Three-Dimensional 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. Fig. 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 nodosa bend into a saddle shape (Configuration III) whose two principal curvatures feature opposite signs. The long petals of phragmipedium brasiliense feature wavy patterns along petal margins (Configuration IV). More complex petal morphologies resulting from the combination of these configurations are also present in nature (SI Appendix, Fig. S1).

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Fig. 1.

Observations and experiments on live plant leaves. (A) Three-dimensional morphologies of long orchid petals (images from www.speciesorchids.com/CeretobeDendrobiums.html; https://queenslandorchid.files.wordpress.com/2014/03/prosthechea-cochleata.jpg; https://commons.wikimedia.org/wiki/File:Brassavola_nodosa_-_Flicker_003.jpg; and https://www.orchidweb.com/orchids/phragmipedium/species/phrag-brasiliense-clone-a) with the corresponding basic buckling configurations. From Left to Right: d. helix in twisting (Configuration I), p. cochleata in helical twisting (Configuration II), b. nodosa in saddle bending (Configuration III), and phragmipedium brasiliense in edge waving (Configuration IV). Configuration I image courtesy of Paul Zorn (photographer). Configuration II image courtesy of SoundEagle/Queensland Orchids International. Configuration III image courtesy of Wikimedia Commons/Elena Gaillard. Configuration IV image courtesy of Orchids Limited. (B and 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.

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 Fig. 1 B and 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 εg=(l−l0)/l0, where l0 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, asεg(y)=β(yW)n,[1]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. Fig. 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 with leaves with twisting or edge-waving configuration, leaves with saddle-bending configuration feature much smaller maximum growth strain level (β < 0.1).

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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; edge waving: B, s1) and croton “interruptum” (c. variegatum; edge waving: B, s2).

Mechanistic Modeling of a Growing Leaf.

To investigate the effect of growth strain on morphogenesis, we conducted finite-element method (FEM) simulations of a long rectangular leaf with a nonuniform 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 nonzero thermal expansion coefficient but a nonuniform 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.

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Fig. 3.

FEM 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 FEM. (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.

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 (24⇓–26). 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 (27), 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 (28). 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. Young’s modulus and Poisson ratio values were set at E = 350 MPa and ν = 0.25, respectively (27). We considered a full range of exponent n and maximum strain β. Fig. 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). SI Appendix, Fig. 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 SI Appendix for theory). As shown in Fig. 3C and SI Appendix, 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 byεg(y)=1+(αy)2−1,[2]where α is the twist angle per unit length along the stem. Taylor expansion of Eq. 2 givesεg(y)=12(αy)2−18(αy)4+116(αy)6+⋯≈12α2W2(yW)2,[3]when the fourth-order term is one order of magnitude smaller than the second-order term, the maximum strain at edge β=(1/2)α2W2≤0.2. Eq. 3 indicates that, at a low maximum strain level (β≤0.2), 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 asα=2β+β2W.[4]Fig. 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% (SI Appendix, Fig. S4). Fig. 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 β=0.2. This transition occurs with the deviation of center axis away from a straight line and a nonuniform distribution of the von Mises effective stress (SI Appendix, Fig. S3). A sudden release (drop) of total strain energy is observed as a result of the mode transition, as clearly shown in SI Appendix, Fig. S2B.

For steeper change of growth strain, n = 10 (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 λ∼β−0.2. 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 (29⇓–31), 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. Fig. 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 SI Appendix, 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).

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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.

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 β≈0.2. 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 β≈0.7, Fig. 2B). This apparent discrepancy can readily be rationalized through considerations of a stiff midvein 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 midvein. Therefore, the twisting-to-helical twisting transition is expected to occur at a much larger maximum growth strain in the presence of a stiff midvein. Similarly, the presence of a stiff midvein 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 (32), 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 nongelation region into a polymerized hydrogel network. The continuous polymerization consequently leads to continuous growth of postgelation region (SI Appendix, Fig. S5).

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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.

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 asεg(y)=β|y−α0WW−α0W|n,[5]when |y|≥α0W. 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. Fig. 6A shows that the hydrogel naturally grows into a twisting configuration (I), and then turns into helical twisting (II) at higher strain level.

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Fig. 6.

Reproduction of the four 3D configurations of leaves: (A) twisting and helical twisting; (B) saddle bending and edge waving. (Insets) Contour plots of normalized growth strain profile (εg/β) in different configurations: n = 2 in A and n = 8 in B. Steps [1] and [2] represent polymerization and hydration, respectively. (Scale bars: 1 cm.)

Other plant morphogenesis configurations can also be engineered by controlling the degree and location of the constraints. Fig. 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. Fig. 6 demonstrates that, with proper design of constraints, our technique is capable of mimicking morphogenesis of plant leaves and flowers.