Leaf growth is conformal

We assess the growth of a total of four leaves of two different species with roughly planar leaves, Petunia and Tobacco, over periods as short as three hours and as long as three days. In these time frames specimens grow between 10% and 42% in overall size. We compare measured displacement fields with those calculated from a conformal map between the initial and final leaf's contour, see figure 1(a). Both measured and calculated displacement fields display the same qualitative characteristics of circular arcs around the leaf tip of ever increasing displacements from leaf base toward leaf tip, see figures 1(d)–(i). To quantify the agreement between measured U and calculated displacement fields V we evaluate the cross-correlation of the vector fields,

$$\begin{eqnarray}&&{\rm{corr}}=\displaystyle \frac{{\displaystyle \sum }_{{i}\in x,y}({U}_{{i}}-\langle U\rangle )({V}_{{i}}-\langle V\rangle )}{\sqrt{{\displaystyle \sum }_{{i}\in x,y}{({U}_{{i}}-\langle U\rangle )}^{2}}\sqrt{{\displaystyle \sum }_{{i}\in x,y}{({V}_{{i}}-\langle V\rangle )}^{2}}}.\end{eqnarray} \tag{ 1 }$$

All of our data sets have very high scores of more than 92% correlation, ranging up to 97% for the top specimen. Also a pixel-by-pixel comparison of the displacement fields in x and y direction normalized by their mean in x and y, respectively, confirm that displacement fields calculated from a conformal map between leaf contours predict the measured displacement fields, see figure 1(b). A detailed mapping of correlation scores per pixel shows that all correlation values are above 85%, see figure 1(c).

Zoom In Zoom Out Reset image size

The displacement field is the most direct assessment of leaf growth from both measurements and evaluating the conformal map within the leaf blade, yet, growth in area at each pixel location is the observable that connects to localized growth of cells and cell groups. The individual growth in area ${\rm{\Delta }}A$ is given by the divergence of the displacement field at each pixel times pixel area $\delta A$: ${\rm{\Delta }}A=\left(\tfrac{\partial {U}_{x}}{\partial x}+\tfrac{\partial {U}_{y}}{\partial y}\right)\delta A$, see figure 2. Taking the derivative, local fluctuations in growth direction and amplitude diminish the absolute correlations between measured and calculated area growth. Correlation scores in the range 40%–60% are significantly smaller than for the displacement field. If we average out local fluctuations in the measured displacement field the correlation in area growth reaches up to 60%–80% at a kernel size about 1/6 of total leaf size, see figure 2(e).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The overall growth patterns are clearly visible in both the measured and the computed growth in area. Growth is largest at the leaf base declining towards the leaf tip in agreement with the growth arrest front that is traveling inward from the leaf tip [14]. Growth is reduced along the midvein, as is seen best close to the leaf base. The Petunia leaf also shows small scale patterns in area growth that are not captured by the area growth calculated from the conformal map. Here, growth is reduced along second order veins with interstitial tissue growing faster.

The conformal map allows us to decompose the overall displacement fields into a series of different modes of growth. To this end we expand the full conformal map

$$\begin{eqnarray}&&f=\displaystyle \sum _{{n}=1,2}{a}_{{n}}{(z-{z}_{{\rm{p}}})}^{{n}}\end{eqnarray} \tag{ 2 }$$

by calculating the best complex Chebyshev approximation of the map f [15] around the position of the petiole z_p, a_n are complex numbers. We neglect higher order modes since their contribution is negligible and the full expansion including the first two orders is already sufficient to reach the same correlation values (see equation (1)) as the full conformal map. The first mode alone which is a simple dilation is not sufficient to describe the observed dynamics of growth. The second mode is necessary to incorporate the low growth rate at the very tip of the leaf. Note that already the combination of two modes of growth allows for a large variety in growth patterns—growth patterns that are captured by a conformal map between initial and final leaf contour.

To investigate when higher orders become important, we theoretically explored the displacement field and growth in area of a conformal map including one additional higher order term $n\gt 2$ in comparison to a map comprising only first and second order growth mode, see figure 4. The first two terms give rise to an elliptically shaped contour starting from an initial disc, while the additional higher order term gives rise to a serrated or multi-lobed shape of the final contour. We therefore expect higher order terms to be of general importance when leaves change their shape to higher undulations of their contour during growth.

Zoom In Zoom Out Reset image size

We now examine the relation between conformal maps and the strain tensor. If the infinitesimal strain is isotropic, then the diagonal strains are equal, ${\partial }_{x}{U}_{x}\,={\partial }_{y}{U}_{y}$, while the shear strain vanishes ${\partial }_{y}{U}_{x}+{\partial }_{x}{U}_{y}=0$. These two relations are exactly the Cauchy−Schwartz equations that are equivalent to the map being conformal. Conversely, if the map is conformal then strain is isotropic.

Many plant cells grow anisotropically, which makes it surprising that conformal maps apply to leaves. However in the later stages of the growth of lateral plant organs, such as leaves, cotyledons, or sepals, it appears either that cellular growth is roughly isotropic or that growth direction fluctuates spatially and temporally [16–19]. Either case yields isotropic growth at the supracellular level that we are considering here (pixel size ∼0.6 mm). Accordingly, we expect our approach to be applicable only to leaves old enough for the blade to have flattened and supracellular growth to have become isotropic.