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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�
YassinRefahi
1
⋆
,Etienne Far ot1
,Yann Guédon1
,Fabri e Besnard2
, Teva Vernoux2
,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
⋆
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 and360
◦
) with highest frequen ies around the anoni al
Fibona i angle of
137.5
◦
. At least four lasses of divergen e angles
•
Shortsegments(i.e.sub-sequen es)ofnon- anoni aldivergen eangles were identied along measured sequen es and were more frequent inthe 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 (seeFigure 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 eS = (µ
1
, ..., µ
ℓ
)
of divergen e anglewill infa tverifyµ
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
ℓ
),
i ∈ {0, ..., ℓ}.
(1)From
V
we deneaseries ontaining theorder ofappearan e oforgansif therstis0
.Wenameitorderindex series ofS
,denotedU (S)
,orsimplyU = (u
0
, u
1
, ..., u
ℓ
)
whenS
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
thenv
J
< 0
sin ev
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 toompletesequen es.Ifthesequen e
S
follows aspiralphyllotaxiswehaveu
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 mostn
su essive organs, results in an ordered sequen e.Denition 1. A sequen e
S = (µ
1
, ..., µ
ℓ
) ∈ (αZ
∗
)
ℓ
is
n
-admissible, for somen ∈ {1, ..., ℓ}
, if and only if its asso iated order index seriesU
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. IfS
isn
-admissible, thenU
is a permutation of(0, ..., ℓ)
. In general, the onverse holdsonly for a ertainn ∈ {2, ..., ℓ}
.U
isapermutation of(0, ..., ℓ) ⇐⇒ ∃n ∈ {1, ..., ℓ}
s.t.S
isn
-admissible.Example 1. The sequen e
S = (−α, 2α, 3α, −α, −α, 3α)
is3
-admissible, butnot2
-admissible.Indeed,itsabsoluteanglesareV = (0, −α, α, 4α, 3α,
2α, 5α)
.Hen ev
J
= v
1
= −α
,andU = (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 most3
, and su e to re onstru t the anoni alsequen e(0, 1, 2, 3, 4, 5, 6)
. Forn
-admissiblesequen es, theµ
i
only belong to anite subset ofαZ
: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 eangleinann
-admissiblesequen e,,weknow fromEq. (1)-(2 ) thatµ
i
= (u
i
− u
i−1
)α
.Inother words, notethat, upto the multipli ative onstantα
,S
isthe rst-order dieren ed sequen e ofU
.There arefour possible ases foru
i−1
andu
i
:1. Neither
u
i−1
noru
i
areinanyshuedblo k,sou
i−1
= (i − 1), u
i
= i
andµ
i
= α
.2.
u
i−1
is ina shuedblo k butu
i
isnot ina shued blo k, sou
i
= i
, andu
i−1
∈ {(i − n), ..., (i − 2)}
thenµ
i
∈ {2α, ..., nα}
.3.
u
i−1
isnot ina shuedblo kbutu
i
isina shued blo k, sou
i−1
=
(i − 1)
andu
i
∈ {i + 1, ..., (i + n − 1)}
thenµ
i
∈ {2α, ..., nα}
. 4. Bothu
i−1
andu
i
areinashued blo k.
u
i−1
andu
i
areinthesame shuedblo ksou
i
, u
i−1
∈ {j, ..., (n +
j − 1)}
,forsomej < i − 1
.Hen eµ
i
∈ {(1 − n)α, ..., (n − 1)α}\{0}
.u
i−1
andu
i
are in two dierent but hained shued blo ks sou
i−1
∈ {(i − n), ..., (i − 2)}
,u
i
∈ {i + 1, ..., (i + n − 1)}
thenµ
i
∈ {3α, ..., (2n − 1)α}
.⊓
⊔
Ingeneral, the on atenation oftwo
n
-admissible sequen es isnotn
-admissible.Howeverthis anbetrueaftertranslating onlytherstangleof the se ond sequen e. As we show now after two preliminary
observa-tions.
Proposition 1. Let
S = (µ
1
, ..., µ
ℓ
)
ben
-admissibleandV (S)
andU (S)
be the sequen e of absolute angles and the order index series respe tively.Then
J < n
and0 ≤ −v
J
< nα
, whereJ
isdened as in (1).Proof. First we prove that
J < n
. By onstru tion of order index series we know thatu
J
= (v
J
− v
J
)/α = 0
.If
J = 0
thenJ < n
. We suppose thatJ 6= 0
, thereforeu
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}
. Thusj = 0
,andk − j + 1 = k + 1 ≤ n
. Hen eJ ≤ k < n
.Now we prove
0 ≤ −v
J
< nα
. Sin eu
0
= −v
J
/α ≥ 0
by denition, this amounts to0 ≤ u
0
< n
, in whi h only the se ond part remains to be proved. It holds obviouslyforu
0
= 0
.Otherwise,u
0
is part of a shued blo k{u
j
...u
k
} = {j...k}
, wherej = 0
,andk < n
,when eu
0
< n
.⊓
⊔
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
)
. Ifi ≥ n
, thenU (S)
is subsequen e ofU (S · P )
whereS · P = (µ
1
, ..., µ
i+k
)
denotes the on atenation ofS
andP
.Proof.
V (S)
issubsequen eofV (S · P )
.ThereforeU (S)
isasubsequen e ofU (S · P )
iv
J
= v
J
′
whereJ
andJ
′
aredenedasinEq.(2)for
S
andS · P
respe tively. Sin eJ < n, J
′
< n
from Proposition 1, the minimal
element of
V
andV
′
appearsamongtheirrst
n − 1
elements, whi htheyshare if
i ≥ n
.⊓
⊔
Proposition 3. Let
S = (µ
1
, ..., µ
i
)
,i ≥ n
, andP = (µ
i+1
, ..., µ
i+k
)
. LetU (S) = (u
0
, ..., u
i
)
andU (P ) = (u
′
0
, u
′
1
, ..., u
′
k
)
bethe orderindexseries ofS
andP
respe tively. Suppose thatS
isn
-admissible.Then,the on atenated sequen e
S · P = (µ
1
, ..., µ
i+k
)
isn
-admissible iP |
u
i
.
=
µ
i+1
+(u
i
−i)α, µ
i+2
, ..., µ
i+k
is n
-admissibleand 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
)
denotetheorderindexseriesS · P
.Sin e fromProposition2weknowthatU (S)
isasubsequen eofU (S ·P )
,we an easilyshowthatU (S · P ) = U (S) · (u
′
0
+ i, u
′
1
+ i, ..., u
′
k
+ i)
.FromProperty 1weknow that{u
0
, ...u
i
} = {0, ..., i}
. Sin eS
isn
-admissibleandu
′
0
= 0
then it is lear that
S · P
isn
-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
isn
-admissible i(u
i+1
− i, ..., u
i+k
− i)
is the order index series of ann
-admissible sequen e. From the initial remark, the divergen e angle sequen e leading to this order index seriesan 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 tlyP |
u
i
.⊓
⊔
Itwill beuseful inthe last se tion to s ansequen es ba kwards. One
Property3. Let
S = (µ
1
, ..., µ
ℓ
)
be a sequen e of divergen e angles.S
inn
-admissiblei the reversed sequen eS
′
= (µ
ℓ
, ..., µ
1
)
isn
-admissible. Proof. LetU (S) = (u
0
, u
1
, ..., u
ℓ
)
betheorderindex seriesofS
,we know thatS = ((u
1
− u
0
)α, ..., (u
ℓ
− u
ℓ−1
)α)
andS
′
= ((u
ℓ
− u
ℓ−1
)α, ..., (u
1
−
u
0
)α)
.Moreover,V (S)
andV (S
′
)
obviouslyhave thesame minimum,say
v
J
.Then,U (S
′
) = (−v
J
, (u
ℓ
− u
ℓ−1
) − v
J
, (u
ℓ
− u
ℓ−2
) − v
J
, ..., (u
ℓ
− u
0
) − v
J
)
= u
ℓ
− v
J
− (U (S))
′
,
where
(U (S))
′
= (u
ℓ
, ..., u
0
)
isthe reversed orderindex sequen e ofS
. It is lear that thelatter an be de omposed into shued blo ks of length≤ n
iU (S)
itself an. Sin eU (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 inD
n
, and ould orrespond to two or more of these theoreti al angles. Thismayleadto several
n
-admissiblesequen es. Thiswilllater bestored asa suxtree.Denition 2. A labelled tree
T = (V, E, L)
, whereL : V → D
n
, is alled ann
-admissible tree if all leaves have a ommon depthℓ ∈ N
, and every pathfrom the root toa leaf is labelled by ann
-admissible sequen e. LetΓ
be a mapping that for ea h measured angle proposes andidate theoreti al angles amongthose inD
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 levelω(x
i
, µ
q
)
typi ally a probabilityfor ea h(x
i
, µ
q
)
.3 Dete ting
n
-admissibility in Noisy Sequen es3.1 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
ℓ
) ∈ [0, 360
◦
)
ℓ
be measuredangles, and
C =
Q
ℓ
i=1
C
i
⊂ D
n
ℓ
whereC
i
= Γ (x
i
)
. The task is to nd alln
-admissibleIn 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 whetherS
isn
-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. LetU
be a permutationof{1, ..., ℓ}
. Thenfor all1 ≤ i < j ≤
ℓ
,{u
j
, ..., u
k
} = {j, ..., k}
imin{u
j
, ..., u
k
} = j
andmax{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
ℓ
} 6= {0, 1, ..., ℓ}
inlineartime,using a Parikhmapping [8℄. Then, it determines whether an input sequen e isn
-admissiblebya singles anofthesequen efromleft toright.Thetime omplexityofthealgorithmisO(ℓ)
whereℓ
islengthoftheinputsequen e.n-admissiblealgorithm:
input:
n, S
#S
:asequen eoflengthℓ
apriori omposedoftheoreti aldivergen eangles output: Boolean (true or false)Begin
if
S
0
not inD
n
:return False #sin e
(µ
i+1
+(u
i
− i)α
inP
|
u
i
inproposition3 ould be not inD
n
Constru t the order index seriesU
(S)
if
{u
0
, ..., u
ℓ
} 6= {0, 1, ..., ℓ} :
return False #usingParikhmapping i:=0while
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+1else: i:=i+1
return true #theorderindexseries
U
(S)
analsobereturnedifneeded EndRemark 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
Now we an deal with Problem 1. A naive algorithm would
on-stru t all andidate sequen es
S = µ
1
, µ
2
, ..., µ
ℓ
, and then apply the n-admissible algorithm . The number of andidate sequen es equalsQ
ℓ
i=1
|C
i
|
, where the|C
i
|
is the ardinality ofC
i
. Sin eC
i
are typi ally not singletons,|C|
in reases exponentially withℓ
. Therefore we propose alookahead algorithmto explorethesear hspa eavoidingnonne essarypaths,assket hedbelow.Thesour e odeisavailableformoredetails[10 ℄.
n-admissibletree algorithm:
input:
Γ
,n
,(x
1
, ..., x
ℓ
)
#asequen eofmeasureddivergen eanglesoflengthℓ
output: n-admissible tree#withnodeslabelledbybothdivergen eangles
µ
i
andorderindi esu
i
BeginT
:={root}
;µ(root)
:=0
;u(root)
:=0
find:= falsewhile True:
nLeaves:=
nonterminal
_leaves(T )
#leavesofdepth< ℓ
if nLeaves is empty: return Tfor
leaf
in nLeaves:d
:=depth(leaf
) ;m
:=min(n
,ℓ
− d
); fork
∈ {1, ..., m}
:for
P
∈ Γ (x
d+1
) × ... × Γ (x
d+k
)
:Compute
P
|
u(leaf )
# f.(3),Proposition3if n-admissible(
P
|
u(leaf )
): #alsoreturnsµ
andu
fornodesinP
append all nodes onP
toleaf
End
Thanksto theuseofProposition3,
n
-admissibility an be tested on sub-sequen es only. The time omplexity of then
-admissible tree algorithm in reases exponentially with the lookahead limitn
. When the returned tree ontainsonlyonen
-admissiblesequen e(aswasgenerallythe asein pra ti e), itismore pre iselyO(l × (k)
n
)
where
k = max(|C
i
|)
.Proposition 4. Let us all
A
n
(C)
the set ofn
-admissible sequen es inC
, andπ(T )
the set of all (labels of) paths inT
, from the root to the leaves.Thenπ(T ) = A
n
(C)
,i.e.the tree builtinthe n-admissible tree algorithm ontainsexa tly then
-admissible sequen es in C.Proof. The in lusion
π(T ) ⊂ A
n
(C)
is lear, sin e in the 2nd for loop onlypathswhi haren
-admissible anbeadded,thankstoProposition3. To see that the onverse holds, it su es to remark that given ann
-admissible sequen e(µ
1
, ..., µ
ℓ
)
,one ofthen
subsequen esmust 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 ann
-admissible sequen e that is not dete ted by the algorithm, and thusπ(T ) ⊃ A
n
(C)
.⊓
⊔
Further Pruning.
In the ase where
A
n
(C)
is large, one may use the weightsω(x
i
, µ
j
)
to sort the sequen es a ording to a onden e level. A tually, to a givenpath with labels
µ
1
, ..., µ
ℓ
in the omputedn
-admissible tree, one may naturally assignthe weightQ
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 thepooledwild-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, withparametersµ
q
(mean dire tion) andκ
( on entration pa-rameter).Themainoutputofthisrststepofanalysiswastheestimatedommon on entration parameter (inverse varian e)
κ
. This parameter orresponds to a standard deviation of approximatelyσ = 18
◦
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
andσ
orrespond typi ally to a umulative probability of 0.9975 withrespe t to the angle distributionenteredat
µ
q
.Forea hpossibletheoreti alangleofindexq
,theposterior probabilityω(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 einC
asdenedinProblem1. Thiswasoftenduetoeithersomeerrorinthemeasurement ofdivergen eangles,or totoolargedeviationsbetween anobservedangleand any
x
i
isnon-explained, ifnotheoreti al angleinΓ (x
i
)
leads toa non-empty outputofthen-admissible treealgorithm.Duetothedependen iesin-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 andU (S)
be the order index seriesofS
,u
i
isa splittingpoint iu
i
= i
andu
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
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
◦
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).
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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).
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2000.
8. ParidaL.,PatternDis overyinBioinformati s:Theory&Algorithms,Chapman& Hall/CRCMathemati alComputationalBiology series,2008.
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