Orientations of Cells on Compliant Substrates under Biaxial Stretches: A Theoretical Study  Guang-Kui Xu, Xi-Qiao Feng, Huajian Gao  Biophysical Journal  Volume 114, Issue 3, Pages 701-710 (February 2018) DOI: 10.1016/j.bpj.2017.12.002 Copyright © 2017 Biophysical Society Terms and Conditions

Figure 1 Schematic of a polarized cell adhered on (a) a soft or (b) a stiff substrate subjected to biaxial strains εxx and εyy. The cell is oriented at angle θ measured from the x direction. The tensegrity model, capturing the essential cell structure on (c) soft or (d) stiff substrates, consists of four bars, corresponding to the longitudinal SFs and lateral supports of the cytoskeleton, and eight strings, representing the microfilaments. To see this figure in color, go online. Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 2 Schematic of major elements of the mechanosensing device at the front of a polarized cell. Actin polymerization occurs at the cell front. The sarcomeric unit of an SF includes actin filaments, myosin II motors, and cross-linker (α-actinin) proteins. The myosin II motors provide the contractive force with a retrograde flow velocity Vmyo of actins. The relative movement of actin filaments and FAs generates a cell-substrate tangential force that can be determined by the binding dynamics of the receptors (i.e., integrins) and their complementary ligands. To see this figure in color, go online. Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 3 Traction stress with respect to the substrate stiffness. The theoretical predictions (Eq. 5) fit well with the experimental data reproduced from (43) and (42), respectively. To see this figure in color, go online. Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 4 Angular distributions of cells under static stretches. The theoretical predictions (red lines) fit well with the experimental data of (a) mammalian skeletal myocytes (26) and (b) mesenchymal stem cells (23). Here, we have taken the strain amplitude ε0 = 0.1 and the angle interval Δθ = 100, as in the experiments (23,26). Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 5 Order parameter S = 〈cos2θ〉 (red square) and width σ of the cellular angular distribution (blue round) as a function of the strain amplitude ε0 under static stretches. The width calculated by Eq. 16 can be fitted well by the function σ = m/(ε0 + n), indicating that the width scales inversely with the strain amplitude at relatively large strain amplitude. Here, we used the parameters in Fig. 4 b. To see this figure in color, go online. Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 6 Angular distributions of cells for different scaled effective stiffnesses ξ¯C1U under ultralow-frequency cyclic stretches. The cells concentrate at the stretching direction for a large value of ξ¯C1U, whereas they align randomly for a small value of ξ¯C1U. Here, we have taken ξ¯C1U=1. Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 7 Angular distribution of cells under high-frequency cyclic stretches. The theoretical prediction (red lines) fits well with the experimental data of endothelial cells (11). Here, we have taken ε0 = 0.1 and Δθ = 50, as in the experiments. Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions

Figure 8 Order parameter S = 〈cos2θ 〉 (red square) and width σ of the cell angular distribution (blue round) as a function of the strain amplitude ε0 under high-frequency cyclic stretches. The width obtained by Eq. 16 can be fitted well by the function σ = m/ε0, indicating that the width scales inversely with the strain amplitude. Here, we used the parameters in Fig. 7. To see this figure in color, go online. Biophysical Journal 2018 114, 701-710DOI: (10.1016/j.bpj.2017.12.002) Copyright © 2017 Biophysical Society Terms and Conditions