Composition Vendor Newgen KnowledgeWorks (P) Ltd. PLoS One plos plosone PLOS One 1932-6203 Public Library of Science San Francisco, CA USA 10.1371/journal.pone.0349246 PONE-D-25-65525 Research Article Physical sciencesMathematicsGeometryCurvature Biology and life sciencesPlant sciencePlant anatomyLeaves Physical sciencesPhysicsClassical mechanicsDeformation Physical sciencesPhysicsClassical mechanicsDamage mechanicsDeformation Physical sciencesMaterials scienceMaterial propertiesElasticity Physical sciencesPhysicsClassical mechanicsDeformationBending Physical sciencesPhysicsClassical mechanicsDamage mechanicsDeformationBending Research and analysis methodsMathematical and statistical techniquesFourier analysis Physical sciencesPhysicsClassical mechanicsMotionVelocity Physical sciencesPhysicsClassical mechanicsPressureHydrostatic pressure Size–curvature constraint in the closing motion of Venus flytrap leaves Size–curvature constraint in Venus flytrap Hirata Michiko Data curation Investigation Methodology Validation Visualization 1 Kang Zichen Formal analysis Data curation 1 ¤ https://orcid.org/0009-0005-4543-6507 Asakawa Hiroki Data curation 2 Suda Hiraku Data curation Writing – review & editing 2 * Toyota Masatsugu Data curation Supervision 2 3 4 Ohashi Yuji Conceptualization Supervision 1 https://orcid.org/0000-0003-3587-5635 Tsugawa Satoru Conceptualization Formal analysis Funding acquisition Investigation Supervision Validation Visualization Writing – original draft Writing – review & editing 1 * ¤ Department of Mechanical Engineering, Faculty of Systems Science and Technology, Akita Prefectural University, Yurihonjo, Akita, Japan Department of Biochemistry and Molecular Biology, Saitama University, Saitama, Japan Suntory Rising Stars Encouragement Program in Life Sciences (SunRiSE), Suntory Foundation for Life Sciences, Soraku-gun, Kyoto, Japan College of Plant Science and Technology, Huazhong Agricultural University, Wuhan, Hubei, China Berardo Alice Editor University of Padova: Universita degli Studi di Padova, ITALY

The authors declare no competing interests.

Current address: Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido, Japan

 * E-mail: suda222@mail.saitama-u.ac.jp (HS); tsugawa@eng.hokudai.ac.jp (ST) 2652026 2026 21 5 e0349246 8122025 2742026 2026 Hirata et al This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Among carnivorous plants, the Venus flytrap (Dionaea muscipula) is known for its rapid (<1 s) trap closure. Although buckling instability, hydrostatic pressure, and hydroelastic coupling have all been proposed to be involved, the nature of this process and the relationship between trap size and curvature remain elusive. Here, we monitored the closure of Venus flytraps and performed micro–CT scanning and 3D reconstruction, revealing that increasing angular velocity was correlated with higher values of a non-dimensional shape index. Based on these experimental data, we constructed a geometric model of the trap that takes leaf orientation into account. We found that leaf curvature is dependent on leaf size, a relationship we denote as a size–curvature constraint. We further propose a curvature design derived from differential deformations of a two-layer model of the leaf, which could be a powerful tool to control the curvatures of soft and bending surface structures in the field of biomimetics.

 http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science JP23H01143 https://orcid.org/0000-0003-3587-5635 TsugawaSatoru http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science JP22J00902 SudaHiraku http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science JP25KJ0714 https://orcid.org/0009-0005-4543-6507 AsakawaHiroki http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science JP24H00565 ToyotaMasatsugu http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science JP25K18499 KangZichen http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science JP25K18427 SudaHiraku http://dx.doi.org/10.13039/501100002241 Japan Science and Technology Agency JPMJCR2121 https://orcid.org/0000-0003-3587-5635 TsugawaSatoru http://dx.doi.org/10.13039/501100002241 Japan Science and Technology Agency JPMJER2403 ToyotaMasatsugu This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI grant numbers JP23H01143, JP22J00902, JP25KJ0714, JP24H00565, JP25K18499, JP25K18427, JST CREST grant number JPMJCR2121, and JST ERATO grant number JPMJER2403. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Data Availability All data files and related rendering files are available from the github (https://satorutsugawa.github.io/flytrap_geometric_model_datashare/). Introduction

Charles Darwin described the Venus flytrap as “one of the most wonderful plants in the world,” perhaps more because of the rapid closing (within 1 s) of its component leaves, which is initiated by a slight touch stimulus, than because of its function of catching insects [1]. This fast closure typically occurs when Venus flytrap leaves receive a second stimulation within 30 s after the first, which can look to observers as if the plants are waiting and watching the insect [2,3]. Given that the plant body does not have an equivalent to animal muscles, how Venus flytraps achieve high closure speeds has been a subject of research and debate [4–6]. Also of interest is how the plants, lacking a nervous system, transmit touch-induced signals; the regulation of Ca2+ concentrations and action potentials have received attention in this regard [7–12]. Unresolved questions include whether there is a condition that controls the ability of the leaves to close and whether closure is accelerated by some form of stress release.

Early experiments demonstrated the trap’s ability to close and the conditions under which this occurs. Darwin observed that the open state is very stable: the trap does not close spontaneously, even under the influence of raindrops and wind gusts [1,13]. The flytrap can close both in air and under water conditions [14]. In addition, seedling traps <1 cm in length can close, but their closure time (also called snapping time) is > 5 s, longer than typical times for adult traps [14]. Uncouplers and blockers of membrane channels inhibit trap closure, making closing very slow [15]. Trap closure is not triggered by sustained displacement of a sensory hair or by deflection that is sufficiently slow, and only one touch could initiate the motion of the leaves if sensory hair deflection is above a threshold [9]. These results indicate that fast closure requires certain conditions: leaf maturity, rapid deflection of the sensory hair, internal biological processes that occur after stimulation, and appropriate regulation of membrane channels.

The outer surface of the trap expands after closure [3,4,16] and the plastic extensibility of the outer surface increases [17], indicating that the crucial factors in fast closure may relate to properties of the outer surface. Therefore, it is tempting to think that the leaves undergo an elastic buckling mechanism, as proposed previously, resulting in a smooth snapping transition from the open to the closed state [4,18]. However, we showed that traps lacking outward curvature also close, and the stretching energy and the curvature energy of the outer surface increase simultaneously, thereby suggesting the possibility of an additional effect separate from the elastic snap–buckling instability [19].

A recently proposed model, the hydrostatic pressure coupling model [15,16,20], posits that the curvatures of the multiple layers of leaves composing a trap may be driven by the independent turgor pressures of the layers. The changes in the trap might thus reflect the time scales of membrane channel opening and subsequent water flow. However, in this model the closure speed does not depend on the trap shape, and thus the model does not take into account the experimental data showing higher closure speeds for larger traps [4]. In the context of plant tropism, there appears to be a geometric relationship between curvature of the material elements and differential growth rates [21]. Therefore, we reasoned that a basic geometric model might explain the relationship between the shape and the deformational effects of the multiple layers.

In this study, we experimentally monitored the closure of Venus flytraps and quantified their non–dimensional geometric parameters, finding a trend relating the shape and the closing speed. We then built geometrical models of the open and closed states of the trap based on micro–CT scanning followed by 3D reconstruction. Analyzing these data, we determined that the trap closure is governed by a size–curvature constraint. We propose that the trap features a curvature-controlling design that underlies trap motion.

 Materials and methods Plant material and imaging conditions

Venus flytrap plants were purchased from Hanadonya Associe in Japan and cultivated in the laboratory with a natural light environment. For three-dimensional quantification of trap closure, two cameras (DC-G9L-K and DC-GH6L, Panasonic). The cameras were located at stereo angles of about 45° and –45° toward the specimen. Calibration was performed using a calibrator of a precisely fabricated cuboid (30 mm × 30 mm × 8 mm) with two slits (6 mm × 3 mm × 30 mm). The recording speed was 60 fps. Trap closure was stimulated by careful human touch manipulation. ImageJ multi-point tool tracking was used to detect and track the feature points. Surface mesh construction from the point cloud was performed using Meshlab software.

 Micro–CT

A Venus flytrap trap leaf was cut at the petiole, and micro–CT data were acquired using a Skyscan1276 (Bruker, USA) under the following conditions: resolution, 1008 × 672 pixels; source voltage, 40 kV; source current, 200 µA; image pixel size, 30.000639 µm; depth, 16 bits; exposure, 90 ms; rotation step, 0.600 degree; frame averaging, 2. Data were obtained from the same trap leaf in both the open and closed states. Three-dimensional reconstructions from the CT imaging data were generated using NRecon (Bruker, USA) and InVesalius (Centro de Tecnologia da Informação Renato Archer, Brazil). The reconstructed models were then converted into solid objects using Meshmixer (Autodesk, USA).

 Elliptic Fourier transformation

We applied elliptic Fourier transformation to the 21 cross-sectional contour data. The contour data contains n-th number of x-coordinates xn and y-coordinates yn of the cross-sectional contour. We calculated the Fourier series expansion for xn and for yn independently.

The x- and y-coordinates can be rewritten as

x(t)=∑n=1N[Ancos(2πntT)+Bnsin(2πntT)], y(t)=∑n=1N[Cncos(2πntT)+Dnsin(2πntT)]

where n is the harmonic number, N is the maximum harmonic number, t is the displacement along the contour, and T is the total displacement. The elliptic Fourier coefficients are

An=T2n2π2∑p=1kΔxpΔtp[cos(2πntpT)−cos(2πntp−1T)], Bn=T2n2π2∑p=1kΔxpΔtp[cos(2πntpT)−cos(2πntp−1T)] Cn=T2n2π2∑p=1kΔypΔtp[cos(2πntpT)−cos(2πntp−1T)], Dn=T2n2π2∑p=1kΔypΔtp[cos(2πntpT)−cos(2πntp−1T)]

where k is the total number of steps around the contour, Δx and Δy are the displacements along the x-axis and the y-axis between points p and p+1, Δt is the length of the step between points p and p+1, and tp is the accumulated length of step segments at point p. We used about 200 data points along the contour and performed the elliptic Fourier transformation with n=40 harmonics.

 3D reconstruction

We used a 3D reconstruction method, direct linear transformation (DLT) based on the detailed description in [20].

 Calculation of the elastic strain energy

We performed elastic energy estimation based on the detailed description in ref. [20].

 Results and discussion The speed of trap closure increases with trap size

To assess the relationship between Venus flytrap trap shape and speed, we experimentally observed the closing motions of traps of various sizes and shapes (n=15). We measured the width W, and height H (Fig. 1a) and the leaf opening angle θ (Fig 1c). θ was measured as the angle between the lines connecting the leaf joint and the base of the trap teeth (Supplementary S1 Fig). The 95% confidence interval for W was 10.8–16.5 (mm), and that for angle was 0.77–1.15 (rad). We observed that traps with width <6 mm or >21 mm did not move, possibly due to leaf immaturity or senescence, respectively (Fig 1b) [14]. Furthermore, the positive angular velocity ωangle=(θopen−θclosed)/2Δt  was distributed within a certain range of leaf widths (defined by the dotted rectangle in Fig 1b). We detected an increasing trend in leaf angular velocity with greater initial leaf angle (Fig 1d); notably, traps with ωangle near π/2 rad closed rapidly (dotted rectangle). To visualize this trend in a size-independent fashion, we adopted a non-dimensional geometric parameter, the stretching–bending ratio α=S4κ2/h2, (where S=(W+H)/2), which is the ratio between the stretching energy flattening the leaf flat and the bending energy resisting that flattening ([4]; Fig 1e). Here, S/h and Sκ represent size–thickness anisotropy and size–warping anisotropy, respectively; the curvature κ was measured at the leaf midline (Supplementary S2 Fig). The speed of closure increased as the shape index (stretching–bending ratio) increased (Fig 1f), consistent with the relationship observed in previous work [4]. These results indicate that rapid closure requires a certain size and angle.

10.1371/journal.pone.0349246.g001Relationships between trap speed and size, leaf angle, and stretching–bending ratio.

(a) Definitions of trap width 𝐖, height 𝐇 and size 𝐒. (b) Leaf angular velocity versus leaf width. (c) Definition of leaf angle θ. (d) Leaf angular velocity versus leaf angle. (e) Schematic defining the stretching–bending ratio. The stretching energy—the energy causing the outside of the leaf to flip outward and become flattened—is denoted by the red vector arrow and the bending energy—the energy causing the inside of the leaf to resist that outward pull and remain curved—by the blue vector arrow. (f) Leaf angular velocity (ωangle) versus stretching–bending ratio (α). Outwardly and inwardly curved leaves are plotted in red and blue, respectively. The p-value of the increasing trend was 0.00373.

The dimensionless index α was adopted as an effective parameter to capture the balance between thickness and elastic modulus. In the context of the flytrap, which consists of hydrated, multilayered tissues, α can be interpreted as reflecting the mechanical response of a bilayer system in which differential strain – potentially arising from turgor, water transport, and cell wall extensibility – drives bending while being constrained by in-plane stretching resistance. Thus, α effectively incorporates the combined influence of tissue hydration, layer thickness, and cell wall mechanics into a single dimensionless parameter.

 Geometric parameters for open and closed trap states can be derived from micro–CT scanning data

Next, to understand the trap shape more clearly, we implemented the following previously published geometric model [18]:

(x0y0z0)=(sin(α)[R0+βH cos(dα)]2Dβ(1−β)cos(π2ααmax)R0cos(α)−R0+βH cos(α)cos(dα))

Here, R0 is the radius of the midrib, H is the leaf height, d is the decreasing degree of height along the midrib, αmax is the degree of leaf waving along the midrib, and D is the degree of leaf bending (i.e., curvature in the height axis). Taking the rotation of the leaf around x-axis at the midrib into account, we introduced an orientational parameter ϕ, which is the rotation angle around the x-axis with the origin at the midrib.

(xyz)=(x0y0cosϕ−(z0−Rcosα+R)sinϕy0sinϕ+(z0−Rcosα+R)cosϕ+Rcos(α)−R)

To extract these geometric parameters, we used micro–CT scanning data fitted to the parameters. From the data for the open (Fig 2a) and closed states (Fig 2b), 21 cross–sectional contours were interpolated by elliptic Fourier transformation (Methods, Fig 2c). Using the interpolations, we aligned the 3D coordinates of the open and closed states (Fig 2d). We estimated R0 from the circle fitting of the midrib, measured H as the leaf height, and estimated d as the difference between the height at the center and the height at edges of the trap. αmax was estimated to be 70π/180 rad. Using these estimates, we obtained the following parameters:

10.1371/journal.pone.0349246.g002Acquisition of geometric parameters from traps in open and closed states based on micro–CT scanning.

(a, b) 3D segmentation (left) and divided cross-sections (right) of the open (a) and closed states (b). (c) Cross-sectional data interpolated by elliptic Fourier transformation for the open (blue) and closed states (red). We defined the cross-sections p1, p2, …, p21 from the petiole to the top of the leaf along the midrib. Stars indicate the points corresponding to the leaf tips. (d) Reconstructed leaves in open (blue) and closed states (red). (e) Geometric models of the open (blue) and closed states (red).

Open: R0=5.1mm, H=6.0mm, d=1.0, D=0.0mm, αmax=70π180rad, ϕ=−π4rad

Closed: R0=5.1mm, H=6.0mm, d=1.0, D=2.6mm, αmax=70π180rad, ϕ=0rad

These results indicated that the parameter D is crucial for distinguishing the open and closed states and that the inclination angle ϕ, which denotes the declination angle of the whole leaf, also differs between the open and closed states.

Assuming that the leaf sizes are the same, the essential parameter D represents the curvature along the leaf perpendicular to the midrib and serves as an geometric descriptor of bending during trap closure. A smaller D indicates a flattened, open state, whereas a larger D corresponds to a strongly curved, closed state. Physiologically, D reflects the differential deformation between the inner and outer tissue layers derived from curvature, driven by asymmetries including those in turgor pressure, water transport, and cell wall mechanical properties. From a biophysical perspective, D can be interpreted as the cumulative effect of strain differences between the two layers translated by curvature change, arising from differential expansion rates. One possible factor underlying these differences is ion fluxes, such as Ca2+-mediated signaling, which may control water movement across membranes and generate rapid volume changes. Thus, D could reflect the coupling between cellular-scale physiological processes and organ-scale mechanical behavior. Further perturbation experiments will clarify the link between the parameters (D and ϕ) and their physiological meanings, which is currently indirect.

 <inline-formula id="pone.0349246.e080"><alternatives><graphic id="pone.0349246.e080g" mimetype="image" position="anchor" xlink:href="info:doi/10.1371/journal.pone.0349246.e080" xlink:type="simple"/><mml:math display="inline" id="M80"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">D</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula id="pone.0349246.e081"><alternatives><graphic id="pone.0349246.e081g" mimetype="image" position="anchor" xlink:href="info:doi/10.1371/journal.pone.0349246.e081" xlink:type="simple"/><mml:math display="inline" id="M81"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">ϕ</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula> are significant parameters for the closing motion

We performed further experiments to reconstruct the 3D deformation dynamics. Using two cameras located to the left and right of the flytraps, we took movies of their closing motions (Methods, Fig 3a, Supplementary S1 Movie). We marked the outer surface of each trap with characteristic black dots using a marker pen and tracked the dots using ImageJ (Fig 3b). This enabled us to quantify the spatiotemporal dynamics of the mean curvature of the traps (Fig 3c).

10.1371/journal.pone.0349246.g003Acquisition of geometric parameters over time using a 3D reconstruction method.

(a) Illustration of the two angles of observation with markers on the outer surface of the trap. (b) Snapshots of the initial state obtained from the left and right cameras. (c) Spatiotemporal mean curvature of the 3D reconstructed data. The color code refers to mean curvature. (d) Spatiotemporal mean curvature of the reconstructed geometric model with fitted geometric parameters.

We then estimated the geometric parameters as follows. We measured H = 16.0 mm based on the distance between top and bottom points and d = 1.10 based on the curved boundary of the leaf top. We calculated the curvature radius R = 10.3 mm using ImageJ. The ranges of α and β were determined to be –45 deg to 45 deg and 0 to 0.78, respectively. We estimated the leaf angle at the initial state ϕ=0.40 rad from the initial leaf angle. The remaining parameter D = 6.93 mm was inferred from the closed morphology. Using these parameters, we were able to reconstruct a geometric model of the spatiotemporal change of the leaf (Fig 3d), which showed large changes in the peripheral region that are consistent with the micro–CT scanning data (Fig 2d).

In summary, we showed that D and ϕ alone are sufficient to reproduce the movement in our simulations, and thus significant parameters for the closing motion, using a 3D reconstruction method.

 Trap closure is subject to a size–curvature constraint

Based on the actual data set, we could evaluate the closing motion from only parameters D and ϕ. Since ϕ represents the rotation of the coordinate system, we fixed parameter ϕ first and assessed the closing motion in a morphospace that characterized the morphological changes using a small number of parameters, i.e., H and D (Fig 4a). In the morphospace, the closing motion can be described by the changes in location from one point to another (vectors). To quantitatively assess the morphology clearly, we demonstrated the mean curvature over time corresponding to the vectors (Fig 4b). Our results showed that the mean curvature increases as the leaf height H decreases, as expected based on the definitions of those parameters (Fig 4b). Here, the important point is that the curvature D changes within 0.5 s from the open state (when Dop=0.0 mm) to the closed state (Dcl=0.7 mm), and there is a correspondence between curvature and time in this geometric model.

10.1371/journal.pone.0349246.g004Morphospace analysis and the relationship between trap shape and speed.

(a) Morphospace in terms of the height 𝐇 and the deflection 𝐃. The color code refers to mean curvature. (b) Temporal change in the mean curvatures of the four models in (a). Boxplot shows the spatial average of the mean curvatures for each model. (c) Schematic illustration of the indices 𝐃cl and 𝐃op. (d) Comparison between the angular velocity with respect to curvature, ωcurv, and that with respect to angle, ωangle. (e) Relationship of index 𝐃cl and height 𝐇. (f) Relationship of 𝐃 to height 𝐇. The arrows denote the vectors from 𝐃op to 𝐃cl. The ratio 𝐃 to 𝐇 is constrained within a limited range (orange area).

To confirm the actual data corresponding to D, we calculated the parameters Dop and Dcl from the measured values for the leaf curvature radius Rl, H, and ϕ using the following geometric relationships between those two parameters and the equation Rl=1/κ (Fig 4c),

Dop≃Rl−Rl2−(H2)2 Dcl=H2tanϕ

We then evaluated the angular velocity with respect to curvature ωcurv as,

ωcurv=(Dcl−Dop)β(1−β2)HΔt, β=0.5

This estimated angular velocity ωcurv was relatively lower than the observed angular velocity in Fig 1 (Fig 4d). Precisely speaking, the angular velocity shown in Fig 1 was measured as the angle difference between the open and closed states, whereas the angular velocity in eq. (5) was measured from the difference in curvature between them. The angle difference reflects the global change of whole curved surface whereas the curvature difference reflects the local change of waving structure of the leaf. As the angular velocity in terms of angle ωangle is comparable to the closing speed of the whole leaf structure, ωangle is better to represent the whole closing motion where the leaves experience both the angle change and curvature change simultaneously. In addition, we found that the measured Dcl increased as a function of H, even though ϕ can be variable in eq. (4) (Fig 4e). Thus, the morphospace of H and D with actual data reveals the existence of a size–curvature constraint (orange area in Fig 4f), as all of the data are found within a certain range of the ratio D/H. This implies that small traps cannot bend faster than larger traps.

In summary, we discovered that the effect of ωangle is dominant in the closing motion compared to that of ωcurv (Fig 4d). In addition, we confirmed that there is a size–curvature constraint in the closing motion (Fig 4f).

In our observations, the sign of curvature reverses during motion in many leaves, suggesting a contribution of elastic instability (Fig 4f). However, we also found several cases in which initially flat surfaces bent during closure (Fig 4f), suggesting that even leaves lacking elastic energy storage can sometimes exhibit leaf movement. These exceptional motions may instead be accounted for by the hydrostatic pressure model [15,16,20]. When considered within a multilayer framework, this model offers an informative basis for interpreting differential deformation. These mechanisms likely provide the underlying driving forces are manifested at the organ scale. Our findings suggest that size-dependent geometric constraints may be applicable to any of these existing models and will be essential for understanding of the trap mechanics. An important direction for future work is to integrate turgor-driven deformation and elastic instability within a size-aware framework and experimentally test their relative contributions.

 Size-dependent curvature derives from differential deformations of different layers

As indicated by the results described in previous sections, trap shape and size are subject to certain constraints, making it possible to construct a curvature formula based on differential deformation dynamics. In doing so, we use a two–layer model of the trap (with inner and outer layers) and denote their expansion rates as ε˙in and ε˙out, respectively. The thickness of the layers is denoted h. The temporal derivative of the curvature in the height direction κ can be written as follows (Fig 5a, see also [21]).

10.1371/journal.pone.0349246.g005Dynamic relationship between curvature and strain rate.

(a) Illustrations of the two-layer model. (b) Simulated results of κ(t)~tanh(t) with different ϵ˙out.

 ∂κ(t)∂t=1h(1−κ(t)2h2)E(t)Δ(t)  E˙(t)=ε˙in+ε˙out2 Δ(t)=ε˙out−ε˙inε˙in+ε˙out

Considering that the outer surface expands after trap closure ([3,4,16,19]), the outer deformation is expected to be significantly larger than the inner deformation, i.e., ε˙out(t)≫ε˙in(t). Therefore,

∂κ(t)∂t=12h(1−κ(t)2h2)ε˙out(t)

Assuming that ε˙out(t) is constant (ε˙out) during the period of rapid closure, we can solve the equation, and the solution becomes,

κ(t)=1htanh[0.5ε˙out(t−t1)]

where κ(t) becomes 0 at t=t1. To capture the closure behavior, we set t1=0 and assess the temporal change in curvature. We calculated the temporal dependence of κ(t) for different ε˙out, showing an increase of the change in curvature as ε˙out increases (Fig 5b).

Our model still has several limitations. First, it assumes spatially uniform deformation and therefore does not capture the heterogeneous strain distribution observed across the leaf surface. Second, the model still did not reflect the complex tissue architecture of the trap such as the midrib, marginal teeth, and hinge region, which may play key roles in guiding and stabilizing trap closure. For future studies, it is important to incorporate cell wall mechanics, such as nonlinear elasticity and extensibility, nor potential viscoelastic effects that may contribute to time-dependent deformation during rapid motion. As different parts of the leaf may respond at distinct time scale during closure [22], it is essential to include the regional differences in temporal dynamics. Incorporating these biological and mechanical complexities would improve the realism of the model and make a model accurate representation of the underlying mechanisms governing trap movement.

 Conclusions

Consistent with previous studies such as [4] and [14], size-dependent movement has previously been proposed, but the specific geometric factors constraining trap motion have remained unclear. Our results suggest the presence of a size–curvature constraint and further indicate that traps smaller than 6 mm were unable to deform, and traps larger than 21 mm reached a limit preventing them from deforming fully. These bounds may reflect a balance between driving forces, such as turgor-induced differential strain, and geometric or mechanical constraints including bending stiffness. While this interpretation remains qualitative, it provides a potential physical basis for the observed morphospace. Predicting these limits from first principles may facilitate the biological interpretation of the underlying physical constraints. We did not examine cellular-scale dynamics; however, investigating spatial and temporal variations in cell size, shape, and deformation may help clarify how these multiscale processes contribute to size dependent trap closure.

 Supporting information Three examples of measuring changes in leaf opening angles from the open state to the closed state.

The opening angle was measured as the angle of the lines connecting the leaf joint and the base of the teeth.

(JPG)

 Eight examples of measuring the curvatures of the midlines of the leaves.

Yellow points indicate points detected along the midline.

(JPG)

 Two videos showing the closing movement of a Venus flytrap, filmed from two different angels.

(GIF)

 References Darwin C. Insectivorous Plants. London: John Murray. 1875. Brown WH, Sharp LW. The closing response in Dionaea. Bot Gaz. 1910;49:290–302. Brown WH. The mechanism of movement and the duration of the effect of stimulation in the leaves of Dionaea. Am J Bot. 1916;3:68–90. Forterre Y, Skotheim JM, Dumais J, Mahadevan L. How the Venus flytrap snaps. Nature. 2005;433(7024):421–5. doi: 10.1038/nature03185 15674293 Colombani M, Forterre Y. Biomechanics of rapid movements in plants: poroelastic measurements at the cell scale. Computer Methods in Biomechanics and Biomedical Engineering. 2011;14(sup1):115–7. doi: 10.1080/10255842.2011.593757 Forterre Y. Slow, fast and furious: understanding the physics of plant movements. J Exp Bot. 2013;64(15):4745–60. doi: 10.1093/jxb/ert230 23913956 Böhm J, Scherzer S, Krol E, Kreuzer I, von Meyer K, Lorey C, et al. The Venus Flytrap Dionaea muscipula Counts Prey-Induced Action Potentials to Induce Sodium Uptake. Curr Biol. 2016;26(3):286–95. doi: 10.1016/j.cub.2015.11.057 26804557 Scherzer S, Federle W, Al-Rasheid KAS, Hedrich R. Venus flytrap trigger hairs are micronewton mechano-sensors that can detect small insect prey. Nat Plants. 2019;5(7):670–5. doi: 10.1038/s41477-019-0465-1 31285557 Burri JT, Saikia E, Läubli NF, Vogler H, Wittel FK, Rüggeberg M, et al. A single touch can provide sufficient mechanical stimulation to trigger Venus flytrap closure. PLoS Biol. 2020;18(7):e3000740. doi: 10.1371/journal.pbio.3000740 32649659 Suda H, Mano H, Toyota M, Fukushima K, Mimura T, Tsutsui I, et al. Calcium dynamics during trap closure visualized in transgenic Venus flytrap. Nat Plants. 2020;6(10):1219–24. doi: 10.1038/s41477-020-00773-1 33020606 Scherzer S, Huang S, Iosip A, Kreuzer I, Yokawa K, Al-Rasheid KAS, et al. Ether anesthetics prevents touch-induced trigger hair calcium-electrical signals excite the Venus flytrap. Sci Rep. 2022;12(1):2851. doi: 10.1038/s41598-022-06915-z 35181728 Suda H, Toyota M. Integration of long-range signals in plants: A model for wound-induced Ca2+, electrical, ROS, and glutamate waves. Curr Opin Plant Biol. 2022;69:102270. doi: 10.1016/j.pbi.2022.102270 35926395 Yang R, Lenaghan SC, Zhang M, Xia L. A mathematical model on the closing and opening mechanism for venus flytrap. Plant Signal Behav. 2010;5(8):968–78. doi: 10.4161/psb.5.8.12136 21460610 Poppinga S, Kampowski T, Metzger A, Speck O, Speck T. Comparative kinematical analyses of Venus flytrap (Dionaea muscipula) snap traps. Beilstein J Nanotechnol. 2016;7:664–74. doi: 10.3762/bjnano.7.59 27335756 Volkov AG, Adesina T, Markin VS, Jovanov E. Kinetics and mechanism of Dionaea muscipula trap closing. Plant Physiol. 2008;146(2):694–702. doi: 10.1104/pp.107.108241 18065564 Sachse R, Westermeier A, Mylo M, Nadasdi J, Bischoff M, Speck T, et al. Snapping mechanics of the Venus flytrap (Dionaea muscipula). Proc Natl Acad Sci U S A. 2020;117(27):16035–42. doi: 10.1073/pnas.2002707117 32571929 Hodick D, Sievers A. On the mechanism of trap closure of Venus flytrap (Dionaea muscipula Ellis). Planta. 1989;179(1):32–42. doi: 10.1007/BF00395768 24201419 Poppinga S, Joyeux M. Different mechanics of snap-trapping in the two closely related carnivorous plants Dionaea muscipula and Aldrovanda vesiculosa. Phys Rev E Stat Nonlin Soft Matter Phys. 2011;84(4 Pt 1):041928. doi: 10.1103/PhysRevE.84.041928 22181196 Tsugawa S, Asakawa H, Hirata M, Nonoyama T, Kang Z, Toyota M, et al. Inference of mechanical forces through 3D reconstruction of the closing motion in venus flytrap leaves. Sci Rep. 2025;15(1):24860. doi: 10.1038/s41598-025-10638-2 40721427 Markin VS, Volkov AG, Jovanov E. Active movements in plants. Plant Signaling & Behavior. 2008;3(10):778–83. doi: 10.4161/psb.3.10.6041 Bastien R, Douady S, Moulia B. A unifying modeling of plant shoot gravitropism with an explicit account of the effects of growth. Front Plant Sci. 2014;5:136. doi: 10.3389/fpls.2014.00136 24782876 Wayne RF, Dawn A. A quantitative study of tissue dynamics during closure in the traps of Venus’s flytrap Dionaea muscipula Ellis. Am J Bot. 1991;78:647–57. 10.1371/journal.pone.0349246.r001 Decision Letter 0 Berardo Alice Academic Editor 2026 Alice Berardo This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 0

10 Mar 2026

-->PONE-D-25-65525-->-->Size–Curvature Constraint in the Closing Motion of Venus Flytrap Leaves-->-->PLOS One

Dear Dr. Satoru,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

In particular, you are encouraged to carefully address the reviewers’ questions and to provide a clear justification for the assumptions adopted in the study, supporting your choices with appropriate explanations and references if necessary.

Please submit your revised manuscript by Apr 24 2026 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:-->

A letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Alice Berardo

Academic Editor

PLOS One

Journal Requirements:

When submitting your revision, we need you to address these additional requirements.

1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at

https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2. Please note that PLOS One has specific guidelines on code sharing for submissions in which author-generated code underpins the findings in the manuscript. In these cases, we expect all author-generated code to be made available without restrictions upon publication of the work. Please review our guidelines at https://journals.plos.org/plosone/s/materials-and-software-sharing#loc-sharing-code and ensure that your code is shared in a way that follows best practice and facilitates reproducibility and reuse.

3. We note that the grant information you provided in the ‘Funding Information’ and ‘Financial Disclosure’ sections do not match.

When you resubmit, please ensure that you provide the correct grant numbers for the awards you received for your study in the ‘Funding Information’ section.

4. Thank you for stating the following financial disclosure:

This work was supported by JSPS KAKENHI grant numbers JP23H01143, JP22J00902, JP25KJ0714, JP24H00565, JP25K18499, JP25K18427, JST CREST grant number JPMJCR2121, and JST ERATO grant number JPMJER2403.

Please state what role the funders took in the study. If the funders had no role, please state: "The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript."

If this statement is not correct you must amend it as needed.

Please include this amended Role of Funder statement in your cover letter; we will change the online submission form on your behalf.

5. Please expand the acronym “JSPS” (as indicated in your financial disclosure) so that it states the name of your funders in full.

This information should be included in your cover letter; we will change the online submission form on your behalf.

6. Thank you for stating the following in the Acknowledgments Section of your manuscript:

This work was supported by JSPS KAKENHI grant numbers JP23H01143, JP22J00902, JP25KJ0714, JP24H00565, JP25K18499, JP25K18427, JST CREST grant number JPMJCR2121, and JST ERATO grant number JPMJER2403

We note that you have provided funding information that is not currently declared in your Funding Statement. However, funding information should not appear in the Acknowledgments section or other areas of your manuscript. We will only publish funding information present in the Funding Statement section of the online submission form.

Please remove any funding-related text from the manuscript and let us know how you would like to update your Funding Statement. Currently, your Funding Statement reads as follows:

This work was supported by JSPS KAKENHI grant numbers JP23H01143, JP22J00902, JP25KJ0714, JP24H00565, JP25K18499, JP25K18427, JST CREST grant number JPMJCR2121, and JST ERATO grant number JPMJER2403.

Please include your amended statements within your cover letter; we will change the online submission form on your behalf.

7. We note that Figure(s) 1-3, S1, S2 in your submission contain copyrighted images. All PLOS content is published under the Creative Commons Attribution License (CC BY 4.0), which means that the manuscript, images, and Supporting Information files will be freely available online, and any third party is permitted to access, download, copy, distribute, and use these materials in any way, even commercially, with proper attribution. For more information, see our copyright guidelines: http://journals.plos.org/plosone/s/licenses-and-copyright.

We require you to either (1) present written permission from the copyright holder to publish these figures specifically under the CC BY 4.0 license, or (2) remove the figures from your submission:

a. You may seek permission from the original copyright holder of Figure(s) 1-3, S1, S2 to publish the content specifically under the CC BY 4.0 license.

We recommend that you contact the original copyright holder with the Content Permission Form (http://journals.plos.org/plosone/s/file?id=7c09/content-permission-form.pdf) and the following text:

“I request permission for the open-access journal PLOS ONE to publish XXX under the Creative Commons Attribution License (CCAL) CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). Please be aware that this license allows unrestricted use and distribution, even commercially, by third parties. Please reply and provide explicit written permission to publish XXX under a CC BY license and complete the attached form.”

Please upload the completed Content Permission Form or other proof of granted permissions as an "Other" file with your submission.

In the figure caption of the copyrighted figure, please include the following text: “Reprinted from [ref] under a CC BY license, with permission from [name of publisher], original copyright [original copyright year].”

b. If you are unable to obtain permission from the original copyright holder to publish these figures under the CC BY 4.0 license or if the copyright holder’s requirements are incompatible with the CC BY 4.0 license, please either i) remove the figure or ii) supply a replacement figure that complies with the CC BY 4.0 license. Please check copyright information on all replacement figures and update the figure caption with source information. If applicable, please specify in the figure caption text when a figure is similar but not identical to the original image and is therefore for illustrative purposes only.

8. Please include captions for your Supporting Information files at the end of your manuscript, and update any in-text citations to match accordingly. Please see our Supporting Information guidelines for more information: http://journals.plos.org/plosone/s/supporting-information.

9. If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

-->Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. -->

Reviewer #1: Partly

Reviewer #2: Yes

**********

-->2. Has the statistical analysis been performed appropriately and rigorously? -->

Reviewer #1: No

Reviewer #2: Yes

**********

-->3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.-->

Reviewer #1: Yes

Reviewer #2: Yes

**********

-->4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.-->

Reviewer #1: Yes

Reviewer #2: Yes

**********

-->5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)-->

Reviewer #1: This manuscript investigates the rapid closure mechanism of the Venus flytrap (Dionaea muscipula) by integrating experimental kinematics, micro-CT scanning, 3D reconstruction, and geometric modeling. The authors propose that trap closure is governed by a "size-curvature constraint," where the achievable curvature during closure is limited by the trap size. They further introduce a two-layer differential deformation model to explain the curvature dynamics. The work attempts to bridge plant biomechanics with geometric and physical modeling, and the application of micro-CT and 3D reconstruction to quantify trap morphology. Some suggestions are provided below:

(1) The proposed geometric model, while descriptive, lacks direct linkage to underlying physiological or biophysical mechanisms. The parameters D and ϕ are fitted but not independently validated through perturbation experiments.

(2) The physical meaning of the "deflection" parameter D and its direct correspondence to a measurable tissue property (e.g., differential strain, turgor pressure gradient) is not clearly established, making the model phenomenological rather than explanatory.

(3) The experimental dataset (n=15 traps) appears limited for establishing robust correlations between size, angle, and velocity. No statistical tests (e.g., p-values, confidence intervals) are reported for the trends shown in Fig. 1.

(4) The geometric model is primarily fitted and validated using the same imaging data from which parameters were extracted. There is no independent experimental test to confirm the model predictive power.

(5) The two-layer differential deformation model assumes constant strain rate and a simplified relationship between curvature and strain. It ignores the complex tissue architecture, cell wall mechanics, and potential viscoelastic effects, which are crucial for rapid movements.

(6) The discussion dismisses the hydrostatic pressure and elastic buckling models somewhat superficially. The study fails to reconcile its geometric constraint with these established theories or design experiments to distinguish their contributions.

(7) The claimed size-curvature constraint (Fig. 4f) is presented as an empirical observation without a theoretical derivation of the boundary limits (e.g., why 6"mm" and 21"mm" ?). The physical principles defining this "orange area" remain speculative.

(8) The analysis treats closure as a transition between two static states (open/closed). The dynamics of the curvature change ∂κ/∂t are modeled with a simplistic assumption of constant ϵ _out, which is unlikely to hold true throughout the rapid, non-linear closure process.

(9) The model does not account for the role of the midrib, teeth, or the hinge region in guiding and stabilizing closure, which are known to be structurally important.

(10) The stretching-bending ratio α is adopted from prior work but its direct physical interpretation in the context of the flytrap bilayer, hydrated tissue is not clarified.

Reviewer #2: Authors completed the 3D morphology reconstruction of the Venus flytrap by micro-CT scanning, and deduced a non-dimensional parameter by the geometric value obtained above to feature the motion of the Venus flytrap. The main conclusion is that the non-dimensional index they deduced is correlated to the angular velocity of the snapping. The verification of the relationship between the geometric parameters and dynamic motion is essential for the research of the Venus flytrap. Concerning the novelty and originality, I suggest to accept this paper after minor revision. However, there are several questions I hope to receive response from authors:

1. Page 2, line 45: actually, there is one paper published mentioned that only one trigger could initiate the motion of the plant. For your reference: A single touch can provide sufficient mechanical stimulation to trigger Venus flytrap closure.

2. Page 3, line 103: the non-dimensional parameter α was proposed by Yeol et al., could you explain why you choose this parameter and what new based on your analysis comparing their previous work?

3. Page 4, line 131-132; Page 5, line 169-170: could you explain the value difference of the D, because you mentioned that D for open is 2.6mm, and close is 0mm is page 4, and then D for open is 0.0, and close is 0.7? Besides, please describe the reason this value equals zero.

4. Page 5, line 165-166: It looks like that the closing time is pretty long, why this happen? Did you trigger the motion on site, or you have to cut the leaf for imaging?

5. Page 6, line 181-183: You used two velocity parameters to describe the motion, which should be better to represent the natural notion of the plant?

6. Page 6, line 198-199: Could you verify the two-layer model by your 3D CT data?

**********

-->6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.-->

Reviewer #1: No

Reviewer #2: Yes:Zeng Xiangli

**********

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures

You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation.

NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications.

Submitted filename: review letter.docx

 10.1371/journal.pone.0349246.r002 Author response to Decision Letter 1 Submission Version 1

7 Apr 2026

Dear Alice,

We are very grateful to the editor and the reviewers for taking time to review our manuscript and give us valuable comments. We have taken all the comments into considerations and have made appropriate revisions to the manuscript. We believe that our revised manuscript reaches the quality for a topic of PLOS One.

Our point-by-point response appears below, in which we first repeated the reviewer’s comments (shown in italic) and then responded to them. Our revised text is highlighted in red in our revised manuscript. We also attached an unmarked version of our manuscript.

REVIEWER COMMENTS:

Reviewer: 1

The proposed geometric model, while descriptive, lacks direct linkage to underlying physiological or biophysical mechanisms. The parameters D and ϕ are fitted but not independently validated through perturbation experiments.

Authors’ response:

We thank the reviewer for this constructive comment. Although we were unable to perform perturbation experiments because it is difficult to vary D or ϕ independently without applying mechanical force which alter other parameters, we added the discussion of the physiological interpretation of D since the explanation regarding D was only descriptive. Therefore, we added the following one section in the discussion section.

Line 184: The essential parameter D represents the magnitude of out-of-plane bending of the leaf and serves as an integrated macroscopic indicator of curvature change during trap closure. Physiologically, D reflects the differential deformation between the inner and outer tissue layers, primarily driven by asymmetries such as those in turgor pressure, water transport, and cell wall mechanical properties. A smaller D indicates a flattened, open state, whereas a larger D corresponds to a strongly curved, closed state. From a biophysical perspective, D can be interpreted as the cumulative effect of strain differences between the two layers, arising from differential expansion rates. One possible factor underlying these differences is ion fluxes, such as Ca2+-mediated signaling, which may control water movement across membranes and generate rapid volume changes. Thus, D could reflect the coupling between cellular-scale physiological processes and organ-scale mechanical behavior. Further perturbation experiments will provide a direct link between the parameters (D and ϕ) and their physiological meanings.

The physical meaning of the "deflection" parameter D and its direct correspondence to a measurable tissue property (e.g., differential strain, turgor pressure gradient) is not clearly established, making the model phenomenological rather than explanatory.

Authors’ response:

We thank the reviewer for this clarification. We think that the abovementioned response also answers this question.

The experimental dataset (n=15 traps) appears limited for establishing robust correlations between size, angle, and velocity. No statistical tests (e.g., p-values, confidence intervals) are reported for the trends shown in Fig. 1.

Authors’ response:

We thank the reviewer for this valuable comment. We agree that the original manuscript did not sufficiently support the observed correlations with statistical analysis. The confidence interval for size was 10.9 – 24.5 mm, and that for angle was 0.09 – 0.29 rad. The p-values of the trends shown in Fig.1f is 0.00373, so correlations are statistically significant (p < 0.05). We added the following results in the result section.

Line 139: The 95% confidence interval for W was 10.9 – 24.5, and that for angle was 0.09 – 0.29.

Line 372: The p-value of the increasing trend was 0.00373.

The geometric model is primarily fitted and validated using the same imaging data from which parameters were extracted. There is no independent experimental test to confirm the model predictive power.

Authors’ response:

We thank the reviewer for this comment. We also considered that it is necessary to identify parameters in multiple experiments. We determined the important parameter D with micro–CT scanning data (Fig. 2) and 3D reconstructed data using DLT method (Fig. 3) and the direct measurement of morphological change (Fig. 4) independently. The obtained data of many samples (Fig. 4f) exemplified that the proposed model taking the leaf angle into account properly captured different types of morphological changes. To directly validate the model using an independent dataset, we revised the text as follows to assess the DLT-based model using micro-CT scanning data.

Line207: Using these parameters, we were able to reconstruct a geometric model of the spatiotemporal change of the leaf (Fig. 3d), which showed large changes in the peripheral region that are consistent with the micro–CT scanning data (Fig. 2d).

The two-layer differential deformation model assumes constant strain rate and a simplified relationship between curvature and strain. It ignores the complex tissue architecture, cell wall mechanics, and potential viscoelastic effects, which are crucial for rapid movements.

Authors’ response:

We thank the reviewer for this important comment. We also acknowledged that our model still has limitations as written in the discussion section. Considering that the points from the reviewers (the complex tissue architecture, cell wall mechanics, potential viscoelastic effects), we extended and revised the limitation statements as follows.

Line 282: Our model still has several limitations. First, it assumes spatially uniform deformation and therefore does not capture the heterogeneous strain distribution observed across the leaf surface. Second, the model still did not reflect the complex tissue architecture of the trap such as the marginal teeth, and hinge region, which may play key roles in guiding and stabilizing trap closure. For future studies, it is important to incorporate cell wall mechanics, such as nonlinear elasticity and extensibility, nor potential viscoelastic effects that may contribute to time-dependent deformation during rapid motion. As different parts of the leaf may respond at distinct time scale during closure [22], it is essential to include the regional differences in temporal dynamics. Incorporating these biological and mechanical complexities would improve the realism of the model and make a model accurate representation of the underlying mechanisms governing trap movement.

The discussion dismisses the hydrostatic pressure and elastic buckling models somewhat superficially. The study fails to reconcile its geometric constraint with these established theories or design experiments to distinguish their contributions.

Authors’ response:

We thank the reviewer for this insightful comment. We agree that our original discussion did not sufficiently clarify the relationship between our geometric framework and previously proposed mechanisms, including hydrostatic pressure-driven deformation and elastic buckling.

Importantly, our intention was not to dismiss these established models, but rather to provide a complementary perspective by focusing on geometric constraints that emerge at the organ scale. In our view, hydrostatic pressure differences between tissue layers and elastic instabilities are likely to act as the underlying driving forces, while the size-curvature constraint identified in this study represents a geometric boundary condition that governs how these forces are translated into observable motion. In other words, our model describes what configurations are mechanically accessible, whereas hydrostatic and buckling models describe how forces are generated.

We agree that further work is needed to explicitly reconcile these frameworks. For example, incorporating turgor-driven strain differences into our two-layer model could provide a direct link to hydrostatic pressure mechanisms. Similarly, evaluating whether the system approaches a critical threshold for snap-through instability would help clarify the contribution of elastic buckling. Experimentally, this could be addressed by perturbing turgor pressure (e. g. via osmotic treatments) or altering mechanical stiffness (e.g., through chemical modification of cell walls) and examining how these changes affect the size-curvature relationship.

We have revised the discussion to better articulate this integrative perspective and to explicitly acknowledge these limitations and future directions.

Line 249: In our observations, the sign of curvature reverses during motion in many leaves, suggesting a contribution of elastic instability (Fig. 4f). However, we also found several cases in which initially flat surfaces bent during closure (Fig. 4f), suggesting that even leaves lacking elastic energy storage can sometimes exhibit leaf movement. These exceptional motions may instead be accounted for by the hydrostatic pressure model [15, 16, 20]. When considered within a multilayer framework, this model offers an informative basis for interpreting differential deformation. These mechanisms likely provide the underlying driving forces are manifested at the organ scale. Our findings suggest that size-dependent geometric constraints may be applicable to any of these existing models and will be essential for understanding of the trap mechanics. An important direction for future work is to integrate turgor-driven deformation and elastic instability within a size-aware framework and experimentally test their relative contributions.

The claimed size-curvature constraint (Fig. 4f) is presented as an empirical observation without a theoretical derivation of the boundary limits (e.g., why 6"mm" and 21"mm" ?). The physical principles defining this "orange area" remain speculative.

Authors’ response:

We thank the reviewer for this important comment. We agree that the size-curvature constraint shown in Fig. 4f is currently based on empirical observations, and that a rigorous theoretical deviation of the boundary limits remains to be established.

Original line 94: “We observed that traps with width <6 mm or >21 mm did not move, possibly due to leaf immaturity or senescence, respectively (Fig. 1b) [14].” As explained here, we confirmed that the samples outside the range [6, 21] mm did not move.

At present, we interpret the lower and upper bounds (approximately 6 mm and 21 mm) as reflecting physical constraints arising from the interplay between geometry and mechanics. For smaller traps, insufficient size may limit the generation of curvature due to reduced geometric leverage and lower effective deformation relative to thickness, making rapid bending difficult. Conversely, for larger traps, increased size likely enhances bending resistance and inertial or hydraulic limitations, preventing efficient curvature change within the observed timescale. These considerations suggest that the observed “orange area” may emerge from a balance between driving forces and mechanical resistance.

We acknowledge that this interpretation remains qualitative, and a quantitative framework that predicts these limits from first principles is an important direction for future work. In particular, extending the current model to incorporate thickness, material properties, and fluid transport dynamics may enable deviation of the admissible region in the D-H morphospace. We have revised the manuscript to clarify that the constraint is empirical and to outline these possible physical interpretations and future directions.

Line 295: Consistent with previous studies such as [4] and [14], size-dependent movement has previously been proposed, but the specific geometric factors constraining trap motion have remained unclear. Our results suggest the presence of a size–curvature constraint and further indicate that traps smaller than 6 mm were unable to deform, and traps larger than 21 mm reached a limit preventing them from deforming fully. These bounds may reflect a balance between driving forces, such as turgor-induced differential strain, and geometric or mechanical constraints including bending stiffness. While this interpretation remains qualitative, it provides a potential physical basis for the observed morphospace. Predicting these limits from first principles may facilitate the biological interpretation of the underlying physical constraints. We did not examine cellular-scale dynamics; however, investigating spatial and temporal variations in cell size, shape, and deformation may help clarify how these multiscale processes contribute to size dependent trap closure.

The analysis treats closure as a transition between two static states (open/closed). The dynamics of the curvature change ∂κ/∂t are modeled with a simplistic assumption of constant ϵ _out, which is unlikely to hold true throughout the rapid, non-linear closure process.

Authors’ response:

We thank the reviewer for this comment. Our response to the comment (5) corresponds to answer to this question too.

The model does not account for the role of the midrib, teeth, or the hinge region in guiding and stabilizing closure, which are known to be structurally important.

Authors’ response:

We thank the reviewer for this constructive comment. We added the following statement in the discussion section.

Line 284: the model still did not reflect the complex tissue architecture of the trap such as the midrib, marginal teeth, and hinge region, which may play key roles in guiding and stabilizing trap closure.

The stretching-bending ratio α is adopted from prior work but its direct physical interpretation in the context of the flytrap bilayer, hydrated tissue is not clarified.

Authors’ response:

We thank the reviewer for this important comment. We agree that the physical interpretation of the stretching-bending ratio was not sufficiently clarified in the context of the Venus flytrap.

In general, α represents the relative energetic cost of in-plane stratching versus out-of-plane bending, and is typically determined by geometric and material parameters such as thickness and elastic modulus. In our model, α is adopted as an effective parameter to capture the balance between these two deformation modes at the organ scale.

In the context of the flytrap, which consists of hydrated, multilayered tissues, α can be interpreted as reflecting the mechanical response of a bilayer system in which differential strain – potentially arising from turgor pressure, water transport, and cell wall extensibility – drives bending while being constrained by in-plane stretching resistance. Thus, α effectively incorporates the combined influence of tissue hydration, layer thickness, and cell wall mechanics into a single dimensionless parameter.

We acknowledge that this representation is simplified and does not explicitly account for spatial heterogeneity, viscoelasticity, or fluid-structure coupling. A more rigorous formulation would involve deriving α from measurable physical quantities in a multilayer, poroelastic framework. We have revised the manuscript to clarify this interpretation and to highlight it as an important direction for future work.

Line 153: The dimensionless index α was adopted as an effective parameter to capture the balance between thickness and elastic modulus. In the context of the flytrap, which consists of hydrated, multilayered tissues, α can be interpreted as reflecting the mechanical response of a bilayer system in which differential strain – potentially arising from turgor, water transport, and cell wall extensibility – drives bending while being constrained by in-plane stretching resistance. Thus, α effectively incorporates the combined influence of tissue hydration, layer thickness, and cell wall mechanics into a single dimensionless parameter.

Reviewer: 2

Page 2, line 45: actually, there is one paper published mentioned that only one trigger could initiate the motion of the plant. For your reference: A single touch can provide sufficient mechanical stimulation to trigger Venus flytrap closure.

Authors’ response:

We thank the reviewer for this constructive suggestion. We revised the corresponding part, and added the following statement in the introduction.

Line 59: , and only one touch could initiate the motion of the leaves if

Submitted filename: renamed_203d5.docx

 10.1371/journal.pone.0349246.r003 Decision Letter 1 Berardo Alice Academic Editor 2026 Alice Berardo This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 1

28 Apr 2026

Size–Curvature Constraint in the Closing Motion of Venus Flytrap Leaves

PONE-D-25-65525R1

Dear Dr. Satoru,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Please consider including also the second reviewer's final comments.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. For questions related to billing, please contact billing support.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Alice Berardo

Academic Editor

PLOS One

Additional Editor Comments (optional):

Please consider including also the second reviewer's final comments.

Reviewers' comments:

Reviewer's Responses to Questions

-->Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.-->

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

**********

-->2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. -->

Reviewer #1: Yes

Reviewer #2: Partly

**********

-->3. Has the statistical analysis been performed appropriately and rigorously? -->

Reviewer #1: Yes

Reviewer #2: Yes

**********

-->4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.-->

Reviewer #1: Yes

Reviewer #2: Yes

**********

-->5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.-->

Reviewer #1: Yes

Reviewer #2: Yes

**********

-->6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)-->

Reviewer #1: The authors have addressed most of my concerns. The reviewer do not has further additional comments for the manuscript.

Reviewer #2: 1. Clarification and consistency of parameter D (important)

In the original geometric formulation proposed by Poppinga and Joyeux, the parameter D is defined as a geometric quantity representing the maximum separation between the two lobes in the reference (closed) configuration, and is used to construct a lifelike initial shape of the trap.Importantly, in that framework, D is treated as a fixed geometric parameter, rather than a dynamic variable describing the temporal evolution of the motion.

In this paper, the parameter D plays a central role throughout the manuscript, as it is used:

1. to distinguish open and closed states (e.g., D = 0 vs D > 0),

2. to describe the deformation process over time,

3. to construct the size–curvature relationship (e.g., D–H morpho space).

Given this central importance, the definition and interpretation of D should be as precise and consistent as possible. At present, D appears to serve multiple roles: a geometric parameter in the surface model (Eq. 1), a descriptor of deformation state, and, in the discussion, a proxy for underlying physiological processes.

While these interpretations are individually reasonable, their combination can be somewhat ambiguous. Please clearly define D in a single, primary sense (e.g., geometric descriptor vs. state variable), and explicitly state how the other interpretations relate to this definition (e.g., whether they are conceptual or derived).

Even a short clarifying paragraph would greatly improve the conceptual clarity of the manuscript.

2. Physical interpretation of D

The added physiological interpretation of D is interesting and provides useful intuition. However, because D is introduced through a geometric model and extracted from morphological fitting, its connection to quantities such as: differential strain, turgor pressure, or ion-mediated processes remains indirect.

To avoid overinterpretation, I suggest: explicitly framing this connection as qualitative or interpretive, unless a direct quantitative relationship is established. This will strengthen the rigor of the manuscript while preserving its interdisciplinary perspective.

**********

-->7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.-->

Reviewer #1: No

Reviewer #2: Yes

**********

Submitted filename: review letter_round2.docx

 10.1371/journal.pone.0349246.r004 Acceptance letter Berardo Alice Academic Editor 2026 Alice Berardo This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

PONE-D-25-65525R1

PLOS One

Dear Dr. Satoru,

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS One. Congratulations! Your manuscript is now being handed over to our production team.

At this stage, our production department will prepare your paper for publication. This includes ensuring the following:

* All references, tables, and figures are properly cited

* All relevant supporting information is included in the manuscript submission,

* There are no issues that prevent the paper from being properly typeset

You will receive further instructions from the production team, including instructions on how to review your proof when it is ready. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few days to review your paper and let you know the next and final steps.

Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

You will receive an invoice from PLOS for your publication fee after your manuscript has reached the completed accept phase. If you receive an email requesting payment before acceptance or for any other service, this may be a phishing scheme. Learn how to identify phishing emails and protect your accounts at https://explore.plos.org/phishing.

If we can help with anything else, please email us at customercare@plos.org.

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Dr. Alice Berardo

Academic Editor

PLOS One