Differential Growth and Shape Formation in Plant Organs
Short Title: 3D Shape Formation in Plant Organs
Morphogenesis is a phenomenon by which a wide variety of functional organs are formed in biological systems. In plants, morphogenesis is primarily driven by differential growth of tissues. Much effort has been devoted to identifying the role of genetic and biomolecular pathways in regulating cell division and cell expansion and in influencing shape formation in plant organs. However, general principles dictating how differential growth controls the formation of complex three-dimensional (3D) shapes in plant leaves and flower petals remain largely unknown. Through quantitative measurements on live plant organs and detailed finite element simulations, we show how the morphology of a growing leaf is determined by both the maximum value and the spatial distribution of growth strain. With this understanding, we develop a broad scientific framework for a morphological phase diagram that is capable of rationalizing four configurations commonly found in plant organs: twisting, helical twisting, saddle bending, and edge waving. We demonstrate the robustness of these findings and analyses by recourse to synthetic reproduction of all four configurations using controlled polymerization of a hydrogel. Our study points to potential new approaches to innovative geometrical design and actuation in such applications as building architecture, soft robotics and flexible electronics.
Plant leaves and flower petals in nature exhibit wide variety of complex 3D shapes. Formation of these shapes has largely been studied from genetic and biomolecular viewpoints, overlooking contributions from biophysical factors such as mechanical stress and deformation. By means of computational simulations and quantitative analyses of the growth strains in live plant organs, we develop fundamental mechanistic insights into how nature invokes mechanics in the evolution of four commonly found shapes in plant organs by differential growth. We also demonstrate how these common shapes can be synthetically reproduced in hydrogel using this mechanistic understanding. Our study provides a broad scientific framework for rationalizing plant organ morphogenesis, but also provides pathways for generating bio-inspired 3D architectures in soft materials.
Biological systems have developed versatile design strategies through evolution over billions of years. Uncovering these design principles not only elucidates the mechanisms underlying the evolution of living systems, but also establishes the scientific basis for biomimetics whereby advanced materials and systems could be developed based on lessons learned from nature. Examples of plant-inspired engineering systems include: super-hydrophobic surfaces designed with strategies drawn from the structure of the lotus leaf (1); self-repairing anti-fouling surfaces inspired by the pitcher plant (2); and actuation systems inspired by the Venus flytrap (3–5). Motivated by the wide variety and diversity of three-dimensional (3D) shapes of plant organs, much effort has been devoted to identifying the genetic and biomolecular pathways that regulate cell division and/or cell expansion as well as shape formation (6–9). Although specific genes dictate cell growth in a programmed manner, biomechanics also plays an essential and significant role in the formation of 3D shapes at the organ level. Compositional inhomogeneity of the constituent materials or the polarized intercellular transport of growth factors leads to inhomogeneous and non-uniform growth in various parts of plant organs such as leaf, stem, root, flower and fruit and, consequently, leads to mismatch of growth strain (10, 11). In contrast to mammalian cells which can migrate to accommodate the need for extra spaces, the physical constraints in adjoining plant cell walls force adjacent cells to deform in a coordinated manner, facilitating the emergence of complex shapes (12–16).
A leaf blade or a flower petal mainly consists of two layers of mesophyll cells sandwiched between two layers of epidermis, with vascular veins lying in the median plane of the mesophyll. Differential cell expansion between the upper and lower sides of the mesophyll in response to environmental stimuli (e.g. temperature, light or humidity) is responsible for the repeated opening and closure of the flower (17). However, excessive cell growth within the marginal region plays a dominant role in the formation of saddle shapes and rippled edges in leaves and flower petals that open only once (18, 19). Besides saddle shapes and localized ripples, helical growth represents another growth pattern commonly seen in nature (20). Direct correlation between the helical growth of stems and the helical pattern of microtubules has also been observed (21, 22). In addition, cell wall composition could dictate the helical growth of plant cells (23). Our current state of understanding of the mechanisms of morphogenesis and shape evolution in plant organs is not sufficiently complete to conclude whether the observed helical pattern of microtubules or cells is the cause or consequence of the helical growth of the organ. In addition, the seemingly random evolution of twisting directions in some species points to the possibility of non-genetic factors influencing the emergence of helical growth of plant organs (20).
In this study, we investigate the origins of 3D shapes driven by biomechanical constraints and differential growth in plant organs by recourse to quantitative experiments on live plant organs and detailed finite element simulations. On this basis, we present here a scientific framework that establishes a morphological phase diagram which is capable of rationalizing four geometric configurations commonly found in plant organs: twisting, helical twisting, saddle bending, and edge waving. Our results show that the morphology of a growing leaf is determined by both the maximum value and the spatial distribution of growth strain. We further demonstrate the robustness of our scientific findings on differential growth-induced morphogenesis by synthetic reproduction of all four common geometric configurations in our phase diagram in a model hydrogel system that mimics plant growth through controlled manipulation of the polymerization process. Our study thus not only provides a broad framework for elucidating the evolution of plant physiology, but also points to potential new approaches for innovative geometrical design and actuation in such applications as building architecture, soft robotics and flexible electronics.
3D shapes of long plant leaves and flower petals
An elevated growth factor level within the marginal region of plant leaves or flower petals often results in an excessive local growth rate and a mismatch of growth strain. The accumulated residual stresses eventually facilitate out-of-plane deformation instability, leading to a large variety of 3D shapes in nature. Figure 1A shows images of four representative 3D shapes of long orchid petals, from which we identify the corresponding four basic configurations. Configuration I shows the twisting growth of long petals of dendrobium helix whose centers remain approximately straight from the base to the distal end. In contrast, the petals of prosthechea cochleata exhibit a helical twisting morphology (Configuration II) where the petals’ centers follow a helical pattern. The petals of brassavola cucullata bend into a saddle shape (Configuration III) whose two principal curvatures feature opposite signs. The long petals of phragmipedium grande feature wavy patterns along petal margins (Configuration IV). More complex petal morphologies resulting from the combination of these configurations are also present in nature (see Fig. S1).
The central question for this study is: how does the differential growth in these leaves influence their morphogenesis? We collected live leaves from several species (from the Phipps Conservatory and Botanical Garden in Pittsburgh, PA). To quantify the growth strain profile of a leaf, we dissected them into individual narrow strips parallel to the stem, and measured the length of each strip, as illustrated in Figs. 1B-C. We observed that dissecting half of the leaf leads to a change in the shape of the other half, suggesting the existence of residual stresses in its natural state. We define the growth strain as: , where and l are the lengths of the center stem and the strip at a distance y from the stem, respectively. As shown in Fig. 2A, the growth strain profiles along the width direction can be approximated by a power-law function, as
where y and W are the distance from the strip to the center and half width of the leaf, respectively. The growth strain monotonically increases from zero at the center to a maximum value, β, at the edge when y = W. The power-law exponent, n, characterizes the steepness of the differential growth strain profile. Figure 2B shows that the leaves with a twisting configuration feature a parabolic growth strain profile (n ≈ 2), and those with edge-waving configuration feature a steeper increase in growth strain near the marginal region, leading to a higher n value. Compared to leaves with twisting or edge-waving configuration, leaves with saddle bending configuration feature much smaller maximum growth strain level (β < 0.1).
Mechanistic modeling of a growing leaf
To investigate the effect of growth strain on morphogenesis, we conducted finite element (FEM) simulations of a long rectangular leaf with a non-uniform growth strain across its width direction (Fig. 3A). The growth strain increases from zero at the center to the maximum value at the edge. As shown in Fig. 3B, increasing the value of n increases the steepness of the strain profile near the leaf margin and reduces the steepness of the strain profile in the interior region. In the present FEM simulations, we modeled the growth process as an equivalent thermal expansion problem. We allow the leaf to grow only along the x-axis direction by prescribing a uniform non-zero thermal expansion coefficient but a non-uniform temperature field, while expansion along the other directions is prevented via a vanished thermal expansion coefficient. The temperature field, in a stress-free state, leads to a strain field which is the same as the growth strain distribution in Eq. (1).
In all simulations, deformation commences as planar at small growth strain; and instability then occurs at a critical level. Like many thin film systems under constraint, the resulting buckled configuration is dictated by local strain fields (25–27). Here, the two critical parameters controlling the shapes of buckled configurations are β and n, which for thin plant organs are dependent on species and age. Therefore, they can vary over a large range (Fig. 2B). Although plant cells exhibit viscoelastic properties in stretch tests (28), we modeled the growing petal as an elastic material for the following reasons. First, the mechanical response of plant cells is captured by a standard linear solid material model (i.e. two Maxwell elements in parallel with an elastic spring), with a characteristic time of relaxation on the order of seconds (29). However, the complete morphogenesis process of a growing leaf occurs over a time scale of several days. Because morphogenesis is primarily driven by cell proliferation and cell growth, growth strain dominates total deformation of the growing petal. The Young’s modulus and Poisson’s ratio values were set at E = 350 MPa and ν = 0.25, respectively (28). We considered a full range of exponent n and maximum strain β. Figure 3 highlights the results for n = 2 and n = 10.
For parabolic growth strain distribution with n = 2, the first instability consistently leads to twisting (Configuration I) at an intermediate maximum growth strain level (β ≈ 0.15), representing the onset of initial instability from planar to 3D mode (Fig. 3C). Figure S2A shows the total strain energy stored in the leaf as a function of maximum strain. As expected, when the strain is low, the total strain energy increase scales with the square of strain, featuring a slope of m = 2.0 in the log-log plot. Initiation of instability is clearly captured by a sudden change in the slope, from quadratic to a nearly linear dependence of m = 1.0 (see Supplementary Information for theory). As shown in Fig. 3C and Fig. S3, a twisted configuration at a low growth strain level is nearly stress-free. Solely from geometric considerations, we note that shape change of a leaf from stress-free planar into stress-free twisting configurations requires a growth strain profile given by
where α is the twist angle per unit length along the stem. Taylor expansion of Eq. (2) gives
when the 4th-order term is one order of magnitude smaller than the 2nd-order term, the maximum strain at edge. Equation (3) indicates that, at a low maximum strain level (), the formation of twisting configuration is energetically favorable for n = 2. The twisting angle per unit length as a function of the maximum growth strain is derived from Eq. (2) as
Figure 3D shows that α increases monotonically with increasing strain. Results from the FEM simulations are in good agreement with theoretical prediction. At a fixed maximum growth strain, wider leaves or petals twist less than narrower ones.
Interestingly, at a low maximum growth strain level (β ≈ 0.05), both twisting and saddle bending configurations emerge as two local energy minimum states. Careful energetic analysis shows that the strain energy of the saddle bending configuration is relatively lower than that of the twisting configuration, but the relative energy difference between these two configurations is less than 10% (Fig. S4). Figure 3C shows that, as growth strain increases (β > 0.15), residual stresses due to in-plane stretch gradually build up as higher order terms in Eq. (3) become more important. Thus maintaining the twisting configuration becomes energetically more unfavorable. Consistent with the theory, our simulations reveal a transition from twisting to helical twisting (Configurations II) when the maximum growth strain approaches a critical level around . This transition occurs with the deviation of center axis away from a straight line and a non-uniform distribution of the von Mises effective stress (Fig. S3). A sudden release (drop) of total strain energy is observed as a result of the mode transition, as clearly shown in Fig. S2B.
For steeper change of growth strain, n = 10 (see Fig. 3B), saddle bending configurations become energetically more favorable than twisting, as shown in Fig. 3E. This is mainly due to the fact that differential growth predominately concentrates in the marginal region of the leaf at higher n values. Consequently, twisting introduces significant in-plane stresses and high strain energy in the interior region. At low growth strains, the leaf grows into a saddle shape (Configuration III). Further increase in growth strain leads to the emergence of an undulating wavy pattern. Eventually, competition between bending in the marginal region and stretching in the interior results in edge waving (Configuration IV). Our simulation (Fig. 3F) shows that the wavelength of edge waves decreases with increasing maximum strain level and follows a power law relationship of . The stress contours in Fig. 3E show that the highest hydrostatic stress levels occur at the peaks and valleys of the edge waves, and that hydrostatic pressure increases with an increase in maximum growth strain. Considering that hydrostatic pressure may alter normal nutrient and growth factor transport and/or cell wall remodeling needed for growth (30–32), the increase in pressure level is expected to influence the leaf’s morphogenesis.
Through parametric simulations we developed a phase diagram of the morphogenesis of a growing leaf. Figure 4 shows the parameter space (n, β) of different morphogenesis configurations (I, II, III, and IV). The experimental results measured from live leaves (Fig. 2B) are superposed as data points on the phase diagram. Note that the boundaries between different zones are approximate. In particular, between the saddle bending (Configuration III) and edge waving (Configuration IV) regions, simulations show a wide transition zone featuring configurations with kinks along the length direction but without regular periodic waves along edges. The emergence of a kink from the saddle bending configuration is a result of a structural instability, accompanied by release of residual stresses in the central region, and by sudden strain energy drop as shown in Fig. S2D. At higher maximum strain β and higher n values, however, this kink transition zone gives way to the edge waving zone (Fig. 4).
Although the general trend of the simulation-generated phase diagram is consistent with the measurements from live leaves, there are some apparent discrepancies in the details. Our simulations predict that a twisting leaf transits into a helical twisting configuration when . However, measurements from twisting croton ‘mammy’ leaves show that this transition occurs at higher maximum growth strains (one leaf even has a maximum strain of, Fig. 2B). This apparent discrepancy can readily be rationalized through considerations of a stiff mid-vein in croton ‘mammy’ leaves. As shown in Fig. 3C, change from twisting to helical twisting requires bending of the centerline, thus a higher driving force for a stiff mid-vein. Therefore, the twisting-to-helical twisting transition is expected to occur at a much larger maximum growth strain in the presence of a stiff mid-vein. Similarly, the presence of a stiff mid-vein in the fern tree leaves rationalizes why data points obtained from edge-waving leaves fall within the ‘intermediate state’ region.
Engineering morphogenesis with directed polymerization of hydrogel
With the foregoing quantitative knowledge of the role of mechanics in plant morphogenesis, we now explore synthetic reproduction of such processes and shapes in engineered systems. We resort to our development on the polyacrylamide (PA) hydrogel (33), in which we control the oxygen concentration to modulate the polymerization of PA hydrogel. As shown in Fig. 5A, oxygen diffusion from the surrounding air into the gel solution prevents gelation within the vicinity of liquid-air interface. Polymerization in the region away from the liquid-air interface consumes free monomers and cross-linkers, leading to a concentration gradient of free monomers and cross-linkers, and driving their diffusion from the non-gelation region into a polymerized hydrogel network. The continuous polymerization consequently leads to continuous growth of post-gelation region (Fig. S5).
We induce differential growth of hydrogel by introducing geometric constraints. An example of such a constraint is a string (e.g. cotton thread) placed parallel to the centerline of the reaction chamber. The thread is relatively inextensible and imposes limited resistance to the out-of-plane bending deformation of the hydrogel sheet. Consequently, local growth of the hydrogel near the thread is inhibited as the edges grow, giving rise to a growth strain profile similar to that given by Eq. (1). The growth profile can be controlled by adjusting the location and stiffness of the constraining wire within the specimen. When we introduce a pair of cotton threads, separated by a distance of 2α0W, as demonstrated in Fig. 5A, the modified growth strain profile can be approximated as
when . As shown in Fig. 5B, increasing the separation distance is essentially equivalent to increasing the value of n. It is difficult to quantify the exact value of the exponent experimentally. However, the configurations of the hydrogel in the presence of the thread suggests that the value is close to n = 2 when a single string is placed along the centerline. Figure 6A shows that the hydrogel naturally grows into a twisting configuration (I), and then turns into helical twisting (II) at higher strain level.
Other plant morphogenesis configurations can also be engineered by controlling the degree and location of the constraints. Figure 6B shows that, when placing a pair of constraining threads with a separation distance of 0.68W, the differential strain mismatch induced by polymerization transforms the hydrogel into a saddle bending shape (III). Further swelling the hydrogel leads to a configuration with wavy edges (IV).
The unique feature of the present technique is that the continuous polymerization process controlled by oxygen concentration, albeit chemically and mechanistically different, closely mimics the cell growth process in plants controlled by growth factor levels. Figure 6 demonstrates that, with proper design of constraints, our technique is capable of mimicking morphogenesis of plant leaves and flowers.
Morphogenesis is achieved by differential growth of tissues in response to controlled distribution of the level of growth factors, and/or different growth rates among different constituent parts (e.g. vascular network vs. leaf lamina). Dictated by genes, the differential growth pattern varies in a plant species- and age-dependent manner. Our study highlights that shape evolution in plant systems is strongly influenced by the mechanics of plant organs – the deformation of tissues to accommodate differential growth, the instabilities that give rise to a cascade of different shapes, and the build-up of strain energy to maintain the shapes. By recourse to quantitative measurements of live leaves, finite-element simulations of differential growth-induced mechanical deformation of leaf-shaped structures, and experimental models to mimic leaf growth, we elucidate a number of mechanisms underlying plant leaf morphogenesis. More importantly, we have established a general framework to quantify the role of mechanical deformation and geometrical constraints in influencing the evolution of stable shapes commonly found in a number of plant organs. Future work in this area would inevitably call for investigations of a much wider spectrum of plant species as well as considerations of the details of plant leaf shapes, including the cross-sectional shapes and the presence of vascular networks. Besides the in-plane differential growth, the growth across the thickness direction may also vary and play an important role in shaping the leaves (34). A more comprehensive investigation will establish the critical relationship among the spatiotemporal pattern of genetic and molecular factors at molecular level, the resultant differential growth patterns and 3D shapes at tissue level. Such effort will also require more detailed constitutive models of the mechanics of plant organs than the ones considered in this work.
Besides providing insights into morphogenesis of plant organs in nature, our present work also points to novel pathways to create bio-inspired complex 3D shapes that could find applications in such areas as tissue engineering, flexible electronics and soft robotics. Biomimetics of the differential growth of thin plant organs in vitro has primarily involved differentially stretching a plastic sheet or rubber, or swelling a patterned hydrogel (4, 12, 35–37). More specifically, material anisotropy has been routinely incorporated in order to generate twisting and helical twisting configurations (38–41). Besides the fundamental differences between living tissue growth in nature and hydrogel swelling or stretching in a plastic sheet (33), anisotropic material designs currently invoked in biomimetic systems do not faithfully reproduce the morphogenesis of most biological systems. Our findings point to a more biologically relevant approach to generating twisting and helical twisting configurations.
Finite element analysis
The morphogenesis of a growing leaf was simulated using commercial software ABAQUS. We modeled the growing process as an equivalent thermal expansion problem, where a uniform non-zero thermal expansion coefficient was prescribed along the longitudinal direction and a non-uniform temperature field was assigned as an analytical field. The leaf was modeled as an elastic solid discretized by 4-node general purpose shell elements with reduced integration and accounting for finite membrane strains. Prior to post-buckling analyses, a linear buckling analysis was performed to identify the buckling modes. A linear combination of first 10 buckling modes was introduced as geometric imperfection into the post-buckling analyses to initiate the out-of-plane buckling. The level of the geometric imperfection was systematically tested to make sure that it was big enough to initiate the out-of-plane deformation but had no influence on the resulting configurations. Constraints were applied to the two short sides (at y = 0 and y = L in Fig. 3A) to make sure these two edges remain within an x-z plane throughout the simulation (i.e. same displacement along longitudinal direction). The Young’s modulus and Poisson’s ratio are E = 350 kPa and v = 0.25, respectively.
Polymerization of polyacrylamide hydrogel
Rectangular reaction chambers were constructed by gently pressing a rectangular polydimethylsiloxane (PDMS) ring onto a glass slide. To prepare the mold for making the PDMS ring, rectangular patterns were cut out of a 6.35 mm thick acrylic sheet using a laser cutter (Epilog Laser) and then glued onto the bottom surface of a Petri dish using a two-part epoxy (Gorilla Epoxy; Gorilla Glue Inc.). PDMS rings were prepared using a 10:1 weight mixture of base to curing agent (Sylgard 184; Dow Corning). Prior to being introduced into the reaction chamber, hydrogel solution of 10% acrylamide and 0.1% bis-acrylamide (Amersco) was degassed for 30 min in a vacuum desiccator to completely remove dissolved oxygen, and cotton wires pre-wetted with hydrogel solution were placed into the reaction chamber as a way to introduce differential growth. Polymerization was then initiated by introducing tetramethylethylenediamine (TEMED; Teknova) and ammonium persulfate (APS; Promega) into the gel solution with final concentrations of 0.3% and 0.05%, respectively. The gel solution of a desired volume was immediately pipetted into the polymerization chamber with minimal disturbance so as to avoid introducing oxygen. The solution was allowed to react for 1 h before disassembling the reaction chamber to release hydrogel structures.
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Fig. 1. Observations and experiments on live plant leaves. (A) 3D morphologies of long orchid petals (images from the internet (24)) with the corresponding basic buckling configurations. From left to right: dendrobium helix in twisting (Configuration I), prosthechea cochleata in helical twisting (Configuration II), brassavola cucullata in saddle bending (Configuration III), and phragmipedium brasiliense in edge waving (Configuration IV). (B-C) Dissection of twisting croton ‘mammy’ (codiaeum variegatum) leaf (B) and edge-waving fern tree (filicium decipiens) leaf (C) into thin strips for growth strain quantification.
Fig. 2. Quantitative experimental characterization of growth strain profile of live plant leaves. (A) Representative growth strain profiles of leaves with twisting and edge waving configurations normalized by maximum growth strain. (B) Plot of n and β values of twisting, edge waving and saddle bending leaves. Leaves with edge waving configuration are taken from two different species: fern tree (filicium decipiens; s1) and croton ‘interruptum’ (codiaeum variegatum; s2).
Fig. 3. Finite element modeling of a growing leaf. (A) Representative growth strain distribution normalized by maximum growth strain at the edge within a long leaf (strain increases from center to edge, but remains uniform along length direction). (B) Differential growth strain profiles with various n values normalized by the maximum growth strain. (C) Emergence of twisting and helical twisting configurations at n = 2. Contours of hydrostatic stress at various maximum strain levels are plotted. (D) Normalized twist angle per unit length as a function of the maximum strain at n = 2. The theoretical prediction is from Eq. (4), and the simulation results are from finite-element modeling. (E) Emergence of saddle bending and edge waving configurations at n = 10. Color contours of hydrostatic stress at various maximum strain levels are plotted. (F) Wavelength of the edge waves and maximum hydrostatic stress level as a function of the maximum strain at n = 10.
Fig. 4. Phase diagram of morphogenesis of thin plant organs as functions of n and β. Experimental data points are same as those in Fig. 2B, where triangles (blue), squares (black) and circles (green) denote leaves with twisting, edge waving and saddle bending configurations, respectively.
Fig. 5.In vitro reproduction of leaf morphogenesis. (A) Schematics of the working principles of the in vitro hydrogel system. (B) Manipulation of the growth strain profile in PA hydrogel.
Fig. 6.Reproduction of the four 3D configurations of leaves: (A) twisting and helical twisting; (B) saddle bending and edge waving. Insets are contour plots of normalized growth strain profile () in different configurations: n = 2 in panel (A) and n = 8 in panel (B). Steps  and  represent polymerization and hydration, respectively. Scale bars: 1 cm.