Climbing plants -- Wrapping elastic plant stems around a cylindrical stake

Henri Gouin (AMU; Iusti)

REVIEW 3 major objections 4 minor 16 references

Climbing plants -- Wrapping elastic plant stems around a cylindrical stake

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A static energy minimum predicts that thin climbing-plant stems coil as circular helices with pitch 0.05–0.22, and that stems stop climbing once Poisson's ratio reaches about 0.25.

desk verdict The paper's pitch-vs-Poisson relation is new but rests on a strain measure that includes rigid rotation, so the prediction is an artifact. read the letter →

arxiv 2507.15414 v1 pith:FHJQFT5A submitted 2025-07-21 physics.bio-ph

Henri Gouin (AMU , IUSTI) This is my paper · ORCID

classification physics.bio-ph MSC 74B0574K1092C80

keywords climbingplantshelicalcoilingelasticrodminimumenergyPoissonratioturgorpressurecylindricalstakevariationalprinciple

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks why climbing plant stems wrap around a vertical stake in a helix instead of some other shape. It models the stem as a thin elastic rod coiled on a rigid cylinder and chooses the configuration that minimizes total potential energy — elastic deformation, gravity, and turgor pressure — for fixed stem volume and stake radius. In the thin-stem limit gravity and turgor are negligible, and the minimizer is a circular helix with pitch $a$ in the interval $[0.05, 0.22]$ when Poisson's ratio $\nu$ lies in $[0.05, 0.22]$. For $\nu \gtrsim 0.25$ the minimum moves to $a=0$, so the stem does not climb. The paper's contribution is a parameter-free derivation of helix coiling from a minimal-energy principle, with stem fineness and stake radius entering through the geometry of the helix.

What carries the argument

The central object is the deformation matrix $D$ obtained from the tangent map $\partial x/\partial X^0$ between the straight reference cylinder and the helical configuration, used in the linear-elastic energy density $2E_D = \lambda (\operatorname{Tr}D)^2 + 2\mu \operatorname{Tr}(D^2)$. The Frenet frame of the helix gives curvature $\rho = 1/(R_1(1+a^2))$ and torsion $\gamma = a/(R_1(1+a^2))$, and the energy integrals over $k$ full turns reduce to functions of the pitch $a$. The variational principle of minimum total energy — with gravity and turgor terms kept in the general formula but neglected in the numerical application — selects the pitch. The developable nature of the cylinder justifies the circular-helix ansatz as an extremum of length.

What would settle it

Coil elastic rods of fixed radius and Poisson's ratio around cylinders of several different diameters and measure the helix pitch; the paper's energy is independent of stake radius, so a systematic pitch change with cylinder diameter would falsify the model. A second check is to rotate a coiled stem rigidly as a whole and evaluate the paper's energy formula — a nonzero energy change would show the strain measure is not physically acceptable.

Watch

Extended reading notes

Core claim

The central claim is that the equilibrium shape of a thin elastic stem on a cylindrical stake is the circular helix that minimizes the elastic deformation energy, with gravity and turgor contributions negligible in the thin-stem limit. The paper computes the deformation matrix from the mapping between the straight reference cylinder and the helical configuration, integrates the linear-elastic energy density over an integer number of turns, and obtains the total energy per unit volume as a function of helix pitch $a$ and the Lamé coefficients. Plots of this energy show a minimum at pitch $a \in [0.05, 0.22]$ for Poisson ratio $\nu \in [0.05, 0.22]$; for $\nu \gtrsim 0.25$ the minimum is at $a=0$, meaning the stem straightens. For a fixed stem volume $V$ and cross-section radius $r_0$, the climbed height is $z = (V/(\pi r_0^2))\, a/\sqrt{1+a^2}$, so thinner stems climb higher, while a larger stake radius $R_0$ reduces the winding angle and the number of turns.

Load-bearing premise

The prediction depends on treating the deformation matrix $D$, which contains the rigid turning of the stem's cross-sections, as a small elastic strain; if that is not legitimate, the energy formula and the predicted pitch range do not follow.

Editorial extensions

If this is right

For thin stems with $\nu \in [0.05, 0.22]$, the predicted helix pitch lies in $[0.05, 0.22]$, connecting material elasticity to observed climbing geometry without fitted parameters.
For $\nu \gtrsim 0.25$, the model predicts no coiling ($a=0$), so stems with Poisson ratio above this threshold should fail to climb by wrapping.
For fixed stem volume $V$, the reached height scales as $(V/(\pi r_0^2))\, a/\sqrt{1+a^2}$; decreasing the stem radius $r_0$ increases both stem length and climbed height.
For fixed stem length and pitch, a larger stake radius $R_0$ reduces the winding angle and number of turns, so the stem wraps less around thick stakes.
Gravity and turgor energies are orders of magnitude smaller than elastic energy in the thin-stem regime, so helical coiling is essentially an elastic phenomenon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: separating $D$ into a pure strain plus a rigid rotation would introduce a bending energy that depends on $1/R_1$; this could be tested by coiling identical rods around cylinders of different diameters and checking whether the pitch changes.
Editorial inference: the same energy-minimization scheme could be applied to a stem coiling with no stake or to non-circular supports by replacing the circular-helix ansatz with the appropriate geodesic on the support surface.
Editorial inference: because the model predicts pitch depends mainly on Poisson's ratio, an experiment with elastomer rods of controlled Poisson ratio and fixed radius would give a direct quantitative test of the predicted interval.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper is a simple variational model for the pitch of twining stems around a cylindrical stake. The new thing is a predicted relation between helix pitch and the stem's Poisson ratio, including a threshold near nu=0.25 beyond which the stem stops climbing. That parametric result is not in the cited literature, and the paper presents the geometry clearly. The order-of-magnitude argument that gravity and turgor are negligible for thin stems is fine.

The trouble is the elastic energy. The matrix D built in Section 2.1 is essentially the deformation gradient, not the infinitesimal strain tensor. The formula 2ED = lambda (TrD)^2 + 2 mu Tr(D^2) is only valid for a strain measure that is symmetric and objective under rigid rotation. D here is neither; it is assembled from the rotated coordinate expressions and includes the rigid rotation of the cross-sections. The resulting elastic energy density is independent of the stake radius R1. That is impossible for a rod of fixed length coiled on stakes of different diameters: bending energy per unit length scales as 1/(R1^2 (1+a^2)^2), so R1 must appear. The paper misses the curvature of the helical centerline entirely, so the pitch minimum is an artifact of the chosen strain measure, not a property of the elastic stem.

There is also a circularity problem. The stem is assumed to follow the shortest path on the cylinder, which is a circular helix. So the model does not demonstrate why stems form helices; it only selects a pitch within an already-assumed helical shape. That would be survivable if the energy were correct, but combined with the strain error it leaves no reliable new result.

What the paper does do well is the geometry and the scaling arguments. The statements that a larger stake reduces the number of turns for a given stem length, and that a thinner stem reaches greater height, are correct if not original. But the central pitch-vs-Poisson curve should not be taken seriously in this form.

I would not send this to peer review as is. A competent mechanics referee will see the strain-measure error immediately, and the paper would be rejected. The author could recast the problem with a proper rod theory (Kirchhoff or Cosserat) and the pitch-vs-Poisson idea might then be worth testing. As it stands, it is a desk reject.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a static energy-minimization model for a plant stem wrapped around a rigid cylindrical stake. The centerline is assumed to be a circular helix, justified by a geodesic/developable-surface argument, and the total potential energy is assembled from an elastic deformation energy, gravitational potential, and turgor-pressure work. For thin stems, gravity and turgor are neglected, and minimizing the remaining elastic energy with respect to the dimensionless pitch a leads to the central quantitative claim: an optimal pitch a in [0.05, 0.22] for Poisson ratios in [0.05, 0.22], with no climbing for approximately ν ≥ 0.25.

Significance. The paper is attractively simple and, if the energy functional were correct, it would offer a closed-form explanation of the observed pitch of twining stems and of the roles of stake radius and stem slenderness. The variational setup and the ordering of elastic versus gravitational and turgor energies are clearly presented. However, the central elastic-energy calculation is not sound: the deformation matrix used in the strain-energy density is not the strain of the stated deformation, and the resulting energy is independent of the stake radius R1. Since the curvature and torsion of the helix both scale as 1/R1, this independence is physically impossible for a bent and twisted elastic rod. The numerical pitch predictions in Section 3 are therefore artifacts of an incorrect strain measure rather than robust physical predictions. Because the main quantitative result rests on this error, the paper cannot be accepted in its present form.

major comments (3)

[Section 2.1] The matrix D displayed after the Eulerian coordinate expressions is asserted to be the linear tangent application ∂x/∂X0, but direct differentiation of the mapping in Section 2.1 does not yield this matrix. Derivatives with respect to z0 contain the factor dθ/dz0 = 1/(R1√(1+a²)) and depend on the transverse coordinates x0 and y0, whereas the displayed D contains neither R1 nor x0 or y0. Thus D is not the deformation gradient, and the later treatment of this matrix as a strain tensor is not justified.
[Section 2.2] The elastic energy expression obtained after integration, 4W/SL = ..., contains no R1. For the assumed helix, the centerline curvature is κ = 1/(R1(1+a²)) and the torsion is τ = a/(R1(1+a²)), so a standard rod bending and twisting energy per unit length, e.g. (EI/2)κ² + (GJ/2)τ², scales as 1/R1². An energy independent of R1 cannot represent the cost of wrapping a fixed-length stem around stakes of different diameters; indeed, for a=0 (a horizontal circular loop) the model gives a nonzero energy with no R1 dependence, whereas the bending energy of a circular rod loop is EI/(2R1²) per unit length. The minimization over a in Section 3 is therefore performed on an energy that does not describe elastic rod bending.
[Section 1.2 / Conclusion] Condition (a) assumes the stem lies on a minimum-length curve on the cylinder, and the text infers from developability that the curve is a circular helix. This is not a derivation of helical coiling: a developable surface can be unfolded isometrically, but an arbitrary curve on it does not become a straight line; only geodesics do. The paper therefore assumes the helix and then minimizes energy only over the pitch. The abstract's claim that the model 'demonstrates why plant stems climb mainly on their circular helix-shaped stakes' overstates what is actually derived. The pitch-selection result could still be meaningful if the energy were correct, but the geometric claim itself is not new.

minor comments (4)

[Section 3] In the text, ν is called 'Young's modulus' ('For approximately ν ⪰ 0.25'), while Eq. (2.4) defines ν as Poisson's ratio and Figure 3 labels it 'Poisson's number'. Please correct the terminology consistently.
[Section 2.4] The turgor pressure is modeled as a force PT k applied at the top end, with PT defined as a pressure; the notation switches between Pτ and PT in the same section, and no mechanical justification is given for representing an internal turgor pressure by an end force.
[Section 2.1] The dimensionless parameter a is called the 'pitch' of the helix, but the physical pitch is 2π R1 a; please define the term explicitly so that Figures 2 and 3 are not misleading.
[Section 2.2] The assumption of an exact integer number of revolutions k for a stem of given length L is not generally valid; a stem of arbitrary length will not in general complete an integer number of turns, and this restriction may exclude relevant configurations from the minimization.

Circularity Check

1 steps flagged · score 4.0 of 10

Helix geometry is assumed via the minimum-length postulate, so the qualitative helical-coiling result is an input; only the pitch selection is genuinely derived.

self definitional [Section 1.2, assumptions (a)-(b) and subsequent paragraph on developable surfaces]
"We assume that the ascent of the rod on its support satisfies the following two conditions corresponding to an extremum of length and an extremum of energy as it is done in [9]: a) The plant stem connects two points of the vertical stake along an extremal (minimum) length on its support. ... Since circular cylinders are developable surfaces, the planar development of the (Γ)–curve is a straight line segment. As a result, the plant stem deformed by coiling on the stake is such that the (Γ)–curve, which must be of minimum length, is a circular helix of axis O−→k ."

The stated aim is to 'demonstrat[e] why plant stems climb mainly on their circular helix-shaped stakes,' but the circular-helix form is not an output of the energy minimization: it follows directly from assumption (a), because geodesics on a circular cylinder are exactly circular helices (including degenerate pitches). The energy calculation is then performed only over this pre-imposed family, minimizing with respect to the pitch parameter a. Thus the qualitative conclusion that stems coil in helices is equivalent to the variational ansatz, even though the quantitative pitch range a in [0.05, 0.22] and the threshold at nu >= 0.25 are genuinely derived, independent results.

full rationale

The quantitative core of the paper is not circular: the elastic energy, gravity, and turgor terms are written as functions of the helix pitch a, material parameters (E, nu) are taken from literature rather than fitted to the target pitch, and the minimization over a produces the claimed pitch range and nu threshold. No fitted parameter is relabeled as a prediction, and the self-citation [9] is not load-bearing because the variational postulates are stated explicitly. The partial circularity is limited to the qualitative claim about helical coiling itself: the circular-helix shape is imposed by the minimum-length postulate in Section 1.2, so the model demonstrates the pitch of an assumed helix, not the origin of helical coiling. The objective-strain issue raised by a reader would be a modeling-correctness concern, not a circularity of the derivation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model relies on the assumption that the stem follows a geodesic (a helix) on the cylinder, and on a linear elasticity treatment that misidentifies the deformation gradient as strain. These are domain assumptions that are not independently justified. No new entities are introduced, and no parameters are fitted to data; material constants are taken from the literature.

assumptions (4)

domain assumption The deformed stem centerline is a circular helix because it minimizes length on the cylinder (geodesic).
Section 1.2 states that since the cylinder is developable, the planar development of the curve is a straight line, so the curve is a circular helix. This assumes the stem always follows a shortest path, which puts the helical shape in by assumption.
domain assumption Linear elasticity with infinitesimal strains applies to the helical deformation.
Section 2.2 uses 2ED = λ(TrD)² + 2μTr(D²) with D from Section 2.1, but D has O(1) entries and includes rotation, so the strain is not small.
ad hoc to paper Turgor pressure acts only at the top end as PT k.
Section 2.4 models turgidity as a point force on the stem tip, a very crude representation of internal pressure.
domain assumption The rod has no attachment points to the stake.
Section 1.2 says the stem has no point attaching it to the post; real twining plants often have contact and friction.

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Cite this review

Pith. "Pith review of Climbing plants -- Wrapping elastic plant stems around a cylindrical stake." pith.science (2026). https://pith.science/paper/FHJQFT5A

@misc{pith2026250715414,
  author       = {Pith},
  title        = {Pith review of: Climbing plants -- Wrapping elastic plant stems around a cylindrical stake},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FHJQFT5A}},
  note         = {Machine review of arXiv:2507.15414}
}

read the original abstract

Since Charles Darwin's time, the study of climbing plants on a cylindrical stake has been the subject of numerous articles in plant biology. One of the main ideas for studying the coiling of an elastic plant stem is to consider the growth of the plant stem in terms of evolution over time. However, as this development takes place over a long time scale, the static study alone has not been studied independently. Our static approach requires us to take into account elasticity, turgor pressure and gravity forces in a first analysis. The aim of this article is to present a simplified model demonstrating why plant stems climb mainly on their circular helix-shaped stakes, with the diameter of the stake playing an important role in plant stem ascent, as does the fineness of the stem. To perform this calculation, for a given mass density, we consider the variational principle of minimum energy. For thin plant stems, we can see, in first approximation, that the effect of gravity and turgor pressure can be neglected with respect to the energy of elasticity, and that the bulk of the calculation concerns elasticity terms.

Figures

Figures reproduced from arXiv: 2507.15414 by the authors.

**Figure 2.** From left to right, we have plotted the graphs for cases (a) to (f). [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** The x-axis plots Poisson’s number ν of elasticity in the interval [0, 0.25]; the y-axis plots the corresponding value of pitch a of helix. unit in (2.4)1 and ν is considered with different values: (a): ν = 0.05, (b): ν = 0.1, (c): ν = 0.15, (d): ν = 0.2, (e): ν = 0.22, (f): ν = 0.25. We obtain graphs on [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

O. Ali, I. Cheddadi, B.T. Landrein, and Y. Long. Revisiting the relationship between turgor pressure and plant cell growth. New Phytologist, 238(1):62– 69, 2023

work page 2023

[2] [2]

Berdichevsky

V. Berdichevsky. Variational Principles of Continuum Mechanics: I. Fun- damentals. Interaction of Mechanics and Mathematics. Springer Berlin Heidelberg, 2009

work page 2009

[3] [3]

C. Darwin. The movements and habits of climbing plants . John Murray, 1875

work page

[4] [4]

E. Gianoli. The behavioural ecology of climbing plants. AoB plants , 7:plv013, 2015

work page 2015

[5] [5]

A. Goriely. The mathematics and mechanics of biological growth, volume 45. Springer, 2017

work page 2017

[6] [6]

Goriely and S

A. Goriely and S. Neukirch. Mechanics of climbing and attachment in twining plants. Physical review letters , 97(18):184302, 2006

work page 2006

[7] [7]

Goriely, M

A. Goriely, M. Robertson-Tessi, M. Tabor, and R. Vandiver. Elastic growth models in: Mathematical modelling of biosystems. edited by R.P. Mondaini and P.M. Pardalos, vol. 112, Springer , pages 1–44, 2008. 11

work page 2008

[8] [8]

H. Gouin. The d’Alembert–Lagrange principle for gradient theories and boundary conditions. Asymptotic Methods in Nonlinear Wave Phenomena, p.p. 79-95. Eds. T. Ruggeri and M. Sammartino and A.M. Greco. World Scientific, 2007

work page 2007