Mathematical Modeling and Design for Alignment of Active Nematic Liquid Crystals and Its Application to Alignment Control of Cellular Sheets on a Large Scale

Cardiac and skeletal muscular cells have been widely applied to driving forces of biohybrid actuators and biohybrid robots. Since these cells are elongated along specific directions at a confluent state and can generate contractile forces along their cell alignment, it is important to predict and control the alignment of the muscular cells in preparation for cell culture experiments because of the high cost of the experiments. One theoretical approach for the prediction of the cell alignment is to model the elongated muscular cells as living nematic liquid crystals, which are known as “active nematics” in the fields of biophysics and non-equilibrium physics.<br/><br/>For the mathematical prediction and control of the muscular cell alignment, our research group has developed explicit formulas of nematic cell alignment based on complex function theories [1-3] and confirmed the validity of the developed formulas by doing cell culturing experiments for mouse myoblast cells (C2C12) [4]. However, our previous studies are limited to the prediction of cell alignment on single and confined domains. To achieve reproducible and large-scale fabrication of biohybrid actuators based on muscular cells, it is necessary to extend the formulas for two-dimensional sheets.<br/><br/>In this study, we propose a new design method for controlling cell alignment of two-dimensional cellular sheets on a large scale based on our developed formulas for nematic cell alignment. In the proposed method, we consider unit circular triangles, which have positive and negative curvatures, and divide the two-dimensional sheets by the unit circular triangles. Such division is experimentally achieved by making micro-grooves on the sheets. Since the cell alignment in each unit circular triangle can be easily predicted based on the explicit formulas of the nematic cell alignment, we can achieve the control and prediction of the cell alignment of the entire cell sheets. In this presentation, we first show how to calculate the cell alignment on the unit circular triangles and then optimize the geometry of the unit circular triangles by numerical calculations and optimization. The optimality of the geometry was also experimentally verified by culturing C2C12 cells on PDMS substrates which have micro-grooves.<br/><br/><br/>[1] H. Miyazako and T. Nara, “Explicit calculation method for cell alignment in non-circular geometries,” R. Soc. Open Sci., 92, 11663 (2022).<br/>[2] H. Miyazako and T. Sakajo, “Defect pairs in nematic cell alignment on doubly connected domains,” Proc. R. Soc. A, 480, 20230879 (2024).<br/>[3] H. Miyoshi, H. Miyazako and T. Nara, “Free energy formulas for confined nematic liquid crystals based on analogies with Kirchhoff-Routh theory in vortex dynamics,” arXiv:2405.16742 [physics.flu-dyn]<br/>[4] H. Miyazako, K. Tsuchiyama and T. Nara, “Predictive model of spatial nematic order<br/>in confined cell populations” (in revision).