# Internal Stress Controls Tendril Writhing Dynamics in Climbing Plants

Émilien Dilly, Julien Derr, Dražen Zanchi

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Émilien Dilly, Julien Derr, Dražen Zanchi. Internal Stress Controls Tendril Writhing Dynamics in Climbing Plants. 2026. hal-05441223

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# Internal Stress Controls Tendril Writhing Dynamics in Climbing Plants

Émilien Dilly<sup>a,b,c</sup>, Julien Derr<sup>c</sup>, and Dražen Zanchi<sup>b,1</sup>

<sup>a</sup>LiPhy, UMR5588, Université Grenoble Alpes, CNRS, 140 Rue de la Physique, 38402 Saint-Martin-d'Hères; <sup>b</sup>Laboratoire Matière et Systèmes Complexes, UMR 7057, Université Paris Cité, CNRS, F-75205 Paris Cedex 13, France; <sup>c</sup>Laboratoire Reproduction et Développement des Plantes, École Normale Supérieure de Lyon, CNRS, INRAE, Inria, 69364 Lyon Cedex 07, France

This manuscript was compiled on January 5, 2026

We present experimental results on the writhing dynamics of cucum-
ber plant tendrils subjected to an axial traction force. At low forces,
the tendrils' curvature saturates exponentially in time at a limiting
value. At forces higher than the critical threshold, the tendril fails
to coil but develops an intrinsic coiling-prone stress, correspond-
ing to an intrinsic residual curvature, visible if the load is released.
Growth-field assessments indicate that curvature arises from differ-
ential growth.

To understand these experiments we develop an autotropic morphoe-
lastic growth (AMG) model with a bi-strip geometry, in which one strip
undergoes mechanosensitive growth. In the absence of an external
load, curvature generation is solely limited by internal stresses. To
link the actual curvature to a finite external force, the tendril is approx-
imated as a Kirchhoff rod. The resulting model reproduces the two
observed regimes: below the critical axial force, the tendril coils, and
above this force, the tendril remains straight but still exhibits intrinsic
curvature.

Tendrils | Curvature | Morphoelasticity | Kirchhoff rod | Helix | Perversions | Writhing | Instability | Nonlinear | Elasticity | Growth

Cyrano had no admiration for climbing plants in his fa-
mous monologue: "As a dank ivy, which feeds upon a
tree, Climbing by cunning, not by strength? No, thanks!"
Yet, is there not beauty in the ingenuity of climbing plants?
Darwin, with his deep fascination for them, would certainly
have disagreed. Indeed, he wrote a full book dedicated to the
climbing plants (1) where he describes the behavior of the
twining plants like ivy which mechanisms were explained only
recently (2-4).

In his treatise, Darwin mention also at length another kind
of climbers: the tendril-bearing plants. Tendrils are depicted
as slender, irritable structures derived from stems, leaves, or
flower peduncles (5), enabling plants to climb. Examples in-
clude vines, cucumbers, squash, melons, Bryonia dioica, and
passionflowers. Tendrils undergo three main phases: first,
they exhibit nutation (4), exploring their surroundings. Upon
touching a support, tendrils enter a second phase—contact
coiling—during which they curve and wrap around the sub-
strate (4, 6). This phase has been studied extensively (7),
particularly with regard to the physiological cascade linking
contact to growth. The deformation and curvature generated
in the tendril during contact coiling are reversible: if contact
with the support is lost, the tendril can lose its curvature
and straighten itself. The third phase, known as free-coiling
(8) or writhing (9), follows: the tendril segment between the
attachment point and the main stem coils, irreversibly forming
two helices of opposite chiralities linked by a perversion (see
Fig. 1 and Movie S1 available at (10)), a term attributed
to Listing (11). Darwin (1) had already described this phe-

nomenon: “when a tendril has caught a support and is spirally
contracted, there are always as many turns in one direction as
in the other; so that the twisting of the axis in one direction is
exactly compensated by the twisting in the opposite direction.”

In recent decades, perversions and their mechanical behav-
ior have attracted growing interest, as they appear in diverse
systems such as combed hair (12), spider webs (13), and he-
logical elastic rods under rotation (14, 15). (16) and later (9)
demonstrated, through studies of Kirchhoff equations, that per-
versions and helices of opposite chiralities naturally emerge in
intrinsically curved elastic rods. Intrinsic curvature—defined
as the curvature of a filament in its unconstrained state—plays
a crucial role in this instability, leading to helices linked by
a perversion. Goriely and Tabor showed that an elastic rod
with a given intrinsic curvature and zero net twist can either
remain straight or form two helices with a perversion, depend-
ing on the applied axial force at its end (16). This effect can
be replicated by unwinding a helical elastic rod, pulling it
straight, and then gradually releasing tension, allowing helices
of opposite chiralities to form through the so-called *writhing
transition* (15) (see Movie S2). In tendrils, however, intrinsic
curvature—not external tension—is the key evolving factor.
Initially straight, tendrils develop curvature along the seg-
ment between the attachment point and the stem, triggering
the writhing instability (9). Understanding tendril writhing
therefore requires investigating the mechanisms of intrinsic

Significance Statement

Plant tendrils are mechanosensitive and highly motile or-
gans known for touch-induced differential growth. The result-
ing coiling dynamics under external traction reveal that non-
homogeneous internal stress over the tendril cross-section is
fundamental to understanding the system. External loading
inhibits curvature generation and can fully suppress it above a
threshold force. Remarkably, however, the internal differential
stress onset persists even under the highest applied traction
forces.

We develop an autotropic morphoelastic growth (AMG) model,
grounded in a bi-strip geometry and Kirchhoff rod theory, which
capture these main features. In particular, the AMG model re-
produces the observed 1/4 ratio between the generated intrinsic
curvature at high force and that at zero force. According to the
AMG model, this ratio depends solely on the twist-to-bend ratio,
which is a parameter determined by the plant species.

The authors declare no conflict of interest.

<footnote n="1">¹To whom correspondence should be addressed. E-mail: drazen.zanchi@u-paris.fr</footnote>
curvature generation within the tendril.

In various systems, curvature arises through differential
growth, expansion, or contraction—as observed in bimetal-
lic strips (17) and plant stems (18). In tendrils, it has been
hypothesised (19, 20) that curvature develops via asymmet-
ric longitudinal contraction, where the presence of gelatinous
fibers ("g-fibers") and lignified structures induces asymmetric
internal stresses over the tendril cross section. However, to
our knowledge, no direct experimental measurements of ten-
dril growth have been carried out. Investigating curvature
generation in tendrils thus necessitates examining differential
growth dynamics.

Tendril writhing introduces further complexity: both intrinsic curvature and the force imposed at its ends evolve over time. As seen in Fig. 1, tendril writhing deforms the principal stem, pulling it closer to the support. Consequently, the observed curvature and shape imply both elastic deformation and intrinsic curvature evolution. If intrinsic curvature stems from growth, this raises fundamental questions about the interplay between external forces, observed shape, and growth. Mechanical constraints are known to influence growth rates, as observed in maize roots (21) and at the cellular level through the Lockhart model, which inherently links growth to mechanical constraints (22), thereby highlighting the importance of the relationship between differential growth, curvature generation, and external force.

Even in the absence of external force, internal residual
stresses within growing tissues may affect curvature generation
(23), as they influence growth itself. Modeling plant curvature
generation must account for such effects, as demonstrated in
(24), where an auto-tropic term countering curvature genera-
tion was necessary to match experimental observations. This
suggests that even without external constraints, limitations
on curvature generation—and consequently on differential
growth—persist. Therefore, understanding tendril writhing
also requires studying cases where external constraints are
negligible. Only then, a study including the effects of the
external force on tendril writhing can be performed. Rec-
ognizing that external force is a key parameter to tune the
system, MacDougal devised an experiment in which tendrils
were subjected to varying weights (8). He found that beyond
a critical weight, tendrils failed to coil. MacDougal's century-
old experiments remain foundational in modern research on
plant growth, and his findings still prompt the question of
the intricate relationship between mechanics and plant growth
during development.

With its simple geometry, the tendril provides a convenient
system to study the interplay between growth, curvature gen-
eration, and external constraints. In this paper, our approach
is to manipulate experimentally the tendrils in varying con-
trolled conditions in order to infer a biophysical model for
curvature generation in growing stems like tendrils. Three
natural experimental configurations emerge: first, fixing the
distance between the tendril's attachment point and its base to
monitor the axial force generated by tendrils. Second, study-
ing curvature generation in the absence of external constraints,
to record the *basal* response of the system. Finally, perturbing
the system by tuning external force and examine subsequent
curvature generation.

Our results indicate that tendrils are able to generate,
via writhing, forces of magnitudes much higher than their

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Successive views of tendril writhing in cucumber. After grasping a support, the tendril segment between the attachment point and the principal stem coils. Two helices of opposite chiralities appear linked by a perversion, pulling the plant stem closer to the support.</figcaption></figure>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Experimental results on tendril writhing. (A) Writhing at a fixed clamping distance: axial force over normalized time time ($t_{max}$ = 20 h) for five tendrils, normalized to a maximal force $T_{max}$, whose average value reads $\bar{T}_{max}$ = 32 mN, and time $t_{max}$ whose average value reads $\bar{t}_{max}$ = 20 hours. Pink: averaged force; blue: data dispersion. (B) Writhing at low force: curvature evolution in the first coil, normalized by $\kappa_g$ = (1.08 ± 0.10) × 10³ m⁻¹ for six tendrils. Exponential fit $\kappa_g$ × [1 − exp(−$t/\tau$)], with normalized time $\bar{t}$ = $t/\tau$, where $\tau$ = (5.7 ± 5.1) h. Pink: 1 − exp(−$t$); blue: data dispersion. Inset: semi-log plot of 1 − $\kappa_0/\kappa_g$ vs. 1 − exp(−$\bar{t}$). (C) Writhing at fixed force: intrinsic (with mass) and actual (after mass removal) curvatures vs. force. Blue: bi-strip model prediction, scaled with $\kappa_g$ (obtained from experiments in (B)) and measured bending rigidity $B$ = (2.2 ± 0.9) × 10⁻⁷ Pa·m⁴ and twist-to-bend ratio $\Gamma$ = $C/B$ = 0.51 ± 0.04 (See S.I. section 4). Blue shaded area: theoretical error estimation based on experimental dispersion of $\kappa_g$, $B$, and $\Gamma$ (See S.I. section 4).</figcaption></figure>

<figure><img src="image_6.png" /></figure>

own weight. The *basal* (without external force) dynamics of curvature generation is exponential, and so saturates to an asymptotic *basal* value. By tuning the external force, we recover the qualitative findings of MacDougal, but we show that even above the threshold where writhing is not observed, the tendrils still generates intrinsic curvature. We are able to explain the quantified experiments by a biophysical model without any fitting parameter. The tendril is modeled as a bi-strip system governed by Kirchhoff's rod equations, and where local growth is dependent on local stress in the tissue. Our model yields a general equation for curvature generation under applied force. Although our model is phenomenological, its strong predictive power allows to discuss its assumptions and potential applications in understanding the role of residual stresses in limiting curvature generation.

## Materials and Methods

Cucumbers (*Cucumis Sativus*) were cultivated from seeds in 1000 cc pots in a temperature-regulated room (26 ± 3 °C), isolated from natural daylight, with a day/night cycle of 16 hours of daylight on a 24-hour cycle. Cucumbers were watered daily with 10cc of water.

**Writhing at fixed distance.** To monitor writhing behavior under a fixed elongation between the cucumber branch and the tendril's attachment point, a cucumber branch was clamped along with the base of a fresh tendril—one that had sprouted from the apex less than 24 hours before the experiment (see Fig. 2A). At the opposite end, near its tip, the tendril was positioned to make contact with a cylindrical wooden rod, initiating contact coiling upon touch. This wooden rod was fixed to a scale that measured the apparent weight of the clamp, enabling the assessment of the axial force generated as the tendril writhed. Axial forces were continuously monitored

throughout the writhing process, and photographs of the tendrils were taken at regular intervals—approximately every five minutes.

**Writhing at zero force.** For experiments investigating writhing under low force, the entire lower apparatus was mounted on a translation stage, allowing the position of the tendril's attachment point to be adjusted relative to the upper clamping point at its base, thereby preventing axial force buildup. As writhing began, the tendril exerted a pulling force on the support, causing the translation stage to shift—approximately every ten minutes (see Movie S3). This adjustment ensured that the axial force recorded on the scale remained constant at its initial value, with a precision of 2 mN. The curvature of the tendril was measured over time in the coil nearest to the attachment point. Specifically, the helix radius $R$ and pitch angle $\theta$ were recorded, and curvature was computed using $\kappa = \sin^2 \theta/R$ (see Supporting Information -SI- section 1).

**Writhing under constant force.** Experiments with non-zero force were conducted by attaching a mass $m$ to the tip of the tendril using a clothespin and paper clips, thereby applying a vertical force $T = mg$ (see Fig. 2C and Movie S4). The clothespin triggered contact coiling, causing the tendril to wrap around the clamp. This experiment was repeated with different masses $m$, ranging from 0.9 g to 20 g for 61 tendrils, clamped at the tip of the tendril. Experiments were stopped after 48 hours. At this point, the curvature of the tendril in the first coil was measured using the previously described method. The clamped mass was then removed, and the helix contracted over a period of 10 minutes, corresponding to the visco-elastic relaxation time of the tissue. The radius of curvature of the tendril was then measured at the point where it was lowest, or equivalently, where the curvature was maximal.

**Anatomy of tendrils.** According to histological cuts in S.I. section 5) we approximate the tendril's cross-section as a rectangle. Measurements of the typical width and thickness (see Fig. 3 for notations) of cucumber tendrils were conducted using a caliper on five different straight tendrils at two curvilinear positions: approximately 2 cm
from the tip and approximately 6 cm from the tip. These measure-
ments yielded $h = 407 \pm 53$ µm for thickness and $w = 696 \pm 105$ µm
for width, respectively. Upon writhing, cucumber tendrils develop
curvature along the direction corresponding to the minimal size
of their cross-section (see SI section 5 for detailed notations and
Fig.3B for a schematic representation).

**Longitudinal growth measurements.** Using the same apparatus, tendrils were subjected to writhing under a fixed elongation. Blue ink was sprayed onto the tendril, creating distinct ink spots on the tissue (see Fig. 3A). Photos were taken approximately every five minutes using a camera. In Fig. 3A, two ink spots (circled in red) are visible at different time points. The center-to-center distance between each pair of ink spots was manually measured to track growth dynamics.

Results

**The tendril is able, by writhing, to generate a force much higher than its own weight in less than a day.** When performing an experiment at fixed distance, as observed previously, upon the writhing instability, two helices of opposite chiralities appear linked by a perversion (Fig. 2A). As writhing continues, the number of coils in both helix continues to increase, causing the perversion to rotate around the helical axis. After approximately 48 hours, the tendril ceased coiling, and the number of turns in each helix remained constant. Fig. 2A illustrates the axial force curve obtained from five different writhing experiments. To enhance readability, the writhing transition was set at $t = 0$, and the axial force was normalized by its maximal value $T_{max}$ for each tendril. Time was also normalized such that the maximal value of the force occurred at $t = t_{max}$. At $t = 0$, there is a sharp increase in force corresponding to the initiation of writhing instability, reaching an average maximum value of 32 mN over approximately 20 hours. This force is remarkable as it is more than one order of magnitude higher than the own weight of the tendril ($\approx 10^{-1}$ mN). Subsequently, the axial force undergoes a gradual decrease over a timescale at least two or three times longer than the timescale of force increase due to writhing.

**At "zero" force, the plant builds up an intrinsic curvature, exponentially saturated in time.** The force was set to zero with a precision of 2 mN, small in front of the encountered axial force in the previous experiments, therefore justifying that these experiments are called at low force. For writhing at low force, Fig. 2B displays the obtained curvature as a function of time averaged over seven tendrils. For readability, curvatures were normalized by their asymptotic values $\kappa_g$, taken as the mean value of the plateau in curvature for $\hat{t} > 3$, where $\hat{t}$ stands for normalized time, which was obtained by fitting an exponential function of the form $1 - \exp{-t/\tau}$ to the curvature data for each tendril. Time was subsequently normalized as $\hat{t} = t/\tau$. The mean asymptotic value of curvature reads $\kappa_g = (1.08 \pm 0.10) \times 10^3$ m$^{-1}$, over a mean timescale of $<\tau> = 5.7$ hours. Timescales present a high dispersion, ranging from 2 to 11 hrs. At time $\hat{t} = 0$, there is an increase in curvature, corresponding to the appearance of the helix due to the writhing transition. Subsequently, the rate of curvature generation slows down and ultimately shrinks to zero. This curve reveals an exponential profile of curvature as a function of time, as illustrated in the inset of Fig. 2C by a semi-log plot of the obtained values of curvature.

**There exists a threshold in imposed force, above which, the tendril doesn't writh, but still generates intrinsic curvature.** Fig. 2C

presents the asymptotic actual and intrinsic curvatures observed during writhing as a function of an imposed force. Two distinct behaviors were identified: for lower force values, between 0 g and 3 g, successful writhing occurred, leading to the formation of a helix ($\kappa \neq 0$). However, beyond a certain force threshold, the tendril remained straight during writhing ($\kappa = 0$) but exhibited significant intrinsic curvature upon mass removal ($\kappa_0 \approx 0.2$ mm$^{-1} > 0$, corresponding to several coils in the tendril). In this regime, intrinsic curvature appeared independent of the axial force imposed between 50 mN and 200 mN. The transition between these two regimes is marked by a sharp discontinuity in actual curvature, with the measured intrinsic curvature being approximately five times higher in the successful coiling regime.

**Writhing is accompanied by differential growth.** Fig. 3B shows the relative variation in length between ink spots on the inner (ventral, red) and outer (dorsal, blue) parts of the helix. Ink-spot pairs were selected based on their longitudinal alignment to ensure that the measured changes reflected tendril longitudinal growth. As the tendril rotates around the perversion, some ink-spot pairs become temporarily obscured. We define $L_i$ as the center-to-center distance between two ink spots at their first appearance in the photos, which does not necessarily coincide with the onset of writhing, and $L_f$ as their last recorded distance. Fig. 3B reports the relative elongation or contraction of these distances for two tendrils, with data from 9 dorsal and 11 ventral ink-spot pairs, yielding an average elongation of 0.24 for the dorsal side and a contraction of 0.03 for the ventral side. While measurements show significant dispersion due to variations in the timing of measurements of $L_i$ and $L_f$, a clear trend emerges: the dorsal side experiences noticeable elongation, while the ventral side undergoes only minor length changes.

Autotropic morphoelastic growth (AMG) model

Motivated by the experimental results, we model curvature generation in tendrils, by starting to construct an elastic bi-strip representation of the tendril, where one of the two strips undergoes growth. Given the prescribed lengths of both strips, we compute the resulting curvature of the system in the absence of external constraints. Next, this equivalent bi-strip model is considered as a Kirchhoff rod, incorporating the intrinsic curvature obtained in the first step. Finally, by introducing a mechanosensitive phenomenological growth model for the dorsal strip, we solve for curvature evolution under various external force conditions. We will see that this model exhibit autotropic properties, in the sense that curvature generation depends on the actual generation.

**The bistrip description.** We model the tendril as two rectangular strips (see Fig. 3C): one grows to a total length $L_d(t)$, while the other remains fixed at $L_0$. Both strips have equal thickness $h/2$ and width $w$. They are assumed to be homogeneous, with Young's modulus $E$ and Poisson's ratio $\nu$, set to zero for simplicity. When it is free of external constraints, the strips form a rod with center-line length $L_*$ and intrinsic curvature $\kappa_0$, governed by the dorsi-ventral mismatch $\chi = L_d(t)/L_0 - 1$ (25). An orthonormal frame $(\vec{d}_1, \vec{d}_2, \vec{d}_3)$ is defined, where $\vec{d}_3$ is the tangent vector to the rod. $\vec{d}_2$ is defined as the unit vector pointing in the direction of thickness $h$. Therefore, when the

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<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Differential growth and bi-strip model (A) Longitudinal growth upon writhing measured using ink spots. (B) Box plot of relative ink spot distance evolution shows outer helix (dorsal, blue in C) growth of 0.24 ± 0.15 and inner helix (ventral, red in C) contraction of 0.03 ± 0.05. Hard lines: Bi-strip model predictions of dorsal and ventral side growth for measured $h\kappa_g = 0.44$ at zero force, predicting a relative final growth of the dorsal side of 0.4 and contraction of ventral side: 0.1. (C) Tendril modeled as two strips: the dorsal strip grows to length $L_d(t)$, ventral remains $L_0$. Growth induces curvature and residual stresses, black arrows indicate the residual elastic stress in the absence of external constraints on the bi-strip. The dorsal strip is globally compressed, and ventral strip extended.</figcaption></figure>

two strips are glued together, they curve around $\vec{d}_1$. The curvilinear length at position $x_2$ on the cross-section in the direction $\vec{d}_2$ is noted $L(x_2, t)$. When placed at some curvature $\kappa$, to preserve the integrity, the fiber length at position $x_2$ varies linearly across the section:

$$L(x_2) = L_*(1 + \kappa x_2), \quad [1]$$

where $L_*$ stands for the length of the center-line of the rod. The elastic longitudinal stress at position $x_2$ can then be expressed as follows:

$$\sigma(x_2) = \begin{cases} E \left[ \frac{L_*(1 + \kappa x_2)}{L_d} - 1 \right], & \text{if } x_2 > 0, \\ E \left[ \frac{L_*(1 + \kappa x_2)}{L_0} - 1 \right], & \text{if } x_2 < 0. \end{cases} \quad [2]$$

In the absence of external load the curvature equals the intrinsic one, $\kappa = \kappa_0$. By integrating the stress given in Eq. (2) over the cross-section, and enforcing global force and torque to vanish, one can solve for $\kappa_0$ and $L_*$ as functions of the dorsiventral mismatch $\chi$ (see Supporting Information, Section 6, for the full derivation), yielding:

$$\left\{ \begin{aligned} L_* &= L_d \frac{8(2 + \chi)}{\chi^2 + 16\chi + 16} \\ \kappa_0 &= \frac{3}{\hbar} \frac{\chi}{2 + \chi}. \end{aligned} \right. \quad [3]$$

A representation of the corresponding stress profile across the section is shown in Fig. 3C, where black arrows indicate that the dorsal strip is predominantly under residual compression. For a given dorsal strip length, we determined the intrinsic curvature of the bi-strip system. The next section addresses the effect of an external force.

**Tendrils as Kirchhoff rods and elastic writhing instability.** The bi-strip is modeled as a Kirchhoff rod with bending rigidity $B$, torsional rigidity $C$, intrinsic curvature $\kappa_0$, and center-line length $L_*$, as described in (25). Under an axial force $T$ and no applied torque—as is the case in experiments presented in Fig. 2C where the clothespin is free to rotate—, at time $t$ the rod adopts a helical shape with curvature $\kappa$ (see Fig. 4A). The relationship between the axial force, curvature, and intrinsic curvature at time $t$ is given by (9):

$$\Gamma^2 B^2 T^2 = [\kappa_0(t) - \kappa(t)(1 - \Gamma)]^3 (\kappa_0(t) - \kappa(t)), \quad [4]$$

for $T < B\kappa_0(t)^2/\Gamma$, while $\kappa(t) = 0$ otherwise. $\Gamma = C/B$ is the twist-to-bend ratio. Equation 4 can be seen as a constitutive equation $\kappa(\kappa_0, T)$ at time $t$. The resulting curve, linking $\kappa$ with $T$ and $\kappa_0$ is shown in Fig. 4A. When the axial force exceeds the critical threshold $T_c = B\kappa_0(t)^2/\Gamma$, the rod straightens, marking the onset of what we hereafter refer to as the elastic writhing instability. Similarly, if the tendril is placed at a fixed axial force $T$, the generation of intrinsic curvature $\kappa_0(t)$ can induce the elastic writhing instability, if the intrinsic curvature of the tendril remains below the critical value $\sqrt{T/(B\Gamma)}$ (blue shaded area of Fig. 4B), the tendril remains straight -even though it is intrinsically curved, i.e. curved in the absence of external force-, but if the generated intrinsic curvature exceeds this critical value, the tendril will be coiled at curvature $\kappa$ under axial force $T$.

For a given length of the dorsal strip $L_d(t)$, we have derived the expression for the intrinsic curvature and the relationship between the axial force and the actual curvature. The next step is to establish an evolution equation for the length of the dorsal strip, depending on external force.

**Growth dynamics and mechanosensitivity hypothesis.** We suppose that passive visco-elastic relaxation in tendrils occurs on a timescale and much shorter than the timescale of writhing,
which is on the order of 10 hours. Thus, we assume that the system is at elastic equilibrium at any given time. Corroborating standard assumptions as in the context of e.g. Lockart-Ortega approach at the cell scale (22), we assume that the growth rate of the dorsal side, at the tissue scale, is stress-dependent—if the dorsal side is compressed, its growth rate is reduced. However, as shown in Fig. 3C, under a prescribed dorsi-ventral mismatch, the dorsal side may be either under compression or extension, depending on the position across the cross-section. Since we are concerned with the overall growth of the dorsal side, we hypothesize that this growth is sensitive to the stress averaged across the cross-section. In its simplest form, we assume that compression leads to a linear decrease in growth rate, corroborating standard assumptions and observations in the context of plant growth (21, 22). Assuming constant growth rate in the absence of stress, the equation governing the dynamics of $L_d(t)$ reads:

$$ \frac{1}{L_d(t)} \frac{dL_d}{dt} = a_0 + a_1 \langle \sigma \rangle_{x_2>0}, \quad [5] $$

where $a_0$ and $a_1$ are constants, and $\langle \sigma \rangle_{x_2>0}$ represents the mean elastic stress acting on the cross-section of the dorsal strip. This stress is expressed in next section as a function of the intrinsic curvature, the actual curvature, and the external axial force.

**Curvature generation dynamics.** In the limit of small growth $\chi \ll 1$, by linearizing Eq. (3), the intrinsic curvature is given, at a prescribed dorsi-ventral mismatch $L_d(t)$, by

$$ \left\{ \begin{array}{l} L_* = L_d \left(1 - \frac{\chi}{2} + o(\chi)\right) \\ \kappa_0 h = \frac{3}{2} \chi + o(\chi). \end{array} \right. \quad [6] $$

If the tendril is placed at curvature $\kappa$, with corresponding axial force $T$, the local stress at position $x_2$ can be expressed using Eq. (2). Inserting the linearized expressions of Eq. (6) in the stress on the dorsal strip yields at low curvature, $(\kappa h \lesssim \kappa_0 h \ll 1)$,

$$ \langle \sigma \rangle_{x_2>0} = -\frac{E\kappa_0 h}{3} + \frac{E\kappa h}{4} + o(\kappa_0 h), \quad [7] $$

A detailed derivation is given in section 6 of S.I. and in section 7 for a bistrip of with different thickness. The mean stress acting on the cross-section of the dorsal strip depends on both $\kappa_0$ and $\kappa$. In the absence of external constraints ($T=0$ or equivalently $\kappa = \kappa_0$), the dorsal strip still experiences mean compression, indicating residual internal stress even without external forces. When an axial force is applied, $\kappa$ becomes smaller than $\kappa_0$, and the mean compression in the dorsal strip increases as $\kappa$ decreases. This means that if the helix is stretched, the compression in the dorsal strip intensifies. Consequently, under axial force, the mean compression of the dorsal strip increases, leading to a reduction in growth.

Inserting the obtained mean compression in Eq. (5), at linear order in curvature, yields:

$$ \frac{d\kappa_0}{d\hat{t}} = \kappa_g - \kappa_0 + 3(\kappa(T, \kappa_0) - \kappa_0), \quad [8] $$

where $\kappa_g = \frac{12a_0}{Eh a_1}$, $\hat{t} = tEa_1/8$, and $\kappa$ is a function of the axial force at time $t$ through eq. 4 (see S.I. section 8). Interestingly, this equation describing the dynamics of intrinsic curvature

generation in units of $\kappa_g$ is parameter-independent, with the factor 3 arising purely from the geometry of the model, as discussed later and detailed in the S.I. section 8.

**The model predicts the two regimes observed experimentally.** We begin by considering two extreme cases.

First, in the absence of external force ($T=0$), the model predicts that the curvature remains at $\kappa = \kappa_0$ for all times. In this scenario, the dynamics of curvature generation follows an exponential growth, eventually saturating at a value of $\kappa_g$. This behavior reproduces the one observed experimentally (see Fig. 2B). This allows us to infer a value of $\kappa_g$. Similarly, the temporal profiles of the dorsal and ventral strip lengths also exhibit an exponential saturation, with the dorsal side growing and the ventral side contracting, in agreement with experimental measurements, as shown in Fig. 3B. This saturation can be understood as follows: growth on the dorsal side generates intrinsic curvature through the presence of the ventral side. As a result, the dorsal side experiences net residual compression as mentioned above, which slows its growth and limits further curvature development, producing the observed linear saturation.

A similar reasoning applies under axial force. When an axial force is present, the helix's actual curvature is lower than its intrinsic curvature, reaching zero when the rod is fully straight. Unbending an intrinsically curved rod induces compression on the dorsal side and extension on the ventral side. From Eq. (8), when $\kappa < \kappa_0$, an extra term, $3(\kappa - \kappa_0)$, appears compared to the zero-force case, representing the additional slowing of curvature generation due to dorsal compression induced by the tendril's unbending under external force. This effect is naturally maximal when the tendril is completely unbent.

The second extreme case therefore corresponds to $\kappa = 0$ at all times. If a sufficiently large axial force is applied, the rod remains straight. Setting $\kappa = 0$ in eq. 8 results in an exponential generation of curvature, which asymptotically approaches $\kappa_M = \kappa_g/4$, which is the maximal curvature of the tendril without observable coiling. For all times, according to Eq. (4), the rod stays straight provided the axial force exceeds the elastic writhing critical force $T_c(\kappa_0) = B/\Gamma\kappa_0^2$, implying the condition $T > T_c^M = T_c(\kappa_M) = \frac{B}{16\Gamma}\kappa_g^2$, where $T_c^M$ is the critical tendril coiling force. Indeed, for axial forces greater than $T_c^M$, the intrinsic curvature $\kappa_0$ converges to $\kappa_g/4$, and the rod remains straight without observable coiling.

For intermediate cases ($0 < T < T_c^M$), the evolution of the system under three different axial forces is computed numerically and plotted in Fig. 4B. The green ($T/(B\kappa_g^2) = 0.03$) and red ($T/(B\kappa_g^2) = 0.3$) illustrate the aforementioned extreme cases. In the intermediate case (blue line, $T/(B\kappa_g^2) = 0.11$), intrinsic curvature is generated for a period of time, during which the rod remains straight. At a certain time, the intrinsic curvature reaches a threshold that causes the system to cross the elastic writhing instability: the actual curvature $\kappa$ becomes non-zero, resulting in the appearance of coils. As time progresses, the curvatures asymptotically approach values lower than $\kappa_g$ but significantly higher than $\kappa_M$. The bi-strip coils, and the final values of both intrinsic and actual curvatures lie between $\kappa_g$ and $\kappa_M$. As in the experimental observations, one can examine the final values of intrinsic and actual curvatures in the asymptotic state, which correspond to the fixed points of the system described by Eq. 8, for different axial forces.
<figure><img src="image_8.png" /><figcaption>**Fig. 4.** (A) The rod formed by gluing the dorsal and ventral strips is modeled as a Kirchhoff rod with bending rigidity *B*, torsional rigidity *C*, and intrinsic curvature κ₀(*t*). Under axial force *T*, the rod adopts a helical shape. The force-curvature relationship is shown by the black curve (equation 4). Above the critical force *T*<sub>c</sub> = *B*/Γ, the rod is straight (κ = 0), while below this threshold, writhing instability occurs. (B) Curvature dynamics within the bi-strip model: Left: Intrinsic curvature as a function of time for three axial forces (green: *T*/(Bκ<sub>g</sub><sup>2</sup>) = 0.3; red: 0.03 and blue: 0.11). Middle: Actual curvature for the same axial forces. Right: Final intrinsic curvature after writhing as a function of axial force for twist-to-bend ratio Γ = 0.5 measured in cucumber tendrils (Blue line). Above the critical tendril writhing force *T*<sub>c</sub><sup>M</sup> (part of the blue curve in the blue shaded area), the tendril remains straight, while below it, the tendril coils. Black line corresponds to the elastic writhing instability line. The obtained asymptotic curvature is also plotted for varying Γ (red: 1/3 and green 0.3)</figcaption></figure>
**The theoretical bifurcation diagram of the model reproduces quantitatively the experiments without fitting parameter.** The fixed points of Eq. 8 also define the asymptotic values of intrinsic $\kappa_0^\infty$ and actual curvature $\kappa^\infty$ at given axial force $T$, and therefore satisfy

$$ 
\left\{ 
\begin{array}{l}
\kappa_g - \kappa_0^\infty + 3(\kappa^\infty - \kappa_0^\infty) = 0, \\
\text{if } T < B(\kappa_0^\infty)^2 / \Gamma : \\
\Gamma^2 B^2 T^2 = [\kappa_0^\infty - \kappa^\infty(1-\Gamma)]^3 (\kappa_0^\infty - \kappa^\infty), \\
\text{if } T > B(\kappa_0^\infty)^2 / \Gamma : \\
\kappa^\infty = 0.
\end{array}
\right. 
$$

The obtained asymptotic intrinsic curvature curve is shown in the right panel of Fig. 4B for the twist-to-bend ratio $\Gamma = 0.5$ measured in cucumber tendrils (see Supporting Information). The diagram displays two stable branches (solid blue) separated by an unstable one (dashed). The upper stable branches ($\kappa_0^\infty \neq \kappa_M$) correspond to coiled conformations, whereas the lower horizontal branch ($\kappa_0^\infty = \kappa_M$) represents straight conformations ($T > T_c^M$) with nonzero intrinsic curvature.

As expected, the bifurcation diagram intersects the elastic writhing instability line at $(\kappa_M, T_c^M)$. For sufficiently high forces ($T > T_c^M$), the elastic writhing instability is not reached, and the intrinsic curvature remains too small to induce coiling. At lower forces, the elastic instability threshold is crossed, and the tendril coils (black arrows indicate the corresponding trajectories in the $(\kappa_0, T)$ space).

The unstable branch leads to a discontinuous jump in the asymptotic intrinsic curvature. Starting from a straight tendril, intrinsic curvature develops progressively from zero; if $T > T_c^M$, it converges to the fixed point $(\kappa = 0, \kappa_0 = \kappa_M)$. By contrast, for values just below the threshold ($T < T_c^M$), the trajectory switches to the upper branch. This sub-critical bifurcation thus produces a finite jump in intrinsic curvature as the axial force crosses $T_c^M$. In our model, the super-critical or sub-critical behavior of the tendril solely depends on its twist-to-bend ratio $\Gamma$.

The bifurcation diagram is also shown for different twist-to-bend ratios $\Gamma = C/B$ (blue: 0.5, red: 1/3, green: 0.3). For values of $\Gamma$ less than 1/3, as the axial force increases, the asymptotic value of intrinsic curvature decreases monotonically, resulting in a supercritical diagram. This can be demonstrated by linearizing eq. 8 around the critical point $(\kappa_M, T_c^M)$ for the coiled conformation. Setting $T = T_c^M(1+\mu)$ and $\kappa_0 = \kappa_g(\frac{1}{4} + \epsilon)$, with $\mu \ll 1$ and $\epsilon \ll 1$, expanding the axial force (Eq. 4 at quadratic order in both $\epsilon$ and $\kappa$, keeping only linear terms in $\mu$) leads to the following normal form:

$$ \frac{d\epsilon}{dt} = \frac{3\mu}{2(3\Gamma - 4)} + 4\frac{1 - 3\Gamma}{3\Gamma - 4}\epsilon + \frac{72\Gamma^2}{(3\Gamma - 4)^3}\epsilon^2, \quad [9] $$

showing the transition from supercritical to sub-critical behavior at $\Gamma = 1/3$.

The obtained bifurcation diagram for intrinsic and actual curvatures, plotted alongside experimental data for $\kappa_g$ (measured at zero force), $B$, and $\Gamma$ (through cantilevered vibration experiments see Supporting information), shows good agreement with the experiments. It is worthy of noting that no fitting parameters were used for this curve. The tendril writhing critical force $T_c^M$, above which the tendril remains straight, is accurately predicted. The model predicts that the ratio of generated intrinsic curvature between the zero-force

$\kappa_g$ and straight regimes $\kappa_M$, which is quantitatively agreeing experiments. Importantly, according to the model, intrinsic curvature is generated even when the tendril remains straight under force, as observed in experiments.

## Discussion

As highlighted in previous studies (26–28), the twist-to-bend ratio $\Gamma$ plays a crucial role in biological structures, including in our model. This ratio directly influences writhing behavior under external force. In cucumber tendrils, experiments reveal a finite jump in intrinsic curvature as a function of imposed axial force—a phenomenon our model predicts when $\Gamma$ exceeds 1/3. This is consistent with experimental measurements of $\Gamma \approx 0.5$ for cucumber tendrils, as well as with observations of perversion overwinding behavior (19). In contrast, MacDougal’s study (8) on Passiflora tendrils under force does not report such a curvature jump, though it does confirm the existence of a critical force above which no coiling occurs. If the twist-to-bend ratio of Passiflora tendrils is indeed below the critical threshold of 1/3, as suggested by their perversion overwinding behavior (19), our model would predict a continuous variation of asymptotic curvature with axial force rather than a discontinuous jump. Further experiments measuring intrinsic curvature and twist-to-bend ratios in Passiflora tendrils could help confirm whether they belong to a supercritical writhing regime.

Over longer timescales (greater than two days), the axial force exerted on the attachment point due to curvature generation gradually decreases (see Fig. 2A). Eventually, the configuration under axial force relaxes into the intrinsic conformation, a process described as intrinsic curvature relaxation in (29). This long-term fixation could be associated with the formation of longitudinal lignified bundle structures, reported in cucumber tendrils (19) and other Cucurbitaceae species (30), which are initially absent before writhing. The appearance of such stiff bundles may act as a reinforcing substructure, fixating the tendril in its intrinsic conformation and thereby eliminating external force. Whether lignified structures develop simultaneously with writhing or arise afterwards remains uncertain. The gradual decrease in axial force over long timescales suggests that fixation occurs after writhing. In this case, lignification would not participate directly in writhing but instead provide a long-term fixation process.

However, (19) offered an alternative interpretation, suggesting that lignification may itself act as a driver of differential growth. They reported that lignified fibers emerge predominantly in the ventral side of the tendril. This asymmetric stiffening could constrain expansion in the ventral region, while growth continues unimpeded in the dorsal region, where such bundles are absent. Consequently, the dorsal side would, relatively to the ventral, side elongate, thereby producing the differential growth pattern observed in our experiments.

Our model simplifies the tendril’s structure by assuming two strips of equal thickness and identical elastic moduli. However, this could be refined to account for more complex internal structures, such as the aforementioned lignified bundles. These could be represented as a thin, high-modulus ventral strip, which would further restrict dorsal growth while still producing qualitatively similar effects. Notably, the curvature ratio $\kappa_g/\kappa_M = 4$ between the two regimes arises from a geometrical factor, which depends on the relative thickness and elastic
moduli of the two strips. Computations with varying thick-
nesses (see Supporting Information ) yielded curvature jump
ratios ranging from 0 to 5, but this ratio stays robustly be-
tween 4 and 5 as long as the growing tissue is larger than the
other one. Despite the tendril's heterogeneous and composite
internal structure, our anatomical cuts confirm that the dorsal
strip is indeed larger than the ventral strip. (See S.I. section
5 and 7)

In our model, curvature generation slows due to the passive
elastic response of non-growing tissue, which leads to residual
stresses. While this is a simplification, it underscores how
non-growing tissue influences and potentially limits curvature
generation. This entirely passive mechanism raises an intrigu-
ing question: in other curvature-generating plants, could the
autotropic response—where the stem "senses" its own curva-
ture (24)—also emerge from nonlinear growth profiles that
induce residual stresses, as shown in (23). Our model ac-
counts for both residual stresses and those induced by external
forces—here, in the context of helical coiling. However, the
rate of curvature generation could also be expressed under
different force fields, under gravity for instance, with a poten-
tial dependence on the arc-length along the rod. Testing the
robustness of Eq. 8 across different tendril-bearing species
and other curvature-generating plant systems would be an
interesting avenue for future research.

Conclusion

We investigated tendril writhing under controlled axial forces
and clamping conditions, both experimentally and theoreti-
cally. At low forces, intrinsic curvature saturates exponen-
tially. At higher forces, two regimes emerge. Below the critical
writhing force, the tendrils coil successfully. Above this force,
however, they remain straight, although differential growth
occurs and intrinsic curvature is generated.

We propose the autotropic morphoelastic growth (AMG)
model. Based on a simple bi-strip model that incorporates
differential and stress-dependent growth, the AMG model
captures the dynamics of writhing and reduces to a one-
dimensional dynamical system. The internal stress generated
within the tendril explains why curvature saturates, even in
the absence of an external force. Without fitting parame-
ters, the model reproduces the abrupt transition between the
coiling and non-coiling regimes, as well as the observed cur-
vatures. The system's behaviour is solely governed by two
tendril parameters: twist-to-bend ratio and the asymmetry
of the bi-strip thickness. Experiments on other species, with
different tendril parameters, could help determine whether the
AMG mechanisms are universal.

**ACKNOWLEDGMENTS.** The present research is a part of Dy-
naVine action supported by the IdEX Université Paris Cité, ANR-18-
INDEX-0001. This project has received financial support from the
CNRS through the MITI interdisciplinary programs through its ex-
ploratory research program. The authors would like to acknowledge
S. Neukirch, B. Moulia and P. Durand for fruitful discussions.

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