PROJECT The Emergence of Geometric Order in Proliferating Cells When some cells

Question

PROJECT The Emergence of Geometric Order in Proliferating Cells When some cells divide they form thin sheets in which the majority of cells are each adjacent to six other cells. This gives the sheet a hexagonal pattern as shown in Figure 1 In this project we will see how a mathematical model predicts the emergence of this pattern simply by the way cells divide. Each cell in a sheet can be viewed as a polygon having a certain number of edges and vertices (see Figure 2). We assume that each cell has a minimum of four edges and that, when a cell division occurs (that is, when a cell splits into two daughter cells), the line of cell division always connects two edges, subdividing each edge (see Figure 3). The cells divide asynchronously and we take a single time step to be the time after which all cells in the sheet have divided once. Let cr. e, and e, denote the total number of cells, vertices, and edges after cell divisions. Since each cell divides into two in a single time step, we have c,+1-2c," FIGURE FIGURE 2 The shaded cell has four vertices and edges 1. Explain why the total number of vertices in the cell sheet obeys the recur- sion , 2c, and why the total number of edges obeys the recursion ee 3c,. (Figure 3 might be helpful for this.) FIGURE3 Numerals in each cell indicate the number of edges

Explanation / Answer

++++In the proliferation of T cells shown above in fig. 1 we can take c1 as the first cell, v1 as the first vertice and e1 as the first edge. A cell first divides into two and has some edges and vertices. In figure 2 the first cell has four edges and four vertices. Hence the second cell, as well as the rest of the cells, will develop according to the number of edges and vertices. The cells sheet obeys the recursion v1 + 1= v1+ 2cv as each vertex can proliferate into 2 cell or support at least two cell and the edges obey the recursion e1 + 1 = e1 + 3cv as each edge of a cell can proliferate into 3 cells as shown in figure 3, one perpendicular to the edge another two from the two sides(the right edge and the left edge).

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