Predictive Design of Orientational Order in Confined Active Nematic Materials

Elongated biological materials can be modeled as nematic liquid crystals which can actively move by energy consumption, known as “active nematics [1].” Recent studies have shown that such active nematic behaviors are observed in many biological systems, such as cytoskeleton-motor protein compounds and cell monolayers. In active nematic materials, the orientational order and the active behaviors are affected by topological defects, at which the orientation angle cannot be defined. Therefore, many studies have demonstrated spatial control of defect positions and movement by topographic guides or confinement to achieve the reproducible generation of targeted orientational patterns or defect movement. However, the geometrical effects of the guides or confinement have not been fully understood. For precise control of the orientational order, it is necessary to establish calculation methods to predict defect positions in various geometries.<br/><br/>To this end, this study aims to establish a calculation method for predicting and designing defect positions of confined active nematic materials. In our previous study [2], we derived an explicit formula of orientational angles and Frank elastic energy of nematic materials confined in a two-dimensional space. The formula was expressed by complex potentials defined on a unit disk and a conformal mapping from the unit disk to a simply-connected domain. Thus, the derived formula can be applied to any simply-connected domain, which is an advantage of the derived formula. However, our previous study cannot predict stochastic fluctuations of defect positions caused by random movement of the active nematic materials.<br/><br/>To quantify the stochasticity of the defect positions, we propose a numerical method to calculate equilibrium distributions of defects. By using the explicit formula of the Frank elastic energy described as a function of defect positions, the equilibrium distributions can be efficiently calculated based on Markov chain Monte Carlo methods. The computed distributions quantify spatial localization and stochastic fluctuations of defect positions in a confined space. Since the geometry of the confinement is expressed as a complex function in our approach, we can easily evaluate geometric effects on the spatial localization of the defects. Therefore, we can predict an optimal geometry for suppressing stochastic fluctuations of defects by tuning the parameters of the geometry.<br/><br/>The proposed method was experimentally verified by culturing mouse myoblast (C2C12) cells, which were used as a model of active nematic materials. The C2C12 cells were cultured on PDMS microgrooves whose geometries were parametrically expressed as f(z)=z+ az³. The cell alignment and the defects were observed by phase-contrast microscopy. The distributions of the defects were localized as the geometry became asymmetric, which were consistent with our proposed calculation method. The fluctuations of the defect positions were also evaluated from the experimental data by maximum likelihood methods. The evaluation suggested the spatial localization of the defects was mainly due to the geometric effects rather than the randomness of the cell migration and alignment. The proposed method will be applied to the high-reproducible and dependable fabrication of nematic orientational order in active nematic materials.<br/><br/>[1] A. Doostmohammadi<i> et al., Nature Communications, </i>Vol. 9, 3246, 2018.<br/>[2] H. Miyazako <i>et al., Royal Society Open Science, </i>Vol.9, No.1, 211663, 2022.