Pattern identification and characterization reveal permutations of organs as a key genetically...

Pattern identification and characterization reveal permutationsof organs as a key genetically controlled propertyof post-meristematic phyllotaxis

Yann Guédon a,n, Yassin Refahi a, Fabrice Besnard b, Etienne Farcot a,Christophe Godin a, Teva Vernoux b

a CIRAD/INRA/Inria Virtual Plants Team, UMR AGAP, F-34095 Montpellier, Franceb Laboratoire Reproduction et Développement des Plantes, ENS/CNRS/INRA/Université Lyon I, F-69342 Lyon, France

H I G H L I G H T S

� We study phyllotaxis, the geometric arrangement of organs around the stem.� We design a pipeline of methods for analyzing stem phyllotaxis.� We identify phyllotaxis perturbations in wild-type and mutant Arabidopsis plants.� These perturbations can be explained by permutations in the order of organ insertion.� Permutations are an intrinsic property of spiral phyllotaxis.

a r t i c l e i n f o

Article history:Received 10 July 2013Accepted 24 July 2013Available online 13 August 2013

Keywords:Combinatorial modelHidden Markov modelMixture modelPlant phenotypingVariable-order Markov chain

a b s t r a c t

In vascular plants, the arrangement of organs around the stem generates geometric patterns calledphyllotaxis. In the model plant, Arabidopsis thaliana, as in the majority of species, single organs areinitiated successively at a divergence angle from the previous organ close to the canonical angle of 137.51,producing a Fibonacci spiral. Given that little is known about the robustness of these geometricarrangements, we undertook to characterize phyllotaxis by measuring divergence angles between organsalong the stems of wild-type and specific mutant plants with obvious defects in phyllotaxis. Sequences ofmeasured divergence angles exhibit segments of non-canonical angles in both genotypes, albeit to a fargreater extent in the mutant. We thus designed a pipeline of methods for analyzing these perturbations.The latent structure models used in this pipeline combine a non-observable model representingperturbation patterns (either a variable-order Markov chain or a combinatorial model) with von Misesdistributions representing divergence angle uncertainty. We show that the segments of non-canonicalangles in both wild-type and mutant plants can be explained by permutations in the order of insertionalong the stem of two or three consecutive organs. The number of successive organs between twopermutations reveals specific patterns that depend on the nature of the preceding permutation (2- or3-permutation). We also highlight significant individual deviations from 137.51 in the level of baselinesegments and a marked relationship between permutation of organs and defects in the elongation of theinternodes between these organs. These results demonstrate that permutations are an intrinsic propertyof spiral phyllotaxis and that their occurrence is genetically regulated.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In vascular plants, the arrangement of leaves or flowers aroundstems generates regular patterns, called phyllotaxis. Phyllotaxisoriginates in a tissue located at the tip of stems that often exhibits

a dome-shaped structure, the shoot apical meristem (SAM). Organsare produced iteratively at precise positions on the flanks of the SAM,determining the final phyllotactic pattern. The architecture of thestem is thus primarily controlled at the SAM, although post-meristematic events such as differential growth can modify themeristematic pattern (Peaucelle et al., 2007; Landrein et al., 2013).The most widespread type of phyllotaxis found in nature is a spiral,where the mean divergence angle between consecutive organs isclose to 137.51, the golden angle. This apparent regularity, arising

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from the tight spatial and temporal control of organogenesis, haslong fascinated not only biologists but also mathematicians andphysicists (Adler et al., 1997).

Most models explaining the developmental mechanisms behindphyllotaxis at the SAM are based on the idea that each organproduces an inhibitory field in the SAM that prevents the formationof a new organ nearby (Douady and Couder, 1996a–c; Smith et al.,2006; Vernoux et al., 2010). This view is supported by pioneeringmicrosurgery experiments (Snow and Snow, 1932a,b; Wardlaw,1949) or more recently by laser ablation experiments (Reinhardtet al., 2005) in meristems. A large body of theoretical works,including physical experiments and numerical simulations, haveshown that highly regular phyllotactic patterns similar to thosefound in nature can emerge from a dynamical system involvingrepulsive interactions between neighboring elements, see Douadyand Couder (1996a–c); Smith et al. (2006), and Jönsson et al. (2006)and references therein. Despite their conceptual contributions, themodels proposed have led to an idealistic view of phyllotaxis focusedon regularity. However, high variability in divergence angles has beenobserved on the stem (Barabé, 2006; Peaucelle et al., 2007), indicat-ing that these models likely account only partially for the dynamicsof phyllotaxis. Indeed a biological system is expected to be subject tovarious sources of developmental and environmental variability thatcan induce developmental defects at a frequency depending on therobustness of the system (Lander, 2011). Therefore, an accuratemodel of a developmental system should be able to capture notonly the main patterning mechanism but also variations of patternsobserved in nature.

In the case of phyllotaxis, only few studies have attempted toquantify in details the real dynamics of phyllotaxis at the SAM. Thisis mostly due to the difficulty to access living SAM and to followorgan initiation over long period of time. A classical alternativeapproach consisted of analyzing stem phyllotaxis, as a proxy of theSAM dynamics. Two main approaches have been used to analyzestem phyllotaxis and in particular perturbations in phyllotaxis:

� A qualitative and structural approach which consists in identi-fying perturbed segments (i.e. sub-sequences) within asequence of divergence angles measured for an individualusing simple exploratory tools (Couder, 1998).

� A quantitative and nonstructural approach which consistsbasically in characterizing the frequency distribution of eitherabsolute angular positions or divergence angles using probabil-istic and statistical tools, see Itoh et al. (2000), Barabé (2006)and references therein.

Both approaches have some shortcomings: the lack of systema-tic sampling and statistical validation for the first and the forget-ting of potential dependencies between successive divergenceangles in the case of perturbations for the second. Conservingthe sequential information is mandatory for analyzing dependen-cies between successive organ positions and inferring informationrelative to the ontogenetic processes underlying phyllotaxis. Allthe models proposed for explaining phyllotactic patterns assumethat the angular position of a new organ depends on the positionsof older organs, see Douady and Couder (1996a–c), Smith et al.(2006), and Jönsson et al. (2006) and references therein. Conse-quently, following Douady and Couder (1996a) and Couder (1998),we assume that the post-meristematic expression of phyllotaxisshould reflect primarily these dependencies between successiveorgans within the SAM (but without forgetting that it can also bemodified by post-meristematic events). Following Barabé (2006),we consider that a phenotyping approach with an appropriatesampling strategy is required to investigate phyllotactic patterns.

The objective of this work was therefore to analyze a large sampleof stem phyllotaxis in order to characterize perturbations in post-

meristematic phyllotactic patterns and to deduce information aboutthe dynamics of organogenesis at the SAM.We chose to analyze stemphyllotaxis of Arabidopsis thaliana, a model species exhibiting aFibonacci spiral phyllotaxis. In this context, it can also be helpful tostudy mutants exhibiting obvious perturbations in their phyllotaxis(Golz and Hudson, 2002; Barabé, 2006). These phyllotactic mutantscan first be used to identify new genes involved in the control ofphyllotaxis. Moreover, the greatly enhanced frequency of perturba-tions in phyllotactic mutants helps the statistical characterization ofperturbation patterns. We thus analyzed the phyllotaxis of wild-typeArabidopsis and of a mutant line exhibiting obvious phyllotacticdefects. This line bears a null mutation in the gene encoding thepseudo phosphotransfer protein AHP6 (Mähönen et al., 2006), aninhibitor of cytokinin signaling. We selected this mutant in order toinvestigate a possible role of this regulator in the control ofphyllotaxis in Arabidopsis.

We designed a pipeline of methods to identify and characterizecomplex patterns resulting from perturbations in phyllotaxis. Thispipeline relies on models combining perturbation patterns withspecific distributions representing divergence angle uncertainty.We adopted an inductive inference approach whose objective wasto infer both perturbation patterns and uncertainty parameters.This inference paradigm only requires minimal a priori assump-tions regarding the data. The pipeline of methods we propose foranalyzing perturbations in Arabidopsis phyllotaxis decomposesinto the three following steps:

� Model building.� Labeling of the observed divergence angle sequences using the

models built.� Pattern analysis within the labeled divergence angle sequences.

The phenotyping results obtained using the proposed pipelineof analysis methods provide new elements for assessing thecurrent dynamical models of phyllotaxis.

2. Results

We measured divergence angles between siliques (Arabidopsisfruits) along fully elongated stems in 82 wild-type and 89 ahp6-1mutant plants, see details of the measurement protocol in Section 4.1.In the measured sample, 39 of the 82 wild-type individuals and 45 ofthe 89 mutant individuals were right-winding. Hence, left- and right-winding spirals were in roughly equal proportions in both the wild-type and mutant samples. This in accordance with previous results forthe wild type (Beal, 1873; Allard, 1946) and these roughly equalproportions were unaffected by the AHP6 loss of function mutation.

2.1. Exploratory analysis reveals specific perturbation patternscommon to the wild type and mutant

The divergence angle sequences measured for each plant tookthe form of a baseline where the successive measured anglesfluctuated around the canonical Fibonacci angle α¼ 137:51 inter-spersed by segments of non-canonical angles. The pointwise meandirections of divergence angles and the associated circular stan-dard deviations (measures of location and dispersion for circulardata, see Mardia and Jupp, 2000) computed for the wild type andthe mutant (Supplementary material Fig. S1) showed that therewere no trend or successive phases common to the individuals.

The exploratory analysis highlighted two characteristics of themeasured divergence angle sequences:

� Short segments of non-canonical divergence angles (see anexample in Fig. 1) which were more frequent in the mutant.

Y. Guédon et al. / Journal of Theoretical Biology 338 (2013) 94–110 95

These segments, corresponding to phyllotaxis perturbations,seemed to be highly structured.

� The divergence angles measured covered almost all possiblevalues (between 0 and 3601) with highest frequencies aroundthe canonical Fibonacci angle, see the frequency distribution inFig. 2. At least four classes of divergence angles were apparent,but were not unambiguously separated.

In agreement with previous studies (Peaucelle et al., 2007; Ragniet al., 2008), the phyllotaxis of wild-type plants appeared to besignificantly perturbed. As expected, mutant plants exhibited farmore phyllotaxis perturbations, with higher frequencies of non-canonical divergence angles and longer and more complex segmentsof non-canonical divergence angles. A motif corresponding approxi-mately to ½2α �α 2α� was frequently observed in the wild type(Fig. 1) and even more often in the mutant. This motif can beexplained simply by a permutation in the order of insertion along thestem of two consecutive organs without changing their angularpositions (Fig. 3). The reasoning is the following. In the absence ofperturbations, the sequence of absolute angular positions of thesuccessive organs with reference to a given origin is frα; r¼ 1;2;…gwhere r is the organ rank along the stem. In case of perturbations,the divergence angle sequence which is the first-order differencedsequence alternates baseline segments at the value α with segmentsof non-canonical angles corresponding to perturbations. Hence, it ismore efficient to search for perturbation patterns in this divergenceangle sequence (Couder, 1998; Peaucelle et al., 2007) rather than inthe sequence of absolute angular positions because the linearincreasing trend is removed. First-order differencing can be viewedas a specific linear filter that removes the trend and highlights localfluctuations (Chatfield, 2003). Consider now the sequence of absoluteangular positions 2α α 3α corresponding to a permutation betweenorgans 1 and 2. The divergence angle sequence is 2α �α 2α (the firstangular position is considered as a divergence angle with respect tothe origin). The motif ½2α �α 2α� was predicted by theoreticalmodels (Douady and Couder, 1996a) and has been observed in asunflower stem (Couder, 1998). Consider now the three situationsshown in Table 1 and corresponding to 2, 1 and zero organ betweentwo permutations of two consecutive organs. The last situationgenerates a new pattern with the divergence angles 3α correspond-ing to the chaining of two 2-permutations. This can be viewed as aside effect of filtering. This elementary example highlights the factthat the patterns potentially found in divergence angle sequenceswill correspond to chaining of permutations. In theory, permutationsin the order of insertion of n organs (n-permutation), withoutchanging their angular positions, generate divergence angles that

are multiples of the canonical divergence angle α, see Refahi et al.(2011) for a formal justification. The permutation of two consecutiveorgans (2-permutation) generates the divergence angles 2α, �α and3α.

Assume now that a permutation may involve not only twoconsecutive organs but also three consecutive organs (3-permuta-tion). Among the ð3!�1Þ proper permutations of 1 2 3 (the multi-plicative factor α is here omitted), only the 3-permutations3 21, 3 1 2 and 23 1 will be valid permutations in our context sincethe two other possible permutations 2 13 and 1 32 contain a2-permutation of consecutive organs (of the first two organs in thefirst case and of the last two organs in the second case). Thisillustrates the notion of indecomposable permutations (i.e. the factthat an indecomposable permutation cannot contain a permuta-tion of lower order of consecutive organs with a common end),see Comtet (1974) and Section 4.2.

2.2. Model building and labeling of the observed divergence anglesequences

Beyond the simple motif [2α �α 2α] corresponding to an isolated2-permutation, no empirical evidence of other patterns emerged fromthe exploratory analysis. We thus adopted an inductive inference

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Fig. 1. Optimal labeling of a wild-type divergence angle sequence. The observeddivergence angles are figured by red squares for canonical angles within baselinesegments and by green squares for divergence angles within permuted segments.The predicted divergence angle sequence is figured in blue. The predicted organ orderis given below (2-permutations in red). (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

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Fig. 2. Fit of the divergence angle frequency distribution by the mixture ofobservation distributions of the five-state hidden variable-order Markov chain.The fit is visually indiscernible from that obtained with the observation distribu-tions of the five-state hidden first-order Markov chain.

Fig. 3. Schematic representation of a 2-permutation: (a) normal succession oforgans, (b) permutation of two consecutive organs, and (c) sequence of divergenceangles corresponding to an isolated 2-permutation.

Table 1Absolute angular positions and divergence angles for different numbers of organsbetween two 2-permutations.

No. organs Angular positions Divergence angles

2 2α α 3α 4α 6α 5α 7α 2α �α 2α α 2α �α 2α1 2α α 3α 5α 4α 6α 2α �α 2α 2α �α 2α0 2α α 4α 3α 5α 2α �α 3α �α 2α

Y. Guédon et al. / Journal of Theoretical Biology 338 (2013) 94–11096

approach whose objective was to build models for labeling divergenceangle sequences. In this setting, the permutation of organs was not anassumption made a priori but potentially resulted from modelinference. The inference was thus not restricted to parameter estima-tion in the usual sense (e.g., estimation of a divergence angleuncertainty parameter) but encompassed pattern inference usingmodel selection methods. We applied successively two model selec-tion methods:

1. selection of the number of classes of measured divergenceangles;

2. identification of specific dependencies between “theroreti-cal” (i.e. denoised) successive divergence angles.

Complementary explanations regarding model building and vali-dation are given in Supplementary material Appendices S1 to S5.

2.2.1. The selection of the number of classes of measured divergenceangles suggests the occurrence of both 2- and 3-permutations

In a first step, stationary hidden first-order Markov chains wereestimated on the basis of pooled wild-type and mutant measureddivergence angle sequences (171 sequences of cumulative length5220), see Section 4.3 for a formal definition of stationary hiddenMarkov chains and Section 4.5 for a presentation of the methodused to estimate such models. The stationarity assumption enablesto properly model the fact that measured sequences start at thefirst silique just above the rosette of leaves, not at the base of thestem. We therefore assumed that the transient phase correspond-ing to the transition from the decussate to the spiral phyllotaxisdid not overlap the part of stem measured. This was confirmed bythe roughly constant pointwise mean directions of divergenceangles, see Supplementary material Fig. S1. In these hidden first-order Markov chains, the states of the non-observable Markovchain represent “theoretical” divergence angles while the vonMises observation distributions attached to each state of thenon-observable Markov chain represent divergence angle uncer-tainty. The von Mises distribution is a univariate Gaussian-likeperiodic distribution for a circular variable xA ½0;3601Þ (Mardiaand Jupp, 2000).

The objective of the model selection method was to determinethe number of states of the non-observable Markov chain corre-sponding to the number of classes of measured divergence angles.The Bayesian information criterion (BIC, see Definition (1)) favoredthe five-state model incorporating von Mises observation distribu-tions with a common concentration parameter (inverse dispersion).The models of highest BIC values were the four-state model withdifferent concentration parameters and the five-state model with acommon concentration parameter consistently with the divergenceangle frequency distribution shown in Fig. 2, see Table 2 andSupplementary material Appendix S1 for more details. This choicewas a posteriori validated by the fit of the divergence angle frequencydistribution by the mixture of von Mises observation distributions,see Fig. 2. The von Mises observation distributions estimated for thefive-state model were centered on the multiples of the canonicaldivergence angle α, 2α, �α, 3α and �2α, see Table 1. Theidentification of �2α using this five-state model suggested theoccurrence of permutations involving three organs in the measuredsequences. If 3-permutations are considered in addition to 2-permu-tations, the divergence angles �2α, 4α and 5α are expected to beobserved. It should be noted that out of the seven possible theoreticalstates α, 2α, �α, 3α, �2α, 4α and 5α, it was not possible to identify4α and 5α which are involved only in the chaining of permutations,including a 3-permutation, and were thus very rare in the measureddivergence angle sequences, see illustrations with the permutationpatterns described in Section 2.3.

2.2.2. The identification of specific dependencies between successivedivergence angles reinforces the assumption of permutation of organs

Having selected five classes of divergence angles, the next stepconsisted in identifying dependencies between successive “theoreti-cal” divergence angles that could correspond to frequent motifs inthe denoised divergence angle sequences. The optimally labeleddivergence angle sequence (i.e. discrete sequence with five possiblevalues chosen from among α, 2α, �α, 3α and �2α) was computedfor each observed sequence using the estimated hidden first-orderMarkov chain incorporating von Mises observation distributions witha common concentration parameter. The memories of a variable-order Markov chain were then selected (Csiszár and Talata, 2006) onthe basis of these labeled divergence angle sequences, see Section 4.3for a formal definition of variable-order Markov chains and Section4.4 for the method used to select these memories. This can beinterpreted as a way to identify local dependencies between succes-sive “theoretical” divergence angles. For the selection of the mem-ories of the variable-order Markov chain, we chose to discard theindividuals which were very poorly explained by the estimatedhidden first-order Markov chain (10 individuals out of 171 whoseposterior probability of the optimally labeled divergence anglesequence – i.e. weight of the optimal labeling among all the possiblelabelings of a given observed sequence – o0:13). The selectedvariable-order Markov chainwas a mixed first-/second-order Markovchain where the first-order memory 2α was replaced by the foursecond-order memories α2α, 2α2α, �α2α, �2α2α (the memory3α2α was not observed) with respect to a simple first-order Markovchain, see Fig. 4 and Supplementary material Appendix S2 for moredetails. This means that in order to predict accurately the mostfrequent permutation patterns, only the divergence angle thatprecedes 2α needs to be taken into account.

The selected mixed first-/second-order Markov chain restrictedto the modeling of 2-permutations only generates valid divergenceangle sequences according to the formal language induced by2-permutations (i.e. the set of possible sequences corresponding tothis assumption and defined on the alphabet α, 2α, �α, 3α). This isa direct consequence of the fact that the finite-state automatondeduced from the transition graph in Fig. 4c (for possible transi-tions, the probabilities are no longer considered) recognizes theformal language induced by 2-permutations, see typical diver-gence angle sub-sequences in Table 1 and a formal definition ofthe regular language induced by 2- and 3-permutations in Sup-plementary material Appendix S3.

This first step of inference based on model selection methodsenabled us to (i) identify five classes of divergence angles centered onthe multiples of the canonical divergence angle α, 2α, �α, 3α and

Table 2Characteristics of von Mises observation distributions. Characteristics (meandirection μ and circular standard deviation ν) estimated within the five-statehidden first-order Markov chains (with different or common concentration para-meters) and within the five-state hidden variable-order Markov chain. Since theweights of the different states are similar for the two estimated hidden first-orderMarkov chains, they are only given once. Log-likelihoods and BIC values for thethree estimated models.

First order Variable order

μ ν μ ðν¼ 18:5Þ Weight μ ðν¼ 18:5Þ Weight

α (137.5) 136.5 18.2 136.5 0.71 136.6 0.72α (275) 273.3 18.7 273.4 0.14 273.1 0.15�α (222.5) 221.9 17.7 222.1 0.1 221.5 0.13α (52.5) 50.3 20.1 50.8 0.03 49.7 0.03

�2α (85) 80.9 14.3 83 0.02 88.9 0.02

Log-likelihood �17,041.3 �17,034.4 �16,694.3BIC �34,339.4 �34,282.9 �33,705.4


�2α (this latter angle cannot be explained by 2-permutations butonly by 3-permutations), (ii) estimate a common concentrationparameter for the von Mises distributions, reflecting divergenceangle uncertainty, and (iii) identify a remarkable dependency struc-ture that can be interpreted with regards to 2-permutations. This ledus to hypothesize that the segments of non-canonical angles could beexplained by permutations involving two or three organs.

2.2.3. Building of three models corresponding to differentcompromises between labeling capabilities and the permutationassumption

The models built for labeling the observed divergence anglesequences were generalizations of mixture models including up toseven states corresponding to the theoretical divergence angles α,2α, �α, 3α, �2α, 4α and 5α. One potential difficulty in this analysislay in the particularly unfavorable mixture structure with up to sevenvon Mises distributions with strong overlaps (Fig. 5) and verycontrasted weights for overlapping distributions (see Table 2). Thusfor labeling the divergence angle sequences, we chose to build threedifferent models and to deduce a consensus from the three inde-pendent labelings. These three models correspond to differentcompromises between labeling capabilities and the permutationassumption. The first model was a five-state hidden variable-orderMarkov chain, the second was a combinatorial mixture model basedon the permutation assumption while the third was intermediatebetween the first two:

1 Stationary five-state hidden mixed first-/second-order Markov chainwith the memories (α, α2α, 2α2α, �α2α, �2α2α, �α, 3α, �2α)

previously selected and a concentration parameter common tothe five von Mises observation distributions, see Section 4.3 for aformal definition of stationary hidden variable-order Markovchains and Section 4.5 for a presentation of the method used toestimate such models. This parsimonious model was the directoutput of the model selection methods and was expected to havethe best labeling capabilities. But the divergence angle seque-nces labeled using this model were not guaranteed to respectthe permutation assumption in the case of 3-permutations inparticular because the divergence angles 4α and 5α were not

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Fig. 4. Hidden Markov chains restricted to the modeling of 2-permutations. (a) Hidden first-order Markov chain: transition graph of the underlying Markov chain; hiddenvariable-order Markov chain. (b) Memory tree. (c) Transition graph. In (a) and (c), each vertex represents a possible memory. Possible transitions are represented by arcs.The associated probabilities are noted nearby. The transition probabilities leaving a given state do not sum to 1 since only the transition subgraphs corresponding to themodeling of 2-permutations are shown.

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Fig. 5. Von Mises observation distributions centered on multiples of the canonicalFibonacci angle α¼ 137:5○ with a common concentration parameter correspondingto a circular standard deviation ν¼ 18:5○ .


represented in this model. The validation of this model isdiscussed in Supplementary material Appendix S4.

2. Seven-state combinatorial mixture model representing explicitly2- and 3-permutations. This model can be viewed as a latentstructure model where the underlying Markov chain of thehidden Markov chain is replaced by a combinatorial modelbased on the assumption that the permutations involve at mostthree successive organs (e.g. ½3α �α �α 3α� corresponding tothe organ order 3 2 14). The von Mises observation distribu-tions were centered on the multiples of the canonical diver-gence angle α, 2α, �α, 3α, �2α, 4α and 5α. The concentrationparameter common to the seven von Mises observation dis-tributions was that estimated within the hidden variable-orderMarkov chain. This combinatorial mixture model can also beviewed as the coupling of a deterministic combinatorial model,based on a permutation assumption, with an independentmixture model. The algorithm for labeling divergence anglesequences using this combinatorial mixture model is presentedin Section 4.6. This “saturated” model was overparameterizedaccording to model selection criteria and was expected to besub-optimal in terms of labeling capabilities. In particularcertain rare permuted patterns associated with many strongdeviations between measured and predicted angles weresometimes predicted. This is a direct consequence of theimportant overlaps between the seven von Mises observationdistributions, see Fig. 5 and Supplementary material Table S1.

3. Stationary seven-state hidden mixed first-/second-order Markov chainwithmemories (α, α2α, 2α2α,�α2α, 3α2α,�α, 3α,�2α, 4α, 5α)and a concentration parameter common to the seven von Misesobservation distributions. This intermediate model was built toease a majority vote when comparing the divergence anglesequences labeled using the different models. The building of thismodel is presented in Supplementary material Appendix S5.

Since the divergence angle sequences labeled using hiddenvariable-order Markov chains were not guaranteed to respect thepermutation assumption in the case of 3-permutation, the simplecombinatorial model (i.e. without the von Mises observation distribu-tions modeling divergence angle uncertainty) was applied as a post-processing to the divergence angle sequences labeled using hiddenvariable-order Markov chains in order to invalidate segments of non-canonical angles that did not fulfill the permutation assumption.

One advantage of hidden Markov chains is the capability tocompute an absolute measure of the relevance of the optimallylabeled divergence angle sequence in the form of a posteriorprobability. Contrary to the hidden Markov chain case, it is notpossible to compute an absolute measure of the relevance of theoptimally labeled divergence angle sequence in the form a poster-ior probability since the combinatorial mixture model is not anintegrated probabilistic model. One advantage of the combinator-ial mixture model is the possibility to label divergence anglesequences following other spiral phyllotaxies than the Fibonacciphyllotaxis (canonical divergence angle of 137.51). In particular,we used it to test the Lucas phyllotaxis assumption simply by settingthe mean directions of the seven von Mises observation distributionsat the multiples of the canonical divergence angle of 99.51.

2.2.4. Consensus labeling of the divergence angle sequencesWhen the labelings of an observed sequence by the five-state

hidden variable-order Markov chain, the seven-state hiddenvariable-order Markov chain and the combinatorial mixture modeldid not coincide, an expert examination of the three labelings wasneeded to chose the most likely solution. The final result was thus aconsensus deduced from the divergence angle sequence labeled bythe hidden variable-order Markov chains and the combinatorial

mixture model. This approach is illustrated in Table 3 by the countsof matches and mismatches between the optimal labeling of thedivergence angle sequences using the estimated hidden variable-order Markov chains and the combinatorial mixture model, and thefinal consensus labeling. Mismatch counts for the hidden variable-order Markov chains should be interpreted as upper bounds since inpractice, alternative labelings were considered when the optimallabeling was not valid in terms of 2- and 3-permutation of organs.The model ranking in terms of labeling performance (Table 3) isconsistent with the outputs of model selection criteria. These resultsillustrate the complementarity of the two modeling approaches andthe strong accordance of the estimated hidden variable-order Mar-kov chains with the permutation assumption.

2.3. The frequency and complexity of permutation patterns aregenetically regulated

On the basis of the permutation assumption, we were able tolabel a very large proportion of the sequences measured for boththe wild type and the mutant (98% and 95% of explaineddivergence angles respectively, see Table 4). The propagation ofmeasurement errors within permuted segments (which are morefrequent and longer in the mutant) likely explains the higherproportion of unexplained angles in the mutant. The specificity ofthe permutation assumption in conjunction with the high fre-quencies of non-canonical angles (Table 4) and the high confi-dence in the explanation of the observed sequences by theestimated hidden variable-order Markov chains (given by theposterior probabilities of the optimal labeling of the observedsequences, see Supplementary material Fig. S2) render veryimprobable other structural hypotheses for explaining the phyllo-taxis perturbations such as a reversal in the orientation of the

Table 3Counts of matches and mismatches between the optimal labeling of the divergenceangle sequences using the 5-state hidden variable-order Markov chain (HMC), the7-state hidden variable-order Markov chain, the combinatorial mixture model(CMM), and the final consensus labeling taken as reference (the four Lucasphyllotaxis individuals were excluded from the sample). Unexplained angles aregiven in the last row.

5-state HMC 7-state HMC CMM

Match Mismatch Match Mismatch Match Mismatch Consensus

α 3620 21 3618 23 3579 62 36412α 695 5 690 10 675 25 700�α 440 6 439 7 433 13 4463α 117 8 115 10 120 5 125

�2α 23 10 20 13 33 0 334α 10 8 2 10 0 105α 1 0 1 1 0 1

Total 4895 61 4890 66 4851 105 4956

? 99 50 149

Table 4Characteristics of the optimally labeled divergence angle sequences.

Wild type Mutant

No. sequences/no. angles 82/2405 89/2815% of non-canonical angles 15 37% of unexplained angles 2 5No. individuals, Lucas phyllotaxis 2 2No. 2-permutations 123 297No. 3-permutations 3 53% of permuted organs 10.3 25.9


generative spiral advocated in previous studies (Peaucelle et al.,2007; Ragni et al., 2008).

The proportion of permuted organs increased from 10.3% to 25.9%between the wild type and the mutant (Table 4) and corresponded toa marked increase in the occurrence of 2-permutations and a burst inthe occurrence of 3-permutations. Note that, proportionally,3-permutations increased far more than 2-permutations in the ahp6genetic background, suggesting that the developmental defects lead-ing to 3-permutations are very sensitive to the loss of AHP6 function.However, these overall proportions of permuted organs hide a greatinter-individual heterogeneity within each genotype, with a strongoverlap of the two populations (Fig. 6). In particular, 18 of the 82 wild-type individuals but only 1 of the 89 mutant individuals wereunaffected by organ permutations. However, most of the individualswith more than 20% of permuted organs were mutant. This hetero-geneity is illustrated by the typical wild-type individual in Fig. 1 to becompared with the highly perturbed wild-type individual in Fig. 7.The highly perturbed mutant individuals were characterized by theoccurrence of long and complex permuted segments correspondingto the chaining of 2- and 3-permutations, see examples in Fig. 8.The permuted segments detected in the measured sequences arelisted with their respective frequencies for each genotype inTables 5 and 6. The similar frequencies of the three possible permutedsegments corresponding to an isolated 3-permutation (Table 5) shouldbe noticed. The similar ratios of the frequency of nþ 1 chained2-permutations to the frequency of n chained 2-permutations (16/90for n¼1 for the wild type and 32/193, 5/32 and 1/5 for n¼ 1;2;3 forthe mutant, see Tables 5 and 6) suggest that the chaining of

permutations is memoryless. This is further supported by the occur-rence with low frequencies of all the possible except one permutedsegments corresponding to the chaining of a 2-permutation with a3-permutation (Table 5).

The canonical angle α is mainly involved in baseline segments butalso in 3-permutations while 2α and �α are involved in both 2- and3-permutations, 3α in chainings of 2-permutation and in 3-permu-tation, �2α in 3-permutations, 4α in chainings involving at least a3-permutation and 5α in chainings of two 3-permutations, seeTables 5 and 6. This in part explains the very contrasted frequenciesof the different multiples of α in the samples (see Table 7). It shouldbe noted that the permuted segments enumerated exhaustively inTable 5 and corresponding to isolated 2- and 3-permutations,chainings of two 2- or 3-permutations, and chainings of a 2-permutation with a 3-permutation, provide a complete picture ofthe formal language induced by 2- and 3-permutations. Permutedsegments corresponding to the chaining of more than two permuta-tions are simply built by applying the rules illustrated by the chainingof two permutations, see illustrations in Table 6.

2.4. Properties of baseline segments

2.4.1. The number of successive organs between two permutationssuggests that permutations propagate along contact parastichies

The intervals corresponding to successive organs between twopermutations were extracted and categorized according to thepermutation that precedes the interval (2- or 3-permutation).In the interval length frequency distributions (Fig. 9), the zerovalue corresponds to the chaining of permutations and thecorresponding frequencies were extracted from Tables 5 and 6. A

0

5

10

15

20

25

30

35

40

45

0.05 0.15 0.25 0.35 0.45 0.55 0.65

Permuted organ proportion

Freq

uenc

y

wild type

mutant

Fig. 6. Histograms of the proportions of permuted organs per individual.

0

50

100

150

200

250

300

Div

erge

nce

angl

e

2 1 3 4 5 6 7 8 10 9 12 11 13 15 14 16 17 18 19 20 21 23 22 24 25 26 27 28

Organ order

2α−α 2α2α−α 3α−α 2α 2α−α 2α2α −α 3α −α 2α

truncation

Fig. 7. Optimal labeling of a highly perturbed wild-type divergence anglesequence. The observed divergence angles are figured by red squares for canonicalangles within baseline segments and by green squares for divergence angles withinpermuted segments. The predicted divergence angle sequence is figured in blue.The predicted organ order is given below (2-permutations in red). The left-truncation of a permuted segment is hypothesized at the beginning of thesequence. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

Organ order

Div

erge

nce

angl

eD

iver

genc

e an

gle

Div

erge

nce

angl

e

Fig. 8. Optimal labeling of mutant divergence angle sequences. The observeddivergence angles are figured by red squares for canonical angles within baselinesegments and by green squares for divergence angles within permuted segments.The predicted divergence angle sequences are figured in blue. The predicted organorders are given below (2-permutations in red and 3-permutations in blue). Theleft-truncation of a permuted segment is hypothesized at the beginning of the lasttwo sequences. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)


value u40 corresponds to u�1 successive canonical angle αbetween two permuted segments. When the zero value isexcluded, the “�1-shifted” frequency distributions are thus thefrequency distributions of baseline segment lengths between two

permuted segments. For intervals following a 2-permutation, thelength frequency distribution is highly structured with highfrequencies for 0 (chaining of two permutations), 1 and 3, andcomparatively low frequencies for 2 and 4, see Fig. 9. The intervallength frequency distributions are fairly similar for the wild typeand mutant, see Supplementary material Fig. S3. For intervalsfollowing a 3-permutation, the length frequency distribution isalso highly structured despite overall lower frequencies than forthe 2-permutation case. In both cases, the interval length fre-quency distributions differ markedly from geometric distributionswhich are the only “memoryless” discrete distributions. Hence, theoccurrence of a permutation affects the occurrence of a subse-quent permutation, at least on the four positions that follow it.

In addition to the generative spiral, phyllotaxis generatessecondary geometric structures in the form of two sets of inter-secting spirals winding in the two opposite directions that aredefined by linking each organ to its contact neighbors. In the caseof a Fibonacci spiral phyllotaxis, the numbers of these spirals,called parastichies, are two successive terms of the Fibonacciseries 1, 1, 2, 3, 5, 8, 13, 21, … in which each term is the sum ofthe two previous terms (the first two terms being 1 and 1). Thus,in a Fibonacci spiral with m and n parastichies (m¼3 and n¼5 inour case), the contact neighbors of new organ i are older organsi�m and i�n. In configurations corresponding to high u frequen-cies, permuted organs of the same rank in the two permutationsare direct neighbors in contact parastichies. This can be illustratedin the simple case of two consecutive isolated 2-permutations. Inthe sequence of organs 2 135 46 corresponding to a single organbetween two 2-permutations (u¼1), organs 2 and 5 and organs1 and 4 are, respectively, neighbors along parastichies of order 3.Hence, permuted organs align along two parastichies of order3 and non-permuted organs along the third. In the sequence oforgans 21 345 768 910 corresponding to u¼3, organs 2 and7 and organs 1 and 6 are, respectively, neighbors along parasti-chies of order 5. Hence, permuted organs align along two para-stichies of order 5 and non-permuted organs along the otherthree. In the sequence of organs 213 465 78 corresponding tou¼2, neither organs 2 and 6 nor organs 1 and 5 are neighborsalong parastichies of order 3 or 5. This suggests that permutationspropagate preferentially along contact parastichies. These geo-metric rules can be illustrated using centric representation ofmeristems, see Fig. 10. The chaining situation (sequence of organs214 35) cannot be interpreted in this framework since theinterval between permuted organs of the same rank of length2 is below the parastichy orders. Wild-type and mutant plants hadsimilar interval length frequency distributions (Supplementary

Table 5Permuted segments up to length 7. These segments are delimited by two splittingpoints. The divergence angle sequence is the first-order differenced organsequence. By convention, the origin of the organ sequence is 0 (not indicated).Palindromes (a segment and its reverse in terms of divergence angles are the same)are given at the beginning of each category. A segment and its reverse are thengiven on two successive rows. Frequencies of segments are given for the wild typeand the mutant.

Table 6Permuted segments of lengths 8–10 (only the observed permuted segments aregiven). These segments are delimited by two splitting points. The divergence anglesequence is the first-order differenced organ sequence. By convention, the origin ofthe organ sequence is 0 (not indicated). Frequencies of segments are given (allthese long permuted segments were detected in mutant individuals).

Table 7Counts of divergence angles and short internodes for each category of divergenceangle. The α within baseline segments are distinguished from the α within3-permutations (*). Expected counts of short internodes are given for each categoryof divergence angle under the independence assumption.

Wild type Mutant

Angle Internode Expected Angle Internode Expected

α 2003 100 144.6 1655 140 226.5αn 2 1 0.1 37 11 5.1

2α 214 5 15.5 499 46 68.3�α 126 58 9.1 330 120 45.23α 20 5 1.4 111 28 15.2

�2α 2 2 0.1 37 21 5.14α 1 0 0.1 10 1 1.45α 3 0 0.4

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8

No. organs

Freq

uenc

y

2-permutation

3-permutation

Fig. 9. Frequency distributions of the number of successive organs between twopermutations. These runs of organs are distinguished as a function of thepermutation that precedes them.


material Fig. S3). Therefore, whereas the loss of AHP6 functionpromotes permutations, its presence in the wild-type does notaffect the propagation of permutations along contact parastichies.

2.4.2. Baseline segment mean directions can significantly differ fromthe golden angle

The statistical summaries computed for baseline segments(weighted mean absolute deviation and sample autocorrelationfunction) summarize deviations of small magnitude for which thecircularity of the data can be ignored and standard statisticalsummaries can be computed. The divergence angles measured forsome long baseline segments appeared to be mostly above ormostly below the canonical angle of 137.51. We thus computed theweighted mean absolute deviation between the average baselinesegment level for an individual and the canonical angle of 137.51.The sample weighted mean absolute deviation is given by

∑anajxa�137:5j=∑

ana;

where xa is the mean direction of the divergence angles withinbaseline segments and na is the cumulative length of the baselinesegments for the ath individual.

We obtained a mean absolute deviation of 8.11 for the wildtype, 5.71 for the mutant and 71 for the pooled sample. It should benoted that the baseline segment effect was slightly larger than thisindividual effect (8.41 for the wild type, 7.21 for the mutant and7.81 for the pooled sample). Hence, the baseline segment meandirection can differ significantly from 137.51 for an individual andthe concentration parameter estimated for the von Mises distribu-tions in part reflects this inter-individual heterogeneity.

2.4.3. Sample autocorrelation function reveals putative localdependencies within baseline segments

We next investigated possible correlations between successivedivergence angles within baseline segments. The divergence anglesequences were deduced from the sequences of measured angularpositions by first-order differencing, see the measurement proto-col in Section 4.1. The application of linear filters induces anautocorrelation structure, see Diggle (1990). In the case of first-order differencing, the induced autocorrelation function is

ρðkÞ ¼1 k¼ 0;�0:5 k¼ 1;0 k41:

8><>:

We checked that the sample autocorrelation function for thebaseline segments was similar to the autocorrelation functioninduced by first-order differencing a white noise sequence

(Fig. 11). The sample autocorrelation coefficients which measurethe correlation between divergence angles at different distancesapart within baseline segments (pooled sample autocorrelationfunction) are given by

rðkÞ ¼∑a∑na�k�1t ¼ 0 ðxa;t�xaÞðxa;tþk�xaÞ=∑aðna�kÞ

∑a∑na�1t ¼ 0 ðxa;t�xaÞ2=∑ana

; k¼ 0;1;…

where xa ¼∑na�1t ¼ 0xa;t=na is the mean direction of the divergence

angles within the ath baseline segment. The baseline segmentmean direction xa can be replaced by the mean direction of thedivergence angles within baseline segments for an individual butnot by the overall mean direction because of the inter-individualheterogeneity of the baseline segment levels. Because of the use ofdifferent mean estimates for the different segments, the standarderrors shown in Fig. 11 may be fairly rough approximations.

It should be noted that the correlation coefficients for lags 3, 5,8 are positive and, for lags 3 and 5, are slightly above theconfidence limit and that the correlation coefficients for lags 4,6 and 9 are negative and slightly below the confidence limit. Theoccurrence of 3, 5, 8 - three consecutive terms of the Fibonacciseries – reminds us that Arabidopsis phyllotaxis is organized withthe parastichy pair (3, 5), and the triplet (3, 5, 8) can be interpretedas an intermediate between the parastichy pairs (3, 5) and (5, 8).Positive correlation coefficients could thus be interpreted as apropagation of each deviation from the baseline segment meandirection along the contact parastichies and may be related to thepropagation of the permutation patterns along the contact para-stichies (see above). However, it should be noted that an increase

Fig. 10. Illustration of the propagation of permutations along contact parastichies using a centric view of the shoot apical meristem (SAM). Organs are represented by circlesand are numbered from the oldest to the youngest. Organs involved in a given permutation are colored using a given shade of red, while organs not involved in apermutation are in green. The meristem is the central white circle: at its periphery (orange), two small red circles indicate the next organs to form (13 and 14) that will bepermuted. Contact parastichies linking permuted organs are represented by plain lines while contact parastichies linking non-permuted organs are represented by dashedlines. The sequence of organs is indicated below each SAM view. (For interpretation of the references to color in this figure legend, the reader is referred to the web version ofthis article.)

-0.6

-0.4

-0.2

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1

0 1 2 3 4 5 6 7 8 9 10

Lag

Aut

ocor

rela

tion

coef

ficie

ntdatawhite noiseconfidence limit

Fig. 11. Sample autocorrelation function of divergence angles within baselinesegments. The randomness 95% confidence limits are given.


in the measurement precision (measurement step of 51 in thisstudy) would be necessary to confirm this result.

2.5. Short internodes are often but not systematically associatedwith permutations

To further explore the impact of permutations on the architectureof the stem, we analyzed the effect of a permutation on internodelength. Short internodes were associated with each of the six possibleangles α, 2α, �α, 3α, �2α and 4α, but the association was far frombeing uniform, see Table 7. To investigate the relation between shortinternodes and divergence angles, we distinguished between angle αin the baseline segments and angle α involved in 3-permutations(and in this case systematically associated by pair with �2α, seebelow). In the case of a 2-permutation, the internode between thetwo permuted organs always corresponds to �α (e.g. within½2α �α 2α� corresponding to the organ order 213 in the case of anisolated 2-permutation) while in the case of a 3-permutation, thetwo successive internodes between the permuted organs taken bysuccessive pairs correspond to �α �α, α �2α or �2α α (e.g. within½3α �α �α 3α� corresponding to the organ order 3214,½3α �2α α 2α� corresponding to the organ order 3124 and½2α α �2α 3α� corresponding to the organ order 2314 in the caseof isolated 3-permutations), see Tables 5 and 6. It should also benoted that �α and �2α only occur in such contexts. The number ofshort internodes associated with a permuted organs was 58 (�α) +2(�2α) +1 (α in 3-permutations) to be compared with 100 shortinternodes corresponding to two successive non-permuted organs inthe wild type and 120 (�α) +21 (�2α) +10 (α in 3-permutations) tobe compared with 141 in the mutant, see Table 7. Hence the orders ofmagnitude of the number of short internodes involved or notinvolved in permutations were fairly similar: 61 and 100 for thewild type and 151 and 141 for the mutant.

A large proportion of �α (46% for the wild type and 36% for themutant) and an even higher proportion of �2α corresponded to shortinternodes, while proportions were far lower for the other angles, seeTable 7. But the association between �α (respectively �2α) and shortinternodes was far from being systematic. We checked that most ofthe short internodes were isolated (i.e. surrounded by two elongatedinternodes), which excluded long-range dependencies concerning theoccurrence of short internodes, see the frequency distributions of thenumber of successive short internodes in Supplementary materialTable S2. In the mutant, 15 of the 35 runs (or clumps) of shortinternodes of length 41 corresponded to 3-permutations. Concerning3-permutations such that the two successive internodes between thepermuted organs taken by successive pairs correspond to α �2α or�2α α, we found that 23 of the 39 internodes corresponding to �2αwere short while only 11 of the 39 internodes corresponding to αwere short in the pooled sample (Table 7). According to the χ2 test forcontingency table, the short internodes were preferentially associatedwith �2α in these 3-permutation configurations. These two config-urations of 3-permutations can be interpreted as a simple permutationbetween a pair of ordered organs (the corresponding divergence angleis thus α) and another organ such that short internodes are preferen-tially associated with the permutation. In nine of the 11 3-permutations where the internode corresponding to α was short,the internode associated with �2α was also short. In conclusion,short internodes were often but not systematically associated withpermutations indicating that organ permutations and internodeelongation are only partially coupled.

2.6. Classification of individuals belonging to different geneticbackgrounds

To confirm the role played by the AHP6 gene in controllingorgan permutations, we analyzed phyllotaxis in different genetic

backgrounds in which AHP6 function was genetically modified.We selected ahp6-3, another independent mutant allele of AHP6and the ahp6-1/3 trans-heterozygous, the direct progeny of a crossbetween ahp6-1 and ahp6-3 parents (thus bearing one null ahp6-1allele and one null ahp6-3 allele). We also selected ahp6-1 mutantsinto which a functional AHP6 transgene had been re-introduced(complemented lines). In this case, we obtained ahp6-1 plantsexpressing either AHP6-GFP or AHP6-HA, both protein fusionsbeing expressed from the native AHP6 promoter and thus showingthe same expression patterns as the native AHP6 protein. The newahp6 mutant lines were expected to exhibit the same phenotype asthe ahp6-1 line (many permutations) while the complemented lineswere expected to exhibit a wild-type phenotype (few permutations)because the transgene provides a functional version of AHP6. Wethen applied a model-based classification approach where eachsequence to be classified was assigned to the model, chosen amongthe wild-type and the ahp6-1mutant models, that best explained it.

We adopted the following conservative strategy to build thewild-type and ahp6-1 mutant models. The two models shared thesame memories, i.e. (α, α2α, 2α2α, �α2α, �2α2α, �α, 3α, �2α).This was supported by the selection of the memories on the basisof the optimally labeled divergence angle sequences correspond-ing to the wild-type (respectively mutant) individuals computedusing the globally estimated hidden variable-order Markov chain.They also shared the same von Mises observation distributionswith common concentration parameter estimated on the basis ofthe pooled sample of wild-type and mutant individuals (withinthe globally estimated hidden variable-order Markov chain). Onthe basis of the wild-type (respectively ahp6-1 mutant) sample,we applied the iterative estimation algorithm (ECM algorithm)where only the transition probabilities were re-estimated. Finally,we thresholded the lowest transition probabilities at 10�2 in orderto avoid misclassification of individuals due to unobserved pat-terns in the learning samples.

The log-likelihoods of the wild-type and ahp6-1 mutant modelswere computed for each observed divergence angle sequence to beclassified using the forward or filtering algorithm, see Section 4.5.The observed divergence angle sequence was then assigned to themodel that best explained it. One difficulty in this classification taskarises from class overlap. The wild-type model is actually nestedwithin the ahp6-1 mutant model since both wild-type and mutantindividuals exhibit permutations, the difference stemming mainlyfrom the more frequent occurrence of permutations in the mutantcase. Hence, less perturbed mutant individual can be classified in thewild-type class. This particular class configuration is illustrated by theclassification of the individuals used to build the two models where21 of the 89 ahp6-1 mutants were misclassified while only nine ofthe 82 wild-type individuals were misclassified (Table 8).

Most of the individuals of the other ahp6 mutant lines (i.e.ahp6-3 and ahp6-1/3) were assigned to the ahp6-1 model (recallthat the classification results are biased toward the wild-typemodel because of model nestedness). Concerning the complemen-ted lines, most of the ahp6-1 plants transformed to express AHP6-GFP were assigned to the wild-type model, indicating clearcomplementation of the AHP6 function in this line. Regarding theplants transformed to express AHP6-HA, only half were assigned tothe wild-type model, suggesting less effective complementation(Table 8). The capacity of AHP6-GFP and AHP6-HA to rescue thewild-type phenotype, even partially, is a strong argument support-ing the causal link between AHP6 loss of function and the increasein permutations. Despite the bias induced by model nestedness, itwas therefore possible, using this model-based classificationapproach, to characterize the phenotype of various genetic lineswith reference to target previously modeled phenotypes. Overall,these results strongly support a role for AHP6 as a negativeregulator of permutations.


3. Discussion

Phyllotaxis was originally defined as the arrangement of organsaround the plant stem. However, since the first observations under amicroscope by Hofmeister (1868), most studies of phyllotaxis havefocused on meristems where organ initiation takes place (Adler et al.,1997). It is now well-established that organ founder cells differenti-ate into organs at very precise locations determining for a large partthe final phyllotactic pattern. However, analysis at the meristem levelin a plant such as Arabidopsis using either static pictures or liveimaging (see e.g. Heisler et al., 2005) only allows to follow the spatialarrangement of no more than 10 organs. By contrast, a post-meristematic retrospective analysis allows to analyze full sequencesof divergence angles (and associated internode lengths) along a stem.It thus offers the possibility not only of analyzing more easily a largepopulation of plants (thus giving statistical relevance) but also ofidentifying long patterns in a single individual.

In this study we developed a new pipeline of methods foridentifying and characterizing patterns in post-meristematic spiralphyllotaxis and applied it to the model plant Arabidopsis thaliana.As a main result we demonstrated that a very high proportion of theperturbations of post-meristematic phyllotaxis we observed in bothwild-type and mutant can be explained by permutations in the orderof insertion along the stem of two or three consecutive organs. Wealso showed that the cytokinin inhibitor AHP6 (Mähönen et al., 2006)regulates the frequency of permutations, demonstrating a geneticcontrol of this phyllotactic perturbation. Therefore, permutationsrepresent an intrinsic characteristic of phyllotaxis even in low-order phyllotactic systems such as the one of Arabidopsis. We discussbelow the relevance of the proposed methodology and the biologicalimplications of the results obtained.

3.1. Pattern theory approaches reveal that permutations are agenetically controlled property of the stem architecture

Approaches previously proposed for analyzing post-meristematicphyllotactic data were either non-structural (Barabé, 2006) or non-probabilistic (Couder, 1998). Here we applied pattern theoryapproaches combining structure with probabilities (Grenander andMiller, 2007) to analyze phyllotactic patterns and possibly inferinformation on the underlying ontogenetic processes, see Section3.2 for a discussion on this point. The fact that permutation patternswere inferred instead of being a priori assumed makes the proposedapproach robust and easily applicable to other post-meristematicphyllotactic data. This study can be viewed as a contribution to asuite of studies (Guédon et al., 2001, 2007; Chaubert-Pereira et al.,2009) whose aimwas to apply pattern theory approaches at differentscales of plant description to infer information related to differentgrowth processes.

One of the main findings of our analyses is that perturbations inpost-meristematic phyllotactic sequences can be explained bypermutations in the order of insertion along the stem of two or

three consecutive organs without changing their angular posi-tions. In addition we found that the AHP6 gene negatively andspecifically regulates the frequency of these permutations. AHP6 isa small pseudo-Histidine–Phosphotransfer protein (pseudo HPt),which has been characterized as an inhibitor of cytokinin signaling(Mähönen et al., 2006). This suggests that AHP6 buffers permuta-tions in Arabidopsis by regulating cytokinin signaling and demon-strates that permutations are genetically controlled.

Permutations are likely to be a common pattern in post-meristematic spiral phyllotaxis (whatever the order of the phyllotacticsystem) and may in part explain the frequent occurrence of non-canonical divergence angles observed on the stem in various plantspecies (e.g. Fujita, 1938; Barabé, 2006; Peaucelle et al., 2007).Supporting this idea, frequency distributions of divergence anglespublished in previous studies with another Arabidopsis ecotype,Wassileskija (Peaucelle et al., 2007; Ragni et al., 2008), were similarto the frequency distribution shown in Fig. 2, suggesting that permuta-tions are not specific to the Columbia ecotype of Arabidopsis growingin our culture conditions. An example of a permutation between twoleaves along a sunflower stem following Lucas phyllotaxis was alsoreported by Couder (1998), indicating that permutations are notrestricted to Arabidopsis. However, further phenotyping of differentspecies following an approach similar to ours is required to determinehow frequent these defects are in nature. In this context it should bestressed that in principle permutations that involves more than two orthree consecutive organs could exist, notably in higher-orderphyllotactic systems where complex perturbation motifs have beenreported (see e.g Tucker, 1961). Investigating whether similar motifsoccur frequently in other species and whether they could beexplained by permutations involving more than three consecutiveorgans would provide an interesting avenue for further research.

3.2. The origin of permutations could be either meristematic or post-meristematic

A key question raised by this work is the origin of permutationsobserved in post-meristematic phyllotaxis. Both a change in thedynamics of organ initiation at the SAM and a change in post-meristematic growth pattern could in theory generate permuta-tions of the order of organs on the stem. This would implicate verydifferent biological mechanisms. To try to discriminate betweenthese hypotheses, we carefully searched for possible growthdefects on the stem. We observed that the stemwas not perturbedby any twisting at the position of permutations (Supplementarymaterial Fig. S4) or anywhere else, and pedicels linking flowers tothe stem were normal in either wild-type or ahp6-1 mutant.Permutations must appear earlier, possibly in the meristem itselfor in regions close to the meristem. Nevertheless, our analysisrestricts the possible origin of permutations – and hence theprocess controlled by AHP6 – to three simple scenarios that can beviewed as predictions from our analysis:

1. Successive organs could be initiated in the meristem with acorrect divergence angle but in an inverted order of appearancecompared to the normal order generating a spiral phyllotaxis.

2. Successive organs could be initiated in the meristem with acorrect divergence angle but roughly at the same time, thenlater be either positioned in the normal order or permuted as aresult of post-meristematic growth. This would result from anannulation of the time interval between the initiation of twoconsecutive organs, the plastochron.

3. Successive organs could be initiated in the meristem with acorrect divergence angle and following the normal order.However, in their early development, successive organs wouldbe permuted as a result of post-meristematic growth. Forexample, the younger organ could grow faster than the older

Table 8Classification of divergence angle sequences.

Genotype HMC

Wild type ahp6-1

Leaning Wild type 73 9samples ahp6-1 21 68

ahp6-3 5 9ahp6-1/3 5 9

Test L91GFP 14 5samples L94GFP 12 3

T314HA 8 11T32HA 11 8


one and would connect lower on the stem, fixing the permuta-tion post-meristematically.

The interesting common point between these scenarios is thatthey all postulate the absence of strict coupling between the timingof organ development and their angular and longitudinal position onthe stem. Importantly in scenarios (1) and (2), permutations originatein the SAM and their observation on the stem informs on thedynamics of organ initiation. In contrast, in scenario (3), permuta-tions are unrelated to the mechanisms generating phyllotacticpatterns in the SAM. A multiscale approach coupling meristematicdynamics to the post-meristematic phyllotaxis is needed to differ-entiate between these three scenarios. High-resolution X-ray tomo-graphy recently emerged as a powerful non-destructive approach toanalyze plant architecture and could be used to address this issue(Dhondt et al., 2010; Mairhofer et al., 2012), in addition to live-imaging analysis of the meristem. However, it should be noted thatAHP6 is expressed in the meristem specifically during organ initia-tion (Bartrina et al., 2011; Gordon et al., 2009) and our workidentifies it as a genetic regulator of permutations. This observationrather supports scenarios (1) and (2), suggesting that permutationshave a meristematic origin. In this view, AHP6 would be a factor thatenhances the coupling between the timing of organ initiation andorgan positioning during the establishment of phyllotaxis at themeristem. A thorough molecular analysis of the molecular mechan-isms underlying AHP6 function will be needed to confirm thesehypotheses.

3.3. The frequent occurrence of permutations needs to be taken intoaccount in dynamical models of phyllotaxis

As mentioned earlier, inhibitory field dynamical models havebeen widely used to explain the dynamics of phyllotaxis – seeDouady and Couder (1996a–c) for an in-depth presentation and alsoAtela et al. (2003) – and are sometimes considered as a paradigm forphyllotaxis (Shipman and Newell, 2005). Briefly, this type of modelassumes that the formation of regular phyllotactic patterns is a self-organizing process that emerges from local repulsive interactionsbetween direct neighboring organs and the growth of the apex. Moreprecisely, these models predict that the closest organs positionedalong contact parastichies are key determinants of the system self-organization because they propagate the geometry of phyllotaxis.The original experimental data supporting the theory of localrepulsive interactions between neighboring organs came frommicrosurgery or laser ablation experiments in meristems (Snowand Snow, 1932a,b; Wardlaw, 1949; Reinhardt et al., 2005). Theseinvasive techniques have often been criticized because of possibleartifacts due to side effects resulting from tissue wounding. However,recent experimental and modeling works have lead to propose thatpolar auxin transport could generate auxin-based inhibitory fields inthe meristem (for a review see Besnard et al., 2011), stronglysupporting the inhibitory field model. Considering that permutationmight have a meristematic origin as discussed above, it is interestingto ask whether our data are compatible with the inhibitory fieldmodel for phyllotaxis. In our analysis of Arabidopsis phyllotaxis, weobtained the following results (summarized in Fig. 12):

1. Left- and right-winding generative spirals were in roughlyequal proportions.

2. Fibonacci phyllotaxis coexisted with Lucas phyllotaxis, thislatter being rare (four individuals out of 171).

3. The overall mean direction of the divergence angles withinbaseline segments was very close to 137.51 in a population ofArabidopsis plants.

4. There were significant individual deviations of the level ofbaseline segments with reference to 137.51.

5. The segments of non-canonical angles in both wild-type andmutant plants can be explained by permutations involving twoor three consecutive organs.

6. The number of successive organs between two permutationsshowed that permutations propagate preferentially along con-tact parastichies. This result illustrates the advantages ofpattern theory approaches that take full account of the depen-dencies between successive organs along the stem for analyz-ing phyllotaxic patterns.

Results 1, 2 and 3 are in accordance with many biologicalobservations in other species (Beal, 1873; Weisse, 1894; Allard,1946; Schoute, 1938; Jean, 1994; Couder, 1998; Kuhlemeier, 2007;Peaucelle et al., 2007; Prasad et al., 2011) and have been used tovalidate model predictions. For instance, dynamical models predictequal proportions of left- and right-winding generative spirals as aresult of an initial symmetry-breaking event after the initiation ofthe first two leaves. These models also predict the coexistenceof Fibonacci and Lucas spiral phyllotaxies, and the prevalence ofFibonacci spiral phyllotaxis.

Result 4 also provides experimental support for a prediction ofdynamical models of phyllotaxis. In their work, Douady andCouder (1996a) predict that to each parastichy pair (m,n) corre-sponds a range of possible divergence angle values around thegolden angle: the lower the parastichy order, the broader therange. The range predicted for the parastichy pair (3, 5) iscompatible with the range we estimated. Interestingly, the meandirection of the divergence angles was stable over the baselinesegments for a given individual. Given the possible inter-individual variability in the structure of the Arabidopsis meristem,

Fig. 12. Schematic summary of the phyllotactic pattern. Permuted organs are in redand non-permuted organs in light green. Organs are numbered from the oldest tothe youngest and the divergence angles are indicated. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)


this suggests that the golden angle is not necessarily the diver-gence angle established in a stable regime. Instead, the goldenangle is the mean direction of angles towards which such adynamical system converges.

Importantly, while results 5 and 6 could provide experimentalsupport for several predictions of dynamical models of phyllotaxis,they also question the predictive capacity of these models. Con-cerning the dynamical models of Douady and Couder (1996a,1996c), simulations suggest that organ permutations can occurwithout disrupting the global pattern, as long as parastichies aremaintained. The results further predicted that such permutationare more likely to occur in high-order phyllotaxis, such as that ofthe sunflower capitulum. Couder (1998) provided the first bota-nical evidence for a permutation along the stem of sunflower.Here, we further show not only that permutations occur inArabidopsis, an unrelated species, but also that these permutationshave unpredicted properties (result 5): their high frequency evenin the low-order (3, 5) Arabidopsis phyllotaxis, the occurrence ofpermutations of three organs in addition to permutations of twoorgans, and possible complex patterns resulting from the chainingof permutations. Therefore, our analyses question the capacity ofdynamical models to generate realistic phyllotactic sequences thatinclude frequent and complex permutation patterns. Along thisline of idea, Mirabet et al. (2012) recently showed that introducingnoise in a dynamical model similar to the one developed byDouady and Couder (1996b) could lead to a more frequentoccurrence of permutations in the model. While this analysisindicates that biological noise might trigger organ permutationsin the meristem, whether realistic frequencies and motifs ofpermutations can indeed be obtained with this model remainsto be demonstrated. Finally, the pattern of permutation propaga-tion and the correlation between divergence angles at differentdistances apart within baseline segments (Figs. 10 and 11, result 6)can be explained if one considers that the position of an organ isunder the control of its direct neighbors along contact parastichies.These results might provide the first evidence, obtained withoutinvasive techniques, for the dynamic role played by parastichies inphyllotaxis. However, whether this result can be explained usingdynamical models also remain to be established. We have focusedhere on dynamical models since they are currently the one thathave the strongest experimental support. However, it is importantto stress that our phenotyping results do not provide support to aspecific model. If permutations indeed originate from the meris-tem, any model that aim at explaining phyllotaxis should beconsistent with our phenotyping results.

4. Material and methods

4.1. Measurement protocol

We designed a measurement protocol inspired from previousstudies (Peaucelle et al., 2007; Ragni et al., 2008) and built adedicated device consisting of a mobile protractor fixed on avertical structure (Supplementary material Fig. S5). The main inflor-escence of fully grown Arabidopsis plants was placed in the center ofthe protractor and the angular position of each silique was notedfrom bottom to top. The inflorescence was cut at its base and wasfirmly taped at the center of both the top and the base of the device.To ensure that the stem remains fully stretched, the top of the devicecan be moved along the structure in order to accommodate plants ofdifferent heights. Note that the stretching and positioning of theinflorescences at the center of the protractor (indicated by a piece ofcupboard) were carefully monitored throughout the measurements.If necessary the inflorescence was further stretched before measure-ment by moving the top of the device upward. Angles were

measured using the orientation of the first 2 mm of the pedicels ofsiliques, so that possible twisting of the siliques during flowerdevelopment did not bias the angle value. Note also that plants weresystematically tutored early during their growth in order to ensureminimal occurrence of twisting both for the inflorescence stem andfor the siliques.

Sequences always started with the first silique just above therosette of leaves. The last siliques were discarded when internodeswere not sufficiently elongated or when the height of the stemexceeded the height of the measuring device. Consequently,perturbation patterns can be truncated at the beginning or theend of the sequence. We then deduced by first-order differencingthe successive divergence angles between pairs of consecutivesiliques according to the two possible orientations (left- or right-winding) of the generative spiral. Spiral orientation was deter-mined in an initial exploratory analysis of the divergence anglesequence mainly by identifying segments of divergence anglesclose to the canonical Fibonacci angle of 137.51. In the case ofhighly perturbed mutant individuals, the optimal labeling of thedivergence angle sequence using a previously estimated modelwas found to be useful for selecting the most likely spiralorientation, see below for details. For each divergence angle, thecorresponding internode was classified as short or long. Aninternode was considered as short if its lengtho2 mm. Westudied wild-type and ahp6-1 mutant plants belonging to thesame Columbia ecotype, and thus sharing the same genetic back-ground. Eighty-two wild-type plants of cumulative length (innumber of divergence angles or internodes) 2405 and 89 ahp6-1mutant plants of cumulative length 2815 were measured.

4.2. Properties of theoretical divergence angle sequences in the caseof permutations

In the absence of permutations, the sequence of absoluteangular positions of the successive organs with reference to agiven origin is frα; r¼ 1;2;…g where r is the organ rank along thestem. In our context, the appropriate notion to describe permuta-tions that affect successive organs is that of indecomposablepermutation introduced by Comtet (1974): a permutationa1 a2 … an of 1 2 … n (the multiplicative factor α is here omitted)is said to be indecomposable if there does not exist pon such thata1 a2 … ap is a permutation of 1 2… p and p41 such thatap apþ1 … an is a permutation of p p+1… n. For instance, 3 2 1 isan indecomposable permutation of 1 2 3, whereas 2 13 is not,being decomposable into 2 1 and 3.

A divergence angle sequence is called n-admissible if it is thefirst-order differencing of an original sequence of absolute angularpositions than can be decomposed into segments which areindecomposable permutations of length at most n. For a canonicalangle α, the set of all n-admissible divergence angle sequences canbe described by a regular language (Hopcroft et al., 2006) definedon the alphabet Dn ¼ fiαj 1�nr ir2n�1; ia0g (e.g. D3 ¼ fα;2α;�α;3α;�2α;4α;5αg, see Refahi et al. (2011) for a formal justifica-tion and Supplementary material Appendix S3 for the regulargrammars that generates 2- and 3-admissible sequences. Two-admissible sequences can be recognized by a finite-state automatonsimilar to that shown in Fig. 4c.

Indecomposable permutations induce specific patterns throughfirst-order differencing which affects not only the divergenceangles between permuted organs but also the divergence anglebefore the first permuted organ and after the last permuted organ.For instance, the sequence of absolute angular positions 2α α 4α3α 5α with two successive 2-permutations gives the permutedsegment 2α �α 3α �α 2α by first-order differencing (the firstangular position is considered as a divergence angle with respectto the origin). Therefore, divergence angle sequences take the form


of a concatenation of baseline and permuted segments. A per-muted segment is defined as a sub-sequence of divergence anglescorresponding to the chaining of permutations. A permutedsegment is either a palindrome (a segment and its reverse arethe same) or is reversible, i.e. the reverse of the segment is also apermuted segment (a direct consequence of the fact that apermutation is a reversible operation).

Proof of the reversibility property (see also Refahi et al.,2011). The permuted segment x1-x2-x3-x4 in a forward passbecomes �x1←�x2←�x3←�x4 in a backward pass. The divergenceangle is then the difference between the rank of the current organand the rank of the subsequent organ. Re-numbering the organsfrom right to left, we obtain the reverse of the permuted segmentx4-x3-x2-x1. □

A permuted segment is delimited by two splitting points. At asplitting point, the sequence can be split and this does not affectthe identification of permutations. The organ rank at a splittingpoint coincides with the sequence index. As a direct consequenceof the ending of a permuted segment at a splitting point, the set ofpermuted segments is prefix free i.e. a permuted segment cannotbe a proper prefix of another permuted segment. Because of thereversibility property, the set of permuted segments is also suffixfree i.e. a permuted segment cannot be a proper suffix of anotherpermuted segment, see Table 5 for illustrations of these properties.Properties of splitting points are detailed in Supplementarymaterial Appendix S6.

4.3. Definition of stationary (hidden) variable-order Markov chains

In the following, we first introduce high-order Markov chainsbefore defining variable-order Markov chains and hidden Markovmodels based on first- and variable-order Markov chains, that arethe stochastic models used in this study. In the case of an rth-orderMarkov chain fSt ; t ¼ 0;1;…g, the conditional distribution of Stgiven S0;…; St�1 depends only on the values of St�r ;…; St�1 but notfurther on S0;…; St�r�1

PðSt ¼ st jSt�1 ¼ st�1;…; S0 ¼ s0Þ ¼ PðSt ¼ st jSt�1 ¼ st�1;…; St�r ¼ st�rÞ:In our context, the random variables represent theoretical

divergence angles and can take the five possible values α, 2α,�α, 3α or �2α (or the seven possible values α, 2α, �α, 3α, �2α, 4αor 5α). These possible values correspond to the Markov chainstates. A J-state rth-order Markov chain has JrðJ�1Þ independenttransition probabilities if all the transitions are possible. Therefore,the number of free parameters of a Markov chain increasesexponentially with the order. Let the transition probabilities of asecond-order Markov chain be given by

phij ¼ PðSt ¼ jjSt�1 ¼ i; St�2 ¼ hÞ with ∑jphij ¼ 1:

These transition probabilities can be arranged as a J2 � J matrixwhere the row ðphi0;…; phiJ�1Þ corresponds to the transitiondistribution attached to the [state h, state i] memory. If for a givenstate i and for all pairs of states ðh;h′Þ with hah′, phij ¼ ph′ ij foreach state j, i.e. once St�1 is known, St�2 conveys no furtherinformation about St, the J memories of length 2 [state h, state i]with h¼ 0;…; J�1 can be grouped together and replaced by thesingle [state i] memory of length 1 with associated transitiondistribution ðpi0;…; piJ�1Þ. This illustrates the principle used tobuild a variable-order Markov chain where the order (or memorylength) is variable and depends on the “context” within thesequence.

The memories of a Markov chain can be arranged as a memorytree such that each vertex (i.e. element of a tree graph) is either aterminal vertex or has exactly J “offspring” vertices. In practice, the

memories corresponding to unobserved contexts are not includedin the memory tree (this is the case for the memory 3α2α that wasnot observed). The memories associated with the J vertices(memories of length r þ 1) deriving from a given vertex (memoryof length r) are obtained by prefixing the parent memory witheach possible state. For instance, in Fig. 4b, the three second-ordermemories α2α, 2α2α and �α2α derive from the first-ordermemory 2α. A transition distribution is associated with eachterminal vertex of this memory tree.

A stationary Markov chain starts from its stationary distribu-tion and will continue to have that distribution at all subsequenttime points. In the case of a variable-order Markov chain, thestationary distribution – which is the implicit initial distribution –

is defined on the possible terminal memories.Because of the noisy character of the measurements, we built a

noisy or hidden Markov model (Ephraim and Merhav, 2002; Zucchiniand MacDonald, 2009) based on a variable-order Markov chain.In this hidden Markov model, the non-observable variable-orderMarkov chain represents the succession of theoretical divergenceangles along the stems while the von Mises observation distributionsrepresent divergence angle uncertainty. A hidden variable-orderMarkov chain can be viewed as a two-level stochastic process, i.e.a pair of stochastic processes fSt ;Xtg where the “output” process fXtgis related to the “state” process fStg – which is a finite-state variable-order Markov chain – by the observation distributions. The prob-ability density function of the von Mises observation distribution(in degrees) for state j is given by

gjðxt ;μj; κÞ ¼1

360I0ðκÞexp κ cos ðxt�μjÞ

π180

n oh i;

where I0ðκÞ is the modified zeroth-order Bessel function of the firstkind, μj the mean direction (location parameter) and κ the concen-tration parameter (precision or inverse dispersion) common to thestates in the above definition (Mardia and Jupp, 2000). The von Misesdistribution, also known as the circular Gaussian distribution, is aunivariate Gaussian-like periodic distribution for a variablexA ½0;3601Þ. The distribution has period 3601 so that gjðxt þ360;μj; κÞ ¼ gjðxt ;μj; κÞ. The circular standard deviation (in degrees)of the von Mises distribution of concentration parameter κ is

υ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 log fI1ðκÞ=I0ðκÞg

q 180π

where I1ðκÞ is the modified first-order Bessel function of the firstkind and I1ðκÞ=I0ðκÞ is the mean resultant length. The definition ofthe observation distributions expresses the assumption that theoutput process at time t depends only on the non-observable Markovchain at time t.

4.4. Selection of the memories of a variable-order Markov chain

The order of a Markov chain can be estimated using theBayesian information criterion (BIC). For each possible order r,the following quantity is computed:

BICðrÞ ¼ 2 log Lr � dr log n; ð1Þwhere Lr is the likelihood of the rth-order estimated Markov chainfor the observed sequences, dr is the number of free parameters ofthe rth-order estimated Markov chain and n is the cumulativelength of the observed sequences. The principle of this penalizedlikelihood criterion consists in making a trade-off between anadequate fitting of the model to the data (given by the first term in(1)) and a reasonable number of parameters to be estimated(controlled by the second term, the penalty term). In practice,it is infeasible to compute a BIC value for each possible variable-order Markov chain of maximum order rrR since the number ofhypothetical memory trees is very large. An initial maximal


memory tree is thus built combining criteria relative to themaximum order (four in our case) and to the minimum count ofmemory occurrences in the observed sequences. This memory treeis then pruned using a two-pass algorithm that is an adaptation ofthe Context-tree maximizing algorithm (Csiszár and Talata, 2006):a first dynamic programming pass, starting from the terminalvertices and progressing towards the root vertex for computingthe maximum BIC value attached to each sub-tree rooted in agiven vertex, is followed by a second tracking pass starting fromthe root vertex and progressing towards the terminal vertices forbuilding the memory tree.

4.5. Estimation method and algorithms for stationary hiddenMarkov chains

The maximum likelihood estimation of the parameters of ahidden Markov chain requires an iterative optimization technique,which is an application of the Expectation–Maximization (EM)algorithm (Ephraim and Merhav, 2002; Zucchini and MacDonald,2009). The M-step corresponding to the re-estimation of vonMises observation distributions with a common concentrationparameter is detailed in the appendix. The direct application ofthe EM algorithm to hidden Markov chains such that the under-lying Markov chain is stationary leads to an M-step which requiresa numerical solution (Zucchini and MacDonald, 2009) because ofthe inclusion of a term that depends on the stationary distributionπ in the function to be maximized for re-estimating the transitionprobabilities. We chose instead to apply a variant of the EMalgorithm called the Expectation Conditional Maximization algo-rithm (McLachlan and Krishnan, 2008). The transition probabilitieswere re-estimated at iteration k using standard closed-formsolution (i.e. taking into account πðkÞ instead of πðkþ1Þ). Then,πðkþ1Þ was deduced from the transition probability matrix Pðkþ1Þ.

In the following, S¼ s designates the whole state sequenceS0 ¼ s0;…; ST�1 ¼ sT�1 (this convention transposes to the observedsequence X¼ x). Once a hidden Markov chain has been estimated,the most probable state sequence sn can be computed for agiven observed sequence x using the Viterbi algorithm. Thisdynamic programming algorithm solves the following optimizationproblem:

sn ¼ argmaxs

PðS¼ sjX¼ x;θÞ;

where θ designates the model parameters i.e. the transitionprobabilities of the Markov chain and the mean directions andconcentration parameters of the von Mises observation distribu-tions, and where PðS¼ sjX¼ x;θÞ is the posterior probability of thestate sequence s. The restored state sequence sn can be interpretedas the optimal labeling of the observed sequence. The top N mostprobable state sequences can be computed by a direct general-ization of the Viterbi algorithm (Foreman, 1993). The log-likelihoodof a hidden Markov chain of parameter θ for a given observedsequence log PðX¼ x;θÞ can be computed using the forward orfiltering algorithm. This quantity was used as a score to assignindividuals belonging to different genetic backgrounds to the model(chosen among wild-type and ahp6-1 mutant models) that bestexplained them, see Section 2.6.

The algorithms for hidden first-order Markov chains applydirectly to hidden variable-order Markov chains by the followingstandard device. A variable-order Markov chain can be viewed as afirst-order Markov chain defined on an extended state spacecorresponding to the possible memories. In the case of a stationaryvariable-order Markov chain, the implicit initial distribution,which is the stationary distribution π, is defined on the possiblememories.

4.6. Definition of combinatorial mixture models and sketch of thealgorithm for labeling divergence angle sequences using these models

A combinatorial mixture model can be viewed as the couplingof a deterministic combinatorial model, based on a permutationassumption, with an independent mixture model (McLachlan andPeel, 2000). The non-observable combinatorial model representsthe succession of theoretical divergence angles along the stemswhile the von Mises observation distributions represent diver-gence angle uncertainty. The non-observable combinatorial modelis based on the assumption that any permutation of organs canoccur, provided the permuted organs can be arranged in disjointbounded blocks of consecutive organs. In practice, a bound of atmost three permuted organs was sufficient for the analysis of thedivergence angle sequences, but the model and algorithms weredesigned in more general terms (Refahi et al., 2011).

The probability density function of a mixture of von Misesdistributions (in degrees) with a common concentration para-meter κ (J components corresponding to the possible theoreticalangles) is given by

f ðxt ;θÞ ¼∑jηjgjðxt ;μj; κÞ ¼∑

jηj

1360I0ðκÞ

exp κ cos ðxt�μjÞπ

180

n oh i:

where the parameters are the J weights ηj ¼ PðSt ¼ jÞ such that∑jηj ¼ 1, the J mean directions μj, and the concentration para-meter κ. We assumed that the weights were all equal i.e. ηj ¼ 1=J.In fact, the frequencies of occurrence of the possible angles arevery different in the combinatorial model and this implicitlyweights the angles in the overall combinatorial mixture model.In the same way, the different angles are weighted by thecorresponding state probabilities within the underlying variable-order Markov chain in the hidden Markov chain case. Hence, itwould be irrelevant to include explicit weights ηj with contrastedvalues reflecting the frequencies shown in Tables 2 and 7. The Jmean directions μj were the theoretical angles corresponding tothe seven states α, 2α, �α, 3α, �2α, 4α and 5α while theconcentration parameter κ common to the seven von Misesobservation distributions was the concentration parameter esti-mated within the hidden variable-order Markov chain.

The algorithm for labeling the observed divergence anglesequences includes the following steps:

� Selection of a set of candidate theoretical angles independentlyfor each measured divergence angle using the von Misesobservation distributions.

� In the set of sequences of candidate theoretical angles, selection ofthe set of valid sequences according to the permutation assump-tion using a branch and bound algorithm (Refahi et al., 2011).

� When this set is not empty, the valid sequences of theoreticaldivergence angles are sorted according to their log-likelihoodsfor the observed divergence angle sequence computed usingthe von Mises observation distributions.

� When this set is empty, segments that could not be explainedby 2- and 3-permutations are invalidated and the valid seg-ments of theoretical divergence angles are sorted according totheir log-likelihoods for the corresponding observed diver-gence angle segment.

To limit the number of potential labelings, ranges of possiblemeasurements for a given theoretical angle were defined usingtwo alternative criteria: (i) a quantile criterion (e.g. between the0.005th and the 0.995th quantiles corresponding to interval of96.91 with the estimated concentration parameter κ̂) that does nottake account of the overlap between distributions, and (ii) aposterior probability criterion that takes account of this overlapbetween distributions. In this latter case, the range of possible


values for state j is such that

PðSt ¼ jjXt ¼ xt ;θÞ ¼gjðxt ;μj; κÞ

∑igiðxt ;μi; κÞZλ;

where λ is a predefined threshold on the posterior probabilities.Different thresholds were tested for the two criteria. Since theseven von Mises distributions were roughly regularly spaced (seeFig. 5), the two criteria gave similar labeling rates for similarranges of possible measurements for a given theoretical angle, seeSupplementary material Tables S3 and S4. This pruning step isuseful for limiting the search space of the branch and boundalgorithm since the support of each von Mises observationdistribution is ½0;360○Þ. It should be noted that there wereimportant overlaps between ranges of possible measurementsfor adjacent angles (e.g., between 2α and �α, see Fig. 5) and thusa measured angle could be assigned to different theoretical angles(generally 2, rarely 3) resulting in a large set of sequences ofcandidate theoretical angles for a given observed sequence.

The algorithm described above takes an observed sequence asinput and returns the set of n-admissible sequences of candidatetheoretical angles according to the quantile or the posteriorprobability criterion defined above. When this set is empty, thealgorithm returns sets of n-admissible segments ending at split-ting points separated by invalidated segments. The algorithmscans a measured sequence x¼ x0;…; xT�1 from left to right,building all n-admissible sub-sequences of candidate theoreticalangles starting at 0. When the algorithm fails to find ann-admissible sub-sequence at a given step, a backtracking proce-dure is applied to invalidate all the preceding angles up to asplitting point. Finally, thanks to the reversibility property, thesame series of operations is applied to the reversed sequencex′ ¼ xT�1;…; x0, leading to another set of n-admissible segmentsand invalidated segments. This enables to reject only the segmentsof angles that have been invalidated in both directions, and topropose an n-admissible labeling for all the other segments.Alternative labelings of an observed sequence (or of parts of thesequence when some segments cannot be explained by permuta-tions) were finally compared on the basis of log-likelihoods

log PðS¼ s;X¼ x;θÞ ¼∑tlog PðSt ¼ st ;Xt ¼ xt ;θÞ ð2Þ

p∑tlog gðxt ;μst ; κÞ: ð3Þ

All the models and algorithms presented in this paper areintegrated in the OpenAlea software platform (http://openalea.gforge.inria.fr/wiki/doku.php?id=openalea).

Appendix A. Estimating hidden Markov chains with von Misesobservation distributions with a common concentrationparameter κ

At each iteration k, the M-step of the EM algorithm consists inmaximizing the different components of the conditional expecta-tion of the complete-data log-likelihood, each term depending ona given subset of θ. For an observed sequence x, the component ofthe conditional expectation of the complete-data log-likelihoodcorresponding to the von Mises observation distributions is given by

∑j∑tPðSt ¼ jjX¼ x;θðkÞÞ log gjðy;μj; κÞ ¼�T log 360

þκ∑j∑tPðSt ¼ jjX¼ x;θðkÞÞ cos ðxt�μjÞ

π180

n o�T log I0ðκÞ

¼�T log 360þ κ∑j

∑tPðSt ¼ jjX¼ x;θðkÞÞ

� �Rj cos ðxj�μjÞ

π180

n o

�T log I0 κð Þ: ð4Þ

The mean direction xj is solution of the equation (see Mardiaand Jupp, 2000)

Cj ¼ Rj cos xjπ

180

� �; Sj ¼ Rj sin xj

π180

� �;

where the mean resultant length is given by

Rj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2j þ S

2j

r;

with

Cj ¼∑tPðSt ¼ jjX¼ x;θðkÞÞ cos ðxt π=180Þ

∑tPðSt ¼ jjX¼ x;θðkÞÞ;

Sj ¼∑tPðSt ¼ jjX¼ x;θðkÞÞ sin ðxt π=180Þ

∑tPðSt ¼ jjX¼ x;θðkÞÞ:

Since cos x has its maximum at x¼0, the re-estimated meandirection is μðkþ1Þ

j ¼ xj

μðkþ1Þj ¼

ðπ=180ÞarctanðSj=CjÞ if CjZ0; SjZ0;

ðπ=180ÞarctanðSj=CjÞ þ 180 if Cjo0;

ðπ=180ÞarctanðSj=CjÞ þ 360 if CjZ0; Sjo0:

8>><>>:

Differentiating (4) with respect to κ and using I′0ðκÞ ¼ I1ðκÞ gives

∑j


� �Rj cos ðxj�μjÞ

π180

n o�TAðκÞ;

where AðκÞ ¼ I0ðκÞ=I1ðκÞ. Hence

κðkþ1Þ ¼ A�1 1T∑j


� �Rj

!:

Appendix B. Supplementary material

Supplementary material associated with this article can befound in the online version at http://dx.doi.org/10.1016/j.jtbi.2013.07.026.

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