A stochastic multicellular model identifies biological watermarks from disorders in self-organized patterns of phyllotaxis

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.

DOI: http://dx.doi.org/10.7554/eLife.14093.001

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.10.7554/eLife.14093.003

Figure 1.

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).

DOI: http://dx.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.10.7554/eLife.14093.004

Figure 2.

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.

DOI: http://dx.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)=∑q∈QE(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 j, i<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:i≤l≤j: (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.10.7554/eLife.14093.005

Figure 3.

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. Ek−E1 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 Ek−E1 in inhibition levels is higher than in A. (C) Variation of the distance E2−E1 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 q_n), 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.

DOI: http://dx.doi.org/10.7554/eLife.14093.005

10.7554/eLife.14093.006

Figure 3—figure supplement 1.

Divergence angle of a series of simulations of the classical model with control parameter Γ1=0.975 and for which the choice of the jth local minimum (instead of the global minimum, i.e. j=1) has been forced at a given time-point (red arrow).

After the forcing, the reaction of the classical model is observed. 1. divergence angles (left column) 2. corresponding plastochrons (right column) (A,B) j=2. (C,D) j=3. (E,F) j=3 and j=2 are imposed in this order.

DOI: http://dx.doi.org/10.7554/eLife.14093.006

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).

DOI: http://dx.doi.org/10.7554/eLife.14093.007

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.

DOI: http://dx.doi.org/10.7554/eLife.14093.008

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.

DOI: http://dx.doi.org/10.7554/eLife.14093.009

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 0≤λ(Ek(t))δt≤1; 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))=e−β(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)=e−β(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)=δt∑k=1Ke−β(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=∑k∈Klλk(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))=δt∑l=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=maxk∈Kl 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.10.7554/eLife.14093.010

Figure 4.

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.

DOI: http://dx.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.10.7554/eLife.14093.011

Figure 5.

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.

DOI: http://dx.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.10.7554/eLife.14093.013

Figure 6.

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=Γ1β1/6E∗1/2 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)

DOI: http://dx.doi.org/10.7554/eLife.14093.013

10.7554/eLife.14093.014

Figure 6—figure supplement 1.

New control parameter ΓD for divergence angle and plastochrons.

Each graph is made up of points that correspond to different values of the parameters Γ,β,E∗ of the stochastic model. Left column: different trials to define a control parameter for divergence angles α. Right column: different tries to define a control parameter for the plastochrons. For the parameter ΓD=Γ1(β1/3E∗)1/2, both clouds of points collapse on a single curve (we assume here that s = 3, see Appendix 1—figure 6 for more details).

DOI: http://dx.doi.org/10.7554/eLife.14093.014

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/31β1/6E∗1/2 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=ΓβE∗ΓD=Γs/31β1/6E∗1/2

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 (ΓD, ΓP) 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).10.7554/eLife.14093.015

Figure 7.

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].

DOI: http://dx.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).10.7554/eLife.14093.016

Figure 8.

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/sβ1/6E∗1/2 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….

DOI: http://dx.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.