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Table 4 For the perplexed: how to determine the expected dynamics in any CSM model

From: The biophysical nature of cells: potential cell behaviours revealed by analytical and computational studies of cell surface mechanics

Step 1. Write the Hamiltonian or energy function in the standard form as used in this paper (i.e. E=J p+λ p (pP)2+λ a (aA)2 for 2D; E=J s+λ s (sS)2+λ v (vV)2 for 3D).
Step 2. When using CPM, first the effective values of J, and P and λ p in 2D, or S and λ s in 3D, have to be determined, or inversely the correct values of \(J_{_{\mathit {CPM}}}\) etc. have to be assigned. This is done by calculating the perimeter scaling factor ξ, which depends on the neighbourhood radius, in its turn depending on the neighbourhood level. The value of ξ can be obtained either by using the theoretical formulae Eq. 29, Eq. 33, for 2D and 3D respectively, in the case that the neighbourhood is sufficiently large, or by using the numerical estimates given in Table 2, for small neighbourhoods that can present significant deviations from those theoretical values. Transformations between the effective and CPM parameter values are then given by Eq. 31, Eq. 32 for 2D, and Eq. 34 for 3D. Note that because of the way interfacial tension is implemented in the CPM, \(J=\xi J_{_{\mathit {CPM}}}\) for cell-medium interfaces and \(\frac {\xi }{2}J_{_{\mathit {CPM}}}\) for cell-cell interfaces.
Step 3. Calculate k p , k a (for 2D) cq. k s , k v (for 3D) for the cell shape of interest. The values for typical shapes in 2D and 3D, namely circle and hexagon, cq. sphere and rhombic dodecahedron, are given in Table 3 and Table 6.
Step 4. Calculate τ and ε (for 2D), ν and μ (for 3D when λ s ≠0), or ν and μ (for 3D when λ s =0), using the formulae given in Table 3 (for 2D) and Table 6 (for 3D).
Step 5. The expected behaviour of a single cell or tissue is given by Figure 3; the bifurcation lines which form the transitions between the different zones are given in Table 3 (for 2D) and Table 6 (for 3D).