Mathematical modeling of cellular response to mechanical and geometrical cues on curved substrates

The active response of cells to mechanical stimuli has been a subject of extensive research over the past two decades. Initially, the focus was on cells cultured on flat substrates and subjected to periodic stretching. Several theoretical models were proposed to account for various physical factors influencing cellular response. However, in the last decade, there has been growing interest in how geometrical cues, such as substrate curvature, affect cell behavior. This interest is driven by experimental findings that highlight the importance of curvature in cell mechanical response due to the interplay between cellular contractility and substrate geometry. Research has shown that cells respond to mechanical influences, not just chemical signals. This understanding has evolved since the late 1980s, with a focus on how cells behave on flat, two-dimensional surfaces. Key observations include how cells align and reorient their internal structure when stretched, influenced by factors like cell type and the characteristics of the deformation. Recent studies reveal that cells behave differently on curved surfaces. Here, cells align their stress fibers (SFs) based on geometrical cues such as the curvature of the surface. For example, muscle cells align with lesser curvature on cylindrical surfaces, while epithelial cells orient towards the maximum curvature. This surprising behavior is due to the interaction between the cell’s contractility and the surface curvature, leading to bending of the SFs. This work expands on the mathematical model by Biton and Safran, which explains cell behavior on curved substrates. The model hypothesizes a competition between cellular contractility and bending due to substrate curvature. A more structured model is developed to explain cell reorientation, considering recent findings. The interplay of bending and contractility forces creates a complex energy landscape that governs cell stability and configuration. Stability analysis reveals critical configurations where cells undergo significant changes. These configurations result from the competition between bending energies and contractile forces, leading to equilibrium states. The stable configuration depends on a parameter that includes both contractility and curvature information. The model developed in this work aligns with Biton and Safran’s results in a limiting case. Inspired by biology, this work investigates cell reorientation on inflated cylindrical substrates, considering both isotropic and anisotropic mechanical responses. It studies the kinematics of cylinder inflation and its impact on cell orientation. However, the model only accurately predicts a few experimental results, indicating a need for further consideration of cellular-level interactions. The analysis is then extended to materials with initial stress, using hyperelastic theory to describe residual stresses in orthotropic materials. The study explores the effect of residual stress on cylinder inflation and cell reorientation, offering a deeper understanding of the mechanical environment that cells experience in vivo. This comprehensive study provides a theoretical framework for understanding the mechanical response of cells on curved substrates, which is critical for processes such as tissue engineering.