Minimizing the Elastic Energy of Growing Leaves by Conformal Mapping

During morphogenesis, the shape of living species results from growth, stress relaxation, and remodeling. When the growth does not generate any stress, the body shape only reflects the growth density. In two dimensions, we show that stress free configurations are simply determined by the time evolution of a conformal mapping which concerns not only the boundary but also the displacement field during an arbitrary period of time inside the sample. Fresh planar leaves are good examples for our study: they have no elastic stress, almost no weight, and their shape can be easily represented by holomorphic functions. The growth factor, isotropic or anisotropic, is related to the metrics between the initial and current conformal maps. By adjusting the mathematical shape function, main characteristics such as tips (convex or concave or sharp-pointed), undulating borders, and veins can be mathematically recovered, which are in good agreement with observations. It is worth mentioning that this flexible method allows us to study complex morphologies of growing leaves such as the fenestration process in Monstera deliciosa, and can also shed light on many other 2D biological patterns.

Figure 1

Schematic diagram: (a) $\mu$ and $\eta$ are the curvilinear coordinates, $\mu ={\mu }_0$ corresponds to a green curve and $\eta ={\eta }_0$ to a blue curve. (b) Parameters $b_{k0}$ depend on the leaf contour at $t=0$, ($F_1$). (c) Parameters $b_k(t)$ are determined by the contour at time $t$, ($F_2$).

Figure 2

Natural leaves: (a) Jujube leaf with sharp tip and oscillatory border. (b) Redbud leaf with rounded tip and smooth border. (c) Robinia pseudoacacia leaf with concave tip and smooth border. Mathematical images: (d)–(f) simulated by shape function (see Supplemental Material [27], Sec. SII). Level of green colors shows the local growth density in different regions, lighter color indicating more growth intensity. Jujube leaf [(a) and (d)] grows more at the tip while the top and bottom of Robinia pseudoacacia leaf [(c) and (f)] have a weak level of growth. Redbud leaf [(b) and (e)] shows a relatively uniform growth in the entire area, except near the petiole.

Figure 3

Different leaf borders, shape function of all contours: $z=-i{b}_{1}S+S_c$, where ${\mu }_0=0.6$, $a=2$ and $b_1=2$. (a)–(c): Boundary shapes changing with decreasing values of $\alpha (t)$ and $\beta (t)$. (d) Jujube leaves showing similar edges with (c). (e)–(g) Boundary shapes evolving with only decreasing values of $\alpha (t)$. (h) White Sapote leaves show similar edges as (g).

Figure 4

(a) On left, natural Monstera deliciosa leaf with holes located either at the center (on top right) or connected to the border. On the same level on top, initial and current state of a mathematical leaf with one hole, located at the center in a chosen area. (b) A rectangle including a hole on left mapped into a specific curvilinear rectangle with boundaries defined by $\mu$ and $\eta$. The selected conformal mapping is a solution of the periodic Darcy flow with periodic and symmetric bubbles. Selection of constant $k$ and $c_2$ are made in relation with (a). (c) Natural Monstera leaf with “fingerslike” shape holes. The initial and current state of a mathematical leaf with asymmetrical hole. (d) Conformal mapping generating an asymmetric bubble. The hole is closer to the outer boundary. (e) Natural Mexico Monstera leaf with holes and fingers and the initial and current state of mathematical leaf with two holes. (f) Anisotropic growth of the mathematical leaves with two holes, the functions $k(\mu )$ and $l(\eta )$ are detailed in [49], notice the changes of vein position between (e) and (f).