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Table 5 For the perplexed: how to correctly rescale a CPM model

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

Rescale spatial resolution When the spatial resolution of a CPM model is k-fold increased, the required changes in the standard set of kinetic parameters are given in Eq. 38 and Eq. 39.
Resizing neighbourhood When the neighbourhood used in a simulation is changed, the value of \(J_{_{\mathit {CPM}}}\phantom {\dot {i}\!}\), and \(P_{_{\mathit {CPM}}}\phantom {\dot {i}\!}\) and \(\lambda _{p\,{}_{\mathit {CPM}}}\phantom {\dot {i}\!}\) in 2D, or \(S_{_{\mathit {CPM}}}\phantom {\dot {i}\!}\) and \(\lambda _{s\,{}_{\mathit {CPM}}}\phantom {\dot {i}\!}\) in 3D, have to be modified, such that the effective values remain the same (Eq. 31, Eq. 32, Eq. 34). This can be achieved by setting \(J_{_{\mathit {CPM}}}'=\frac {\xi _{\textit {old}}}{\xi _{\textit {new}}}J_{_{\mathit {CPM}}}\phantom {\dot {i}\!}\), where ξ old and ξ new are the perimeter scaling factor before and after resizing the neighbourhood, respectively. Likewise, \(P_{_{\mathit {CPM}}}'=\frac {\xi _{\textit {new}}}{\xi _{\textit {old}}}P_{_{\mathit {CPM}}}\phantom {\dot {i}\!}\), \(\lambda _{p\,{}_{\mathit {CPM}}}'=\frac {\xi _{\textit {old}}^{2}}{\xi _{\textit {new}}^{2}}\lambda _{p\,{}_{\mathit {CPM}}}\phantom {\dot {i}\!}\), \(S_{_{\mathit {CPM}}}'=\frac {\xi _{\textit {new}}}{\xi _{\textit {old}}}S_{_{\mathit {CPM}}}\phantom {\dot {i}\!}\), and \(\lambda _{s\,{}_{\mathit {CPM}}}'=\frac {\xi _{\textit {old}}^{2}}{\xi _{\textit {new}}^{2}}\lambda _{s\,{}_{\mathit {CPM}}}\phantom {\dot {i}\!}\). Details on calculating ξ are in Step 2 of Table 4.
Concurrent rescaling of J and P It is possible to concurrently change all J values (for example from all being positive to all being negative) in a CPM, or in fact any CSM simulation, without causing any change in the dynamics of the model (Figure 6), by means of well-chosen shifts in the membrane rest lengths of all cells (P σ and S σ in 2D and 3D, respectively). The required shifts in the rest lengths are given in Eq. 47 (for 2D) and Eq. 49 (for 3D). In contrast, it is not possible to change only a subset of the J values without causing changes in dynamics. Specifically, in such a case, it is still possible to keep for a specific configuration the weighted mean adhesion-driven interfacial tension constant (using Eq. 46 for 2D and Eq. 48 for 3D), but those weighted means are expected to change over time (for example due to cell sorting), generating an imbalance and hence a change of dynamics on the long run.