Helices, spirals and phyllotaxis

MAE 545: Lecture 21 (12/8)

Helices, spirals and phyllotaxis

2

Shaping of gel membrane properties by halftone lithography

of Ω(r) at each lattice point according to Eqs. 3and 4, determining the corresponding value offlow from the fit of Eq. 2 to the data in Fig. 1H,and finally setting the size of the dot at thatlattice point according to Eq. 1. Because thepower-law metrics in Eq. 3 diverge or vanish atthe origin, it is necessary to cut out a small re-gion around the center of each of the two cones.

The shapes adopted by the corresponding gelsheets (Fig. 2, A to D) are measured by laser scan-ning confocal fluorescence microscopy (LSCM)and analyzed as described in the SOM. Each ofthe four surfaces shows only small deviationsabout an average Gaussian curvature, with theexception of the regions near the free edges,where our analysis yields artifactual curvatures(due to the finite thickness of the gel sheets, thesurface meshing procedure used yields addition-al points on the edges that do not accuratelyreflect the 2D geometries of the sheets). Afterexcluding regions of the surface within 2h of theedges to avoid these artifacts, we find the aver-age Gaussian curvatures of the spherical cap andsaddle to be 6.2 mm−2 and –20.6 mm−2, respec-tively, with nearly axisymmetric distributionsof curvature (fig. S2A). Both values are in rea-sonable agreement with the target values, al-though the tendency of disks with uniform dotsizes to show slight curvatures (with radii of 2mm) suggests the presence of slight through-thickness variations in swelling (see SOM fordetails) that may contribute to the observed de-viations from the programmed curvature. Inter-estingly, we do not observe a boundary layerwith negative Gaussian curvature around theedge of the spherical cap as has been reported

for truly smooth metrics (17, 18), possibly re-flecting the influence of the through-thicknessvariations in swelling. For both cones, the av-erage Gaussian curvatures, excluding regions atthe free edges, are close to zero. Further, Fig. 2Eshows a plot of the deficit angle d measured forfive different cone metrics with power law ex-ponents −1 ≤ b < 0, which agrees closely withthe programmed value d = −pb.

We next consider metrics of the form

WðrÞ ¼ c½1þ ðr=RÞ2ðn−1Þ&2 ð5Þ

corresponding to Enneper’s minimal surfaceswith n nodes. These surfaces all have zero meancurvature and so are expected to minimize theelastic energy for these metrics at vanishingthickness (18). Although Eq. 5 is axisymmetric,Enneper's surfaces spontaneously break axialsymmetry by forming n wrinkles. In Fig. 2, Gto J, we demonstrate patterned surfaces with n =3 to 6, each of which reproduces the targetednumber of wrinkles. As shown in the maps ofcurvature in Fig. 2 (and azimuthally averagedplots in fig. S2B), each surface has small meancurvature and negative Gaussian curvature thatmatches closely with the target profile. For agiven film thickness, increasing n eventuallyleads to a saturation in the number of wrinkles,because the bending energy arising from Gaussiancurvature increases with n (for the films withh ≈ 7 mm in Fig. 4, a metric with n = 8 yieldedonly six wrinkles). However, given the subtledifferences between the metrics plotted in Fig.2F, the ability to accurately reproduce the pro-grammed number of wrinkles for n = 3 to 6 is a

strong testament to the fidelity of the metricspatterned by this technique.

The true power of our approach lies in thefabrication of nonaxisymmetric swelling pat-terns. As a simple demonstration, we first con-sider the problem of how to form a spherethrough growth. For the axisymmetric metricdescribed in Eq. 4, the maximum value of r/Rto which this metric can be experimentally pat-terned is restricted by the accessible range ofswelling. In our case, this range is Ωhigh/Ωlow ≈3.7, limiting the maximum portion of a spherethat can be obtained to slightly less than half.Although further improvements in the materialsystem are likely to increase the available range,the axisymmetric metric is inherently an ineffi-cient way to form a sphere, because as one seeksto go beyond a hemisphere and toward a closedshape, the required swelling contrast divergesrapidly. Given access to 2D metrics, however, anumber of well-established conformal mappingsof the sphere onto flat surfaces are known fromthe field of map projections. For example, thePeirce quincuncial projection (27) maps a sphereof radius R onto a square using the metric

Wðx; yÞ ¼ 2jdn xþiy

R j 1ffiffi2

p" #

sn xþiyR j 1ffiffi

2p

" #j2

1þ jcn xþiyR j 1ffiffi

2p

" #j2

h i2 ð6Þ

where sn, cn, and dn are Jacobi elliptic func-tions, and x and y are the components of r. Thismetric still has four cusp-like singularities whereΩ(r) = 0; however, one of its useful propertiesas a map projection is that only a small portion

Fig. 2. Halftoned diskswith axisymmetric met-rics. Patterned sheets pro-grammed to generate (A)a piece of saddle surface(Sa), (B) a cone with anexcess angle (Ce), (C) aspherical cap (Sp), and(D) a cone with a deficitangle (Cd). (Top) 3D re-constructed images ofswollen hydrogel sheetsand (bottom) top-viewsurface plots of Gaussiancurvature. Initial thick-nesses and disk diame-ters are 9 and 390 mm,respectively, althoughthe apparent thicknessof sheets is enlarged dueto the resolution of theLSCM. (E) Measured val-ues of deficit angle dfor cones with five dif-ferent exponents b (see Eq. 3) (black solid circles) and the programmedvalues (blue dashed line). (F) Swelling factors for the target metrics as afunction of normalized radial position on the unswelled disks r/R, with pointsplotted at values corresponding to lattice points to indicate the resolu-tion with which Ω is patterned. (G to J) Patterned sheets programmed to

generate Enneper’s minimal surfaces with n = (G) 3, (H) 4, (I) 5, and (J) 6wrinkles upon swelling as dictated by Eq. 5. 3D reconstructed images (top)and top-view surface plots of squared mean curvature H2 and Gaussiancurvature K (bottom). Initial thicknesses and disk diameters are 7 and 390 mm,respectively.

www.sciencemag.org SCIENCE VOL 335 9 MARCH 2012 1203

REPORTS










R j 1ffiffi2

p" #

sn xþiyR j 1ffiffi

2p

" #j2


2p

" #j2

h i2 ð6Þ





REPORTS

J. Kim et al., Science 335, 1201 (2012)










R j 1ffiffi2

p" #

sn xþiyR j 1ffiffi

2p

" #j2


2p

" #j2

h i2 ð6Þ





REPORTS

saddle (Sa) cone with deficit angle (Cd)

cone with excess angle (Ce)

sphericalcap (Sp)

K - Gauss curvature

H - meancurvature

Enneper’s minimal surfaces (H=0)

swelling profiles

3

Helices in plants

How are helices formed?

4

Differential growth or differential shrinking produces spontaneous curvature

faster growthof the top layer

L

L(1 + )

R R

W

L(1 + )

L=

R+W

R

Differential growth (shrinking) of the two layers produces spontaneous curvature

K =1

R=

W

rectangular cross-section. The two strips are then glued togetherside-by-side along their length. At this stage, the bi-strips are flatand the red strip is under a uniaxial pre-strain, defined asx~(LL’)=L’. Being elastomers, volume conservation requires

that the heights are related by h’~hffiffiffiffiffiffiffiffiffiffi1zxp

. Then, in the finaloperation, the force stretching the ends of the bi-strip is graduallyreleased, with the ends free to rotate. More details of themanufacturing and experimental procedures are given in Materialsand Methods.

Upon release, the initially flat bistrips start to bend and twist outof plane and evolve towards either a helical or hemihelical shape,depending on the cross-sectional aspect ratio. As indicated by theimages in Fig. 1, when the aspect ratio h=w is small, we observe theformation of periodic perversions, separating helical segments ofalternating chiralities, whereas when the bi-strips have a largeaspect ratio, they spontaneously twist along their length to form aregular helix. Significantly, these three-dimensional shapes formspontaneously and do so irrespective of whether the release isabrupt or the ends are slowly brought together. Furthermore, it isalso observed that after release, the bi-strip can be stretched

straight again and released many times and each time the sameshape, complete with the same number of perversions, reforms.Experiments were also performed under water to minimizegravitational effects and dampen vibrations. Video recordings,reproduced in File S1, capture the evolution of the 3D shapes,several transient features including how perversions move alongthe bi-strip to form a regular arrangement as well as how an initialtwisting motion is reversed.

The experimental observations indicate that the number ofperversions n is the critical geometric parameter that distinguisheswhich shape forms upon release. Assuming that the perversionsare uniformly distributed along the length of the bistrip, thenumber that form can be expected to depend on the prestrainratio, the cross-sectional aspect ratio and the length of the bi-strip.Dimensional arguments then suggest that the number is given by:wn=L~g(x,h=w). To establish how the number of perversionsdepends on these variables, a series of experiments wereperformed with different values of pre-strain and cross-sectionalaspect ratio. The results of these experiments are shown in thestructural phase diagram in Fig. 3 where the numbers associatedwith the symbols indicate the number of perversions observed.The boundary between the formation of helices and hemihelices isshown shaded. The data in Fig. 3 indicates that increasing the h=wratio drives the strip from the hemihelical configurations to helices.On the other hand, the prestrain ratio x has only a weak influenceon both the helix-to-hemihelix transition and the number ofperversions. This phase diagram (Fig. 3) was established underexperimental conditions that allowed both ends to freely rotate asthe stretching force was reduced. A similar phase diagram (Fig. S5in File S1) but notable by the absence of any helices was obtainedupon unloading when the ends were constrained from rotating (seeFile S1 for details).

Finite element simulations

Numerical simulations to explore the morphological changesoccurring during the release in the bi-strip system were conductedusing detailed dynamic finite element simulations. In our analysis,

Figure 1. Illustration of a helix (top), a hemihelix with oneperversion marked by an arrow (middle) and a hemihelix withmultiple perversions (bottom). The scale bar is 5 cm, and is thesame for each image. These different shapes were all produced in thesame way as shown in figure 2 with the same value of pre-strain x~1:5but with decreasing values of the height-to-width ratio of the bi-strip’scross-section. L~50cm, w~3mm, h~12,8,2:5mm).doi:10.1371/journal.pone.0093183.g001

Figure 2. Sequence of operations leading to the spontaneouscreation of hemihelices and helices. Starting with two longelastomer strips of different lengths, the shorter one is stretched tobe the same length as the other. While the stretching force, P, ismaintained, the two strips are joined side-by-side. Then, as the force isslowly released, the bi-strip twists and bends to create either a helix or ahemihelix.doi:10.1371/journal.pone.0093183.g002

Figure 3. The number of perversions observed as a function ofboth the prestrain and the cross-section aspect ratio, h=w. Thedata indicates that there is a transition between the formation of helixesat larger aspect ratios and hemihelices at smaller aspect ratios. Theprecise phase boundary cannot be determined with any precisionexperimentally and so is shown shaded but there is evidently only aweak dependence on the value of the pre-strain. In some cases, bistripsmade the same way produce either one or the other of the twoperversion numbers indicated.doi:10.1371/journal.pone.0093183.g003

Structural Transition from Helices to Hemihelices

PLOS ONE | www.plosone.org 2 April 2014 | Volume 9 | Issue 4 | e93183

Filaments that are longer than , form helices to avoid steric interactions.

L > 2R

more shrinkingof the bottom layer

5

Helixx

yz

2r0

p

~t

~n1

~n2

pitc

h

diameter

Mathematical description

~r(s) =

r0 cos(s/), r0 sin(s/),

p

2s

~t(s) =d~r

ds=

r0

sin(s/),

r0

cos(s/),p

2

g = ~t · ~t = r202

+p2

422= 1

=qr20 + (p/2)2

Set to fix the metric

6

Helix

x

yz

2r0

p

~t

~n1

~n2

pitc

h

diameter


~r(s) =


p

2s

Tangent and normal vectors

~t(s) =d~r

ds=

r0

sin(s/),

r0

cos(s/),p

2

=qr20 + (p/2)2

~n1(s) = cos(s/), sin(s/), 0

Helix curvatures

~n2 ·d2~r

ds2= 0

~n1 ·d2~r

ds2=

r02

=r0

r20 + (p/2)2= K

~n2(s) =

p

2sin(s/), p

2cos(s/),

r0

7

Cucumber tendril climbing via helical coiling

S. J. Gerbode et al., Science 337, 1087 (2012)

Cucumber tendrils want to pull

themselves up above other plants in order to get more sunlight.

8

Helical coiling of cucumber tendril

youngtendril

oldtendril

extractedfiber ribbon

tendril cross-section

theoretical treatments have incorporated in-trinsic curvature or differential growth withoutaddressing its origin or mechanical consequences(6, 15, 16). Recent studies of tendril anatomy(17, 18) have provided a new twist by revealingan interior layer of specialized cells similar tothe stiff, lignified gelatinous fiber (g-fiber) cellsfound in reaction wood (19). These cells providestructural support in reaction wood via tissuemorphosis driven by cell-wall lignification, waterflux, and oriented stiff cellulose microfibrils. Thepresence of a similar ribbon-like strip of g-fibercells in tendrils suggests that the coiling of thesoft tendril tissuemay be driven by the shaping ofthis stiff, internal “fiber ribbon” (18).

We investigated the role of the fiber ribbonduring tendril coiling in both Cucumis sativus(cucumber) and Echinocystis lobata (wild cu-cumber) (20). The g-fiber cells, identified in wildcucumber by using xylan antibodies in (18), areeasily distinguished as a band of morphologicallydifferentiated cells consistently positioned alongthe inner side of the helical tendril that lignifyduring coiling (17, 18). In straight tendrils thathave not yet attached to a support (Fig. 1A), a faintband of immature g-fiber cells is barely visible byusing darkfield microscopy (Fig. 1B), with noultraviolet (UV) illumination signature, indicatingthe absence of lignification (Fig. 1C). In coiledtendrils (Fig. 1D), g-fiber cells are clearly visible(Fig. 1E) and lignified (Fig. 1F). The fiber ribbonconsists of two cell layers, with the ventral layer

on the inside of the helix showing increased lig-nification relative to the dorsal outer layer (Fig. 1,G andH), which is consistent with earlier observa-tions of increased lignification on the stimulatedside of the tendril (17, 18). When a fiber ribbon isextracted from the coiled tendril by using fungalcarbohydrolases [Driselase (Sigma-Aldrich, St.Louis, MO)] to break down the nonlignified epi-dermal tendril tissue (20), it retains the helicalmorphology of a coiled tendril, and furthermore,lengthwise cuts do not change its shape (Fig. 1Iand fig. S2).

These observations suggest that tendril coil-ing occurs via asymmetric contraction of the fiberribbon; the ventral side shrinks longitudinally rel-ative to the dorsal side, giving the fiber ribbon itsintrinsic curvature. The asymmetric contractionmay be generated by a variety of dorsiventralasymmetries, including the observed differentiallignification (Fig. 1H), variations in cellulose mi-crofibril orientation as in reaction wood, or dif-ferential water affinities. For example, becauselignin is hydrophobic the ventral cells may expelmore water during lignification, driving increasedcell contraction. This would be consistent withobservations of extracted fiber ribbons that pas-sively shrink and coil even further when dried butregain their original shape when rehydrated(movie S2). Dehydrated tendrils also exhibit thisbehavior because they are dominated by the stifffiber ribbon (movie S3). Together, these factssuggest that the biophysical mechanism for

tendril coiling is provided by the asymmetric con-traction of the stiff fiber ribbon, whose resultingcurvature is imposed on the surrounding softtendril tissue. The perversions in a doubly sup-ported tendril follow naturally from the topo-logical constraint imposed by the prevention oftwist at its ends.

To better understand the origin of curvature infiber ribbons, we reconstituted the underlyingmechanism using a physical model composed oftwo bonded, differentially prestrained silicone rub-ber sheets, similar to rubber models for shapingsheets (21–23). The first silicone sheet was uni-axially stretched, and an equally thick layer ofsilicone sealant was spread onto the stretchedsheet. After the sealant was fully cured, thin stripswere cut along the prestrained direction, yieldingbilayer ribbons (Fig. 2A) with intrinsic curvatureset by the relative prestrain, thickness, and stiff-ness of the two layers (fig. S3) (20). Like fiberribbons, the initially straight physical models spon-taneously form coiled configurations with twoopposite-handed helices connected by a helicalperversion (Fig. 2A, left).

However, there is an unexpected difference inmechanical behavior between the physical mod-els and tendril fiber ribbons. When clamped atboth ends and pulled axially, the physical modelsimply unwinds to its original uncoiled state (Fig.2A and movie S4). In contrast, in fiber ribbonswe observed a counterintuitive “overwinding”behavior in which the ribbon coils even further

A B

E

G H

C

FD

I

Fig. 1. Tendril coiling via asymmetric contraction. During coiling, a strip ofspecialized structural gelatinous fiber cells (the fiber ribbon) becomes lignifiedand contracts asymmetrically and longitudinally. (A to C) A straight tendrilthat has never coiled (A) lacks lignified g-fiber cells. In the tendril crosssection, darkfield (B) and UV autofluorescence (C) show no lignin signal. (D toH) In coiled tendrils (D), the fully developed fiber ribbon consists of ∼2 layersof highly lignified cells extending along the length of the tendril. In the tendril

cross section, darkfield (E) and UV autofluorescence (F) show strong lig-nification in the fiber ribbon. In (G) and (H), increased magnification revealsthat ventral cells (top left) are more lignified than dorsal cells. (I) The extractedfiber ribbon retains the helical morphology of the coiled tendril. (Inset) Highermagnification shows the orientation of g-fiber cells along the fiber ribbon.Scale bars, (B) and (C) 0.5 mm, (E) and (F) 100 mm, (G) and (H) 10 mm, (I)1 mm.

31 AUGUST 2012 VOL 337 SCIENCE www.sciencemag.org1088

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on

Augu

st 3

0, 2

012

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

0.5mm0.5mm

100µm100µm

10µm10µm1mm

Coiling in older tendrils is due to a thin layer of stiff, lignified gelatinous fiber cells, which are also found in wood.


lignified g-fiber cells

9

Helical coiling of cucumber tendrilDrying of fibber ribbon

increases coilingDrying of tendrilincreases coiling

Rehydrating of tendrilincreases coiling









A B

E

G H

C

FD

I




REPORTS

on

Augu

st 3

0, 2

012

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

During the lignification of g-fiber cells water is expelled, which causes shrinking.

The inside layer is more lignified and therefore shrinks more and is also

stiffer than the outside layer.

insidelayer

outsidelayer

10

Coiling of tendrils in opposite directions

right-handedhelix

left-handedhelix

perversion








A B

E

G H

C

FD

I




REPORTS

on

Augu

st 3

0, 2

012

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

perversion

Ends of the tendril are fixed and cannot rotate. This constraints

the linking number.

Link = Twist + Writhe

Coiling in the same direction increases Writhe, which needs to

be compensated by the twist.

Note: there is no bending energy when the curvature of two helices correspond to the spontaneous curvature due to the differential

shrinking of fiber.

In order to minimize the twisting energy tendrils combine two helical

coils of opposite handedness(=opposite Writhe).

11

Twist, Writhe and Linking numbersLn=Tw+Wr linking number: total number of turns of a particular end

twist: number of turns due to twisting the beamWr writhe: number of crossings when curve is projected on a plane Tw

12

Overwinding of tendril coils

when

pulled,adding

turnson

bothsides

ofthe

perversion(Fig.2A

,right,andmovie

S5).Even-

tuallythough,

underhigh

enoughtension

thefiberribbon

unwinds,returning

toaflat,uncoiled

stateas

expected(m

ovieS5).

Inspiredby

ourobservations

ofasym

metric

lignificationin

fiberribbons,w

hichsuggestthat

theinner

layeris

lessextensible,

weadded

arelatively

inextensiblefabric

ribbonto

theinside

ofacoiled

physicalmodel.

Tomim

iclignified

cellsthat

resistcom

pression,weadded

anin-

compressible

copperwire

tothe

exteriorof

thehelix.

The

internalfabric

ribbonprevents

elon-gation,w

hereastheexternalcopperw

ireprevents

contraction.Together,thesemodificationsincrease

Fig.3.Mechanicalconsequencesofoverwinding.(A

andB)

Forceextension

curvesfor

oneyoung

tendrilthatdoesnot

overwind(red

curves)and

oneold

tendrilthat

exhibitssubstantial

overwinding(blue

curves).Each

tendrilwas

separatedinto

asegm

entcontainingthe

helicalperversion(dotted

curvesindicate

perverted)and

asegm

entwith

noperversion

(solidcurves

indicateclam

ped).Thedim

ension-less

forceF ∼isplotted

againstthe

scaleddisplacem

ent∆l

(detaileddefinitions

areavailable

inthe

supplementary

materials)

in(A).The

differencein

scaledforce

dueto

thehelicalperversion

∆f=

f(perverted)−f(clam

ped)isplotted

against∆lin

(B).Theshaded

rangein(B)indicatesvariations

inthe

fittedinitial

slopevalue.

(C)Dim

ensionlessforce-

extensioncurvesare

plottedfornum

ericalfilamentswith

B/Cvalues

1/5(red),1

(green),5(blue).(Inset)Log-linear

plotof

thesam

edata.

(D)The

differencein

force∆F ∼

=F ∼(perverted)−

F ∼(clamped)highlightsthe

mechanicaleffect

ofthe

helicalperversion.ForB<C,the

perversionalways

decreasestheforce

neededtoaxiallyextend

thefilam

ent;forB>C,the

perversioninitially

decreasestheforce

neededbut

eventuallyincreases

thisnecessary

forceat

higherexten-

sions.(Inset)∆fis

plottedagainst∆lfor

directcomparison

withthe

experimentaldata.

Fig.2.

Twistlesssprings

unwindingand

overwinding.(A)Asilicone

twistlessspring

withlower

bendingstiffness

Bthan

twistingstiffness

Cunwinds

whenpulled,returning

toitsoriginalflatshape.(B)W

henafiberribbon

ispulled,itinitiallyoverwinds,adding

oneextra

turntoeach

sideofthe

perversion(num

berofturnsare

indicatedinwhite).(C)Overwinding

isinducedinthe

siliconemodel

byaddingarelativelyinextensible

(undertension)fabricribbontothe

interiorofthe

helixand

aninextensible

(undercompression)copperwire

tothe

exterior.

Together,theseincrease

theratio

B/C.(D)W

henB/C

>1,num

ericalsimulations

ofelastic

helicalfilam

entsrecapitulate

thisoverwinding

behavior,which

isconsistentwith

physicalandbiologicalexperim

ents.(E)Changeinthe

number

ofturnsineach

helix∆Nisplotted

versusscaleddisplacem

ent∆lforB/C

values1/5

(red),1(green),and

5(blue).Overwinding

becomesm

orepronounced

withincreasing

B/C.(F)Overwindingisalso

observedinold

tendrils,whichhave

driedand

flattenedinto

aribbon-like

shapewith

B/C>1.Scale

bars,1cm

.

www.sciencem

ag.orgSC

IENCE

VOL337

31AUGUST

20121089

REPORTS

on August 30, 2012www.sciencemag.orgDownloaded from

when

pulled,adding

turnson

bothsides

ofthe

perversion(Fig.2A

,right,andmovie

S5).Even-

tuallythough,

underhigh

enoughtension

thefiberribbon

unwinds,returning

toaflat,uncoiled

stateas

expected(m

ovieS5).

Inspiredby

ourobservations

ofasym

metric

lignificationin

fiberribbons,w

hichsuggestthat

theinner

layeris

lessextensible,

weadded

arelatively

inextensiblefabric

ribbonto

theinside

ofacoiled

physicalmodel.

Tomim

iclignified

cellsthat

resistcom

pression,weadded

anin-

compressible

copperwire

tothe

exteriorof

thehelix.

The

internalfabric

ribbonprevents

elon-gation,w


ireprevents



andB)

Forceextension

curvesfor

oneyoung

tendrilthatdoesnot

overwind(red

curves)and

oneold

tendrilthat

exhibitssubstantial

overwinding(blue

curves).Each

tendrilwas

separatedinto

asegm

entcontainingthe


curvesindicate

perverted)and

asegm

entwith

noperversion

(solidcurves

indicateclam

ped).Thedim

ension-less

forceF ∼isplotted

againstthe

scaleddisplacem

ent∆l


areavailable

inthe

supplementary

materials)

in(A).The

differencein

scaledforce

dueto


∆f=


ped)isplotted

against∆lin

(B).Theshaded


inthe

fittedinitial

slopevalue.

(C)Dim

ensionlessforce-

extensioncurvesare

plottedfornum

ericalfilamentswith

B/Cvalues

1/5(red),1


plotof

thesam

edata.

(D)The

differencein

force∆F ∼



mechanicaleffect

ofthe


perversionalways

decreasestheforce


thefilam

ent;forB>C,the

perversioninitially

decreasestheforce

neededbut

eventuallyincreases

thisnecessary

forceat

higherexten-

sions.(Inset)∆fis


directcomparison

withthe

experimentaldata.

Fig.2.

Twistlesssprings

unwindingand


twistlessspring

withlower

bendingstiffness

Bthan

twistingstiffness

Cunwinds



henafiberribbon


oneextra

turntoeach

sideofthe

perversion(num

berofturnsare


isinducedinthe

siliconemodel



interiorofthe

helixand

aninextensible


tothe

exterior.


theratio

B/C.(D)W

henB/C

>1,num

ericalsimulations

ofelastic

helicalfilam

entsrecapitulate

thisoverwinding

behavior,which

isconsistentwith


ents.(E)Changeinthe

number

ofturnsineach

helix∆Nisplotted


ent∆lforB/C

values1/5

(red),1(green),and

5(blue).Overwinding

becomesm

orepronounced

withincreasing


observedinold

tendrils,whichhave

driedand

flattenedinto

aribbon-like

shapewith

B/C>1.Scale

bars,1cm

.

www.sciencem

ag.orgSC

IENCE

VOL337

31AUGUST

20121089

REPORTS


Old tendrils overwind when stretched.

rela

xed

stre

tche

d


Rubber model unwinds when stretched.

rela

xed

stre

tche

d

5

5

6

6

13

Overwinding of tendril coils


Preferred curved state Flattened state

In tendrils the red inner layer is much stiffer then the

outside blue layer.

In rubber models both layers have similar stiffness.

High bending energy cost associated with stretching

of the stiff inner layer!

Small bending energy.

theoretical

treatm

ents

have

incorporated

in-

trinsic

curvatureor

differentialgrow

thwithout

addressing

itsoriginormechanicalconsequences

(6,15,16).Recentstudiesof

tendrilanatom

y(17,18)h

aveprovided

anewtwistbyrevealing

aninterior

layerof

specialized

cells

similarto

thestiff,lignified

gelatin

ousfiber(g-fiber)c

ells

foundinreactio

nwood(19).T

hesecells

provide

structural

supportin

reactio

nwoodviatissue

morphosisdriven

bycell-walllignification,water

flux,andoriented

stiff

cellulose

microfib

rils.The

presence

ofasimilarrib

bon-likestrip

ofg-fib

ercells

intendrilssuggeststhat

thecoiling

ofthe

softtendriltissuemay

bedriven

bytheshapingof

thisstiff,internal“fib

errib

bon”

(18).

Weinvestigated

therole

ofthefib

errib

bon

durin

gtendril

coiling

inboth

Cucum

issativus

(cucum

ber)

andEchinocystis

lobata

(wild

cu-

cumber)(20).T

heg-fib

ercells,identified

inwild

cucumberby

usingxylanantibodiesin

(18),are

easilydistinguishedasaband

ofmorphologically

differentiatedcells

consistently

positionedalong

theinnerside

ofthehelical

tendril

that

lignify

durin

gcoiling

(17,

18).In

straight

tendrilsthat

have

notyetattached

toasupport(Fig.1A

),afaint

band

ofimmatureg-fib

ercells

isbarelyvisibleby

usingdarkfield

microscopy(Fig.1B

),with

noultraviolet(UV)illuminationsignature,indicating

theabsenceof

lignification(Fig.1C

).In

coiled

tendrils(Fig.1D),g-fib

ercells

areclearly

visible

(Fig.1E)and

lignified(Fig.1F).T

hefib

errib

bon

consistsof

twocelllayers,w

iththeventrallayer

ontheinside

ofthehelix

show

ingincreasedlig-

nificationrelativetothedorsalouterlayer(Fig.1,

GandH),which

isconsistentwith

earlierobserva-

tions

ofincreasedlignificationon

thestimulated

side

ofthetendril

(17,18).Whenafib

erribbonis

extracted

from

thecoiledtendril

byusingfungal

carbohydrolases[D

riselase(Sigma-Aldrich,

St.

Louis,M

O)]tobreakdownthenonlignifiedepi-

derm

altendril

tissue(20),itretainsthehelical

morphologyof

acoiledtendril,and

furth

ermore,

lengthwisecutsdo

notchangeits

shape(Fig.1

Iandfig

.S2).

These

observations

suggestthat

tendril

coil-

ingoccursviaasym

metric

contractionofthefib

errib

bon;theventralsideshrin

kslongitudinally

rel-

ativetothedorsalside,givingthefib

errib

bonits

intrinsic

curvature.

The

asym

metric

contraction

may

begeneratedby

avariety

ofdorsiventral

asym

metries,includingtheobserved

differential

lignification(Fig.1H),variations

incellulose

mi-

crofibril

orientationas

inreactionwood,

ordif-

ferentialwater

affin

ities.Fo

rexam

ple,

because

ligninishydrophobictheventralcellsmay

expel

morewaterdurin

glignification,drivingincreased

cellcontraction.

Thiswould

beconsistent

with

observations

ofextracted

fiber

ribbons

thatpas-

sivelyshrin

kandcoileven

furth

erwhendriedbut

regain

theiroriginal

shape

when

rehydrated

(movieS2

).Dehydratedtendrilsalso

exhibitthis

behavior

becausethey

aredominated

bythestiff

fiber

ribbon(m

ovie

S3).To

gether,thesefacts

suggestthat

the

biophysicalmechanism

for

tendrilcoiling

isprovided

bytheasym

metric

con-

tractionof

thestiff

fiber

ribbon,

whose

resulting

curvatureis

imposedon

thesurroundingsoft

tendril

tissue.The

perversionsin

adoubly

sup-

portedtendril

follow

naturally

from

thetopo-

logicalconstraintim

posedby

thepreventionof

twistatitsends.

Tobetterunderstandtheoriginofcurvaturein

fiber

ribbons,wereconstituted

theunderly

ing

mechanism

usingaphysicalmodelcomposedof

twobonded,differentially

prestrained

siliconerub-

bersheets,sim

ilarto

rubber

modelsforshaping

sheets(21–23).The

firstsiliconesheetw

asuni-

axially

stretched,

andan

equally

thicklayerof

siliconesealantwas

spread

onto

thestretched

sheet.Afterthe

sealantw

asfully

cured,thinstrip

swerecutalong

theprestrained

direction,yielding

bilayerribbons

(Fig.2A)w

ithintrinsiccurvature

setb

ytherelativeprestrain,thickness,and

stiff-

ness

ofthetwolayers

(fig.S3

)(20).Likefib

erribbons,the

initiallystraightphysicalmodelsspon-

taneouslyform

coiledconfigurations

with

two

opposite-handedhelices

connectedby

ahelical

perversion

(Fig.2A,left).

How

ever,thereisan

unexpected

difference

inmechanicalb

ehaviorbetweenthephysicalmod-

elsandtendril

fiber

ribbons.Whenclam

pedat

both

ends

andpulledaxially,the

physicalmodel

simplyunwinds

toits

originaluncoiledstate(Fig.

2Aandmovie

S4).In

contrast,infib

errib

bons

weobserved

acounterin

tuitive

“overw

inding”

behavior

inwhich

therib

boncoils

even

furth

er

AB E G

HC FD I

Fig.

1.Tendril

coiling

viaasym

metric

contraction.Du

ringcoiling,a

strip

ofspecialized

structuralgelatinousfibercells(th

efiberrib

bon)becomeslignified

andcontractsasym

metricallyandlongitudinally.(Ato

C)Astraight

tendril

that

hasnevercoiled(A)lackslignifiedg-fiber

cells.In

thetendril

cross

section,darkfield(B)and

UVautofluorescence(C)showno

ligninsig

nal.(D

toH)Incoiledtendrils(D),the

fully

developedfiberrib

bonconsistso

f∼2layers

ofhighlylignifiedcells

extendingalongthelengthofthetendril.Inthetendril

crosssection,

darkfield

(E)andUV

autofluorescence(F)show

strong

lig-

nificationinthefiber

ribbon.In(G)and

(H),increasedmagnificationreveals

thatventralcells(topleft)

aremorelignifiedthan

dorsalcells.(I)Theextra

cted

fiberrib

bonretainsthe

helicalmorphologyofthecoiledtendril.(Inset)Higher

magnificationshow

stheorientationof

g-fiber

cells

alongthefiber

ribbon.

Scalebars,(B)

and(C)0

.5mm,(E)

and(F)1

00mm

,(G)

and(H)1

0mm

,(I)

1mm.

31AUGUST

2012

VOL33

7SC

IENCE

www.scien

cemag

.org

1088REPO

RTS

on August 30, 2012 www.sciencemag.org Downloaded from

when

pulled,adding

turnson

bothsides

ofthe

perversion(Fig.2A

,right,andmovie

S5).Even-

tuallythough,

underhigh

enoughtension

thefiberribbon

unwinds,returning

toaflat,uncoiled

stateas

expected(m

ovieS5).

Inspiredby

ourobservations

ofasym

metric

lignificationin

fiberribbons,w

hichsuggestthat

theinner

layeris

lessextensible,

weadded

arelatively

inextensiblefabric

ribbonto

theinside

ofacoiled

physicalmodel.

Tomim

iclignified

cellsthat

resistcom

pression,weadded

anin-

compressible

copperwire

tothe

exteriorof

thehelix.

The

internalfabric

ribbonprevents

elon-gation,w


ireprevents



andB)

Forceextension

curvesfor

oneyoung

tendrilthatdoesnot

overwind(red

curves)and

oneold

tendrilthat

exhibitssubstantial

overwinding(blue

curves).Each

tendrilwas

separatedinto

asegm

entcontainingthe


curvesindicate

perverted)and

asegm

entwith

noperversion

(solidcurves

indicateclam

ped).Thedim

ension-less

forceF ∼isplotted

againstthe

scaleddisplacem

ent∆l


areavailable

inthe

supplementary

materials)

in(A).The

differencein

scaledforce

dueto


∆f=


ped)isplotted

against∆lin

(B).Theshaded


inthe

fittedinitial

slopevalue.

(C)Dim

ensionlessforce-

extensioncurvesare

plottedfornum

ericalfilamentswith

B/Cvalues

1/5(red),1


plotof

thesam

edata.

(D)The

differencein

force∆F ∼



mechanicaleffect

ofthe


perversionalways

decreasestheforce


thefilam

ent;forB>C,the

perversioninitially

decreasestheforce

neededbut

eventuallyincreases

thisnecessary

forceat

higherexten-

sions.(Inset)∆fis


directcomparison

withthe

experimentaldata.

Fig.2.

Twistlesssprings

unwindingand


twistlessspring

withlower

bendingstiffness

Bthan

twistingstiffness

Cunwinds



henafiberribbon


oneextra

turntoeach

sideofthe

perversion(num

berofturnsare


isinducedinthe

siliconemodel



interiorofthe

helixand

aninextensible


tothe

exterior.


theratio

B/C.(D)W

henB/C

>1,num

ericalsimulations

ofelastic

helicalfilam

entsrecapitulate

thisoverwinding

behavior,which

isconsistentwith


ents.(E)Changeinthe

number

ofturnsineach

helix∆Nisplotted


ent∆lforB/C

values1/5

(red),1(green),and

5(blue).Overwinding

becomesm

orepronounced

withincreasing


observedinold

tendrils,whichhave

driedand

flattenedinto

aribbon-like

shapewith

B/C>1.Scale

bars,1cm

.

www.sciencem

ag.orgSC

IENCE

VOL337

31AUGUST

20121089

REPORTS


Tendrils try to keep the preferred curvature when stretched!

14

Overwinding of rubber models with an additional stiff fabric on the inside layers

when

pulled,adding

turnson

bothsides

ofthe

perversion(Fig.2A

,right,andmovie

S5).Even-

tuallythough,

underhigh

enoughtension

thefiberribbon

unwinds,returning

toaflat,uncoiled

stateas

expected(m

ovieS5).

Inspiredby

ourobservations

ofasym

metric

lignificationin

fiberribbons,w

hichsuggestthat

theinner

layeris

lessextensible,

weadded

arelatively

inextensiblefabric

ribbonto

theinside

ofacoiled

physicalmodel.

Tomim

iclignified

cellsthat

resistcom

pression,weadded

anin-

compressible

copperwire

tothe

exteriorof

thehelix.

The

internalfabric

ribbonprevents

elon-gation,w


ireprevents



andB)

Forceextension

curvesfor

oneyoung

tendrilthatdoesnot

overwind(red

curves)and

oneold

tendrilthat

exhibitssubstantial

overwinding(blue

curves).Each

tendrilwas

separatedinto

asegm

entcontainingthe


curvesindicate

perverted)and

asegm

entwith

noperversion

(solidcurves

indicateclam

ped).Thedim

ension-less

forceF ∼isplotted

againstthe

scaleddisplacem

ent∆l


areavailable

inthe

supplementary

materials)

in(A).The

differencein

scaledforce

dueto


∆f=


ped)isplotted

against∆lin

(B).Theshaded


inthe

fittedinitial

slopevalue.

(C)Dim

ensionlessforce-

extensioncurvesare

plottedfornum

ericalfilamentswith

B/Cvalues

1/5(red),1


plotof

thesam

edata.

(D)The

differencein

force∆F ∼



mechanicaleffect

ofthe


perversionalways

decreasestheforce


thefilam

ent;forB>C,the

perversioninitially

decreasestheforce

neededbut

eventuallyincreases

thisnecessary

forceat

higherexten-

sions.(Inset)∆fis


directcomparison

withthe

experimentaldata.

Fig.2.

Twistlesssprings

unwindingand


twistlessspring

withlower

bendingstiffness

Bthan

twistingstiffness

Cunwinds



henafiberribbon


oneextra

turntoeach

sideofthe

perversion(num

berofturnsare


isinducedinthe

siliconemodel



interiorofthe

helixand

aninextensible


tothe

exterior.


theratio

B/C.(D)W

henB/C

>1,num

ericalsimulations

ofelastic

helicalfilam

entsrecapitulate

thisoverwinding

behavior,which

isconsistentwith


ents.(E)Changeinthe

number

ofturnsineach

helix∆Nisplotted


ent∆lforB/C

values1/5

(red),1(green),and

5(blue).Overwinding

becomesm

orepronounced

withincreasing


observedinold

tendrils,whichhave

driedand

flattenedinto

aribbon-like

shapewith

B/C>1.Scale

bars,1cm

.

www.sciencem

ag.orgSC

IENCE

VOL337

31AUGUST

20121089

REPORTS


rela

xed

stre

tche

d

5

5

4

6

stiff fabric

soft rubber

15

Overwinding of helix with infinite bending modulus

x

yz

2r0

p

pitc

h

diameter


~r(s) =


p

2s

=qr20 + (p/2)2

Infinite bending modulus fixes the helix curvature during stretching

Z

length of the helix backboneL

numberof loops

Z = pN = p(L/2)

K =r0

r20 + (p/2)2

N =Z

p

Helix pitch and radius

r0 =1

K

1 Z2

L2

p =2Z

KL

r1 Z2

L2

Number of loops

N =Z

p=

KL

2p

1 (Z/L)2

0 0.2 0.4 0.6 0.8 10

0.51

1.52

2.53

3.54

16

Overwinding of helix with infinite bending modulus

x

yz

2r0

p

pitc

h

diameter

Z

length of the helix backboneL

numberof loops

Helix pitch and radius

N =Z

p

r0 =1

K

1 Z2

L2

p =2Z

KL

r1 Z2

L2

Number of loops

N =Z

p=

KL

2p

1 (Z/L)2

r0K

pK/(2)

Z/L

2N/(KL)

Overwinding

17

Spirals in natureshells beaks claws

horns teeth tusks

What simple mechanism could produce spirals?

↵

↵

↵

↵

↵

↵

18

Equiangular (logarithmic) spiralin polar coordinates radius

grows exponentially

r() = a = exp

( cot↵)

name logarithmic spiral:

=ln r

ln a

↵ = 82

cot↵ = ln a

Ratio between growth velocities in the radial and

azimuthal directions velocities is constant!

cot↵ =

dr

rd=

dr/dt

rd/dt=

vrv

19

Equiangular (logarithmic) spiral↵ = 85 ↵ = 82 ↵ = 80

↵ = 75 ↵ = 60 ↵ = 45

↵ ↵ ↵

↵↵

↵

r() = a = exp

( cot↵)

20

Growth of spiral structures

old structure

newly addedmaterial

W

Lin

Lout

New material is added at a constant ratio of growth velocities, which produces spiral structure with side

lengths and the width in the same proportions.

vout

t

vint

vout

: vin

: vW = Lout

: Lin

: W

W + vwt

Note: growth with constant width (vW=0) produces helices

21

Growth of spiral structures

rin

() = e cot↵

rout

() = e cot↵Assume the following spiral profiles of the outer and inner layers:

=

0.5,

↵=

75

=

0.5,

↵=

86

e2 cot↵ > 1 e2 cot↵ < 1

In some shells the inner layer does not

grow at all

22

3D spirals

3D spiral of ram’s horns is due to the triangular

cross-section of the horn, where each side grows with a different

velocity.

va

vb

vc

Shells of mollusks are often conical

23

PhyllotaxisPhyllotaxis is classification of leaves on a plant stem

distichouspattern

leaves alternating every 1800

decussatepatternpairs of

leaves at 900

whorledpattern

3 or more leavesoriginating from the

same node (1800)

alternate(spiral)patternsuccessive

leaves at 137.50

maize Coleus sp. Veronicastrum virginicum sunflower

http://www.sciteneg.com/PhiTaxis/PHYLLOTAXIS.htm

24

Spiral phyllotaxis

time

schematic description of leaves arrangement

leaves grow from the apical meristem, which

also gives rise to petals, sepals, etc.

leaves

florets(petals)

floral primordia

↵

↵ 137.5

25

Parastichy numbers

21 left-handedspirals

34 right-handedspirals

Parastichy numbers (21,34)

26

Parastichy numbersspiral

phyllotaxis

multijugatephyllotaxis

(e.g. 2 new leaves are added at the same time)

succulent plant (3,5)

Gymnocalycium (10,16)=2(5,8)

27

Parastichy numbers

sunflower (21,34)pince cone (8,13) artichoke (34,55)

aloe (5,8)succulent plant (3,5)aonium (2,3)

Parastichy numbers very often correspond to successive Fibonacci numbers!

28

Fibonacci numbersF1 = 1

F2 = 1

Fn = Fn1 + Fn2

Sequence of Fibonacci numbers1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Fn =1p5['n (1 ')n]

Golden ratio ' =1 +

p5

2

↵

a+ b

a=

a

b

a

b= '

divide perimeter in golden ratio

Golden angle

↵ = 360b

(a+ b)=

360

'2 137.5

In spiral phyllotaxis successive leaves grow at approximately Golden angle!

29

Non-Fibonacci parastichy numbers

Statistics for pine trees in Norway95% Fibonacci numbers 4% Lucas numbers 1% not properly formed

Sequence of Lucas numbers1, 3, 4, 7, 11, 18, 29, 47, 76

Lucas numbersL1 = 1

L2 = 3

Ln = Ln1 + Ln2

30

Spiral phyllotaxis

new primordial

Norway spruceNew primordia start growing at the site where plant hormone auxin is depleted.

meristem Auxin hormones are released by growing primordia. New primordium wants to be as far apart as possible from the existing primordia.

Mechanical analog with magnetic repelling particles

S. Douady and Y. Couder, PRL 68, 2098 (1992)

VOLUME 68, NUMBER 13 PH YSICAL R EVI EW LETTERS

Phyllotaxis as a Physical Self-Organized Growth Process

30 MARCH l 992

S. Douady "' and Y. CouderLaboratoire de Physique Statistique, 24 rue Lhomond, 75231 Paris CEDEL 05, France

(Received 12 November 1991)A specific crystalline order, involving the Fibonacci series, had until now only been observed in plants

(phyllotaxis). Here, these patterns are obtained both in a physics laboratory experiment aud in a numer-ical simulation. They arise from self-organization in an iterative process. They are selected dependingon only one parameter describing the successive appearance of new elements, and on initial conditions.The ordering is explained as due to the system s trend to avoid rational (periodic) organization, thusleading to a convergence towards the golden mean.

PACS numbers: 87. l0.+e, 05.45.+b, 61.50.Cj

The elements of a plant (leaves, sepals, florets, etc. )form very regular lattices, with a crystallinelike order. Inthe most common arrangement (e.g. , on a sunflower heador a pinecone), the eye is attracted to conspicuous spirals(the parastichies) linking each element to its nearestneighbors. The whole surface is covered with a number iof parallel spirals running in one direction, and j in theother. The most striking feature is that (i,j) are nearlyalways two consecutive numbers of the Fibonacci series,[F~]=[1,1,2, 3,5, 8, 13,21,34, . . .] where each new term isthe sum of the two preceding ones. Early works [1-3]showed that such patterns resulted from the successiveappearance of the elements on a uniquely tightly woundspiral, called the generative spira/. The basic quantity isthen the divergence y which is the angle between the ra-dial directions of two consecutive elements. Measure-ments [3] of divergences on mature plants showed thatthey were surprisingly close to the golden section:=2n(I —r )= 137.5', where r =(—I +J5)/2 is thegolden mean.A basic hypothesis is that these phyllotactic patterns

result from the conditions of appearance of the primordianear the tip of the growing shoots (for reviews see Refs.[4] and [5]). The stem tips (the apical meristems) haveaxisymmetric profiles [Fig. 1(a)]. The summit is occu-pied by a stable region: the apex. The primordia (whichwill evolve into leaves, petals, stamens, florets, etc. ) arefirst visible as small protrusions at the periphery of theapex. In the reference frame of the tip, due to thegrowth, the existing primordia are advected away fromthe apex while new ones continue to be formed [6]. Inbotany, it was suggested [7] that a new primordium ap-pears with a periodicity T near the tip in the largest gapleft between the previous primordia and the apex.Altogether this forms an iterative process which we

wish to investigate as a dynamical system. To implementa laboratory experiment and a numerical simulation, weretained from botany the following characteristics: Iden-tical elements are generated with a periodicity T at agiven radius Ro from a center in a plane surface [8].They are radially advected at velocity Vo, and there is arepulsive interaction between them (so that the new ele-ment will appear as far as possible from the preceding

(a)

(b)

R, !I

H

FIG. l. (a) Sketch of the growth in plants. (b) Sketch ofthe experimental apparatus.

ones, i.e., in the largest available place). The results canbe interpreted using only one adimensional parameterG = VoT/Ro.The experimental system [Fig. 1(b)] consists of a hor-

izontal dish filled with silicone oil and placed in a verticalmagnetic field H(r) created by two coils near theHelmholtz position. Drops of ferrofluid of equal volume(i =10 mm ) fall with a tunable periodicity T at thecenter of the cell. The drops are polarized by the fieldand form small magnetic dipoles, which repel each otherwith a force proportional to d (where d is their dis-tance). These dipoles are advected by a radial gradientof the magnetic field (from 2.4&&10 A/m at the center to2.48X10 A/m at the border of the dish), their velocityV(r) being limited by the viscous friction of the oil. Inorder to model the apex, the dish has a small truncatedcone at its center, so that the drop introduced at its tipquickly falls to its periphery. 6 can be tuned by changingeither the periodicity T or the gradient of H (controlling

2098 1992 The American Physical Society

PHYLLOTAXIS AS A DYNAMICAL SELF ORGANIZING PROCESS—PART I 259

FIG. 3. Three photographs (seen from above) of typicalphyllotactic patterns formed by the ferrofluid drops for dierentvalues of the control parameter G. The drops are visible as darkdots. The tube for the ferrofluid supply partially hides the centraltruncated cone. The drops are numbered in their order offormation. (a) For strong advection, G11, each new drop isrepelled only by the previous one and a distichous mode isobtained, 8=180°. (b) Below the first symmetry breakingbifurcation (G10.7) the successive drops move away from eachother with a divergence angle 8=150° (shown between drop threeand four). They define an anti-clockwise generative spiral (dashedline). The parastichy numbers correspond to (1, 2). (c) For smalleradvection (G10.1) higher order Fibonacci modes are obtained.Here 81139° and the parastichy numbers are (5, 8).

the first two (see also Figs 6 and 7). This is a symmetrybreaking bifurcation which selects once for all thedirection of winding of the generative spiral. A steadyregime is reached later with a constant divergence 8

[in Fig. 3(b) 8=150° and two parastichies i=1, j=2are observed]. For smaller T, the new drop becomessensitive to the repulsion of three or more previousdrops, and the divergence gets nearer to F. InFig. 3(c), 8=139° and the Fibonacci numbersare i=5, j=8. The spiral mode obtained in thisnon-biological system is strikingly similar to a veryusual organization observed in botany.

3.3. DISCUSSION

The observed patterns, in fact, do not depend onlyon T, but also on Rc the radius of the circle outsideof which the angular position of the particles is fixed,and on the advection velocity Vo (controlled by themagnetic field gradient). The only relevant parameteris in fact adimensional and defined by G=VoT/Rc . Itis the ratio of the two typical length scales of thesystem, one corresponding to the radial displacementof the elements during one period and the otherdefining the size of the central region. This parameteris directly related to the plastochrone ratio, a, whichwas introduced by Richards (1951). This authorshowed that the apical growth could be characterizedby the ratio a=rn1/rn of the distance of twosuccessive primordia to the centre. This ratio is easilymeasured on transverse sections of apices. Therelation between a and G is simple because the growthnear the apical region is exponential. As V(r) A r andV(Ro )=Vo , the distance of a primordium to thecentre at time t is: r=Ro exp(Vot/Ro ) and its velocityV=Vo exp(Vot/Ro ). The resulting plastochrone ratiois a=rn1/rn=exp(VoT/Ro ) and our parameter G issimply: G=Ln(a). This parameter had been usedpreviously in botanical cases by Meicenheimer (1979)and by Rutishauser (1982).

The only precise way of measuring G both in plantgrowth and in our experiment is to deduce it froma=rn+1/rn . In plants, obtaining G from measurementsof Vo , T and Ro would be more di.cult and lessprecise than deducing it from the geometry oftransverse sections. In our experiment Vo and T areknown with precision, but there is ambiguity on thechoice of Ro because the drops are introduced atthe centre of the cell. If we deduce a value of Ro fromthe measured plastochrone ratio we find that, in allcases, Ro is equal to Rc the radius of the circular zoneout of which there is no further reorganization andchange of 8. This result could have a meaning inbotany. It is well known that the angular positionsof the primordia become rapidly fixed at the apex

decreased, a remarkable evolution of these patternstakes place. Below a threshold value each new dropbecomes sensitive to the repulsion of the two previousdrops and can no longer remain on the radial lineformed by them. In such a case the third depositeddrop slides to one of either side of the line formed by

Parastichynumbers (5,8)

magnetic field drives particles away from the center

particles repel via magnetic dipol interactions

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Energy minimization between repelling particles

L. Levitov, PRL 66, 224 (1991)L. Levitov, EPL 14, 533 (1991)

Local energy minima for repelling particlesL. S. LEVITOV: ENERGETIC APPROACH TO PHYLLOTAXIS 535

1 5 4 3 5 2 1 5 a z l t 3 5 2 s 1 2 9 7 5 0 3 1 0 7 4 9 5 6 7 0 9

Fig. 1. - Trajectories represent positions of minima of the energy (2.2) as a function of y; arcs (2.3) divide the plane (x, y) into triangles (2.4). Since the pattern is symmetric under x + - x and x + x + 1, we show it only for 0.5 < x < 1. Inset: quasi-bifurcation in a triangle [mln, p /q , (m + p)/(n + q)], n c q.

particular case of (2.3): n = q = 1, p = w. The arcs (2.3) divide the upper half-plane into curved triangles with vertices xl, xz, x3 located on the x-axis:

I np - mq I = 1. P m+P x2=- x3=x10x2=- m X] = - n’ q ’ n+q’ (2.4)

The notation we use below is [mln, plql for the arcs (2.3) and [mln, plq, (m + p)/(n + q)] for the triangles (2.4).

By inspecting fig. 1, one can notice that in all triangles the behaviour of the trajectories of the minima is qualitatively similar (see inset of fig. 1). Under decrease of y a trajectory enters the triangle [mln, plq, (m + p)/(n + q)] (0 < n < q) through the arc [mln, plq] and then exits through the arc [plq, (m + p)/(n + a)]. In addition, a new trajectory emerges, separated from the old one, going out of the triangle through [mln, (m + p)/(n + q)]. This configuration (we call it quasi-bifurcation) one finds in every triangle of fig. 1, except for [0/1,1/1,1/21 where a true bifurcation occurs: the trajectory splits into two, symmetrically with respect to the x = 0.5 line. These two .principal)> trajectories correspond to the states dynamically accessible for the system, i.e. to the lattices that can be formed under continuous variation of y starting from + CQ. One can observe that the triangles traced by the principal trajectories form two sequences:

( 2 . 5 ~ )

(2.5b)

where F,, are Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, .... This is exactly the property of phyllotaxis we are tempting to explain. But, instead of

concentrating on the two principal trajectories, we notice that it is sufficient to explain another property of the pattern shared by all trajectories: the arc [plq, (m + p) / (n + a)] , 0 < n < q, chosen to exit from the triangle [mln, plq, (m + p)/(n + q)] corresponds to maximal

radial spacing between particles

1

1

1

2

2

3

3

5

3

44

7

↵↵

1 ↵

2

Fibonacci numbers

Lucas numbers

golden angle

As the plant is growingit is gradually reducing the time delay between formation of new primordia. The spiral patterns

then go sequentially through all the Fibonacci parastichies.

Occasional excursions to the neighbor local minima produce

Lucas parastichy numbers.

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Further reading

Helices, spirals and phyllotaxis

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