.. Consider a flat cell-ECM interface on the boundary of a linear tubule (Fig. 3B). For simplicity, we use a 1-D topology to represent the surface of the tubule. The surface area is then the integral of arc length of the tubule. The length of the tubule is confined as S0 by the surrounding thoroughly matrix environment. After a certain amount of time, the number of cells increases and the proliferation pressure pushes the tubule to reach a new equilibrium length S, which is not allowed due to the confinement, leading to the formation of buckles (i.e., bending cells into the matrix, Fig. 3A). To analyze branching pattern formation, let��s use coordinate r? = (x, y) to describe the surface of the tubule (initial y = 0 everywhere on the tubule surface).
Adapting the equations from reference 52, the simplest 2-D free energy function F that ,takes into account both the bending energy and the confinement of tubule length is: F=��2��x=0x=S0ds(d2r?ds2)2+��2[��x=0x=S0ds?S]2.(1) Here, �� is the bending stiffness, �� is the intracellular pressure to push the extension of the tubule, and ds = (dx2 + dy2)1/2 is the arc length along the tubule surface. Following the variation procedure introduced in reference 52 we can perform a variation of F with respect to the coordinates r?, which gives the equation of motion of r?: ��dr?dt=?��F��r?.(2) Here, �� is the viscosity of the tubule surface when cells move into the ECM space. To investigate how branching patterns emerge on the tubule surface, consider a small perturbation of the tubule surface, i.e., y < < S0. Expanding Eqn.
2 to the first order of y, we have: ��dydt=?��d4ydx4?��[S?S0]d2ydx2+O(y2).(3) Using mode analysis with y(k) = y0(k) exp[��(k)t] where k is the wavenumber, we have a dispersion relation: �Ǧ�(k) = ?��k4 + ��[S ? S0]k2.(4) Eqn. 4 indicates that the tubule surface is marginally stable. It also predicts that instability occurs (i.e., ��(k) > 0) when S > S0. The wavenumber with the maximal growth rate, kmax, and the corresponding spacing between branched sites, L, are: kmax=��[S?S0]2��,(5) L=2��kmax=2��2�ʦ�[S?S0].(6) This model predicts that the spacing between branched sites along the tubule surface, L, decreases with the increment of cell number [Fig. 3C (i)], which is controlled by the dosage of growth factors, while L increases with the bending stiffness (i.e., the tension at cell-ECM interface) [Fig.
3C (ii)]. Branching pattern formation by surface tension and cell scattering Besides bending stiffness, another possible effect due to the tension on the cell-ECM interface is to minimize the surface area of epithelial tubules and to confine the motion Brefeldin_A of cells on the surface. To investigate how such effect can lead to branching pattern formation, consider a group of cells that escape or scatter from a preexisting tubule surface into the surrounding ECM (Fig. 4A). In vivo and In vitro, such scattering can be induced by stimulating cells with scattering factors such as hepatocyte growth factor.