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98 The Geometry of the Gut 腸道為何彎彎曲曲

Small intestines is the main arena for digestion and nutrient absorptions. To fulfill this role, it evolves into a long narrow tube that is about 6 to 7 meters long. The length of small intestines is important because patients with most of their small intestine resected will suffer from an intractable condition called "short bowel syndrome." We really have to have guts! To store this long tube in our body, our small intestines curl into numerous loops to get the most out of the limited space of our abdominal cavity, but how are these loops formed? To answer this question, a research group led by Clifford Tabin from Harvard performed elegant analysis and experiments, and that is what we are going to talk about today.

Suggested Readings:
1. Savin, T., Kurpios, N. A., Shyer. A. R., Florescu, P., Liang, H., Mahadevan, L. & Tabin, C. J. (2011) On the growth and form of the gut. Nature, 476: 57-62.
2. Nerurkar, N. L., Mahadevan L. & Tabin, C. J. (2017) BMP signaling controls buckling forces to modulate looping morphogenesis of the gut. PNAS, 114: 2277-2282.

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Normally, our peritoneum, the thin lining of abdominal cavity, folds itself around our intestinal loops and their feeding arteries, forming a structure called mesentery and attaching our guts to the posterior wall of abdominal cavity. Several hypotheses have been proposed to explain the looping of the gut. Some proposed that the looping results from external compression of abdominal walls, but removing the gut and mesentery from abdominal cavity would not change its geometry. Some believed that the looping comes from the asymmetric growth of gut tube, but Prof. Tabin's team counted the number of mitotic cells and found that the growth of gut tube is symmetric and uniform. Prof. Tabin's team then dissected the intestine from the mesentery, and found that the intestines turns straight once they are separated from the mesentery. This suggest that the mesentery is necessary for loop formation, and the whole system seems to be under tension.

To differentiate the contribution from the mesenteric arteries from the mesentery tissue, the team carefully dissected out the mesenteric arteries and found that the loop structures remained intact. This indicates that it is the mesentery tissues, rather the mesenteric vessels, that folds our intestines. Combining all these information, the team hypothesized that the geometry of the gut is determined by the differential growth between the gut and the mesentery, which brings up tension and the intestines respond by coiling up to reduce the total energy.

To investigate the mechanism, the team made a toy model of gut-mesentery composite by stitching a silicon rubber tube to an extended thin latex sheet. When external forces are released, the rubber tube coils just like intestines do, and the wavelength and amplitude of the loops only depend on the differential strain, the thickness of the latex sheet, the radius of the rubber tube and their material properties.

The team then analyzed the theoretical aspect of this system with scaling laws. As shown in the following figure (Fig 3(a) of the original article), the team defined several parameters that characterize the loop, including the contour length, $\lambda$, the mean radius of curvature, $R$, the inner and outer radii of the gut, $r_{i}$ and $r_{o}$, and the thickness of mesentery, $h$. The Young's modulus of the mesentery sheet and the tube are $E_{m}$ and $E_{t}$, respectively, and the area moment of inertia of the tube is $I_{t}$. The gut tube will buckled into a wavy structure with characteristic amplitude $A$ if the differential strain between the mesentery and the gut tube, $\epsilon$, exceeds a critical strain $\epsilon_{*}$. The mesentery membrane is deformed over a width of $w$, over which the stretching strain exerted by the tube is relaxed. The width is linked to the strain and loop radius of curvature by $\epsilon\propto\frac{w}{R-w}$.
Figure source: Fig.3(a) of Nature476: 57-62.
Before we keep going, let's review about the mechanics of beam deformation. As we stated in episode 3, As shown in the following figure, a beam with Young's modulus $R$ and neural long axis $z$ was deformed to an arc with radius of curvature $R$, and the axis of deformation is $y$.

For a unit volume at location $y$, it would suffer from a shear strain $\epsilon_{y}= \frac{\delta z(y)}{R\theta}=\frac{z\theta}{R\theta}=\frac{z}{R}$. Therefore, the strain energy of a thin slice of beam with width $\Delta s$ would be:
$w_{bend}^{\Delta s} = \int_{0}^{\Delta s}\text{d}y(\int_{A}\frac{1}{2}E\epsilon_{y}^2\text{d}A) = \frac{E\Delta s}{2R^2}\int_{A}z^2\text{d}A = \frac{EI\Delta s}{2R^2}$.
And for the whole beam, the strain energy is $\frac{EI}{2}\int_{0}^{L}\text{d}s\frac{1}{R(s)^2}=\frac{EI}{2}\int_{0}^{L}\text{d}s\kappa (s)^2$. (Recall that $\kappa$, the curvature, is the second derivative of lateral displacement with respect to distance.) 

Back to our mesentery and gut tube, we can write down the elastic energy stored in mesentery ($U_{m}$) and the elastic energy stored in gut tube ($U_{t}$) as the following:
$U_{m}\propto E_{m}\epsilon^2h\lambda^2$
$U_{t}\propto E_{t}I_{t}\kappa^2\lambda$
Note that the strain, curvature, characteristic amplitude and the contour length are mutually dependent: $\kappa\propto\frac{A}{\lambda^2}$, $\epsilon\propto\frac{A}{\lambda}$, $A\propto\lambda$. By substituting these 3 relationships back to the elastic energy term and minimizing the total energy, we can get the scaling law of contour length $\lambda$:
$\lambda\propto(\frac{E_{t}I_{t}}{E_{m}h})^{1/3}$

The authors further derive the scaling law of radius by balancing the internal torques/moments on gut tube ($T_{t}$) and the mesentery ($T_{m}$):
$T_{t}\propto\frac{E_{t}I_{t}}{R}$
$T_{m}\propto E_{m}hw\epsilon R\propto E_{m}h\epsilon^2R^2$
And the scaling law of radius would be:
$R\propto (\frac{E_{t}I_{t}}{E_{m}h\epsilon^2})^{1/3}$.

By measuring the elastic properties of mesentery and gut tube in different developmental stage, the authors gathered enough data to validate their theoretical model. Their formula successfully predicted the changes in curvature and contour length of the gut over time, and they also built a numerical simulation model that can simulate the whole process of gut development. Their theory of minimizing elastic energy and torque balance can also explain the variations in gut morphology across different species.

The structures and functions of biological systems are highly inter-dependent. However, the formation of a highly specialized structure cannot disobey the fundamental physical laws. This study provided an excellent example on how simple physical law can determine the morphology and the dynamical changes of one of the most important organs - our guts.

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