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HAL Id: hal-00828852

https://hal.inria.fr/hal-00828852

Submitted on 31 May 2013

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Arabidopsis thaliana

Yassin Refahi, Etienne Farcot, Yann Guédon, Fabrice Besnard, Teva Vernoux,

Christophe Godin

To cite this version:

Yassin Refahi, Etienne Farcot, Yann Guédon, Fabrice Besnard, Teva Vernoux, et al.. A Combinatorial

Model of Phyllotaxis Perturbations in Arabidopsis thaliana. 22nd Annual Symposium on

Combina-torial Pattern Matching, Palermo, Italy, 2011, Palermo, Italy. pp.323–335. �hal-00828852�

(2)

YassinRefahi

1

,Etienne Far ot

1

,Yann Guédon

1

,Fabri e Besnard

2

, Teva Vernoux

2

,and ChristopheGodin

1

1

CIRAD/INRA/INRIA,VirtualPlantsINRIAteam,UMRAGAP,TAA-108/02, 34398MontpellierCedex5,Fran e

2

RDP,ENS/CNRS/INRA/Univ.Lyon,46, alléed'Italie69364LYON edex07,

Fran e

Abstra t. Phyllotaxisisthegeometri arrangementoforgansinplants, and is knownto be highly regular. However, experimental data (from

Arabidopsisthaliana)showthatthisregularityisinfa tsubje tto spe- i patterns of permutations.In this paperwe introdu e a model for

thesepatterns,aswellasalgorithmsdesigned toidentifythesepatterns innoisyexperimentaldata.Thesealgorithmsthusin orporatea denois-ingstepwhi h isbasedonGaussian-like distributionsfor ir ulardata

forwhi ha ommondispersionparameterhasbeenpreviouslyestimated. Theappli ationoftheproposedalgorithmsallowsusto onrmthe

plau-sibilityoftheproposedmodel,andto hara terizethepatternsobserved inaspe i mutant.ThealgorithmsareavailableintheOpenAlea

soft-wareplatformforplantmodelling[10 ℄.

1 Introdu tion

Vas ular plants produ e new organs at the tip of the stem in a highly

organisedfashion.Thispatterningpro esso ursinsmallgroupsofstem

ells, the so- alled shoot api al meristem (SAM), and generates regular

patterns alled phyllotaxis [6℄. The phyllotaxis of the model plant

Ara-bidopsisthaliana followsaspiral,where singleorgansareinitiated

su es-sively at an approximately onstant divergen e angle from the previous

organ. Themostfrequent anglefound innatureis thegolden angle, lose

to 137.5

,and leadsto the so- alled Fibona i phyllotaxis.

Thegeometri regularityofthisphenomenonhasimpelleds ientiststouse

mathemati al approa hessin e earlystudies, two enturiesago. However

a ompleteunderstandingofthebiologi alpro essesthatdrivephyllotaxis

is still far from omplete. Mostmodels are me hanisti , andallow for an

explanationoftheo urren eofalimitednumberoftheoreti aldivergen e

(3)

angles (in luding 137.5

), aswell as onstrained transitions between

su - essiveanglesinagivenplant,seee.g.[1,3 ℄.Oneleadingprin ipleofthese

models is based on the SAM fun tioning, where the appearan e of new

organs alledprimordiaissupposedtobepre ludedbothinthe enter

of the SAM and in the vi inity of previously formed primordia. This is

explained intermsof aninhibitory eldsurrounding existingprimordia.

In this paper, we are interested inthe variations of angles between

on-se utive organs in real plants. These angles maybe subje tto noise and

perturbations.Onlyfewstudieshavebeendevotedtothisproblem.For

in-stan e,statisti altestshavebeenproposedtodistinguishbetweenrandom

and regular phyllota ti patterns, or ombinations thereof [4,5 ℄.

Pertur-bations in the phyllota ti patterns have also been observed in a study

about transitions between dierent phyllota ti modes inreal plants [2 ℄.

Itwassuggestedthattheseperturbationsmight resultfrompermutations

inthe order ofappearan e oforgans alongthephyllota ti spiral.

In this paper, we build up further on this initial idea. We rst

onsid-eredbothreferen e(wild-type)plantswithspiralphyllotaxis(modelplant

Arabidopsis thaliana) and mutant plants that were markedly perturbed

in their phyllotaxy.We developed a ombinatorial model for the type of

perturbationsobservedinspiralphyllotaxis.Un ertaintyistaken into

a - ount by assuming that ea h measured angle an orrespond to several

theoreti al angles among those predi ted by the model. Algorithms are

proposedto dete t su hpatterns insequen es ofangles, andgenerate all

andidate sequen es from noisy data. For a given theoreti al angle, the

orrespondingobservedanglesaremodeledbyaGaussian-likedistribution

for ir ulardata.Forea h andidatetheoreti alangle,theposterior

prob-abilityof the measured angle is omputed and ompared to a threshold.

Thisallows to redu ethe set of andidate sequen es.

2 Model Formulation

The exploratory analysisof our measured angles highlighted two

hara -teristi s of the measured divergen e angle sequen es. For a given plant,

let

α

denotethe anoni aldivergen e angle:

Themeasureddivergen eangles overedalmostallthepossiblevalues (between 0 and

360

) with highest frequen ies around the anoni al

Fibona i angle of

137.5

. At least four lasses of divergen e angles

(4)

Shortsegments(i.e.sub-sequen es)ofnon- anoni aldivergen eangles were identied along measured sequen es and were more frequent in

the mutant. Theyseemedto follow onstrained patterns, or motifs.

In parti ular, a motif orresponding approximately to

(2α, −α, 2α)

was frequently observed in wild-type and even more often in mutants (see

Figure 1 ). This motif, whi h was already observed in [2 ℄, an be simply

explainedbyapermutationoftwo onse utiveorgansonthestem,without

hangingtheir angularpositions.Thisledusto hypothesizethatthe

seg-mentsofnon- anoni alangles ouldbeexplainedbypermutations

involv-ing2 or even 3 organs (themost realisti numbers given thestru ture of

the SAM). Letus nowformulate thisidea inmore pre iseterms.

An ideal sequen e would simplybe a repetition of the anoni al angle,

0

50

100

150

200

250

300

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

Rank

D

iv

er

g

en

c

e

an

g

le

2

α

-

α

2

α

2

α

-

α

2

α

0

50

100

150

200

250

300

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

Rank

D

iv

er

g

en

c

e

an

g

le

2

α

-

α

2

α

perturbed

2

α

-

α

2

α

2

α

-

α

2

α

segment

(a) Wild type

(b) Mutant

0

50

100

150

200

250

300

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

Rank

D

iv

er

g

en

c

e

an

g

le

2

α

-

α

2

α

2

α

-

α

2

α

0

50

100

150

200

250

300

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

Rank

D

iv

er

g

en

c

e

an

g

le

2

α

-

α

2

α

perturbed

2

α

-

α

2

α

2

α

-

α

2

α

segment

(a) Wild type

(b) Mutant

Fig.1. Identi ation of M-shaped motifs orresponding to isolated 2-permutations. Theperturbedsegments annotbeeasilyexplainedonthemutantindividual.

of the form

(α, α, ..., α)

. Sin e we assume that permutations o ur, all termsina sequen e

S = (µ

1

, ..., µ

)

of divergen e anglewill infa tverify

(5)

µ

j

∈ αZ

= {iα | i ∈ Z, i 6= 0}

. We dene the orresponding absolute angles asfollows:

v

0

= 0,

v

i

=

i

X

j=1

µ

j

,

V (S) = (v

0

, ..., v

),

i ∈ {0, ..., ℓ}.

(1)

From

V

we deneaseries ontaining theorder ofappearan e oforgansif therstis

0

.Wenameitorderindex series of

S

,denoted

U (S)

,orsimply

U = (u

0

, u

1

, ..., u

)

when

S

is lear fromthe ontext:

u

i

=

1

α

(v

i

− v

J

),

0 ≤ i ≤ ℓ

where

J = arg min

j∈{0···ℓ}

v

j

.

(2)

From the denition it is lear that if

J > 0

then

v

J

< 0

sin e

v

0

=

0

. This may o ur when the sequen e starts with permuted angles, a fa t related to the left trun ation of observed sequen e with respe t to

ompletesequen es.Ifthesequen e

S

follows aspiralphyllotaxiswehave

u

i

= i,

∀i ∈ {0, ..., ℓ}.

We all the sequen e

S n

-admissible ifa nite number of permutations, applied to disjoint blo ks of at most

n

su essive organs, results in an ordered sequen e.

Denition 1. A sequen e

S = (µ

1

, ..., µ

) ∈ (αZ

)

is

n

-admissible, for some

n ∈ {1, ..., ℓ}

, if and only if its asso iated order index series

U

satises:

∀i ∈ {1, 2, ..., ℓ}, u

i

6= i ⇒ ∃j, k ∈ {0, 1, ..., ℓ}, j ≤ i ≤ k, k − j + 1 ≤ n

,

(u

j

, ..., u

k

)

isapermutationof

(j, ..., k)

,i.e.theirunderlyingsetsareequal:

{u

j

, ..., u

k

} = {j, ..., k}

.

Su h a

(u

j

, ..., u

k

)

,of length in

{1, ..., n}

, is alled a shued blo k. Property1. If

S

is

n

-admissible, then

U

is a permutation of

(0, ..., ℓ)

. In general, the onverse holdsonly for a ertain

n ∈ {2, ..., ℓ}

.

U

isapermutation of

(0, ..., ℓ) ⇐⇒ ∃n ∈ {1, ..., ℓ}

s.t.

S

is

n

-admissible.

Example 1. The sequen e

S = (−α, 2α, 3α, −α, −α, 3α)

is

3

-admissible, butnot

2

-admissible.Indeed,itsabsoluteanglesare

V = (0, −α, α, 4α, 3α,

2α, 5α)

.Hen e

v

J

= v

1

= −α

,and

U = (1, 0, 2, 5, 4, 3, 6)

.Then,

(u

0

, u

1

) =

(1, 0)

and

(u

3

, u

4

, u

5

) = (5, 4, 3)

are shued blo ks of length at most

3

, and su e to re onstru t the anoni alsequen e

(0, 1, 2, 3, 4, 5, 6)

. For

n

-admissiblesequen es, the

µ

i

only belong to anite subset of

αZ

:

(6)

Property2. Thedivergen e angles of an

n

-admissible sequen e, take val-uesin:

D

n

= {iα | (1 − n) ≤ i ≤ (2n − 1), i 6= 0} .

Proof. Let

µ

i

beadivergen eangleinan

n

-admissiblesequen e,,weknow fromEq. (1)-(2 ) that

µ

i

= (u

i

− u

i−1

.Inother words, notethat, upto the multipli ative onstant

α

,

S

isthe rst-order dieren ed sequen e of

U

.There arefour possible ases for

u

i−1

and

u

i

:

1. Neither

u

i−1

nor

u

i

areinanyshuedblo k,so

u

i−1

= (i − 1), u

i

= i

and

µ

i

= α

.

2.

u

i−1

is ina shuedblo k but

u

i

isnot ina shued blo k, so

u

i

= i

, and

u

i−1

∈ {(i − n), ..., (i − 2)}

then

µ

i

∈ {2α, ..., nα}

.

3.

u

i−1

isnot ina shuedblo kbut

u

i

isina shued blo k, so

u

i−1

=

(i − 1)

and

u

i

∈ {i + 1, ..., (i + n − 1)}

then

µ

i

∈ {2α, ..., nα}

. 4. Both

u

i−1

and

u

i

areinashued blo k.



u

i−1

and

u

i

areinthesame shuedblo kso

u

i

, u

i−1

∈ {j, ..., (n +

j − 1)}

,forsome

j < i − 1

.Hen e

µ

i

∈ {(1 − n)α, ..., (n − 1)α}\{0}

. 

u

i−1

and

u

i

are in two dierent but hained shued blo ks so

u

i−1

∈ {(i − n), ..., (i − 2)}

,

u

i

∈ {i + 1, ..., (i + n − 1)}

then

µ

i

∈ {3α, ..., (2n − 1)α}

.

Ingeneral, the on atenation oftwo

n

-admissible sequen es isnot

n

-admissible.Howeverthis anbetrueaftertranslating onlytherstangle

of the se ond sequen e. As we show now after two preliminary

observa-tions.

Proposition 1. Let

S = (µ

1

, ..., µ

)

be

n

-admissibleand

V (S)

and

U (S)

be the sequen e of absolute angles and the order index series respe tively.

Then

J < n

and

0 ≤ −v

J

< nα

, where

J

isdened as in (1).

Proof. First we prove that

J < n

. By onstru tion of order index series we know that

u

J

= (v

J

− v

J

)/α = 0

.

If

J = 0

then

J < n

. We suppose that

J 6= 0

, therefore

u

J

6= J

and by Denition 1,

∃j, k ∈ {0, 1, ..., ℓ}, j ≤ J ≤ k, k − j + 1 ≤ n

,

u

J

=

0 ∈ {u

j

...u

k

} = {j...k}

. Thus

j = 0

,and

k − j + 1 = k + 1 ≤ n

. Hen e

J ≤ k < n

.

Now we prove

0 ≤ −v

J

< nα

. Sin e

u

0

= −v

J

/α ≥ 0

by denition, this amounts to

0 ≤ u

0

< n

, in whi h only the se ond part remains to be proved. It holds obviouslyfor

u

0

= 0

.Otherwise,

u

0

is part of a shued blo k

{u

j

...u

k

} = {j...k}

, where

j = 0

,and

k < n

,when e

u

0

< n

.

(7)

The order index series of two on atenated sequen es does not always

begin with the order index series of the rst sequen e. However, this is

true ifthe rst sequen e islongenough.

Proposition 2. Let

S = (µ

1

, ..., µ

i

)

,

P = (µ

i+1

, ..., µ

i+k

)

. If

i ≥ n

, then

U (S)

is subsequen e of

U (S · P )

where

S · P = (µ

1

, ..., µ

i+k

)

denotes the on atenation of

S

and

P

.

Proof.

V (S)

issubsequen eof

V (S · P )

.Therefore

U (S)

isasubsequen e of

U (S · P )

i

v

J

= v

J

where

J

and

J

aredenedasinEq.(2)for

S

and

S · P

respe tively. Sin e

J < n, J

< n

from Proposition 1, the minimal

element of

V

and

V

appearsamongtheirrst

n − 1

elements, whi hthey

share if

i ≥ n

.

Proposition 3. Let

S = (µ

1

, ..., µ

i

)

,

i ≥ n

, and

P = (µ

i+1

, ..., µ

i+k

)

. Let

U (S) = (u

0

, ..., u

i

)

and

U (P ) = (u

0

, u

1

, ..., u

k

)

bethe orderindexseries of

S

and

P

respe tively. Suppose that

S

is

n

-admissible.

Then,the on atenated sequen e

S · P = (µ

1

, ..., µ

i+k

)

is

n

-admissible i

P |

u

i

.

=



µ

i+1

+(u

i

−i)α, µ

i+2

, ..., µ

i+k



is n

-admissible

and u

0

= 0.

(3)

Proof. We useagain theidentity

µ

i

= α(u

i

− u

i−1

)

.

Let

U = U (S · P ) = (u

0

, ..., u

i+k

)

denotetheorderindexseries

S · P

.Sin e fromProposition2weknowthat

U (S)

isasubsequen eof

U (S ·P )

,we an easilyshowthat

U (S · P ) = U (S) · (u

0

+ i, u

1

+ i, ..., u

k

+ i)

.FromProperty 1weknow that

{u

0

, ...u

i

} = {0, ..., i}

. Sin e

S

is

n

-admissibleand

u

0

= 0

then it is lear that

S · P

is

n

-admissible if and only if

(u

i+1

, ..., u

i+k

)

is a permutation of

(i + 1, ..., i + k)

that an be de omposed into disjoint shuedblo ksoflength

≤ n

,orequivalentlyfor

(u

i+1

− i, ..., u

i+k

− i)

and

(1, ..., k)

. In other words,

S · P

is

n

-admissible i

(u

i+1

− i, ..., u

i+k

− i)

is the order index series of an

n

-admissible sequen e. From the initial remark, the divergen e angle sequen e leading to this order index series

an bewritten as

α



u

i+1

− i, (u

i+2

− i) − (u

i+1

− i), ..., (u

i+k

− i) − (u

i+k−1

− i)



= α



u

i+1

− i, u

i+2

− u

i+1,

..., u

i+k

− u

i+k−1



,

wherethemultipli ationby

α

isappliedtoea h omponent.Then,the same remarkagain shows thatthis sequen eisexa tly

P |

u

i

.

Itwill beuseful inthe last se tion to s ansequen es ba kwards. One

(8)

Property3. Let

S = (µ

1

, ..., µ

)

be a sequen e of divergen e angles.

S

in

n

-admissiblei the reversed sequen e

S

= (µ

, ..., µ

1

)

is

n

-admissible. Proof. Let

U (S) = (u

0

, u

1

, ..., u

)

betheorderindex seriesof

S

,we know that

S = ((u

1

− u

0

)α, ..., (u

− u

ℓ−1

)α)

and

S

= ((u

− u

ℓ−1

)α, ..., (u

1

u

0

)α)

.Moreover,

V (S)

and

V (S

)

obviouslyhave thesame minimum,say

v

J

.Then,

U (S

) = (−v

J

, (u

− u

ℓ−1

) − v

J

, (u

− u

ℓ−2

) − v

J

, ..., (u

− u

0

) − v

J

)

= u

− v

J

− (U (S))

,

where

(U (S))

= (u

, ..., u

0

)

isthe reversed orderindex sequen e of

S

. It is lear that thelatter an be de omposed into shued blo ks of length

≤ n

i

U (S)

itself an. Sin e

U (S

)

is seen above to be a translation of

this reversed sequen e italso sharesthisproperty.

Property 2 denes the theoreti al angles that may o ur in an

n

-admissible sequen e, but the measured angles are never exa tly in

D

n

, and ould orrespond to two or more of these theoreti al angles. This

mayleadto several

n

-admissiblesequen es. Thiswilllater bestored asa suxtree.

Denition 2. A labelled tree

T = (V, E, L)

, where

L : V → D

n

, is alled an

n

-admissible tree if all leaves have a ommon depth

ℓ ∈ N

, and every pathfrom the root toa leaf is labelled by an

n

-admissible sequen e. Let

Γ

be a mapping that for ea h measured angle proposes andidate theoreti al angles amongthose in

D

n

Γ : [0, 360

) −→ 2

D

n

x

i

7−→ C

i

= {µ

i1

, µ

i2

, , ..., µ

ik

} ⊂ D

n

(4)

We also onsider a fun tion

ω : [0, 360

) × D

n

→ [0, 1]

that returns a onden e level

ω(x

i

, µ

q

)

 typi ally a probabilityfor ea h

(x

i

, µ

q

)

.

3 Dete ting

n

-admissibility in Noisy Sequen es

3.1 Problems

Given the

Γ

fun tion above, a set of measured angles will generate a possiblyhighnumberof andidate sequen es.

Problem 1. Let

x = (x

1

, ..., x

) ∈ [0, 360

)

be measuredangles, and

C =

Q

i=1

C

i

⊂ D

n

where

C

i

= Γ (x

i

)

. The task is to nd all

n

-admissible

(9)

In order to deal with this problem we rst need to know whether a

given sequen eof divergen e angles is

n

-admissible.

Problem 2. Given asequen e

S

ofdivergen e angles,thetaskisto deter-mine whether

S

is

n

-admissible.

Thefollowing is astraightforward observation whi h is a spe ial ase

of lemma10.3in [8℄.Itwill be usedto re ognise

n

-admissiblesequen es. Lemma 1. Let

U

be a permutationof

{1, ..., ℓ}

. Thenfor all

1 ≤ i < j ≤

,

{u

j

, ..., u

k

} = {j, ..., k}

i

min{u

j

, ..., u

k

} = j

and

max{u

j

, ..., u

k

} = k

.

3.2 Algorithms

First we propose an algorithm to solve Problem 2.In order to use

Lem-ma1,itrst he kswhether

{u

0

, ..., u

} 6= {0, 1, ..., ℓ}

inlineartime,using a Parikhmapping [8℄. Then, it determines whether an input sequen e is

n

-admissiblebya singles anofthesequen efromleft toright.Thetime omplexityofthealgorithmis

O(ℓ)

where

islengthoftheinputsequen e.

n-admissiblealgorithm:

input:

n, S

#

S

:asequen eoflength

apriori omposedoftheoreti aldivergen eangles output: Boolean (true or false)

Begin

if

S

0

not in

D

n

:

return False #sin e

i+1

+(u

i

− i)α

in

P

|

u

i

inproposition3 ould be not in

D

n

Constru t the order index series

U

(S)

if

{u

0

, ..., u

} 6= {0, 1, ..., ℓ} :

return False #usingParikhmapping i:=0

while

i

≤ ℓ :

if

u

i

6= i :

lo:=

u

i

; up:=

u

i

; j:=i+1 while true:

if

j > ℓ

:

return false if j - i > n - 1: return false lo:=min(

u

j

,

lo); up:=max(

u

j

,

up)

if (lo = i) & (up = j): i:=j+1 ; break #

(u

i

, ..., u

j

)

shuedblo k j:=j+1

else: i:=i+1

return true #theorderindexseries

U

(S)

analsobereturnedifneeded End

Remark 1. The notion of shued blo k an be seen as a spe ial ase of

interval [8℄, Ch. 10. However, be ause it is mu h more spe i , existing

intervalextra tionalgorithmswouldreturn invalid subsequen es,when e

(10)

Now we an deal with Problem 1. A naive algorithm would

on-stru t all andidate sequen es

S = µ

1

, µ

2

, ..., µ

, and then apply the n-admissible algorithm . The number of andidate sequen es equals

Q

i=1

|C

i

|

, where the

|C

i

|

is the ardinality of

C

i

. Sin e

C

i

are typi ally not singletons,

|C|

in reases exponentially with

. Therefore we propose alookahead algorithmto explorethesear hspa eavoidingnonne essary

paths,assket hedbelow.Thesour e odeisavailableformoredetails[10 ℄.

n-admissibletree algorithm:

input:

Γ

,

n

,

(x

1

, ..., x

)

#asequen eofmeasureddivergen eanglesoflength

output: n-admissible tree

#withnodeslabelledbybothdivergen eangles

µ

i

andorderindi es

u

i

Begin

T

:=

{root}

;

µ(root)

:=

0

;

u(root)

:=

0

find:= false

while True:

nLeaves:=

nonterminal

_

leaves(T )

#leavesofdepth

< ℓ

if nLeaves is empty: return T

for

leaf

in nLeaves:

d

:=depth(

leaf

) ;

m

:=min(

n

,

− d

); for

k

∈ {1, ..., m}

:

for

P

∈ Γ (x

d+1

) × ... × Γ (x

d+k

)

:

Compute

P

|

u(leaf )

# f.(3),Proposition3

if n-admissible(

P

|

u(leaf )

): #alsoreturns

µ

and

u

fornodesin

P

append all nodes on

P

to

leaf

End

Thanksto theuseofProposition3,

n

-admissibility an be tested on sub-sequen es only. The time omplexity of the

n

-admissible tree algorithm in reases exponentially with the lookahead limit

n

. When the returned tree ontainsonlyone

n

-admissiblesequen e(aswasgenerallythe asein pra ti e), itismore pre isely

O(l × (k)

n

)

where

k = max(|C

i

|)

.

Proposition 4. Let us all

A

n

(C)

the set of

n

-admissible sequen es in

C

, and

π(T )

the set of all (labels of) paths in

T

, from the root to the leaves.Then

π(T ) = A

n

(C)

,i.e.the tree builtinthe n-admissible tree algorithm ontainsexa tly the

n

-admissible sequen es in C.

Proof. The in lusion

π(T ) ⊂ A

n

(C)

is lear, sin e in the 2nd for loop onlypathswhi hare

n

-admissible anbeadded,thankstoProposition3. To see that the onverse holds, it su es to remark that given an

n

-admissible sequen e

1

, ..., µ

)

,one ofthe

n

subsequen es

(11)

must be

n

-admissible as well, as follows from the denition. Be ause all these subsequen es are tested in the for loop, there annot be an

n

-admissible sequen e that is not dete ted by the algorithm, and thus

π(T ) ⊃ A

n

(C)

.

Further Pruning.

In the ase where

A

n

(C)

is large, one may use the weights

ω(x

i

, µ

j

)

to sort the sequen es a ording to a onden e level. A tually, to a given

path with labels

µ

1

, ..., µ

in the omputed

n

-admissible tree, one may naturally assignthe weight

Q

j

ω(x

j

, µ

j

)

.Then, all pathsin thetree an beordered a ordingto their weight.

Theseweights analsobeusedtolimitthesizeofthe onstru tedtree,

byrulingoutall andidatepathswhoseweightisbelowa ertainthreshold.

Sin e the weight of ea h node is lower than 1, theweight of a path an

onlybelowerthan thatof anyofitssubpaths.Hen e,itispossibleinthe

forloopofthealgorithmtoprunenot onlythenonadmissiblepaths,but

alsothosehavingaweightbelowathreshold.Thisishowwehavea tually

implementedthealgorithm,using posterior probabilities for weights, and

addinga thresholdasaninputto the algorithm,asexplainedinthenext

se tion.

4 Results

4.1 Assignment of Measured Angles to Theoreti al Angles

We have used the proposed algorithms to analyse the sequen es of

mea-sured angles.The

Γ

fun tion wasparametrised using astatisti almodel. In a rst step, a hidden Markov hain was estimated onthe basisof the

pooledwild-type+mutantmeasureddivergen eanglesequen esinorder

to estimate an angle measurement un ertainty parameter; see more

de-tailsin[9℄.In thishiddenMarkov hain, thestatesof thenon-observable

Markov hain represents theoreti al divergen e angles while the von

Mises observation distributions atta hed to ea h state of the

non-obser-vable Markov hain represents measurement un ertainty. The von Mises

distribution[7℄,alsoknownasthe ir ularGaussiandistribution, isa

uni-variateGaussian-likeperiodi distributionforavariable

x ∈ [0, 360

)

.Let

g (x; µ

q

, κ)

denote the probability density fun tion of the von Mises dis-tribution, withparameters

µ

q

(mean dire tion) and

κ

( on entration pa-rameter).Themainoutputofthisrststepofanalysiswastheestimated

ommon on entration parameter (inverse varian e)

κ

. This parameter orresponds to a standard deviation of approximately

σ = 18

(12)

0

0.005

0.01

0.015

0.02

0.025

0

50

100

150

200

250

300

350

Divergence angle

p

d

f

222.5

-



137.5



275

2



52.5

3



85

-2



327.5

5



190

4



0

0.005

0.01

0.015

0.02

0.025

0

50

100

150

200

250

300

350

Divergence angle

p

d

f

222.5

-



137.5



275

2



52.5

3



85

-2



327.5

5



190

4



Fig.2.TheestimatedvonMisesdistributionsusedforthe

Γ

mapping.

set of measured angles). Using this standard deviation, the

Γ

mapping (4)for our datais dened asfollows:

Γ (x

i

) = {µ

q

∈ D

n

| µ

q

− ρσ ≤ x

i

≤ µ

q

+ ρσ}

The intervals dened by parameters

ρ

and

σ

orrespond typi ally to a umulative probability of 0.9975 withrespe t to the angle distribution

enteredat

µ

q

.Forea hpossibletheoreti alangleofindex

q

,theposterior probability

ω(x

i

, γ) =

g (x

i

; µ

q

, κ)

P

µ

r

∈D

n

g (x

i

; µ

r

, κ)

was al ulated and ompared witha predened threshold (typi al value

0.05) to de ide whether this angle should be kept or reje ted for the

la-belling of the sequen e in the n-admissible tree algorithm. The

im-pli it underlying hypothesis wasthatthe theoreti al angles were a priori

equallyprobable.

4.2 Interpretation on Our Dataset

We applied the modeling approa h to our data set, see Figure 3 for an

example where the predi teddivergen e angle sequen eis in lose

agree-ment with the measured divergen e angle sequen e. For some sequen es

however,therewasno

n

-admissiblesequen ein

C

asdenedinProblem1. Thiswasoftenduetoeithersomeerrorinthemeasurement ofdivergen e

angles,or totoolargedeviationsbetween anobservedangleand any

(13)

x

i

isnon-explained, ifnotheoreti al anglein

Γ (x

i

)

leads toa non-empty outputofthen-admissible treealgorithm.Duetothedependen ies

in-1 2 3 5 4 6 7 8 in-1in-1 9 in-10 in-13 in-12 in-14 in-15 in-16 in-18 in-17 in-19 20 2in-1 22 23 24 25 26 27 28 30 29 3in-1 32 33 34

2

α

-

α

2

α

3

α

-2

α α

3

α

-

α

2

α

2

α

-

α

2

α

2

α

-

α

2

α

Organ order

0

50

100

150

200

250

300

D

iv

er

ge

nc

e

an

gl

e

1 2 3 5 4 6 7 8 11 9 10 13 12 14 15 16 18 17 19 20 21 22 23 24 25 26 27 28 30 29 31 32 33 34

2

α

-

α

2

α

3

α

-2

α α

3

α

-

α

2

α

2

α

-

α

2

α

2

α

-

α

2

α

Organ order

0

50

100

150

200

250

300

D

iv

er

ge

nc

e

an

gl

e

Fig.3.Mutantindividual:predi tionoftheofdivergen eanglesequen e( ontinuous

line)andlabellingofthedivergen eangleswithinthepermutedsegments.

du edbythepermutationpatterns,theangleatwhi hthealgorithmfails

may infa t result from an isolated error, earlier inthe sequen e. Hen e,

all shued blo ks pre eding a non-explained angle should be marked as

not valid.To a hieve this goal, we dene splitting points. The notion of

splittingpoint anbeviewedasadeterministi analogueofaregeneration

point for a sto hasti pro ess. A regeneration point is a time instant at

whi h the future of the pro ess depends only of its state at that instant

andisthusindependentofitspastbeforethatinstant.Thepro essisthus

reborn at a regeneration point.

Denition 3. Let

S

be a sequen e of divergen e angles and

U (S)

be the order index seriesof

S

,

u

i

isa splittingpoint i

u

i

= i

and

u

i

isnot in a shued blo k.

Usingthenotionofsplittingpoint,weimplementedapro edurewhi h

was applied after the n-admissible tree algorithm. It onsisted in a

ba ktra kingstartingat thenon-explainedangleandprogressingtowards

a splitting point. This allowed us to automati ally invalidate blo ks of

angles pre edinga valueat whi hthe algorithm failed.

Torenethisanalysis,we usedreversibility(Property3 ),andappliedthe

wholepro edureto bothmeasuredsequen es andtheir reverse.Then,the

interse tionofangles invalidatedona sequen eandits reverse,wasoften

redu ed to a single angle. Moreoverthese angles were likely due to

mea-surement errors, typi ally the omissionof oneangle intheseries,leading

(14)

expertinvestigationoftheseautomati allydete tedsubsequen esenabled

us to ndwithin reaseda ura y thoseangles whi hwere not explained

byour model. The proposed modeling approa h allows to explain a very

0

50

100

150

200

250

300

D

iv

er

g

en

c

e

an

g

le

0

50

100

150

200

250

300

D

iv

e

rg

en

ce

a

n

g

le

1 2 3 4 5 6 7 8 9 12 11 10 13 14 15 16 17 18 20 19 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Organ order

(a) Forward analysis

(b) Backward analysis

3

α

-

α

-

α

3

α

2

α

-

α

2

α

0

50

100

150

200

250

300

D

iv

er

g

en

c

e

an

g

le

0

50

100

150

200

250

300

D

iv

e

rg

en

ce

a

n

g

le

1 2 3 4 5 6 7 8 9 12 11 10 13 14 15 16 17 18 20 19 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Organ order

(a) Forward analysis

(b) Backward analysis

3

α

-

α

-

α

3

α

2

α

-

α

2

α

Fig.4.Analysisofthesequen einbothdire tionstodete tsegmentsthatareinvalid with respe tto the permutation assumption. Theinvalid segmentsare delimited by

dashedlines.The ontinuousline orrespondstothepredi teddivergen eangles.

large proportion of the non- anoni al divergen e angles despite the

rela-tively high standard deviation (approx.

18

) of the estimated von Mises

distributions. Thisindi ates thattheproposed model orre tly des ribes

the phyllota ti patterns of Arabidopsis thaliana. Wild-type plants were

hara terized byrelatively frequento urren es of2-permutations

gener-allyisolatedwhilemutantswere hara terizedbythefrequento urren es

ofboth2-and3-permutationswhose su essiongenerateshighly omplex

motifs,seeFigure3foranexample.Asummaryoftheresultsisshownin

Table 1,withmore pre ise ountsof patternsinTable 2.

The term Lu as phyllotaxis refers to a spiral phyllotaxis were the

anoni al divergen e angle

α

is99.5

.Although rarer than theFibona i

spiral(

α = 137.5

),itisknowntoo urinnature,andwasabletoexplain

two wild-type andtwo mutant sequen es,for whi h

137.5

(15)

Wild-typeplant Mutant

#sequen es/#organs 82/2405 89/2815

%ofnon- anoni alangles 15% 37%

%ofunexplainedangles 2% 5%

#individuals,Lu asphyllotaxis 2 2

#2-permutations 123 297

#3-permutations 3 53

Table1.Summaryofthepermutationpatternsobservedinbothwildtypeandmutant

plants.

organorderdivergen eangleswild-typemutant

2-permutation 213 2-1 2 90 193 3214 3-1-1 3 1 11 3-permutation 3124 3-2 1 2 1 9 2314 2 1-2 3 13 total 2 33 22-permutations21435 2-1 3-1 2 16 32

Table2.Permutedsegmentsuptolength5.Thesesegmentsaredelimitedbytwo

split-tingpoints.Thedivergen eanglesequen eistherst-orderdieren edorgansequen e. By onvention,theoriginoftheorgansequen eis0(notindi ated).

Referen es

1. AdlerI.,Barabé D.,Jean R.V., AHistory of the Study of Phyllotaxis, Annalsof Botany,80(3):231-244 (1997).

2. CouderY., Initialtransitions, order and disorder inphyllota ti patterns: the

on-togeny of Helianthus annuus.A ase study, A ta So ietatisBotani orum Poloniae, 67:129:150 (1998).

3. DouadyS,CouderY., Phyllotaxis as aself organizing pro ess:I,II,III,Journal of Theoreti alBiology178:255-312 (1996).

4. Jeune B., Barabé J., Statisti al Re ognition of Random and Regular Phyllota ti

Patterns,AnnalsofBotany,94:913-917(2004).

5. JeuneB.,BarabéJ.,Asto hasti approa htophyllota ti patternanalysis,Journal ofTheoreti alBiology238:52-59(2006).

6. KuhlemeierC.,Phyllotaxis,TrendsinPlantS ien e,12:143-150(2007).

7. Mardia,K.V.andJupp,P.E.Dire tionalStatisti s.JohnWiley&Sons:Chi hester,

2000.

8. ParidaL.,PatternDis overyinBioinformati s:Theory&Algorithms,Chapman& Hall/CRCMathemati alComputationalBiology series,2008.

9. Refahi,Y.,Guédon,Y.,Besnard,F.,Far ot,E.,Godin,C.,Vernoux,T.,Analyzing

perturbations inphyllotaxisof Arabidopsisthaliana.In:6thInternationalWorkshop onFun tional-Stru turalPlantModels,Davis,California,170-172(2010).

10. http://openalea.gforge.inria.fr/do /vplants/phyllotaxis_analysis/do /_build/ html/ ontents.html

Figure

Figure 1 ). This motif, whih was already observed in [2 ℄, an be simply
Fig. 2. The estimated von Mises distributions used for the Γ mapping.
Fig. 3. Mutant individual: predition of the of divergene angle sequene (ontinuous
Fig. 4. Analysis of the sequene in both diretions to detet segments that are invalid
+2

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