## 2 Materials and Methods
### Mathematical Model for the Description of Cell Shape Changes  
In order to study the influence of actin filament dynamics on the shape of cells placed on differently charged substrates, we simulated a mathematical vertex model for cell shapes [48; 49; 50]. The two-dimensional cross-section of a cell in this model is represented as a polygon specified by surrounding vertices (Figure 2a) [49; 51; 52]. The cell mechanical properties in stationary conditions are described by an energy function whose functional form is given by:  
\[E=K_{A}(A-A_{0})^{2}+K_{P}(P-P_{0})^{2} \tag{2}\]  
where \(A\) is the cross-sectional area and \(P\) is the perimeter of the cell. \(A_{0}\) is the preferred area that the cell strives to attain such that any deviation from it, i.e., \(A_{0}\), corresponds to cellular pressure, the magnitude of which depends on the deviation \((A-A_{0})\) and the constant \(K_{A}\), which denotes the strength of resistance to cell area changes. The second term represents the tensile properties of the cell membrane and consists of two terms, one quadratic in perimeter and the other linear in perimeter. The term that is quadratic in perimeter describes bond tension due to the actomyosin ring in the subcellular cortex. In contrast, the term that is linear in perimeter describes an effective cell membrane tension due to focal adhesions arising from cell-substrate interaction. The net contribution to cell membrane tension comes from cortical tension, due to actin filaments, that resists an increase in cell perimeter, and cell-substrate adhesion that tends to increase the cell perimeter by growing actin filaments. \(K_{P}\) captures the effective resistance offered by the cell to any changes in the cell perimeter length. By introducing \(L\), which is the typical diameter of an osteoblastic cell and has the dimension of length (\(\mu\)m), and dividing the whole equation by \(K_{P}L^{2}\), Equation (2) can be expressed in the non-dimensional form as:  
\[\widetilde{E}=\widetilde{K}_{A}(\widetilde{A}-\widetilde{A}_{0})^{2}+( \widetilde{P}-\widetilde{P}_{0})^{2} \tag{3}\]  
where \(\widetilde{E}=E/(K_{P}L^{2})\), \(\widetilde{K}_{A}=\frac{K_{A}L^{2}}{K_{P}}\), \(\widetilde{A}=A/L^{2}\), \(\widetilde{A}_{0}=A_{0}/L^{2}\), \(\widetilde{P}=P/L\), and \(\widetilde{P}_{0}=P_{0}/L\). Various parameters of the model, as described by Equation (3), and their meaning are listed in Table 1. The shape of the cell is determined by quasi-static simulations in which, for a given set of parameter values of the energy function given by Equation (3), the system is relaxed to a force-balanced state using a Monte Carlo-based minimization algorithm. The vertex models described by Equation (2) have been successful in predicting cell shapes due to mechanical interactions between cells in a confluent tissue [48; 49; 50]. Here, we use this model for studying shape changes in single cells. In our model, an osteoblastic cell is represented by a polygon with six sides, as shown in Figure 2a. A dimensionless shape index parameter can be defined as \(p_{0}=P_{0}\) / \(\surd\)\(A_{0}\). Earlier studies of the vertex model for confluent tissues given by Equation (2) showed that a transition takes places at a critical value of \(p_{0}=p_{0}^{*}\). When \(p_{0}<p_{0}^{*}\), the stable configuration of a polygon with six sides is given by the ground state and has the shape of a regular hexagon, which is characterized by a higher circularity value (Figure 2b). However, when \(p_{0}>p_{0}^{*}\), the ground state is degenerate, and the polygon has an irregular shape which is characterized by a lower circularity value. For a regular hexagon, this transition has been shown to occur at \(p_{0}^{*}\sim 3.71\)[53; 54; 55]. Based on this knowledge, we use the vertex model