# Curvature in plants

Félix Hartmann, Mélanie Decourteix, Bruno Moulia

► To cite this version:

Félix Hartmann, Mélanie Decourteix, Bruno Moulia. Curvature in plants. Current Biology, 2024, 34 (20), pp.R991-R996. 10.1016/j.cub.2024.07.006. hal-04771849v2

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## Primer

# Curvature in plants

Félix P. Hartmann<sup>1,2,*</sup>,
Mélanie Decourteix<sup>1,2</sup>,
and Bruno Moulia<sup>1,2,*</sup>

Curvatures are ubiquitous in the animal kingdom — the spiral shell of the nautilus and the corkscrew horns of the blackbuck being iconic examples. Dynamic changes in curvature (i.e., curving) are most striking in the locomotion of some animal species — swimming in fishes and mollusks, looping in leeches, undulatory locomotion in snakes and lampreys, and also sperm motility through flagellum beating. When it comes to plants, which are sessile organisms with a rigid body, the terms 'curvature' and 'curving' evoke very different images — leaves of grass swaying in the breeze, a trunk dangerously bent by a powerful gust of wind, a branch sagging under the weight of its own fruits, as well as the frail arabesques of twining plants like the morning glory and the ivy, which were so influential in the Art Nouveau movement and many other artistic traditions. These various vegetal curves not only prompt creative inspiration in the mind of the beholder, they also initiate signaling cascades leading to developmental responses of the plant. Conversely, curvature can result from a biologically active process in response to an internal or external stimulus. Active curving or decurving are indeed important aspects of the plastic interplay of the developmental program of plants and their environment. Although easily accessible to observation, curvature and curving have only recently become the focus of active research in plant development. Lying at the nexus of biology, physics and mathematics, they require an interdisciplinary approach. The aim of this primer is to give readers an intuitive but accurate understanding of what curvature and curving are, as observed in the plant kingdom, then a more formal definition. We will discuss their role in plant development, both as a signal and as a response, and finally the practical issues and solutions involved in measuring plant curvatures.

## Observing and defining curvature in plants

Curvatures can be observed in various situations and at different time scales. Short events, like an animal trampling a tuft of grass or a gust of wind bending a tree, only induce transitory changes in curvature (Figure 1A). As soon as the disturbance ceases, the plant returns elastically to its original shape, provided the deformation has remained below a certain threshold. Otherwise, the induced curvature can be fixed to some extent over a longer time scale. Excessive curving by the wind or animal action can even lead to local cell damage (wounds), and even to the snapping or breakage of the organ.

In temperate climates, tree branches can sag for several months under seasonal loads, such as accumulated snow or hanging fruits. In woody species, curvatures build up over years and can dwindle just as slowly, reflecting the growth history of each individual.

Extreme cases of curvature arising during development can be found in morning glory (Convolvulaceae, Figure 1B) and in dwarf beeches (Fagus sylvatica var. tortuosa), where axes (roots, trunk and branches) are twisted and contorted (Figure 1C). Dwarf beeches are rare and usually grouped in forest patches made up of genetically related trees, suggesting a genetic regulation of curvature.

Curving and decurving can be visualized from time-lapse photographs and movies. From the onset of scientific cinema, the viewing of the complex kinematics of curving and decurving has provided unique insights into plant development (Figure 1E and Video S1). Moreover, 'heat-maps' of curvature as a function of space (s) and time (t) (curvature fields) can reveal patterns of development and shape control (Figure 1E). Just as elongation or enlargement measure changes in organ scales, curving measures changes in organ shapes. They are complementary concepts for quantitatively describing plant development.

However, for assessing the role of curvature in plant development (as in Figure 1E), a formal definition, amenable to quantitative measurement, is needed. Too often, curvature (noted C here, also noted $κ$ in physics) is confused with inclination (i.e. an angle, noted A here, also noted $α$ in physics). Although there
<figure><img src="image_2.png" /><figcaption>**Figure 1. Examples of curved plants, from different phyla and growth habits.**</figcaption></figure>

(A) Reed culms bending in the wind. (B) Painting of a morning glory stem (author Yokoyama, Masana, 1833–1908, artist Hattori, Sessai, 1807–?). (C) A dwarf beech. (D) Time-lapse photographs of an *Arabidopsis* inflorescence stem during its gravitropic movement after a tilting to the horizontal in the dark (s = spatial position along the stem, t = time after tilting). (E) Curvature field C(s,t) quantifying the movement shown in (D). (A) © Sharon Mollerus/Flickr (CC BY 2.0). (B) © Masana Yokoyama/Sessai Hattori/Wikimedia Commons. (C) © C. Lenne. (D) Reused with permission from PNAS. (E) From Bastien *et al*. (2013), re-used with permission by PNAS.

exists a mathematical relation between
these two geometrical concepts, they
are different. The inclination of a line
represents the angle A of this line with
respect to a reference axis, which can be
vertical or horizontal. The inclination of
a straight line is uniform along the line.
If this line is horizontal, its inclination
angle with respect to the vertical is
A = 90°, or A = π/2 radians in SI units.

Curvature, on the other hand,
measures how much a line deviates
from straightness. The curvature of a
straight line is thus zero, whether the
line is horizontal, vertical or oblique. A
line deviating from straightness (i.e., a
curved line) has a non-zero curvature.
A line with uniform curvature is an arc
of circle. However, in general, curvature

in plants is not uniform. Some parts
can be close to straight, while others
are much more curved, as shown in
**Figure 1D**.

Let us now detail a bit more the
relationship between inclination and
curvature. A curved (i.e., non-straight)
line lying in a vertical plane has a
non-uniform inclination and a non-zero
curvature. Curvature, therefore, seems
to be linked to a change in inclination
along a line. Indeed, curvature at a point
on a line is mathematically defined as
the local rate of change of the angle of
inclination with respect to the position
along the line (see Box 1 for a more
formal derivation).

Rotating a curve does not change its
curvature (of course rotating a plant in

the real world may change curvature,
due to mechanical bending under self-
weight, but at this point, we consider
rotating a curve or a picture). Rotation
does shift the inclination at every point
by the same amount, but since curvature
does not depend on the inclination itself
but on inclination differences between
points, it stays invariant. Thus, curvature
is an intrinsic description of a curve's
shape, independent of its orientation
and position. This is one of the reasons
why curvature is a useful concept
for quantitative approaches to plant
morphogenesis.

‘Curvature’ is a spatial concept, noted
C(s) where s is the spatial position
(Box 1). The concept of ‘curving’ adds a
temporal dimension by measuring how
curvature is changing with time, or C(s,t).
It captures all the changes in shape
undergone by a slender organ (Figure 1E)
and is the most complete quantification
of a change in shape. Curving results
from bending, due to either differential
growth or internal or external stresses.
Wind, animals, but also gravity are
primary examples of external stresses
causing organ bending (Figure 1A). In
this case, cells are deformed, or strained
in mechanical jargon (Figure 2; see Tee
et al. for a biologist-friendly introduction
to mechanical terms). Cells in the front
side — with respect to the direction
of bending — are stretched, while
those in the back side are compressed
(Figure 2A,B), and the cells in the middle
are unstrained.

Internally induced bending can be
the product of differential primary
growth, which happens when one side
of an organ elongates at a higher rate
than the opposite side. As a result, the
organ bends away from the side with
the highest elongation rate. This is true
for curving and decurving. In all cases,
there is a direct relation between the
gradient of growth across the organ and
its curving rate. Primary tropic motions
of stems and roots rely on differential
growth (Figure 1D,E).

Consequently, even in the absence
of external stresses, an organ is not
necessarily straight. It can have a rest
curvature, noted C*, which is defined as
its curvature when all external stresses
are removed, even its own weight (i.e.,
in weightless conditions). Fortunately, a
much easier and Earth-bound method
is available for measuring rest curvature
(as well as the change in curvature
due to self-weight) than sending plants into outer space: the upside-down test (Figure 3).

Estimating both the rest curvature C* and the actual curvature C is insightful, as it defines which tissues are compressed or stretched under an external bending (compare Figure 2A and B), and because the rest curvature is the output of internal bending linked to the growth and differentiation of tissues.

Altogether, since external stresses and internally active bending are ubiquitous in plants, the achievement and maintenance of a straight shape by an organ are not... straightforward. It requires advanced biological control that has started to be unraveled only recently.

## Curvature as a biological signal

Plant curvatures are not merely incidental results of tissue growth or external loads. Actually, they often play an informative role during plant development. Let us consider for instance a stem transiently bent by wind (as in Figure 1A). This induces a change in curvature. During this curving of the stem, tissues are deformed: cells on the back side of the curvature are compressed, whereas cells on the front side are stretched (Figure 2). It is well-established now that cells are able to respond to deformations, or 'strains'. This sensory capacity is called mechanosensing.

Trees (and dicots in general) subjected to repeated controlled curving mimicking the effect of wind display an enhanced diameter growth, an ovalization of the stem cross-section along the bending direction, and the formation of a specialized wood called 'flexure wood', with changes in the density of fibers and in their cell walls (thickness, microfibril angle, etc.). All these growth responses improve the margin of mechanical safety of treated plants. Here, curving, through the tissue strains it causes, acts as a signal induced by an environmental perturbation, and leads to significant developmental responses and increased fitness. This whole process, which can be thought of as a developmental feedback loop in which the maximal curvature set by wind-loads is adjusted (acclimated) is termed thigmomorphogenesis (Figure 4A).

At the molecular level, more than 2,000 genes are regulated in poplar after bending. Among them are the *TOUCH*

## Box 1. Illustrating and formally defining the concepts of curvature and curving.

The blue line in the figure below has a steadily increasing curvature. The position along the line is measured by its arc-length s, which ranges from 0 at one end to $S_{max}$ at the other end ($S_{max}$ being the total length of the curve). Over each portion of this line, the local curvature is computed as the difference in inclination angle across the line portion, divided by the length of the line portion. Here, each line portion has the same length $\Delta s$.

To get the exact curvature C(s) at a given position s along the line, we need to make the length of the line portion, $\Delta s$, to approach 0:

$$C(s) = \lim_{\Delta s \to 0} \frac{A(s + \Delta s) - A(s)}{\Delta s} = \frac{\partial A(s)}{\partial s}$$

Curvature C(s) is hence a local spatial variable. To describe curving and decurving, one needs to consider the field of curvature C(s,t), where t is the time (Figure 1E).

<figure><img src="image_3.png" /><figcaption>Accurate but simple and illustrated definition of the curvature and its measure.</figcaption></figure>

(TCH) family and the receptor-like kinase FERONIA. A linear relationship has even been established between the magnitude of the imposed change in curvature and the expression level of a primary mechanosensitive gene, the transcription factor Zinc Finger Protein 2 (ZFP2).

Is it possible that a curvature associated with internally induced bending, for example during tropic motion, could be sensed similarly to an externally induced bending? A combination of clinostat experiments and modeling has demonstrated that plants are indeed endowed with the capacity to sense their own shape, and specifically their curvature, independently of its origin. By analogy with animals, this sense is named proprioception. The molecular mechanisms behind proprioception have yet to be elucidated, but they could involve actin-myosin cables or cortical microtubules. Note that if

mechanosensing of transient bending during thigmomorphogenesis is related to changes in curvature (and to related strains), proprioception seems to be sensitive to the current curvature.

## (De)curving, an active and controlled process

The developmental response associated with proprioception (i.e., curvature sensing) is an intrinsic tendency of plant axes to straighten. It is always present, but is most manifest when all other tropisms are absent. A straightening is a progressive decrease of curvature with time, in other words, a decurving (Figure 4B). This can be an active biological process (unlike bending caused by external perturbations) that has been called autotropism (from auto = self). Besides proprioception, all tropic responses involve an active change of curvature to reorient a growing organ through either curving or decurving. As in nature, all these different
A

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

B

<figure><img src="image_6.png" /><figcaption>**Figure 2.** Relations between curving/decurving and cell/tissue strains (stretching and compression).</figcaption></figure>

(A) Curving from a stem with a straight rest-shape ($C^* = 0$ all along the stem); (B) decurving from an organ with an initially curved shape. Below each transformation, a close-up of the inner and outer cells before and after the transformation is shown, along with the formula relating the cell strain $\epsilon$ to the change in curvature $\delta C$ and the stem radius $R$. Strain magnitudes are color-coded from dark blue (highest negative strain, i.e., compression) to dark red (highest positive strain, i.e., stretch). Note that the tissue strains cannot be inferred only by the actual curvature $C(s)$. They depend on the change of curvature between the actual curvature $C$ and the rest curvature $C^*$. (De)curving of a stem with non-zero rest curvature $C^*$ (panel B) can produce a straight stem with tissue strains.

tropic drivers are concurrently active, this yields a complex choreography of curving and decurving until the organ reaches a new steady-state posture (Figure 1D,E and Video S1).

Depending on the organ and species, active curving/decurving is powered by various mechanisms of actuation (or motors). During primary growth, curvature change originates from a difference in elongation rate between two opposite sides of a stem, due to the action of osmotically-regulated turgor pressure and to differential cell wall loosening. A higher elongation rate on one side of a stem induces an active bending that produces curving or straightening, depending on the initial configuration and the direction of bending (Figures 1D,E and 2). Active bending capacity is not restricted to the subapical zone of primary growth. In plants undergoing secondary growth (e.g., conifers, dicots), active curving and decurving in woody parts of the organ cannot rely on differential elongation rates. Instead, they are mediated by the formation of reaction wood,

which creates asymmetric maturation strains between two opposite sides

of a woody axis. In phyla that lost secondary growth over the course of evolution, such as Poaceae (e.g., maize, wheat, rice), active bending and straightening in large inflorescence stems are performed by flexible joints, called pulvini, that retain a capacity for differential growth and are present at each node all along the stem.

Over the course of its life, a plant is subjected to a variety of fluctuating cues (light, humidity gradients, weight, wind, etc.) which influence its continuous morphogenesis through tropic movements. The long and partly stochastic sequence of associated curving would end up in extremely wavy shapes if there were no feedback mechanism. Proprioception of plant curvature, as a driving cue for autotropism, is precisely the feedback required to ensure that plant axes do not deviate excessively from straightness, and can control their posture (Figure 1D,E). At any given time in its development, the actual shape of a plant organ results from the balance between all past tropic responses and the proprioceptive feedback, integrated throughout its growth history up to that point. This global feedback mechanism ensures posture control (Figure 4B).

<figure><img src="image_7.png" /><figcaption>**Figure 3.** The upside-down test revealing the rest curvature $C^*$ in a lateral branch (Thuja occidentalis), and allowing its simple calculation.</figcaption></figure>

(A) The actual curvature (at each position $s$) is the sum of the rest curvature $C^*$ and of the change in curvature due to the bending under self-weight $\delta C_1$, i.e. $C_1 = C^* + \delta C_1$ (with $C^*$ and $\delta C_1 < 0$). (B) When the branch is turned upside-down, the concavity of the rest curvature is now contrary to the direction of gravity and the actual curvature of the branch $C_2$ differs from $C_1$ and comes as $C_2 = -C^* + C_2$. If the bending under self-weight is moderate, as here and in many cases (i.e. $\delta C_1 \approx \delta C_2 \approx \delta C \ll C^*$), then the difference between the two curvatures yields the rest curvature $C^*$, i.e. $(C_1 - C_2)/2 = C^*$.
<figure><img src="image_8.png" /><figcaption>**Figure 4.** The two developmental feedbacks that control organ curvatures in interaction with the physical environment.</figcaption></figure>

(A) Feedback loop controlling transient curvatures, stature acclimating stem strength to external loads (e.g., wind). (B) Feedback loop controlling lasting curvatures and posture so that the plant can recover from accidental lodging/toppling. Plants from Moulton *et al*., re-used with permission by PNAS.

Note finally that not all plant organs have a tendency to reduce their intrinsic curvature. Lateral branches (Figure 3) may display epinasty (an active curving away from the parent branch) and anti-gravitropic active bending, which balance the proprioceptive driver, leading to organs with steady curvature. This allows for a better spatial display of the foliage. By the same token, many leaves and flower parts have a substantial and controlled curvature. Indeed, the curvature of flower parts (nectar spur, stamen, and pistil) has been shown to be adaptive in plant-pollinator co-evolution.

**Measuring curvature and curving without being 'bent out of shape'**

Since we have seen the relevance of curvature for quantitative approaches to plant development, the question of its measurement in biological studies naturally arises. Compared with more straightforward quantities like lengths or angles, measuring curvatures may seem like a challenging task. Indeed, based on its mathematical definition, this requires first measuring numerous angles along the organ of interest, then performing numerical differentiation,

which is a source of noise. This is even more the case for the estimation of curvature changes and curving rates. Besides, curvature and curving rates are spatially distributed (Box 1), and may change over time during development. Fortunately, user-friendly software solutions exist, which include automatic shape recognition, geometrical analysis, and denoising tools. An early example, devoted to root growth, is KineRoot (https://www.quantitative-plant.org/software/kineroot). Two more recent and versatile programs are KymoRod (https://github.com/ijpb/KymoRod) and Interekt (https://forgemia.inra.fr/felix.hartmann/interekt). By allowing the measurement of curvature field C(s,t) (Figure 1E), and potentially C*(s,t) (Figure 3), these tools have been instrumental in the understanding of the way plants read and control their own shape, and in linking tissue growth and differentiation with morphogenesis.

**Conclusion: bent on considering curvature in biology**

Curvature is under active control all throughout plant development and is a major trait in plant ecology for several

functions (posture control, tropism, pollination, etc.). It is not, however, a notion that biologists typically get familiar with during their academic training. We hope that this primer will ease the mastering of curvature analysis and provide insight into the rich dynamics of curving and decurving in plants. For the sake of simplicity, we have only considered the curvature of slender organs, as it is unidimensional. However, the definition of curvature can be extended to surfaces. Surface curvatures have been shown to be influential at many scales, from the outer tissue of apical meristems to the curvature of membranes in the cell. Curvature research is very active in the animal fields too, further broadening the interest of getting familiar with curvature and its use in current biology.

**Acknowledgments**

The authors thank Catherine Lenne, Marianne Lang, Bamba Niane, Nicolas Dusart and Kaname Teratsuji for reading and commenting the manuscript. This work is dedicated to Pr. Wendy Kuhn Silk, who has been so influential in bringing curvature into consideration in plant developmental and environmental studies.
FURTHER READING

Bastien, R., Bohr, T., Moulia, B., and Douady, S. (2013). Unifying model of shoot gravitropism reveals proprioception as a central feature of posture control in plants. Proc. Natl. Acad. Sci. USA **110**, 755–760. https://doi.org/10.1073/pnas.12143011109.

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Hartmann, F.P., Chauvet-Thiry, H., Franchel, J., Ploquin, S., Moulia, B., Leblanc-Fournier, N., and Decourteix, M. (2022). Methods for a quantitative comparison of gravitropism and posture control over a wide range of herbaceous and woody species. In Plant Gravitropism: Methods and Protocols, E.B. Blancafl or, ed. (New York, NY: Springer), pp. 117–131.

Hartmann, F.P. and Decourteix, M. (2023). Line curvature and plant gravitropic motion [online video] (Cambridge University Press), https://youtu.be/kIBajqnDovE

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<footnote n="1">¹Université Clermont Auvergne, INRAE, PIAF, F-63000 Clermont-Ferrand, France
0000-0001-6225-1268 (F.P.H.); 0000-0002-4485-5065 (M.D.); 0000-0002-3099-0207 (B.M.).</footnote>

<footnote>*E-mail: felix.hartmann@inrae.fr (F.P.H.); bruno.moulia@inrae.fr (B.M.)</footnote>