.

Concerning the added value of our model, Mirabet et al. did exactly what the reviewer suggests here and tried the option to explain permutation with a simple deterministic model + noise. Interestingly, the model is able to capture some features of the observed perturbation (as suggested by the reviewer) but was limited in the capacity to recapitulate satisfactorily the various experimental observations as discussed above. We show here that our model provides a much more plausible interpretation of the observed phenomena.

Speaking of the Smith et al. model, another major source of noise in this system is the fact that the resolution of the possible locations for primordia to initiate is very low. Fate change is a cellular phenomena, and there are really not very many cells around the peripheral zone, maybe 30-40 cells. This gives a resolution of around 10 degrees, a considerable source of noise. Given this it would seem that a perturbation of the possible initiation locations would be very relevant.

As pointed out above, a noise on the spatial resolution has only a limited effect on the dynamics of the system, supporting the idea that it unlikely explains the frequent perturbations of phyllotaxis observed in Arabidopsis. Also note that this paper studies the plausibility of a particular source for the observed phyllotactic perturbations but does not aim to claim that other sources should completely be excluded.

Materials and methods: The common phyllotaxis angles are irrational, and have the property that new primordia will be placed far from existing ones. It is therefore not surprising that if you include higher permutations, you start to obtain a roughly uniform sampling of the entire meristem. Consideration of 5 permutations gives 6 matching points, 52.5, 105, 137.5, 190, 275, 327.5 so random angles will always be close to one of these locations, which could explain the data in Figure 7. I would need to see statistics on this to demonstrate that these angle matches are really closer than random.

We completely agree with the reviewer regarding the fact that multiples of an angle like the golden angle 137.5 will naturally tend to cover the full interval [0,360) densely and indeed match quite well any distribution of randomly drawn angles. However, this viewpoint does not take into account the temporal succession of angles, which is a crucial component of our fit: the data shown in Appendix Figure 7 is not only a fit of individual angles, but also of their sequence. In other words, the fit does not simply consist in finding the “closest match” for each individual angle but the closest match for the whole sequence amongst sequences that can be decomposed into permuted blocks of bounded length n (called n-admissible sequences in our previous publications). In particular, it is likely that a pure distance criterion would yield a quantitatively closer fit using only the closest match of individual angles, but given the variability of angle values observed experimentally (see for instance the spread of the fitted Gaussian-like distributions shown in Figure 1), the sequences we propose are equally plausible: by design, our algorithms only select angles which are within the center part of Gaussian distributions fitting angle distributions. Given that n-admissible sequences have a very low probability of being obtained from a random draw (when n is smaller than the length of the whole sequence) we claim that it is striking to find plausible matches with a high frequency as we did.

In summary, we believe that the answer to this comment was already present in our manuscript, although we apologize that it was not explained with enough clarity. Additional explanations are now provided in the caption for Appendix Figure 7.

Reviewer #2:

I would like to point to three topics that could be addressed by the authors:

1) In Introduction and Abstract, the authors state that developmental disorders are generally not considered as informative. I would not point it so strongly, since in fact studies on mutant phenotypes are often studies on disturbed patterns that lead us to conclusions on pattern generation mechanisms. Also some old literature was devoted to teratology, i.e. disturbed development, in order to speculate on developmental mechanisms.

We agree with the reviewer and reformulated the corresponding passages in a weaker form.

2) From my knowledge on empirical and theoretical papers on phyllotaxis one of the discrepancies between the two is in that in nature, so-called fused or double primordia (or leaves) can be sometimes observed, while they are most often not generated by models. Would it be possible to generate such primordia in the stochastic model (decreasing the epsilon value)? This could provide one more example of pattern perturbation in support of the model.

This is an interesting suggestion. To implement it, we have considered the microscopic, most computationally expensive version of our model, SMPmicro, where the stochastic process of primordium initiation takes place in parallel at each discretization point on the central zone boundary (as opposed to processes only at local inhibition minima, in SMPmacro-max, which effectively precludes fused primordia). In this model, with low values of the sensitivity parameter E*, we indeed observed simultaneous formations of two primordia near the same local minimum. Examples of such simulations are now included in the form of animations (revision_multipleInitiation_slow.wmv and revision_multipleInitiation_fast.wmv), see Appendix section 7 for more details.

3) Most reference to empirical data are done with Arabidopsis, obviously a good choice since some mechanisms on the phyllotaxis regulation are recognized based on the mutation. Nevertheless, the relative size of primordia in Arabidopsis is large, and they are not densely packed around its SAM (low numbers of contact parastichies), while in other species primordia are relatively small and densely packed. I understand that the phyllotaxis parameters used for comparison with model generated phyllotaxis, are not available for these species, but these cases could be to larger extent referred to. For example, shoots with relatively small primordia and short plastochrons (conifer twigs, Asteraceae capitulum) would support the model conclusion on relationships between plastochron duration and frequency of permutations (Results); just have a look on the already dead Christmas tree thicker twigs or trunk, and you will most likely find some permutations in support of the model.

We agree with the reviewer that it is important to show permutations in species other than Arabidopsis. It is for this very reason that we included evidence of permutations in a variety ofspecies that are expected to have different meristem geometries. Optimally we would need to use live-imaging in these species to analyze the dynamics of organ initiation in these species but this is at the moment technically out of reach. Our work is however laying the ground by demonstrating the important of performing such analyses in the future.

Reviewer #3:

My ability to fully interpret this paper is very limited since it is mainly mathematical and my background is biology.

[…]

If the model used does not have this property I am worried that the conclusions are not relevant to the real plant.

We agree with Reviewer #3 and it is important that our model be able to generate phyllotactic patterns from a wide range of initial conditions, including randomly chosen ones. To clarify that this is indeed the case, we have performed simulations with initial conditions where the inhibition field is randomly defined, as well as with a naked meristem with no initial primordium. The results are reported in the form of animations, whose details are shown in Appendix, section 7. In all cases, the system robustly self-organizes and converges to a plausible phyllotactic pattern. We thus believe that this provides further support to the idea that our conclusions are relevant to real plant.

Regarding our paper’s general significance, phyllotactic patterning is a prototypical example of a developmental system driven by lateral inhibitions (the inhibitory fields) as pointed out in our manuscript. Lateral inhibitions have been implicated in a large number of patterning processes in both plant and animals (see before). With the ongoing development of live-imaging in a variety of developmental system, we expect that irregularities in such systems will be reported and we believe that we provide a conceptual framework for understanding how such irregularities can be used to understand the molecular mechanisms at work. We also think that this framework will be useful for the analysis of noise in developmental systems in general by providing a vision of how noise propagates through scale to generate specific developmental signatures.

[Editors' note: further revisions were requested prior to acceptance, as described below.]

The manuscript has been improved but there are some remaining issues that need to be addressed before acceptance, as outlined below:

The reviewers and editors agree that you should change the manuscript title to reflect its more narrow proof of concept nature, for example "…from Disorders in the self-organizing pattern of phyllotaxis".

We have changed the title of the paper as suggested.

Please fix some of the wording, in particular the use of inhibition vs energy (see comments below), to avoid misunderstandings in the context of the wider literature.

We have changed the wording as suggested by you and reviewer 1 throughout the main text, figures and appendixes. In addition, we have corrected a few typos in the manuscript.

Please address the issue of the dynamic definition of inhibitory fields. This appears to be an important limitation of your current model, and although we do not ask you to resolve it at this point, we would like to ask you to explicitly spell out this limitation If it is not captured in the model. That is, specification (or growth of the organ) and the generation of inhibition could occur simultaneously and dynamically rather than in a consecutive, step-wise fashion as is modelled here. Maybe you could also comment on how would this influence the way the system responds to noise?

This is an important point that was not completely clear to us from the original review comments. We thank Reviewer 3 and editors’ new comments that clarify the concern and we do agree with you that our stochastic system, similarly to the classical deterministic model, does not model the details of the molecular processes involved in the formation of the inhibitory fields and that this may impact its ability to capture self-organizing dynamic aspects of field formation. We have thus added a paragraph in the last section of the Discussion stating clearly this potential limit of our model and how to address it in the future:

“Moreover, similarly to the classical deterministic model of phyllotaxis, our stochastic model does not explicitly account for the cascade of molecular processes that participate to the establishment of new inhibitory fields at the location of incipient primordia. This might limit the ability of these models to fully capture the dynamics of the self-organization of the system. To do so, more mechanistic versions of this stochastic model could be developed in the future, combining more detailed cellular models of hormone-based fields, e.g. (Jönsson et al., 2006; Smith et al., 2006; Stoma et al., 2008), and stochastic perception of these hormonal signals in 2D or 3D models with cell resolution.”

https://doi.org/10.7554/eLife.14093.048

Article and author information

Author details

  1. Yassin Refahi

    1. Laboratoire de Reproduction de développement des plantes, Lyon, France
    2. Sainsbury Laboratory, University of Cambridge, Cambridge, United Kingdom
    Contribution
    YR, Implemented the stochastic model, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article
    Competing interests
    The authors declare that no competing interests exist.
  2. Géraldine Brunoud

    Laboratoire de Reproduction de développement des plantes, Lyon, France
    Contribution
    GB, Acquisition of data
    Competing interests
    The authors declare that no competing interests exist.
  3. Etienne Farcot

    1. School of Mathematical Sciences, The University of Nottingham, Nottingham, United Kingdom
    2. Center for Integrative Plant Biology, The University of Nottingham, Notthingam, United Kingdom
    Contribution
    EF, Conception and design, Analysis and interpretation of data
    Competing interests
    The authors declare that no competing interests exist.
  4. Alain Jean-Marie

    INRIA Project-Team Maestro, INRIA Sophia-Antipolis Méditerranée Research Center, Montpellier, France
    Contribution
    AJ-M, Conception and design
    Competing interests
    The authors declare that no competing interests exist.
  5. Minna Pulkkinen

    UMR Lerfob, AgroParisTech, Nancy, France
    Contribution
    MP, Analysis and interpretation of data
    Competing interests
    The authors declare that no competing interests exist.
  6. Teva Vernoux

    Laboratoire de Reproduction de développement des plantes, Lyon, France
    Contribution
    TV, Conception and design, Analysis and interpretation of data, Drafting or revising the article
    For correspondence
    teva.vernoux@ens-lyon.fr
    Competing interests
    The authors declare that no competing interests exist.
    ORCID icon 0000-0002-8257-4088
  7. Christophe Godin

    INRIA Project-Team Virtual Plants, CIRAD, INRA and INRIA Sophia-Antipolis Méditerranée Research Center, Montpellier, France
    Contribution
    CG, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article
    For correspondence
    Christophe.Godin@inria.fr
    Competing interests
    The authors declare that no competing interests exist.
    ORCID icon 0000-0002-1202-8460

Funding

Human Frontier Science Program (RGP0054-2013)

  • Teva Vernoux
  • Christophe Godin

Inria Project-Lab Morphogenetics

  • Yassin Refahi
  • Teva Vernoux
  • Christophe Godin

ANR Institute of Computational Biology

  • Christophe Godin

ERC Morphodynamics

  • Yassin Refahi

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

The authors would like to thank Olivier Hamant and Benoit Landrein for making original data from paper (Landrein et al., 2015) available to them, Arezki Boudaoud, Henrik Jönsson, Olivier Hamant and Jan Traas for their insightful comments on the manuscript, and to acknowledge funding from the following grants: HFSP grant RGP0054-2013 BioSensors (Human Frontier Science Program) to TV and CG, Inria Project Lab Morphogenetics and ANR-funded Institute of Computational Biology (IBC) to CG, Jan Traas’s ERC Morphodynamics & INRA CJS to YR.

Reviewing Editor

  1. Christian S Hardtke, Reviewing Editor, University of Lausanne, Switzerland

Publication history

  1. Received: December 28, 2015
  2. Accepted: May 3, 2016
  3. Version of Record published: July 6, 2016 (version 1)

Copyright

© 2016, Refahi et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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