The mechanics of climbing and attachment in twining plants

Alain Goriely∗Department of Mathematics and Program in Applied Mathematics, University of Arizona, AZ85721

Sebastien Neukirch†Laboratoire de Modelisation en Mecanique, CNRS & Universite Pierre et Marie Curie, Paris, France

(Dated: To be published in Phys. Rev. Lett.)

Twining plants achieve vertical growth by revolving around supports of different sizes on whichthey exert a pressure. These well-known observations raise many intriguing questions that canbe addressed within the framework of elastic filamentary structures. Here, mechanical aspects ofthis problem are investigated by modeling the stem close to the apex as a growing elastic rodwith intrinsic curvature and twist. The analysis reveals that vertical growth is achieved thanks tothe presence of discrete contact points as well as regions with continuous contact. Further, theexperimentally observed fact that the plant stem is under tension is shown to be directly related tothe positive contact pressure. Finally, the maximal radius of the pole around which a twiner canclimb is identified.

PACS numbers: 46.70.Hg,46.32.+x,89.20.-a

Climbing plants have developed a fascinating array ofmechanical strategies to achieve vertical growth with-out being able to support themselves. Hookers, lean-ers, weavers, rooters, stickers, clingers, tendril-bearers,or twiners are just a few realizations of the 30 differentways vines manage to grow by taking advantage of theirsurrounding [1]. Twiners, such as garden peas, climb-ing jasmines, and morning-glories, are perhaps the moststudied of all vines [2]. The growing tip waves around ina circular motion known as circumnutation until it findsan appropriate upright support and then start wrappingaround it to extend upward. The tip of the vine keeps nu-tating and the vine pursue its climbing process by form-ing a spiral around the support. The growth process oftwining plants raises many interesting mechanical ques-tions already noted by 19th century botanists and furtherstudied by Silk, Holbrook and co-workers [3–7].

Can a given twining plant climb around supports ofdifferent sizes? This question was first raised by CharlesDarwin in his book on The movements and habits ofclimbing plants[8]. In there he comments: “Most twin-ing plants are adapted to ascend supports of moderatethough of different thicknesses. Our English twiners, asfar as I have seen, never twine round trees...”. As an ex-ample, Darwin noted that Solanum dulcamara can twinearound supports of 3mm but not on supports of 5 or 6mm(see many other examples in [9]). The natural questionis then to determine the critical cylinder radius abovewhich a plant is no longer able to twine. In the processof establishing themselves on a pole, twining plants relyon friction[2, 7]. As noted by Julius von Sachs [10], thevine may slide off “...if the surface of their vertical sup-port is too smooth to furnish a strong mutual friction”.What is the effect of friction in the vine ability to graspthe pole? Similarly, what is the pressure generated by aplant on the pole? How does it change with its intrin-sic properties and shape? Whereas most plants such as

trees or flowers stems are in compression, a peculiar fea-ture of twining plants is that their stem is in tension [4].How is this tension generated? There is no applied loadat the tip of the growing plant, gravity only increasescompression and although the stem can build compres-sive and tensile domains through differential growth thenet effect, once averaged over the cross-section, vanishes.Therefore, a vine in continuous contact with a pole can-not generate tension. As we will see, tension is actuallyproduced when the plant establishes discrete points ofcontacts which create anchorage points. The purpose ofthis Letter is to identify through simple mechanical ar-guments how twining vines establish themselves, developdiscrete and continuous contacts, and to answer Darwin’squestion on the critical pole radius.

a

b

FIG. 1: Blue Bindweed (Ipomoea purpurea), a typical twiningplant (illustration from Sachs’ physiology of plants [10] ). Theshoot apex spans from a to b.

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Most authors have studied the helical shape of thetwining vine around the pole. By contrast here we focuson the formation of these helices by looking at the waythe tip of the vine manages to grasp the pole. The vinebefore lignification is a long, thin, elastic filament subjectto twisting and bending. Due to the small linear den-sity of the vine and the large stresses developed throughself-contact, the gravitational load on the vine has beenfound experimentally to be negligible by comparison toother forces involved in the problem (for instance the lin-ear weight in Pharbitis nil is about 0.4 mN cm−1 but, itcan exert a contact force of 300 mN cm−1 [4] ). Thereforeit will be omitted in the analysis of the grasping prob-lem. During the circumnutation process, the vine at theapex develops intrinsic curvature and torsion. We as-sume these curvatures to be constant and uniform. Thisassumption is also consistent with the vertical and lateraloscillation of the vine tip observed during growth.

It is therefore reasonable to model the vine as a uni-form inextensible and unshearable elastic rod with circu-lar cross-section, constant intrinsic curvature and twist,in possible contact with a cylindrical support. Sincegrowth is slow with respect to other time scales in theproblem, the attachment problem consists in finding pos-sible equilibria of the rod on the cylinder with appropri-ate boundary conditions. An inextensible and unshear-able rod of length L may be represented by its centerliner(s), where s is the arclength (0 ≤ s ≤ L), and an or-thonormal basis formed by the tangent vector d3(s) = r′,and two vectors d1(s), d2(s) representing the orientationof material cross sections. A complete kinetic descriptionis given by: d′

i = u×di, i = 1, 2, 3 where ( )′ denotes thederivative with respect to s and u is the strain vector.The two first components of u are related to the Frenetcurvature κ =

√u21 + u2

2 and u3 is the twist which com-prises both material twisting and Frenet torsion τ . Thestresses in a cross section at s can be averaged to yield aresultant force n(s) and resultant moment m(s) acting atr(s). The balance of linear and angular momenta yields[11]

n′ + f = 0, m′ + r′ × n = 0, (1)

where f(s) is the body force per unit length applied on thecross section at s (body couples are ignored here). Thebody force is used to model the effect of contact withthe support. To close the system, we assume the lin-ear constitutive relations m = Bu1d1 + B(u2 − u2)d2 +C(u3 − u3)d3, where u2 and u3 are, respectively, the in-trinsic curvature and twist of the rod, and B and C arethe bending and twist rigidities. To gain some insight onthe problem, we first consider a simple two-dimensionalversion where the vine, rather than twining around apole, is restricted in the plane. In this case u2 = 1/Rand u3 = 0. The filament is assumed to be clamped atits base (s = 0), that is r(0) = (R, 0), d3(0) = (0, 1).That is, R, the natural radius of the vine, corresponds to

nL

ere!

y

xs=0

s=L

R

simple support

clamping

"

FIG. 2: Diagram of the apex region of a climbing twiner

the radius of the vine when taken away from its support(see [4] for the experimental procedure that provides thevalue of R). The clamp models the constraint applied bythe lower part of the plant on the shoot apex. The tipat s = L lies on the disc and no moment is applied toit so that its curvature is equal to the intrinsic curvatureu2(L) = u2. We also require the external force at L to beradial (see Fig. 2). For each length L and ratio of radiiρ = R/R, these boundary conditions ensure the existenceof a discrete set of solutions. Therefore solutions can beobtained numerically by traditional shooting methods forboundary-value problems: starting with the initial valuesat s = 0 (r(0) = (R, 0), d3(0) = (0, 1),m(0), and n(0) tobe given by an initial guess), Eq. (1) is integrated witha Runge-Kutta alogrithm, up to s = L where we checkthe end conditions. If they are not satisfied, we adjustthe values for m(0) and n(0) until the computed solu-tion satisfies the boundary conditions. Once a solution isknown, the process of growth on the disk is carried outby finding solutions with increasing length, using param-eter continuation. For each solution, we track the angleα that the tip makes with the tangent to the disk (seeFig. 2). We refer to the portion of the filament off diskas the anchor. For small ρ, a typical bifurcation diagramwith distinct equilibria branches is shown in Fig 3. Onthe first branch (continuous line) and for small enough L,we find stable solutions which can be continued up to a(fold) point where they first penetrate the disk. This is abifurcation point where we identify another branch (thevertical line in Fig. 3) corresponding to solutions havinga segment in continuous contact with the disk in addi-tion to the anchor part. The length of the continuouspart can be extended arbitrarily while the anchor doesnot change shape. The remaining upper part of the firstbranch (dotted curve) corresponds to unphysical configu-rations penetrating the disk. All solutions on the branchcorresponding to large values of α (dash-dotted curve)

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are either unphysical or dynamically unstable (as deter-mined by computing the linearized dynamics around theconfiguration [12]). For larger values of ρ there are alsotwo branches of equilibria. On the lower branch, thereexist stable configurations for 0 < L < Lmax. At Lmax

a fold point is reached with a loss of stability and a dy-namical jump occurs to configurations where the filamentrolls on itself and leaves the disk (see Fig. 4). This rolled-up solutions can have arbitrary length (see vertical linein Fig. 4). These solutions are not viable for a twiningplants since they do not allow the vine to grow aroundthe pole by increasing its length. The bifurcation be-tween these two behaviors defines the maximal radius onwhich a vine can grow and is found to occur at the uniquevalue ρc ' 3.3 where the two branches of equilibria cross.That is, in the plane a vine cannot grow on disks thatare more than 3.3 times larger than its natural radius.

In the section of the filament in continuous contact

0.5 1 1.5 2 2.5 3 3.5 4

α

1

2

3

4

5 L

FIG. 3: Below the critical curvature ratio, 3 = ρ < ρc, fil-aments can grow by first changing its inclination at the tipthen by adding a continuous segment (in red) of arbitrarysize. Dotted and Dash-dotted curves represent curves thatare either unstable (do not remain attached on the pole) orunphysical (where the vine enters the disk). They are howeveruseful to understand the bifurcation.

0.5 1 1.5 2 2.5 3 3.5 4

απ

1

2

3

4

5 L

Lmax

FIG. 4: Above the critical curvature ratio 3.5 = ρ > ρc,filaments attachment on the disk are limited in length. Pasta certain Lmax value, the filaments rolls up on itself and doesnot manage further growth to keep growing around the pole.

with the disk the radial component of the force vanishes.

From Eq. (1), we see that the tension n3 = n·d3 balancesthe contact force p = f · er so that n3 = R p. Since thesupport can only provide repulsive contact force (p > 0),the previous relation implies that the filament must be intension (n3 > 0). This tension is provided by the point-wise contact force at the end s = L of the filament andtransmitted by the anchor to the part [of the filament]in continuous contact with the disk. Further use of themoment balance and of the constitutive relation yields

p =B

R2

(1

R− 1

R

), (2)

from which we see that the condition ρ > 1 must besatisfied for p to be positive.

The analysis so far has been restricted to the friction-less case. However, it has been emphasized by differentauthors [2, 7] that friction plays an important role in theability of a climbing plant to support its weight. Herefriction is modeled by a tangential component of the diskreaction force on the tip of the anchor. To understandits effect, we first consider an equilibrium configurationobtained in the absence of friction. We then change thelength of the filament without moving the tip. This ismade possible by introducing a tangential component inthe reaction force along eθ at s = L due to friction. As Lkeeps increasing (resp. decreasing), the tangential forcecomponent reaches a critical value equal to µ p whereµ < 0 (resp. µ > 0) is the friction coefficient. Pastthis value the tip slips incrementally on the disk to finda new nearby equilibrium configuration where the tan-gential component equals the critical tangential value.Therefore, to identify the equilibrium configurations withfriction we increase the length and find configurationswhere the tangential force equals µ p. For each frictioncoefficient µ we compute, as detailed above, the critcalvalue ρc = ρc(µ) defining the bifurcation between fila-ments that can grow around the disk and filaments thatleave the disk. The results in Fig 5 show that friction canhave a crucial effect on the ability of a plant to remain ona large disk. Note that in the case of a twining plant, oncethe tip is in contact with the pole, friction prevents theplant from sliding back, therefore corresponding to posi-tive values of µ. Intuitively, one can understand Darwin’sproblem as follows: for thin supports the vine tip and thetangent of the support are almost parallel and the vinefollows the support without curling back on itself. Forthicker supports, in order to maintain its grip, the tipneeds to touch the support surface at a larger angle α.If the support becomes too large, so will the angle α andthe vine will curl back on itself. For increasing frictionthe contact angle needed for this to happen will be higherand consequently the plant can wind on thicker supports.

We now turn our attention to the three dimensionalcase. The main properties found in the two-dimensionalcase still hold, namely, discrete points of contacts existbetween plant and support, tension is generated by these

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-1 -0.75 -0.5 -0.25 0.25 0.5

μ

ρc

2

4

6

8

FIG. 5: Change of critical radii ratio as a function of friction.Positive friction coefficient acts when the tangential force pre-vents the filament from sliding back whereas negative frictioncoefficients correspond to situation where friction prevents thetip from slipping forward. Range of µ values are consistentwith experimental data in ref [7]. The prediction of an upperbound of ρc around 8 is consistent with the observation ofBell [13].

contact points, and attachment is possible only by differ-ence between intrinsic curvature and twist and geometryof support. However, there are new subtle difficulties as-sociated with growing a filament on a cylinder. The baseof the apex where the filament is clamped can now havean arbitrary orientation. In our model, this orientation isclosely related to the pitch of the plant helix-like shape onthe pole (it was experimentally observed that the plantadopts an almost uniform helical shape provided thatthe number of gyres is large enough and the support isuniform [13]). Therefore, we investigate the mechanicalorigin of the helical shape on the pole and find that forgiven intrinsic curvature u2 and twist u3, there exists asingle helical equilibrium on a cylinder of radius R withpitch angle β. Solving Eq. (1) in cylindrical geometryyields the trigonometric equation

C(u2 − u2) sin 2β = B(u3 − u3) cos 2β , (3)

where u2 = sin2 βR and u3 = sin β cos β

R are the curvatureand torsion of the helix. As before, this continuous solu-tion lying on the cylinder can only be maintained by thetensile force provided by the anchor. The problem is thento find an anchor starting with the correct pitch angle andtension. In general no such solution can be found. Wethen return to a three dimensional version of the bound-ary value problem: the filament is clamped at s = 0and is simply supported by a radial force at its the tips = L where it contacts the pole. Not surprisingly, the bi-furcation diagram becomes quite complex to study (withmultiple solutions with discrete and continuous contacts).Rather than providing an exhaustive description, we lookfor helix-like solutions, with non-uniform pitch angle, incontinuous contact with the cylinder. The solutions endswith an anchor (see Fig 6). We can now define a ρmax

as the largest value of ρ = R/R such that the contin-uous part of the segment can be extended. Extensivenumerical studies show that the value of ρmax obtained

in the planar case provides an upper bound for the threedimensional case.

FIG. 6: A sequence of three-dimensional solutions to the at-tachment problem. Note the continuous, almost helical, so-lution, followed by the anchor that provides tension in thefilament.

The present analysis shows that a mechanical modelfor anchoring and attachment based on rod theory canprovide a simple explanation for the limitation of twin-ing plants to wind around thick supports and for the roleof friction to boost the plant ability to achieve verticalgrowth. The model also explains how tension is gener-ated and shows that both continuous and discrete con-tact points between stem and pole exist. However, thepresent model does not address the fact that the stem,after it establishes itself on the pole, may adapt its in-trinsic curvature due to stress and contact. We believethis remodeling might be important to obtain an accu-rate picture of the stem shape on the pole, but this nonobvious theoritical treatment must be left for futur work.

This material is based upon work supported by theNational Science Foundation under Grant No. DMS-0604704 and DMS-IGMS-0623989 (A.G.) and made pos-sible by the CNRS, a Fellowship from La Ville de Paris(A.G.) and the Action Concertee Incitative from theFrench Ministry of Research (S.N.).

∗ Electronic address: goriely@math.arizona.edu;URL: http://www.math.arizona.edu/∼goriely

† Electronic address: sebastien.neukirch@lmm.jussieu.

fr; URL: http://www.lmm.jussieu.fr/∼neukirch[1] E. A. Menninger, Flowering vines of the world: An en-

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[2] F. E. Putz and N. M. Holbrook, in The biology of climbingplants, edited by F. E. Putz and H. A. Mooney (Cam-brdige University Press, Cambridge, 1991), pp. 53–78.

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(2005).[8] C. Darwin, The Movements and Habits of Climbing

Plants (Appleton, New York, 1888).

[9] L. Baillaud, in Handbuch des Pflanzenphysiologie, editedby W. Ruhland (1962), pp. 635–715.

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[11] S. S. Antman, Nonlinear problems of elasticity (SpringerVerlag, New York, 1995).

[12] A. Goriely and M. Tabor, Phys. Rev. Lett. 77, 3537(1996).

[13] P. R. Bell, Nature 181, 1009 (1958).