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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 $h_\max = 104 \pm 6 \text{ m } \quad (95\%), \mu = 0.916 \pm 0.004 \quad (95\%)$.

*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 |

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 $h_\max = 104 \pm 6 \text{ m } \quad (95\%), \mu = 0.916 \pm 0.004 \quad (95\%)$.

#### references:

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