Jan 5, 2013

on the size of leaves

Written to Phys. Rev. Lett., Kaare H. Jensen and Maciej A. Zwieniecki try explain the physical limits to leaf size by modeling the fluidity of the botanic vascular system. Their seemingly over-simplified model established the upper and lower limits to leaf size with respect to tree height and surprisingly shows good agreement with a large set of botanical data.

Fig. 1
In their modeling, a plant is treated as three phloem tubes: the leaf, the stem and the root. Each part is assigned hydraulic resistance, \[ \mathcal{R}_\text{leaf} = 1/(2 \pi r l L_p), \quad \mathcal{R}_\text{stem} = 8\eta h/(\pi r^4), \quad \mathcal{R}_\text{root} = 1/(2\pi r s L_p), \] where $l, h, s$ are the length (height) of the leaf, stem and root respectively, $\eta \simeq 5 \text{ mPa}$ the viscosity of the sap, $L_p$ the hydraulic permeability of the membrane, $r$ the phloem radius. Note the difference on resistance between leaf/root and stem: the longer is the leaf/root, the smaller is the resistance, reflecting the role of leaf/root as source/sink; however, the longer is the stem, the larger is the resistance, reflecting the role of stem is to transport. Therefore, the sugar transport speed is, \[
u = \frac{1}{\pi r^2} \frac{\Delta p}{\mathcal{R}} \simeq \frac{2r^2 L_p l}{r^3 + 16 \eta L_p l h}\Delta p, \] where $\Delta p \simeq 1 \text{ MPa}$ is the osmotic pressure difference. The fact that the root is much longer than leaves $s \gg l$, is used for approximation. The transport speed is directly proportional to energy flux $E = k c u$.

First of all, there is a maximum energy flux for fixed $h$, that is achieved when the leaf size $l$ is sufficiently large, \[ E_\max = k c \Delta p \frac{r^2}{8\eta h}. \] Jensen and Zwieniecki argue that, $l$ cannot be unbounded. As the leaf size $l$ increases, the energy flux gain becomes small from increasing the leaf size. Other factors may step in and stop the leaf size from increasing. They model all other factors by setting a energy flux threthold, $E_t = \mu E_\max (\mu \lesssim 1)$. Once the energy flux has reach $E_t$, the increase in leaf size will stop. This implies a upper limit to the leaf size, \[l_\max = \frac{r^3}{16(1-\mu)L_p\eta h} \]
There also exists a lower limit on leaf size for fixed $h$, as if the energy flux is lower than some value $E_\min$, the plant will not survive,
\[ l_\min = \frac{r^3}{16 L_p\eta (h_\max - h)} \] where $h_\max = k c \Delta p r^2/(8\eta E_\min)$ is the constraint on tree height $h$. $h$ large than that, $u_\max < u_\min$ implies $E < E_\min$.

The authors compared their seemingly over-simplified model with a set of botanical data covering 1925 species from 327 genera and 93 families. Surprisingly, their model shows a good agreement with the upper and lower limits of leaf size. With the fitting, they also predicts $h_\max = 104 \pm 6 \text{ m } \quad (95\%), \mu = 0.916 \pm 0.004 \quad (95\%)$.

Fig. 2: Gray triangles show the reported range of leaf sizes for particular species as the longest and shortest leaf lamina length plotted as a function of tree height h. Circles show the five longest (red, dark gray) and five shortest (green, light gray) leaves in each 20 m height bin for trees taller than h = 20 m. Solid lines are fits to theoretical predictions with parameters corresponding to a minimum flow speed of $u_\min = 100 \mu \mathrm{ m/s}$ and energy output efficiency of 90%. Dashed lines indicate 95% confidence intervals.


Kaare H. Jensen and Maciej A. Zwieniecki, Physical Limits to Leaf Size in Tall Trees, PRL 110, 018104 (2013)

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