Nutations in growing plant shoots: The role of elastic deformations due to gravity loading

ContentslistsavailableatScienceDirect

Journal

of

the

Mechanics

and

Physics

of

Solids

journalhomepage:www.elsevier.com/locate/jmps

Nutations

in

growing

plant

shoots:

The

role

of

elastic

deformations

due

to

gravity

loading

Daniele

Agostinelli

a

_,

_Alessandro

_Lucantonio

b

_,

_Giovanni

_Noselli

a

_,

Antonio

DeSimone

a,b,∗

a SISSA–International School for Advanced Studies, Trieste 34136, Italy b The BioRobotics Institute, Scuola Superiore Sant’Anna, Pisa 56127, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 1 July 2019 Revised 24 August 2019 Accepted 25 August 2019 Available online xxx Keywords: Circumnutations Gravitropism Hopf bifurcation Flutter instability

a

b

s

t

r

a

c

t

Theeffectofelasticdeformationsinducedbygravityloadingontheactivecircumnutation movements ofgrowingplant shoots isinvestigated. We consider firstadiscrete model (agravitropicspring-pendulumsystem)and thenacontinuous rod modelwhichis an-alyzedboth analytically(under the assumptionof smalldeformations) and numerically (inthelargedeformationregime).Wefindthat,forachoiceofmaterialparameters con-sistentwithvalues reportedintheavailableliteratureonplantshoots,rodsofsufficient lengthmayexhibitlateraloscillationsofincreasingamplitude,whicheventuallyconverge tolimitcycles.Thisbehaviorstrongly suggeststheoccurrenceofaHopfbifurcation,just asforthe gravitropicspring-pendulumsystem, forwhichthisresult isrigorously estab-lished.Atleastinthisrestricted setofmaterialparameters,ouranalysissupportsaview ofDarwin’scircumnutationsasabiologicalanaloguetostructuralsystemsexhibiting flut-terinstabilities,i.e.,spontaneousoscillationsawayfromequilibriumconfigurationsdriven bynon-conservativeloads.Here,inthecontextofnutation movementsofgrowingplant shoots,theenergyneededtosustainoscillationsiscontinuouslysuppliedtothesystemby theinternalbiochemicalmachinerypresidingthecapabilityofplantstomaintainavertical pose.

© 2019TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Nutationsandcircumnutations,namely,periodicmovementsinrapidlyelongatingplantorganssuchasroots,hypocotyls, shoots, branches, and flower stalks havefascinated scientists for over a century. A large body of literature has followed theseminal workby Darwin(1880),seeSchuster(1996)andMugnaietal.(2015) fora (partial)listofreferences.In one oftheprevailingtheoriescurrentlyaccepted,nutationsare oscillatorymovementsarising fromtheover-compensatory re-sponseofaplanttothechangingorientation ofitsgravisensoryapparatusrelative totheEarth’sgravityvector. Contribu-tionsofparticularrelevanceareIsraelssonandJohnsson(1967),JohnssonandHeathcote(1973)andJohnssonetal.(2009). Veryrecently,quantitativemodelssupportingthishypothesishavebeenproposedandtested, seee.g.,Bastienetal.(2013),

∗ _{Corresponding author at: SISSA–International School for Advanced Studies, Via Bonomea 265, Trieste 34136, Italy.}

E-mail addresses: daniele.agostinelli@sissa.it (D. Agostinelli), alessandro.lucantonio@santannapisa.it (A. Lucantonio), giovanni.noselli@sissa.it (G. Noselli), desimone@sissa.it (A. DeSimone).

https://doi.org/10.1016/j.jmps.2019.103702

0022-5096/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Fig. 1. A 40-day-old sample of Arabidopsis thaliana (ecotype Col-0) grown under normal gravity conditions (1 g) at the SAMBA laboratory of SISSA. (a) A superposition of two snapshots to highlight the amplitude of lateral oscillations (circumnutations) exhibited by the primary shoot. (b) A sequence of snapshots taken at time intervals of 10 minutes. Notice that the characteristic time of circumnutational oscillations is of the order of τc ࣃ 90 min.

Bastienetal.(2014),ChelakkotandMahadevan(2017),Pouliquenetal.(2017)andMerozetal.(2019)andthemicroscopic mechanismsgoverninggravisensinghavebeenidentified(Chauvetetal.,2019).

Inthispaper,buildingontheobservationthatgrowingshootsoftenappearaselongatedbiologicalstructuresof signifi-cantweightrelativetotheirstiffness,wefocusontherolethatelasticdeformationsarisingfromgravityloadingmayhave onthenutationalmovements.WeanalyzethesemovementsasaHopfbifurcationphenomenonintheequationsgoverning thegravitropicdynamicsofa growingshoot, andidentifythe regimeofmaterial andgeometric parametersinwhich this bifurcation,andhencenutationmovements,arelikelytooccur.Asdiscussedinthemanuscript,theparameterswhichcome intoplayarefourcharacteristictimes,namely,thedurationofstatolithsavalancheinthestatocytes

τ

a,thegrowthtime

τ

g, thememorytime

τ

m,thereactiontime

τ

r,andtwodimensionlessloadingparameters,thatis,yand

λ

,determinedbythe shootweightandslenderness,respectively.

Typically, elastic deformations are just one of the concurrent factors promoting oscillatory response. There are how-ever regimesof the governing parameters, identified by values of the fourcharacteristic response times andof the two dimensionlessloadingparameters,inwhichtakingintoaccounttheelasticdeformationsinducedbygravityloadingis nec-essaryto“explain” the emergenceofoscillations.At leastinthisrestrictedsetofgeometric andmaterial parameters,our analysissupportsaviewofDarwin’scircumnutationsasabiologicalanaloguetostructuralsystemsexhibitingflutter insta-bilities,i.e.,spontaneousoscillationsawayfromequilibriumconfigurations.Inrecentpapersappearedinthisjournal,these havebeen analyzedin structuralsystemsloaded by non-conservative(follower)forces,see BigoniandNoselli(2011) and Bigonietal.(2018).Here,inthecontextofnutationmovementsofgrowingplantshoots,theenergyneededtosustain os-cillationsiscontinuouslysuppliedtothesystembytheinternalbiochemicalmachinerypresidingthecapabilityofplantsto maintainaverticalpose.

1.1. Stateoftheart:abriefreview

Movementsofgrowingplantorgansareverycomplexandfarfrombeingcompletelyunderstood.Theobservableshapeof aplantisthespecificresultofitsuniquehistoryofendogenousandexogenousfactors.Discerningwhethercertaindynamics areencodedinthebiologyofthesystemortheyrepresentthemechanicalandphysiologicalreactiontoexternalcues,ora combinationofboth,isafundamentalquestionthathasintriguedandpuzzledmanyscientistsoverthepasttwocenturies. Movementsinplantsaremainlyclassifiedintropismsandnasticmovements.Tropismsarethedirectedgrowthresponses to directional environmental cues, e.g.,light (phototropism), gravity (gravitropism), touch(thigmotropism), etc, while nastic movementsaremotionsrespondingtoexternal stimuliwhichare independentofthepositionofthestimulussource,e.g., temperature(thermonasty),chemicals(chemonasty),touch(thigmonasty),etc.

Thisclassificationfailstoincludecircumnutationmovements,i.e.,circular,ellipticalorpendularoscillationsofelongating plantorgans (exemplified inFig. 1 forArabidopsisthaliana Col-0),which are causedby radially asymmetric growthrates thathaveuncertainorigin.Thefirstreportsaboutcircumnutations canbetracedbacktothe19thcenturyand,sincethen, manydifferent termshavebeen used intheliterature to refer tosuch a phenomenon. Indeed,Palm (1827) talked about

twining,von Sachs(1875)usedthetermrevolving nutation,Darwin(1880)coinedthetermcircumnutation,Noll(1885) re-portedabout rotating nutations, Gradmann (1922) aboutover-bending movements, Rawitscher (1932)about circular move-ments,Bünning(1953)aboutcircumnastyandHammerandGessner(1958)aboutgrowthoscillations.

Itseemswidely acceptedthat circumnutationsare theconsequenceofhelical growthandreversiblecellvolume varia-tions(Mugnaietal.,2015).However,consensusontheregulatorymechanismofcircumnutationshasnotbeenreachedyet, withopinionssplitbetweentwomaintheories:circumnutationscanbeexplainedeitherbyendogenousmechanismsoras internalresponsestoexogenousstimuli.Inthefirstcase, circumnutationsshouldconstituteaclassofmovementsontheir own(CorrellandKiss,2008),whereasinthesecondcase,theyshouldbeincludedamongtropismsornasticmovements.

Darwin(1880)wasprobablythefirstonetosuggestanendogenousfeatureofcircumnutationswhose“amplitude,or di-rection,orbothhaveonlytobemodifiedforthegoodoftheplantinrelationwithinternalorexternalstimuli”.Theories support-ingthispointofviewincludedhormonalandionicoscillationswithgravityactingasanexternalsignal.Arnal(1953) pro-posedperiodic variationsinauxinfluxesfromthetip andJoerrens(1957) hypothesizedperiodicchangesinthesensitivity ofthe elongating cellsto auxin. Heathcote andAston (1970) formulated thetheory ofa nutational oscillator situatedin eachcell.StudyingshootsinPhaseolusVulgarisL.,Milletetal.(1984),Milletetal.(1988)andBadotetal.(1990)proposed aprimarymechanismresidinginthemovingofaturgescencewave aroundtheshoot,alsorelatedtoKdistribution.More recently,Shabalaand Newman(1997) andShabala(2003) observed strongcorrelationsbetween nutationand rhythmical patternsofH+_,K+ andCa2+_{ionfluxesintheelongationregionofcornroots.}

Ontheotherside,Gradmann(1926)introducedtheideathatcircumnutationscouldbetheresultofdelayedgravitropic responses:thedeviationfromthe verticallinetriggers a correctingmovement that, duetoa reactiontime between per-ceptionandactuation,makes theplantovershootgivingriseto self-sustainedoscillations.Buildingonthis, Israelssonand Johnsson(1967)andSomolinos(1978) showedthatamodeldescribing theplantgravitropicresponsebasedon delayand memorycouldexplaincircumnutations.Later,observinganautotropicstraighteningtoaverticalpositioninAvenaseedlings duringweightlessnessexperiments,Chapmanetal.(1994)adaptedthemodelproposedbyIsraelssonandJohnsson(1967)by includingsuchautotropiceffects.

The debateabout therole of gravity forthe inductionor continuation ofcircumnutations persisted and hasbeen fu-eled bynewexperiments onEarthandinspace overthe lastfew decades (Kiss,2006;2009;Kobayashietal., 2019).On the one hand,the studyof agravitropic mutantsin morning glory, peaand arabidopsissupported theidea that gravire-sponseisrequiredinboththeshootsandrootsofdicotyledonousplants(Hatakedaetal.,2003;Kimetal., 2016;Kitazawa etal., 2005; 2008). Ontheother hand, manyexperimental resultshavebeen interpretedas corroboratingthe Darwinian theoryofendogenousoscillationsaffectedbygraviresponses.BasedonexperimentsinmicrogravityreportedbyBrownand Chapman(1984),Brownetal.(1990)andJohnssonetal.(1999)suggesteda two-oscillatormodelbycombiningthe grav-itropic feedback oscillator with an endogenous oscillator. Performing experiments on the gravitropic rice mutant lazy1, YoshiharaandIino(2006)concludedthatcircumnutationandgravitropismareseparate (althoughinterfering)phenomena. Moreover,“minuteoscillatory movements inmicrogravity” have beenobservedby Johnssonetal.(2009)in A.thalianaand, morerecently, alsoKobayashiet al.(2019) reportedminutemovementsin ricecoleoptiles inmicrogravity, although cau-tiouslystressingthefactthatsuchmovementsweredifficulttomeasureanditwas“technicallydifficulttodeterminewhether theyweretrulycircumnutationornot”.

In addition,another important unsolved issueis to which extent the role ofgravity depends on the plantorgans or species.Forinstance,tillersofthericemutantlazy1behavedifferentlyfromcoleoptiles(Abeetal.,1996),andevolutionary benefitsareobviousfortwiningplantsinsearchofmechanicalsupport.Inanycase,itisclearthatconsiderabledeviations fromthe plumb line will causea gravitropicreaction, possibly interfering withan endogenousoscillator (assuming this oscillatorexists). Thisisthe reasonwhygravitropism plays acrucial role whendealingwithcircumnutations. Ingeneral, thestudyofanykindoftropismconsistsinunderstandingthemechanismsofstimulusperception,signaltransductionand response.

The concept of tropism was introduced by Knight in 1806 butonly recently“the data converged to provide a picture of the physiological, molecular and cell biological processes that underlie plant tropisms” (Gilroy and Masson, 2008). About gravitropism,it iswidely acceptedthatplants sense gravityinspecialized cells(statocytes)throughthe sedimentationof starch-filled plastids (statoliths) which are denser than the surrounding cellular fluid. This leads to the development of an asymmetryin auxinconcentration betweentheupperandlower flanks,causingdifferentialcell elongationandhence bending.

Thestandardwaytomodelgravitropismgoesback toVonSachs(1882),whoformulated theso-calledsinelawstating thatthegraviresponseisproportionaltothesineoftheangleformedbytheverticalandtheorganaxis.Animportant modi-ficationtothesinelawhasbeenintroducedbyBastienetal.(2013)andBastienetal.(2014),whoincludedaproprioception termtodescribethephenomenonoforganstraighteningcalledautotropism.However,inthelastfewdecades,anapparent contradictionemergedwhentestingthislaw:underpermanentinclination,theresponseoftheplantappearedtobe inde-pendentofeffectivegravity,whereas,undertransientgravi-stimulation,thegraviresponsewasquantifiedbyadose-response

curve,thedosedependingonthestimulationtimeandtheeffectivegravity(Chauvetetal.,2019).Byintroducingamemory processinthe gravitropicsignallingpathway(usingan approach alreadyproposed by IsraelssonandJohnsson(1967)and similarlytakenbyMerozetal.(2019))andamicroscropicdescriptionofthestatolithsdynamics,Chauvetetal.(2019) identi-fiedfourtime-scalesregulatingthegraviresponse.Inthisunifiedframework,thegravity-independentsine lawisrecovered when the stimulation is long enough compared to the statolith avalanche duration and the memory time, while

dose-Fig. 2. (a) A sketch of the one-degree-of-freedom gravitropic spring-pendulum system discussed in the manuscript. (b) Real (solid lines) and imaginary (dashed lines) part of the roots of the characteristic equation (11) as functions of y ∈ [0, 1] for η= 1 and σ= 0 . 008 . We distinguish three regions cor- responding to three distinct behaviours of the fixed point ˆ θ≡ 0 : (i) a stable node (light blue), (ii) a stable spiral (orange), and (iii) an unstable spiral converging to a stable limit cycle (green). (c) Numerical solutions of the nonlinear discrete delay differential Eq. (6) at increasing values of the parameter y and for ˆ θ( ˆ t) = 0 . 01 for ˆ t ≤ 0 . Specifically, the angular coordinate ˆ θis reported as a function of dimensionless time ˆ t for y = 0 . 9 (light blue curve), y = 0 . 99 (orange curve), and y = 0 . 995 (green curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

responsesareobtainedeitherwhenthetimeofthestimulationisshorterthantheavalanchetimeorwhenitislongerthan thatbutshorterthanthememorytime.

Inthisscenario,elasticdeformationsduetogravity loadinghavebeenoftenoverlooked.Here,we havemadean effort to take them into account buildingon therecent advances in theunderstanding of gravitropicandautotropic responses (Chauvetetal.,2019;Dumais,2013;HamantandMoulia,2016)andinthemodelingoftheadditionaleffectsduetoelastic deflectionsthroughtheseparationofsensedandactuatedvariables(ChelakkotandMahadevan,2017).

2. Agravitropicspring-pendulumsystem

Webeginourstudyoncircumnutationsofgrowingshootsbyexploringthedynamicsofaprototypicalsystemconsisting ofan upward vertical pendulumsupported by a “gravitropic” torsionalspringwhich adaptsits restangle toreorient the penduluminthedirectionoppositetogravity.

Specifically,letusconsiderarigidbaroflength,hingedatthebottomandsupportedbyaspringoftorsionalstiffness

B>0,seeFig.2a.Thebarcarriesits weight,modelledbyaverticaldistributedloadofmagnitudeq,andisconfinedtothe plane{e1,e2}sothatitsconfigurationisdeterminedatanytimetbytheangle

θ

(t)withrespecttothevertical.

Equilibriumofthependulumundertheprescribeddistributedloadrequiresthat m

(

θ

,

θ

0

)

=

1 2 q

2_sin

_θ

_{, (1)}

wherem(

θ

,

θ

0)isthetorqueexertedbythespring.Asforitsconstitutivecharacterization,weconsidertheaffinelaw

m

(

θ

,

θ

0

)

= B

(

θ

−

θ

0

)

, (2)

where

θ

0 isthetime-dependentrestangleforwhichweassumethefollowingtimeevolutionlaw ˙

θ

0

(

t

)

= −

β

τ

g

τ

m  t−τr −∞ e −1 τm(t−τr−τ )sin

θ

(

τ

)

d

τ

, (3)

whereadotdenotesdifferentiationwithrespecttotime.Intheequationabove,

β

≥ 0isthedimensionlessgravitropic sen-sitivity,

τ

g>0 isthecharacteristictime forthe evolutionofthe restangle,

τ

r≥ 0is thegeotropicreaction timeor delay,

whereas

τ

m>0 is a parameter of the exponential weighting function defining the plant’smemory of the stimulus. The evolution law (3) wasinitially proposed by Israelsson and Johnsson(1967) to model the response of growing plants to gravi-stimulation.ThishasbeenrecentlyimprovedbyChauvetetal.(2019)toaccountfor thedynamicsof statoliths sedi-mentation,asdiscussedinthemanuscript.

Forthespecialcaseof

β

=0thetimeevolutionoftherestangleisinhibited.Itfollowsthat

θ

0(t)≡ 0,suchthatthe ver-ticalconfigurationisanequilibriumwhichlosesstabilityasthelengthofthependulumexceedsacriticalvaluec(buckling criticallength).Thiscaneasilybecomputedthroughastabilityanalysisofthespring-pendulumsystem,leadingto

c:=



2 B

q . (4)

Ourgoalis to considerthe more generalcaseof

β

=0and toexplore the behaviour ofthe spring-pendulum system asits length increases.We anticipatethatouranalysiswill revealthat thetrivialequilibrium

(

θ

_,

θ

0

)

=

(

0,0

)

undergoes a supercriticalHopfbifurcationatapendulumlength oftheorderofc.Tothataim,weproceedby takingthederivativeof thebalanceEq.(1).Substitutionof(3)leadsto

˙

θ

(

t

)



1 −q2 2 B cos

θ

(

t

)



= −

β

τ

g

τ

m  t−τr −∞ e −1 τm(t−τr−τ )sin

θ

(

τ

)

d

τ

, (5)

suchthatbyafurtherdifferentiationwithrespecttotimewearriveat



1 −q2 2 B cos

θ

(

t

)



¨

θ

(

t

)

+

_τ

1 m ˙

θ

(

t

)



+ q2 2 B

θ

˙ 2

₍

_t

₎

_sin

_θ

₍

_t

₎

₊

β

τ

g

τ

msin

θ

(

t −

τ

r

)

= 0 . (6) Wenoticethat,intheabsenceofexternalloading(q=0),(6)reducestotheso-called“sunflower” equation.Thismodel wasproposedbyIsraelssonandJohnsson(1967)tointerpretthegeotropiccircumnutationsoftheapicalregionofplants,and hasalreadybeenprovedtoadmitperiodicsolutions(Somolinos,1978)foraspecificrangeofparameters.However,growing shootsoftenappearaselongated,biologicalstructuresofsignificantweightrelativetotheirstiffness, afactthatcannotbe disregarded.Thisis evidentfromthe parameterq_2_{/(2B) intheequation above, measuring therelative magnitudeofthe} plant’sweighttostiffness.

To explore the effect of elastic deformations of the pendulum due to gravity loading, we proceed by linearizing Eq.(6)about

θ

₌0toderivethefollowingsecondorderdiscretedelaydifferentialequation



1 −q2 2 B



¨

θ

(

t

)

+

_τ

1 m ˙

θ

(

t

)



+

_τ

β

g

τ

m

θ

(

t −

τ

r

)

= 0 , (7)

which,assuming

τ

r>0,canberestatedindimensionlessformuponintroductionofthreedimensionlessparameters,i.e., y := q2 2 B,

η

:=

τ

r

τ

m,

σ

:=

β τ

2 r

τ

g

τ

m, (8) leadingto

(

1 − y

)



θ

¨ˆ

(

tˆ

)

+

η

_θ

_ˆ ˙

₍

_tˆ

₎



₊

_σ

_θ

_ˆ

₍

_{tˆ − 1}

₎

_{= 0 , (9)}

where

θ

ˆ

(

tˆ

)

:=

θ

(

tˆ

τ

r

)

andasuperimposeddotdenotesnowdifferentiationwithrespecttothedimensionlesstimetˆ:=t/

τ

r. Here,weprefertoomitthedetailsofouranalysistofocusinsteadonthefollowing,importantresult.Infact,application ofthetheoremsreportedinAppendixAleadstotheconclusionthat(9)exhibitsasupercriticalHopfbifurcationat

y = y := 1 −

σ

η

sin

ξ

ξ

, (10)

where

ξ

isdefinedastheuniquerootof

ξ

=

η

cot

ξ

in(0,

π

/2).Noticethaty=

(

/c

)

2,suchthatweproposethefollowing physicalinterpretationoftheresultabove:inagrowingshootsubjecttogravity,circumnutationsmayariseasaconsequence of a Hopf bifurcation as the shoot’s length attains the critical value  _:=√_{y c}. We infer from Eq. (10) that y <1, so that  _{<c, i.e.,the delayed graviresponse triggers a Hopfbifurcation before thebuckling critical length is reached. This} conclusionisfurthersupportedbytheanalysisoftherootsofthecharacteristicequationrelativeto(9)

(

1 − y

)



ω

ˆ 2₊

_η

_ω

_ˆ

_eωˆ₊

_σ

_{= 0 , (11)}

obtainedby plugging in it therepresentation

θ

ˆ

(

tˆ

)

₌eωˆˆt _{forthe angular coordinate,where}

_ω

_{ˆ is a dimensionlesscircular} frequency.WereportinFig.2bthereal(solidlines)andtheimaginary(dashedlines)partoftherootsofthecharacteristic Eq.(11)asfunctionsoftheloadingparametery∈[0,1]for

η

=1and

σ

=0.008.Thesevaluesweredeterminedbyassuming agravitropicsensitivityof

β

₌0_.8_,

τg

₌ 1200 min,and

τr

₌

τm

₌ 12 min,seeChauvetetal.(2019),andcorrespondto a criticalloadingparameterofy ࣃ0.993.

Inthefigure,wedistinguishthreeregionscorrespondingtodistinctbehavioursofthefixedpoint

θ

ˆ≡ 0:(i)astablenode (lightblueregion,whererootsarerealandnegative),(ii)astablespiral(orangeregion,whererootsarecomplexconjugate

withnegativerealpart),and(iii)anunstablespiralconvergingtoastablelimitcycle(greenregion,whererootsarecomplex conjugate with positive real part) for y>y . The figure also reports numerical solutions of the nonlinear discrete delay differentialEq.(6)atincreasingvaluesoftheparameteryandfortheinitialconditionof

θ

ˆ

(

tˆ

)

=0.01fortˆ≤ 0.Specifically, Fig.2cshowstheangularcoordinate

θ

ˆasafunctionofdimensionlesstimetˆfory₌0_.9(lightbluecurve),y₌0_.99(orange curve),andy=0.995(greencurve).ThenumericalimplementationofthenonlinearEq.(6)wasachievedbyexploitingthe NDSolve functionality ofMathematica v11.3.0.0 andconfirmsthe onset of periodic solutions incorrespondence withthe theoreticalHopfbifurcationpoint.

We notice, in passing, that the distributed delayof (5) is not necessary to give rise to such a qualitative behaviour. Indeed,thefollowingfirstorderdifferentialequationwithdiscretedelay

˙

θ

(

t

)

(

1 − y cos

θ

(

t

)

)

= −

β

τ

g

sin

θ

(

t −

τ

r

)

(12)

hasthesamequalitativefeaturesandundergoesasupercriticalHopfbifurcationat y 0:= 1 −

2

π

σ

η

. (13)

Eq.(12)canberecoveredfromEq.(5)by letting

τ

m0(asthegravitropicmemorykernelin(5)tendstoaDiracdeltain thesenseofdistributions)and,consistently,y y ₀.

Inconcludingthissection,wefinditusefultostressthesignificanceofelasticdeformationsduetogravityindetermining theoscillatorybehavioursometimesexhibitedbygrowingshoots.Infact,byneglectingtheeffectofeitherexternalloading (q→0)orofelasticdeformations(B→∞),onewouldconstraintheparameterytonull,suchthatself-sustainedoscillations wouldbeimpossible,atleastforthechosenvaluesoftheparameters,whicharethemostrealisticonestocharacterizethe biologicalmachineryregulatingthegravitropicresponseofplantorgans.

3. Arodmodelforgrowingshoots

Inspiredby theprototypicalsystemstudiedinSection2,weintroduceaplanarrodmodelthataccountsforgrowthby cell expansion andforthe evolutionof spontaneouscurvature dueto gravitropismand proprioception.Forthe evolution lawsthat governtheactive response oftherodwe relyon recentdevelopmentsfromtheavailable literature. Finally,we deriveareducedmodelthatisappropriateforthestudyofcircumnutationsinyoungplantshoots.

3.1. Mechanics

Wemodeltheplantshootasanunshearableand(elastically)inextensiblerodconfinedtotheplanespannedby{e1,e2}. At eachtime t itscentreline r(s,t) isa planarC2 _{curveoflength (t) parametrizedby thearc-length s∈[0,(t)].Therod} undergoesanextensionalgrowthcharacterizedbyaprescribedtimeevolutionoftheactualstretch

λ

(

S0,t

)

=

∂

s

∂

S0

(

S0,t

)

, (14)

whereS0∈[0,(0)]isthereferencearc-length,ands(S0,t) denotesthechangeofcoordinateinC1 thatatanytimetmaps

S0intothecurrentarc-length.

Wedenoteby

θ

(s,t)inC2_{theanglebetweentheunitvectore}

2pointingintheverticaldirectionandthelocaltangent tothecurve,seeFig.3a.Then,theunittangenttotherodaxisis

r = sin

θ

e1+ cos

θ

e2, (15)

whereaprimedenotesdifferentiationwithrespecttos,suchthattheconfigurationoftherodcanbereconstructedfrom

θ

as

r

(

s,t

)

= r0 +  s

0

r

(

σ

,t

)

d

σ

, (16)

wherer0isthepositionofthebasalend.

Sincewe haveinmindprocesseswherethetime-scaleformechanicalequilibriumismuchshorterthananybiological time-scaleoftheplant,weimposestaticequilibriumatanytime,i.e.,

n + f = 0, m + r × n = 0, (17)

where nandm are the resultant contactforce andcontact couple,whereas fis thebody force per unit currentlength. Fromtheplanarityassumption,n=n1e1+n2e2andm=m3e3 and,fromthephysicsoftheproblem,weprescribef=−qe2 whereqdenotestheplant’sweightperunitcurrentlength.

Asfortheconstitutiveresponseoftherodtobending,wefollowtheapproachbyChelakkotandMahadevan(2017)and assumealinearEuler-Bernoullilawsuchthat

m3

(

s,t

)

= −B

(

s,t

)

Fig. 3. (a) A sketch of the planar rod model for the analysis of periodic oscillations in plant shoots. (b) Real (solid lines) and imaginary (dashed lines) part of the roots of the characteristic Eq. (37) as functions of  /  c ∈ [0, 1] and for the choice of parameters introduced in Section 4.1 . As for the case of the gravitropic spring-pendulum system, we distinguish three regions corresponding to different dynamical responses for ˆ θ( ˆ s , ˆ t) : (i) an exponential decay (light blue), (ii) a damped oscillation (orange), and (iii) an increasing oscillation (green) for  >  _{≈0.895  c . (c) Superimposition of deformed shapes from} the nonlinear rod model as computed for  = 0 . 91  c as the time spans half a period of the limit cycle. (d) Numerical solutions of the nonlinear problem (24) –(26) as obtained for  = 0 . 91  c . Specifically, the transverse displacement and the phase portraits related to the angle and position of the rod’s tip as functions of time are shown. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

whereBisthebendingstiffnessand

κ

isthespontaneouscurvaturewhichmodelsdifferentialgrowthinthecrosssection. Inthefollowing,wewillassume thebendingstiffnesstobeconstant.Morerefined modelsmayaccountforchangesinB

duetolignification.

3.2.Growthlawandtime-evolutionofthespontaneouscurvature

Inthepresentrodmodel,theactiveresponse oftheplantisdescribedbyappropriate evolutionlawsforthestretch

λ

andforthespontaneouscurvature

κ

.Asforthestretch,we assumethatcellsextension occursonlyinthegrowthzoneof constantlength0:=(0),where,followingBastienetal.(2014),weprescribeaconstantelongationrate,i.e.,

˙

λ

λ

(

S0,t

)

=

⎧

⎨

⎩

0 if 0 ≤ s

(

S0,t

)

<

(

t

)

−  0, 1

τ

g if s

(

S0,t

)

≥ 

(

t

)

− 0, (19)

where

τ

g isthe characteristictime ofgrowth induced bycell elongationanda superimposed dotdenotes differentiation withrespecttotime.

TwomainingredientshavebeenidentifiedbyBastienetal.(2014)tomodeltropicmovementsofplants:(i)graviception, thataimstoreorientthestemalongthevertical,and(ii)proprioception,whichtendstoreducethecurvatureoftheorgan. Bothmechanismsdeterminethe timeevolution ofthespontaneous curvature,suchthat, inthegrowthzone, its material time-derivativereads

d

d t

κ

(

s

(

S0,t

)

,t

)

= G

(

s

(

S0,t

)

,t

)

+ P

(

s

(

S0,t

)

,t

)

, (20) whereG(s,t)andP(s,t)arethegravitropicandproprioceptionsignals.

Fromabiologicalstandpoint,theresponseofthegrowthzonetogravi-stimulationistheresultofseveralprocessesatthe cellularlevelwhicharenotyetcompletelyunderstood.Inparticular,differentialgrowthacrosstheplantsectionisattributed toan auxin asymmetrybetween theupperandthe lower partofthe organ, andauxinredistribution istriggered by the sedimentationofstatolithsinspecializedcellscalledstatocytes.FollowingChauvetetal.(2019),wemodelthisdynamicsas

aviscousrelaxationgovernedbytheequation d d t

θ

s

₍

_s

₍

_S 0,t

)

,t

)

= 1

τ

a sin

(

θ

s

₍

_s

₍

_S 0,t

)

,t

)

−

θ

(

s

(

S0,t

)

,t

)

)

, (21)

where

τ

a is thecharacteristic duration ofthe statoliths avalanchesand

θ

s(s, t) is the localorientation of statoliths with respecttothevertical.Moreover,thesameauthorsincludedinthemodeltermsaccountingfordelayandmemoryprocesses thatcharacterizethegravitropicsignallingpathway.Tomodelsuchprocesses,theyproposedthefollowingintegrallaw

G

(

s,t

)

= −

β

R

τ

m

τ

g  t−τr −∞ e −1 τm(t−τr−τ )sin

θ

s

(

s,

τ

)

d

τ

, (22)

where

τ

m isthememorytime,

τ

risthe reactiontime ordelay,

β

isadimensionlessgravitropicsensitivity,whereas Ris theradiusoftheshoot’scrosssection.

Inaddition to thegravitropicresponse,it hasbeen shownthat,after afirst phasewhere theorgan respondsto envi-ronmentalstimuli,anindependentmechanismtakesplacethathampersbendingandfavoursastraightposture.The micro-scopicfunctioningofthisphenomenon,calledautotropismorautostraightening,ismostlyunknown(Okamotoetal.,2015). However,wemodelitasproprioceptionfollowingBastienetal.(2013),suchthat

P

(

s,t

)

= −

γ

τ

g

θ

₍

_s,t

₎

_{, (23)}

where

γ

isadimensionlessproprioceptionsensitivity.

3.3. Areducedmodelforthestudyofcircumnutations:Theregime

τ

a

τ

c

τ

g

Asdiscussedintheprevioussection,fourdifferenttime-scalescontroltheevolutionofgrowth-drivenstretchand spon-taneous curvature,i.e.,

τ

g,

τ

a,

τ

mand

τ

r.Chauvetetal.(2019) determinedtheorderofmagnitudesofthesecharacteristic timesforwheat(Triticumaestivum)coleoptilesasfollows:

τ

aࣃ2min,

τ

mࣃ

τ

rࣃ12min,and

τ

gࣃ1200min.

Sincetheperiod

τ

cofmostcircumnutationaloscillationsisintherangeof20− 300min(Brown,1993),wecanassume thatthestatolithavalanchedynamicsismuchfastercomparedtocircumnutationswhich,inturn,aremuchfasterthanthe dynamicsofcellelongation. Ontheonehand,theassumptionof

τ

g

τ

c impliesthatwecan neglectchangesinlength  andbending stiffnessB.Indeed,at shorttimescompared to

τ

g,the organ length approximatelycoincides with(0),and there is no distinction betweencurrent andreference arc-lengths. Onthe other hand,the condition

τ

a

τ

c yields

θ

s(s,

t)≈

θ

(s,t).Moreover,we neglecttheproprioceptivetermintheevolutionlaw(20)sincecurvaturesarerelativelysmall,as confirmedbytheresultspresentedinthefollowing.Basedontheseassumptions,wearriveatthereducedmodel

B

θ

(

s,t

)

−

κ

₍

_s,t

₎



_{= −q}

₍

_{− s}

₎

_sin

_θ

₍

_s,t

₎

_,

_κ

_˙

₍

_s,t

₎

_{= −}

β

R

τ

m

τ

g  t−τr −∞ e −1 τm(t−τr−τ )sin

θ

(

s,

τ

)

d

τ

(24) inthevariables(

θ

,

κ

)fors∈(0,) andt>0.Specifically,(24)1 wasobtainedby firstsolving forthebalanceofforces(17)1 in the absenceof apical loads andthen plugging the expression for thecontact force in (17)2, while accountingforthe constitutivelawof(18).Asfor(24)2,ittriviallyfollowsfrom(20)withP=0andwiththegravitropicsignalgivenby(22), where

θ

replaces

θ

s_{.ThesystemofEq.(24)issupplementedbythefollowingboundaryandinitialconditions,namely}

θ

(

0 ,t

)

= 0 ,

θ

₍

_,t

₎

₋

_κ

₍

_,t

₎

_{= 0 , (25)}

holding

∀

t>0asthebasalendisclampedandtheapicalendistorquefree,and

θ

(

s,t

)

=

θ

0

(

s,t

)

,

κ

(

s,0

)

=

κ

0

(

s

)

, (26)

prescribing respectively the past history of the angular coordinate and the initial datum for the spontaneous curvature evolution

∀

s∈[0,].

Assumingsufficientregularity,wecancombinethetime-derivativeof(24)1withthespace-derivativeof(24)2,sothatby afurtherdifferentiationwithrespecttotimewearriveat

¨

θ

₊ 1

τ

m ˙

θ

₊ q

(

− s

)

B





¨

θ

+

_τ

1 m ˙

θ



cos

θ

− ˙

θ

2_sin

θ



₊

β

R

τ

m

τ

g

θ

₍

_{s,t −}

_τ

_r

₎

_cos

_θ

₍

_{s,t −}

_τ

_r

₎

_{= 0 , (27)} alongwiththeboundarycondition(25)1and

¨

θ

₍

_,t

₎

₊ 1

τ

m ˙

θ

₍

_,t

₎

₊

β

R

τ

m

τ

g sin

θ

(

,t −

τ

r

)

= 0 , (28)

holding

∀

t>0andresultingfromtimedifferentiationof(25)2.

4. Analysisofcircumnutationsinplantshoots

Guidedbythestudyperformedonthegravitropicspring-pendulumsystem,weproceedbyfirstcarryingoutananalysis of thelinearized version of Eq.(27).Such an analysisreveals theexistence of periodic solutions that are reminiscent of circumnutations.Next,weintroduceacomputationalmodelforthenonlineargoverningequationsthatallowstoexplorethe dynamicsofmodelshootsintheoscillatoryregime,providingevidenceofalimitcycleassociatedwithaHopfbifurcation.

4.1. Linearizationaboutthestraightequilibriumconfiguration

Bylinearizingproblem(27)and(28)abouttheequilibriumsolution

θ

(s,t)≡ 0,wegetthefollowingfourthorderpartial differentialequationwithdiscretedelay

¨

θ

₍

_s,t

₎

₊ 1

τ

m ˙

θ

₍

_s,t

₎

₊ q

(

− s

)

B



¨

θ

(

s,t

)

+ 1

τ

m ˙

θ

(

s,t

)



+

β

R

τ

m

τ

g

θ

₍

_s,t−

_τ

_r

₎

_{= 0 , (29)}

supplementedbytheboundarycondition(25)1and ¨

θ

₍

_,t

₎

₊ 1

τ

m ˙

θ

₍

_,t

₎

₊

β

R

τ

m

τ

g

θ

(

, t −

τ

r

)

= 0 , (30)

holding

∀

t>0andresultingfromthelinearizationof(28).Uponintroducingfourdimensionlessparameters,i.e., y := q3 B ,

η

:=

τ

r

τ

m,

σ

:=

β τ

2 r

τ

g

τ

m,

μ

:=  R, (31)

Eq.(29)canberecastindimensionlessformas ¨ˆ

θ



_{sˆ ,tˆ}

₊

_η

_θ

_ˆ ˙



_{sˆ ,tˆ}

_{+ y}



_{1 − ˆ s}



_θ

¨ˆ



_{sˆ ,}tˆ

+

η

_θ

ˆ ˙



_{sˆ ,}tˆ



+

σμ

_θ

ˆ



_{sˆ ,tˆ − 1}

_{= 0 . (32)} Here,

θ

ˆ

(

sˆ,tˆ

)

:=

θ

(

sˆl,tˆ

τ

r

)

andprimesanddots denotedifferentiationwithrespecttotˆ:=t/

τ

r andsˆ:=s/,respectively. As fortheboundaryconditions,wewrite

ˆ

θ

(

0 ,tˆ

)

= 0 ,

θ

¨ˆ

₍

_{1 ,tˆ}

₎

₊

_η

_θ

_ˆ ˙

₍

_{1 ,tˆ}

₎

₊

_σμ

_θ

ˆ

₍

_{1 ,}tˆ − 1

)

= 0 , (33) holding

∀

tˆ>0,whereastheinitialconditionis

ˆ

θ

(

sˆ ,tˆ

)

=

_θ

ˆ ₀

₍

_{sˆ ,}tˆ

)

, (34)

whichapplies

∀

sˆ∈[0,1],

∀

tˆ∈[−1,0].

We proceed with ouranalysis in the linear regime by seeking time-harmonic solutions ofthe form

θ

ˆ

(

sˆ,tˆ

)

=

ϕ

(

sˆ

)

eωˆtˆ to(32)–(34).Bysubstitutingtheformabovein(32)weobtain

ϕ

₍

_sˆ

₎

_{+ c}

_ϕ

₍

_sˆ

₎

_{+ y}



_{1 − ˆ s}

_ϕ

₍

_sˆ

₎

_{= 0 , (35)}

wherec:=

σμ

e− ˆω/

(

ω

ˆ2₊

_η

_ω

_ˆ

₎

_{.Integrationof(35)leadsto}

ϕ

(

sˆ

)

= e−csˆ/2

_c

1Ai

(

x

(

sˆ

))

+ c2Bi

(

x

(

sˆ

))



, (36)

inwhichc1andc2areconstantsofintegration,whereasAi

(

x

)

andBi

(

x

)

aretheAiryfunctionsofthefirstandsecondkind, respectively,andx

(

sˆ

)

:=[c2_{/4− y}

₍

_{1− ˆs}

₎

_]/y2/3_{.Byimposingtheboundaryconditions(33),andneglectingthetrivialcaseof}

c1=c2=0,wederive Ai

(

x0

)



cAi

(

x1

)

+ 2 √ 3yBi

(

x1

)

− Bi

(

x0

)



cAi

(

x1

)

+ 2 √ 3yAi

(

x1

)

= 0 , (37)

wherex0:=x(0),x1:=x(1),andaprimedenotesdifferentiationoftheAiryfunctionswithrespecttotheirargument. WenumericallycomputedtherootsofthecharacteristicEq.(37) toexplorethestabilityofmodelshootsofincreasing length.Tothisaim,weexploitedthe FindRootfunctionalityofMathematicav11.3.0.0.InagreementwithChelakkotand Mahadevan(2017)andChauvetetal.(2019),wecalibratedthemodelbysetting

β

=0.8,

τ

g=1200min,

τ

m=

τ

r=12min,

R=0.5 mm, E=107 _{Pa forthe Young’s modulus, andq=}

ρ

_{gA where}

ρ

₌₁₀3 _Kgm−3 _{is the mass density, A the} cross-sectional area,and gthe gravitationalacceleration.For such a choiceof modelparameters, we determined valuesof the circularfrequency

ω

ˆ lettingrangein[0,c].Here,cdenotesthecriticallengthatwhichanelasticrodofbendingstiffness

Bsubjecttoadistributed,verticalloadofmagnitudeqlosesstability,thatis c:= 3



α

B/q, (38)

with

α

≈ 7.837,seeGreenhill(1881).

Wereport in Fig.3b thereal (solidlines) andthe imaginary(dashed lines) partof tworoots of(37).As forthe case ofthegravitropicpendulum,wedistinguishinthe figurethreeregions correspondingto differentdynamicalresponses of

ˆ

θ

(

sˆ,tˆ

)

: (i) an exponential decay (lightblue region, where rootsare realand negative),(ii) a damped oscillation (orange region,whererootsarecomplexconjugatewithnegativerealpart),and(iii)an increasingoscillation(greenregion,where rootsarecomplexconjugatewithpositiverealpart)for> _{≈ 0.895c.}

Ouranalysisisrestrictedtosolutionsoftheformintroducedabove,wherespatialandtemporalvariablesareseparated. However,thepresentedresultsclearlyindicatethattherodmodelsuffersan instabilityinthelinearregimeastheshoot’s lengthexceedsthecriticalvalue _{.Thisbehavioursharessimilaritieswiththatexhibitedbythegravitropicspring-pendulum} systemof the previous section. We shall next explore the dynamical response of the active rodmodel in the nonlinear regimebymeansofnumericalcomputations.

4.2. Numericalimplementationoftherodmodel

In thissection, we introduce a numericalimplementation ofthe nonlinearproblem (24)–(26) inorder toexplore the regimeoflargeamplitudeoscillationsexhibitedbymodelshoots.Specifically,wesolveEq.(24)alongwithboundary condi-tions(25)andinitialconditions(26)byusingthefiniteelementmethod.

Asafirststeptowardsthenumericalformulation,itisconvenienttointroducetheauxiliaryfieldw(s,t)representingthe delayintegral,i.e.,

w

(

s,t

)

:=  t−τr

−∞ e −1

τm(t−τr−τ )sin

θ

(

s,

τ

)

d

τ

, (39)

sothatsuchintegralmaybecomputedfromthesolutionofthedifferentialequation ˙

w

(

s,t

)

= −1

τ

m

w

(

s,t

)

+ sin

θ

(

s,t −

τ

r

)

. (40)

Withthedefinition(39),theEq.(24)2 fortheevolutionofthespontaneouscurvaturereads ˙

κ

(

s,t

)

= −

β

R

τ

m

τ

g

w

(

s,t

)

. (41)

Toobtaintheweakformulationofthegoverningequations,wemultiply(24)1,(40),and(41)bythetestfunctions

θ

˜,w˜, and

κ

˜,respectively,suchthatanintegrationinspacealongtheinterval[0,]leadsto

 _ 0



B



θ

(

s,t

)

−

κ

(

s,t

)

θ

˜

(

s,t

)

+ q

(

− s

)

sin

θ

(

s,t

)

θ

˜

(

s,t

)

d s= 0 , (42a)  _ 0



˙ w

(

s,t

)

+

_τ

1 mw

(

s,t

)

+ sin

θ

(

s,t −

τ

r

)



˜ w

(

s,t

)

d s= 0 , (42b)  _ 0



˙

κ

(

s,t

)

+_R

_τ

β

m

τ

g w

(

s,t

)



˜

κ

(

s,t

)

d s= 0 . (42c)

Inperforming theintegrationby partsofthetermsinvolvingcurvatures in(24)1 wehaveaccountedfortheboundary condition(25)2andfortheessentialcondition

θ

˜

(

0,t

)

=0.Followingstandardfiniteelementprocedures,boththeunknowns

θ

,w,

κ

andthecorrespondingtestfieldsarediscretizedinspaceusinglinearLagrangeshapefunctions,whileforthetime discretization we choose the second-order generalized alpha method. The weak form equations were implemented and solvedinCOMSOLMultiphysicsv5.4usingtheWeakFormPDEmode.

Here,wefocusournumericalstudyontheunstableregimepredictedbythelinearanalysisoftheprevioussection,see thegreenregionofFig.3b.Inparticular,wechoosearodoflength_¯:₌0_.91_cthatisabovethethreshold .Astheinitial datum,weset

θ

0

(

s,t

)

= aRe

(

eω¯t

)

ϕ

(

s/¯

)

, (43)

holding

∀

t≤ 0.In theequation, a isthe amplitude,

ω

¯ isthedimensional (complex)eigenvalue thatsolves (37) for=_¯_, and

ϕ

(

s/_¯

₎

isthecorrespondingeigenfunctionasgivenby(36).Consequently,theinitialconditionforthedelayintegral w

canbeestimatedfrom(39)underthehypothesisthata_{ 1,}sothatsin

θ

0(s,t)≈

θ

0(s,t).Toinitializethesimulationwitha zero-stressconfiguration,themagnitudeofthedistributedloadisrampedfromzerotoqoveratimeintervalmuchsmaller than thefastestcharacteristictime scale ofthe system,i.e.,

τ

r, whilethe initialspontaneous curvature

κ

(s,0) equalsthe visiblecurvature

θ

(s,0)with

θ

(

s_,0

)

₌

θ

0

(

s,0

)

=a

ϕ

(

s/¯

)

.

ConsistentlywiththetheoreticalresultsexposedinSection4.1,ourcomputationalstudyrevealstheonsetofalimitcycle astherod’slengthexceeds _{(≈ 7.1cm,forthechosenparameters).Inparticular,wereportinFig.3cseveralconfigurations} oftherodatdifferenttimes,clearlyshowingasymmetricoscillationwithrespecttotheverticallineasthetimespanshalf aperiodofthelimitcycle(≈ 88min,forafullcycle).Inaddition,Fig.3ddepictsthetransversedisplacementandthephase portraitsrelatedtotheangleandpositionofthetip asfunctionsoftime. Theseshow thesignatureofthelimitcycleand provideaquantitativedescriptionofthedynamicsofthesystemduringitsevolutiontowardsthesteady,oscillatoryregime.

5. Conclusionsandoutlook

Inspiredby thefascinatingphenomenon ofcircumnutations,we investigatedtheeffectofelasticdeformationsinduced bygravity loadingontheactive responseofplantshoots, anaspectthat hasbeensofardisregarded.Tothisaim,wefirst introduced asimple prototypicalmodel,the so-calledgravitropicspring-pendulumsystem. Thiswasshowntobe capable ofcapturingthemainfeatures ofplantsresponsetogravity. Thesimplicityofthemodelallowedustoperformarigorous mathematicalanalysistoprovethataHopfbifurcationoccursforacriticallengthofthependulum.

Toaccount forthe complex natureof theinterplay betweenelasticityandgrowthin plantshoots, we then derived a planarrodmodelbuiltuponrecentadvancesintheunderstanding ofgravitropism. Forthismodel,ina linearizedsetting, weprovedtheexistenceofoscillatoryanddivergingsolutionsaboveacriticallengthoftherod.

Finiteelementsimulationsallowedustoextendtheanalysisofthedynamicsofmodelshootsintothenonlinearregime, withcomputational results confirming our theoretical findings. For a choice of material parameters consistent with the availableliteratureonplantshoots,wefoundthatrodsofsufficientlengthmayexhibitlateraloscillationsofincreasing am-plitude,whicheventuallyconvergetolimitcycles.ThisbehaviorstronglysuggeststheoccurrenceofaHopfbifurcation,just asforthegravitropicspring-pendulumsystem,andiscloselyreminiscentoftheperiodicmovementsreportedforelongating plantorgans.

The presentstudysupports the ideathat plantnutations might be theresultof a posturefeedback control based on delayedgraviresponsesofaslenderstructure approachingthebucklingcriticallength.Thisdoesnotexcludethat,inother circumstances such asin the case of climbing plants reaching forsupport, or in the coiling of tendrils, periodic helical motionmaybetheoutcomeofendogenousbiologicalprocesses,leadingtoevolutionaryadvantages.Inthecaseweconsider, however,nutationsmaybe simply aby-product ofgravitropism-the attemptofplantstokeep a verticalposture-andof deflections due to gravity loading. Here gravity plays two distinct roles that should not be confused. On the one hand, gravitycausesthedynamicsofstatolithsandhencetriggerstheplantgravitropicresponse;ontheotherhand,thepresence ofgravityimpliesa loadthatthe plantcarries,withthecorresponding elasticdeflections, andthisstudyshowsthat this effectisnotnegligible.Suchadistinctionmustbekeptinmindwhenalteringtheeffectofgravitybymeansofexperiments inSpaceoron Earth.Interestingly,themain featuresofourself-sustainedoscillations dependonseveraldistinct physical parameters,andthismightexplainthediversityofnutationsobservedinnature,eventhougheachcaseshouldbecarefully examinedtoruleoutdifferentbiologicalmechanismsthatourmodelhasnotyettakenintoaccount.Ageneralconclusion emerging fromour analysis, that is worth emphasizing,is that the energy neededto sustain spontaneous oscillations is continuouslysuppliedtothesystembytheinternalbiochemicalmachinerypresidingthecapabilityofplantstomaintaina verticalpose.

We are awareof the inherentlimitations of theproposed mechanicalmodels, which are simplistic inmany respects. Nevertheless,ourquantitativeresultsseemtobeingoodagreementwithobservationsoncircumnutationsinthesensethat both thecritical length andthe periodof oscillations havethe expectedorder ofmagnitude. Future workwill include a quantitativeassessmentoftheaccuracyofthetheoreticalpredictionsincomparisonwithexperimentalobservations. More-over,thenonlinearrodmodelwillbeextendedtoaccountforafullthree-dimensionaldescriptionofthekinematicsofplant nutationsasrecentlydevelopedbyBastienandMeroz(2016).

Besidetheirrelevanceinabiologicalcontext,studiesoncircumnutationalmovementsinplantsareprovidinginspiration forinnovativedesigns inroboticapplicationsfromwhich,inturn,theybenefitgettingnewinsights.Forinstance,Del Dot-toreetal.(2018)haveshownthatsoilpenetrationstrategiesmimickingcircumnutationsofplantrootsmaybeadvantageous whencomparedtomorestandarddrillingtechniquesinthecontextofroboticsoilexplorationtasks.

Acknowledgments

ADS,DA,ALandGNacknowledgethesupportoftheEuropeanResearchCouncil(AdG-340685-MicroMotility).Partofthis researchwasconductedwhileADSwasvisitingtheUniversityofCambridge,asaVisitingScholaratPembrokeCollege,and asavisitortotheNewtonInstituteofMathematicalSciences.TheauthorswouldliketocongratulateProf.DavideBigonion theoccasionofhis60thbirthday.

AppendixA. Hopfbifurcation:Existenceofperiodicsolutions

Inthissection,weprovethesupercriticalHopfbifurcationresultforthegravitropicspring-pendulumsystem(6)stated inSection2.Tothisend,wefirstrecallsomemathematicalnotionsandresults.

DefinitionA.1 (RFDE orDDE). LetC:=C

(

[−d,0],Rn

₎

_{for d∈[0,∞) anddenotethe normofan element}

_φ

_{inC by}

_|

_φ|

_:= sup_δ∈[−d,0]

|

φ

(

δ

)

|

.Ifx∈C

(

[t0− d,t1],Rn

)

fort1>t0inR,thenforanyt∈[t0,t1],wedefinext∈C by

xt

(

δ

)

:= x

(

t+

δ

)

∀

δ

∈ [ −d, 0] .

GivenD⊂ R× C andf:D→Rn_{,wesaythattherelation} ˙

x

(

t

)

= f

(

t,xt

)

(A.1)

is a Retarded Functional Differential Equation (RFDE), or a Delay Differential Equation (DDE), on D. A function x_∈ C

(

[t0− d,t1],Rn

)

is said to be a solution of (A.1) with initial value

φ

∈C at t0 if (t, xt)∈D for all t∈[t0, t1], xt₀≡

φ

and

x(t)hasacontinuousderivativeon(t0,t1),arighthandderivativeatt0andsatisfies(A.1)on[t0− d,t1

)

.Suchasolutionwill bedenotedbyx(t;

φ

).Moreover,wesaythatEq.(A.1)is

(i) lineariff

(

t,

φ

)

=L

(

t

)

φ

+h

(

t

)

,whereL(t)islinear; (ii) autonomousiff

(

t_,

φ

)

₌g

(

φ

)

wheregdoesnotdependont.

DefinitionA.2(Stabilityofequilibria). Letx beanequilibriumpointofx˙

(

t

)

=f

(

xt

)

,i.e.,f(x )≡ 0.Then,thepointx issaid tobe

(ii) unstableifitisnotstable;

(iii) asymptoticallystableifitisstableandthereisb>0suchthat|

φ

|<bimpliesthat|x(t;

φ

)|→x ast→∞;

(iv) alocalattractorifthereisaneighborhoodUofx suchthatlimt→∞dist

(

x

(

t;U

)

, x

)

=0,i.e.,x attractselementsin

Uuniformly.

DefinitionA.3(Characteristicequation). LetL:C→Rn_{beacontinuouslinearfunctional.Wedefinethecharacteristic} equa-tionofthelinearretardedequationx˙

(

t

)

=L

(

xt

)

as

det

(

ω

I − L ω

)

= 0 , (A.2)

whereL_ω:=[L

(

exp_ωe1

)

|

...

|

L

(

expωen

)

].Here,expω(

δ

):=eωδ and{ei}iisthecanonicalbasisforRn.

ManywellknownresultsforODEscanbe showntobe stillvalidforRFDEs.Inparticular,ifthecharacteristicequation ofalinearsystemhasarootwithpositiverealpart,thentheoriginisunstable.Forasymptoticstability,itisnecessaryand sufficientto haveeachroot

ω

withnegativerealpart.IftheRFDE isnonlinearand0isan equilibriumwithall the char-acteristicroots(resp.acharacteristicroot)ofthelinearizationat0,havingnegative(resp.positive)realpart,thenclassical approachescanbeusedtoshowthat0isasymptoticallystable(resp.unstable)forthenonlinearequation.Alsothestandard HopfbifurcationtheoremisvalidforRFDEsinthefollowingform.

TheoremA.1(Hopfbifurcation). Consideraone-parameterfamilyofautonomousRFDEsoftheform ˙

x

(

t

)

= F

(

μ

,xt

)

, (A.3)

whereF∈C2

₍

_{R× C,R}n

₎

_{suchthat0isanequilibriumpointof(A.3)forall}

_μ

_{.DefineL:R× C→R}n _by

L

(

μ

)

φ

= D_φF

(

μ

,0

)

φ

, (A.4)

whereD_φF(

μ

,0)isthederivativeofF(

μ

,

φ

)withrespectto

φ

at

φ

=0.Assumethat:

(i) thelinearequationx˙

(

t

)

=L

(

0

)

xt hasapairofsimpleimaginarycharacteristicroots

ω

₀±=±iβ0=0andallother

charac-teristicroots

ω

_j=m

ω

₀+foranym_∈Z.

Thenthereisa

μ

0>0andasimplecharacteristicroot

ω

(

μ

)

=

α

(

μ

)

+i

β

(

μ

)

ofequationx˙

(

t

)

=L

(

μ

)

xt s.t.

ω

(

0

)

=

ω

+₀ and

for|

μ

|<

μ

0 itiscontinuouslydifferentiable.Supposethat:

(ii) Re

(

ω

(

0

))

=

α

(

0

)

=0,whereprimedenotesdifferentiationwithrespectto

μ

.

Then,for

μ

closetozero,(A.3)hasnontrivialperiodicsolutions,withperiodcloseto2

π

/

β

0.

Thelinearizationof(6)inthemaintextisasecondorderlinearautonomousRFDEwithdiscetedelayoftheform

¨y

(

t

)

+ ay˙

(

t

)

+ by

(

t − 1

)

= 0 , (A.5)

whichcanberestatedinsystemformas



˙ x1

(

t

)

= x2

(

t

)

˙ x2

(

t

)

= −ax 2

(

t

)

− bx 1

(

t− 1

)

i.e., ˙ x

(

t

)

=



0 1 0 −a



xt

(

0

)

+



0 0 −b 0



xt

(

−1

)

=: L

(

a

)

xt. Sincethecharacteristicequationof(A.5)isequivalentto

(

ω

)

:=



ω

2_{+ a}

_ω

_eω_{+ b= 0 ,}

we are interestedindetermining thebehaviour of itsroots interms ofthe parametera.Tothispurpose,we restate the followingresultshownbySomolinos(1978).

LemmaA.1. Considertheequation



ω

2_{+ a}

_ω

_eω_{+ b = 0 (A.6)}

forb>0andlet

ξ

bbe theuniquesolutionof

ξ

2=bcos

(

ξ

)

in (0,

π

/2)andletab:=sin(

ξ

b)b/

ξb

.Thenthefollowingholdsfor

equation(A.6):

(i) allrootshavenegativerealpartsifandonlyifa_>a_b;

(ii) fora=a_b, ± i

ξb

istheonlypairofsimpleimaginaryroots.Inparticular,nootherrootisanintegermultipleofi

ξb

;

(iii) thereisan



>0andaroot

ω

(a)thatiscontinuouslydifferentiablein

(

ab−



,ab+



)

s.t.

ω

(

ab

)

=i

ξb

andRe

(

ω

(

ab

))

< 0; (iv) foreacha<ab,therearepreciselytworoots

ω

withRe

(

ω

)

>0andIm

(

ω

)

∈

(

−π,

π

)

.