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.

Keywords: A. thaliana; developmental biology; emergence; inhibitory fields; multi-scale modeling; noise; permutations; phyllotaxis; plant biology; stem cells.

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

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

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 ${\displaystyle {\mathrm{\Gamma }}_1}$ = 0.975. ${\displaystyle E_k-E_1}$ 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 (${\displaystyle \alpha =}$137°). (B) Similar inhibition profile for a control parameter ${\displaystyle {\mathrm{\Gamma }}_2}$ = 0.675 < Γ1. The difference ${\displaystyle E_k-E_1}$ in inhibition levels is higher than in A. (C) Variation of the distance ${\displaystyle E_2-E_1}$ 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 ${\displaystyle {\mathrm{\Gamma }}_1}$). 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 ${\displaystyle \alpha }$ 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

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

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 (${\displaystyle \beta }$ = 10.0, ${\displaystyle E^{\ast }}$ = 1.4, ${\displaystyle \mathrm{\Gamma }}$ = 0.625). (E) More complex simulated permutations involving 2- and 3-permutations (${\displaystyle \beta }$ = 10.0, ${\displaystyle E^{\ast }}$= 1.4, ${\displaystyle \mathrm{\Gamma }}$ = 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 ${\displaystyle \beta }$ = 11.0, ${\displaystyle E^{\ast }}$ =1.2, ${\displaystyle \mathrm{\Gamma }}$= 0.8. (G) Histogram of simulated divergence angles and corresponding plastochron distribution for ${\displaystyle \beta }$ = 9.0, ${\displaystyle E^{\ast }}$ = 1.2, ${\displaystyle \mathrm{\Gamma }}$ = 0.8 (H) Histogram of simulated divergence angles and corresponding plastochron distribution for ${\displaystyle \beta }$ = 9.0, ${\displaystyle E^{\ast }}$ = 1.2, ${\displaystyle \mathrm{\Gamma }}$ = 0.625. DOI: http://dx.doi.org/10.7554/eLife.14093.010

Figure 5.. Intensity of 2-permutations as a function of the total amount of perturbations.

As the perturbation intensity ${\displaystyle \pi }$ 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

Figure 6.. Key parameters controlling phyllotaxis phenotypes in the stochastic model.

Phyllotaxis sequences were simulated for a range of values of each parameter ${\displaystyle \beta }$, ${\displaystyle E^{\ast }}$, ${\displaystyle \mathrm{\Gamma }}$. 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 ${\displaystyle \pi }$ as a function of the new control parameter ${\displaystyle {\mathrm{\Gamma }}_P=\mathrm{\Gamma }\beta E^{\ast }}$. (B) Divergence angle ${\displaystyle \alpha }$ as a function of the control parameter ${\displaystyle \mathrm{\Gamma }}$ of the classical model on the Fibonacci branch. (C) Divergence angle ${\displaystyle \alpha }$ as a function of the new control parameter ${\displaystyle {\mathrm{\Gamma }}_D=\mathrm{\Gamma }\frac{1}{{\beta }^{1/6}{E}^{\ast 1/2}}}$ on the Fibonacci branch (here, we assume s = 3, see Appendix 1—figure 6 for more details). (D) Plastochron ${\displaystyle T}$ as a function of control parameter of the classical model ${\displaystyle \mathrm{\Gamma }}$. (E) Plastochron ${\displaystyle T}$ as a function of the new control parameter ${\displaystyle {\mathrm{\Gamma }}_D}$. (F) Parastichy modes ${\displaystyle (i,j)}$ identified in simulated sequences as a function of ${\displaystyle {\mathrm{\Gamma }}_D}$. Modes ${\displaystyle (i,j)}$ are represented by a point ${\displaystyle i+j}$. 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

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 ${\displaystyle \mathrm{\Gamma },\beta ,E^{\ast }}$ of the stochastic model. Left column: different trials to define a control parameter for divergence angles ${\displaystyle \alpha }$. Right column: different tries to define a control parameter for the plastochrons. For the parameter ${\displaystyle {\mathrm{\Gamma }}_D=\mathrm{\Gamma }\frac{1}{({\beta }^{1/3}{E}^{\ast }{)}^{1/2}}}$, 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

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

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 ${\displaystyle E(t)}$ that depends on the azimuthal angle ${\displaystyle \alpha }$ (blue curve). At any time t, and at each intensity minimum of this curve, a primordium can be initiated during a time laps ${\displaystyle \delta t}$ with a probability ${\displaystyle p_k(\delta t)}$ 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 ${\displaystyle \mathrm{\Gamma }}$ 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 ${\displaystyle \mathrm{\Gamma }}$,${\displaystyle \beta }$, ${\displaystyle E^{\ast }}$. The observable variables ${\displaystyle \alpha }$, ${\displaystyle T}$ and ${\displaystyle \pi ,{\pi }_2,{\pi }_3}$… are controlled by two distinct combinations of these parameters: ${\displaystyle {\mathrm{\Gamma }}_D=\frac{{\mathrm{\Gamma }}^{1/s}}{{\beta }^{1/6}{E}^{\ast 1/2}}}$ controls the divergence angle and plastochron while ${\displaystyle {\mathrm{\Gamma }}_P=\mathrm{\Gamma }\beta E^{\ast }}$ controls the global percentage of permutated organs ${\displaystyle \pi }$, which in turns controls the distribution of permutation complexities: ${\displaystyle {\pi }_2,{\pi }_3}$…. DOI: http://dx.doi.org/10.7554/eLife.14093.016

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 $\varphi$. 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 $(8,13)$. DOI: http://dx.doi.org/10.7554/eLife.14093.017

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

We used a sample of the interval $0.04\le {\mathrm{\Gamma }}^{\prime }\le 2.9$ with steps of $0.01$ or less (refinements were performed in areas with higher numbers of branches). For each value of ${\mathrm{\Gamma }}^{\prime }$, we ran simulations of the classical model with (i) spiral initial conditions for divergence angles taking integer values in ${\displaystyle [{20}^{\circ },{180}^{\circ }]}$, 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 $2\le j\le 7$. For each simulation, we estimated the final divergence angle $\varphi$ and phyllotactic mode, and reported these in the graph above (abscissa: ${\mathrm{\Gamma }}^{\prime }$, ordinate: $\varphi$, color code: mode). DOI: http://dx.doi.org/10.7554/eLife.14093.018

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

(A) An exponential model was fitted to the simulated data of $\pi$ and ${\mathrm{\Gamma }}_P$ 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 ${\mathrm{\Gamma }}_P$ values [8.30, 11.49] that could yield the observed $\pi$ value 14.2 was determined by the prediction band. (B) A Gompertz function was fitted to the simulated data of plastochron and ${\mathrm{\Gamma }}_D$ 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 ${\mathrm{\Gamma }}_D$ 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 ${\mathrm{\Gamma }}_D$ 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 ${\mathrm{\Gamma }}_D$ 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. DOI: http://dx.doi.org/10.7554/eLife.14093.019

Appendix 1—figure 4.. Histogram of randomly generated sequences.

(a) Histogram of randomly generated sequences, $\kappa =14.0$,${\pi }_{2,3}=24\%$. (b) Histogram of randomly generated sequences, $\kappa =14.0$, ${\pi }_{2,3}=43\%$. (c) Histogram of randomly generated sequences, $\kappa =10.4$, ${\pi }_{2,3}=24\%$. (d) Histogram of randomly generated sequences, $\kappa =10.4$, ${\pi }_{2,3}=43\%$. DOI: http://dx.doi.org/10.7554/eLife.14093.022

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

(a–b) Average plastochron ratio as a function of $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}_D=\frac{\mathrm{\Gamma }}{{\beta }^{1/6}{E}^{*1/2}}$, respectively. (c) Number of permutations $\pi$ as a function of ${\mathrm{\Gamma }}_P$. DOI: http://dx.doi.org/10.7554/eLife.14093.029

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 $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}_D=\frac{\mathrm{\Gamma }}{{\beta }^{1/6}{E}^{*1/2}}$, respectively. Third column: average plastochron ratio as a function of ${\mathrm{\Gamma }}_D=\frac{{\mathrm{\Gamma }}^{s/3}}{{\beta }^{1/6}{E}^{*1/2}}$. DOI: http://dx.doi.org/10.7554/eLife.14093.030

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. $\phi =-137.5$ 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). DOI: http://dx.doi.org/10.7554/eLife.14093.033