Introduction to force transmission by nonlinear biomaterials

Many-body interactions between contracting living cells

The organization of live cells into tissues and their subsequent biological function involves inter-cell mechanical interactions, which are mediated by their elastic environment. To model this interaction, we consider cells as spherical active force dipoles surrounded by an unbounded elastic matrix. Even though we assume that this elastic medium responds linearly, each cell's regulation of its mechanical activity leads to nonlinearities in the emergent interactions between cells. We study the many-body nature of these interactions by considering several geometries that include three or more cells. We show that for different regulatory behaviors of the cells' activity, the total elastic energy stored in the medium differs from the superposition of all two-body interactions between pairs of cells within the system. Specifically, we find that the many-body interaction energy between cells that regulate their position is smaller than the sum of interactions between all pairs of cells in the system, while for cells that do not regulate their position, the many-body interaction is larger than the superposition prediction. Thus, such higher-order interactions should be considered when studying the mechanics of multiple cells in proximity.

E-cadherin adhesion dynamics as revealed by an accelerated force ramp are dependent upon the presence of α-catenin

Tissue remodeling and shape changes often rely on force-induced cell rearrangements occurring via cell-cell contact dynamics. Epithelial cell-cell contact shape changes are particularly dependent upon E-cadherin adhesion dynamics which are directly influenced by cell-generated and external forces. While both the mobility of E-cadherin adhesions and their adhesion strength have been reported before, it is not clear how these two aspects of E-cadherin adhesion dynamics are related. Here, using magnetic pulling cytometry, we applied an accelerated force ramp on the E-cadherin adhesion between an E-cadherin-coated magnetic microbead and an epithelial cell to ascertain this relationship. Our approach enables the determination of the adhesion strength and force-dependent mobility of individual adhesions, which revealed a direct correlation between these key characteristics. Since α-catenin has previously been reported to play a role in both E-cadherin mobility and adhesion strength when studied independently, we also probed epithelial cells in which α-catenin has been knocked out. We found that, in the absence of α-catenin, E-cadherin adhesions not only had lower adhesion strength, as expected, but were also more mobile. We observed that α-catenin was required for the recovery of strained cell-cell contacts and propose that the adhesion strength and force-dependent mobility of E-cadherin adhesions act in tandem to regulate cell-cell contact homeostasis. Our approach introduces a method which relates the force-dependent adhesion mobility to adhesion strength and highlights the morphological role played by α-catenin in E-cadherin adhesion dynamics.

α-Catenin Dependent E-cadherin Adhesion Dynamics as Revealed by an Accelerated Force Ramp

Tissue remodeling and shape changes often rely on force-induced cell rearrangements occurring via cell-cell contact dynamics. Epithelial cell-cell contact shape changes are particularly dependent upon E-cadherin adhesion dynamics which are directly influenced by cell-generated and external forces. While both the mobility of E-cadherin adhesions and their adhesion strength have been reported before, it is not clear how these two aspects of E-cadherin adhesion dynamics are related. Here, using magnetic pulling cytometry, we applied an accelerated force ramp on the E-cadherin adhesion between an E-cadherin-coated magnetic microbead and an epithelial cell to ascertain this relationship. Our approach enables the determination of the adhesion strength and force-dependent mobility of individual adhesions, which revealed a direct correlation between these key characteristics. Since α-catenin has previously been reported to play a role in both E-cadherin mobility and adhesion strength when studied independently, we also probed epithelial cells in which α-catenin has been knocked out. We found that, in the absence of α-catenin, E-cadherin adhesions not only had lower adhesion strength, as expected, but were also more mobile. We observed that α-catenin was required for the recovery of strained cell-cell contacts and propose that the adhesion strength and force-dependent mobility of E-cadherin adhesions act in tandem to regulate cell-cell contact homeostasis. Our approach introduces a method which relates the force-dependent adhesion mobility to adhesion strength and highlights the morphological role played by α-catenin in E-cadherin adhesion dynamics.

Variational methods and deep Ritz method for active elastic solids

Variational methods have been widely used in soft matter physics for both static and dynamic problems. These methods are mostly based on two variational principles: the variational principle of minimum free energy (MFEVP) and Onsager's variational principle (OVP). Our interests lie in the applications of these variational methods to active matter physics. In our former work [H. Wang, T. Qian and X. Xu, Soft Matter, 2021, 17, 3634-3653], we have explored the applications of OVP-based variational methods for the modeling of active matter dynamics. In the present work, we explore variational (or energy) methods that are based on MFEVP for static problems in active elastic solids. We show that MFEVP can be used not only to derive equilibrium equations, but also to develop approximate solution methods, such as the Ritz method, for active solid statics. Moreover, the power of the Ritz-type method can be further enhanced using deep learning methods if we use deep neural networks to construct the trial functions of the variational problems. We then apply these variational methods and the deep Ritz method to study the spontaneous bending and contraction of a thin active circular plate that is induced by internal asymmetric active contraction. The circular plate is found to be bent towards its contracting side. The study of such a simple toy system gives implications for understanding the morphogenesis of solid-like confluent cell monolayers. In addition, we introduce a so-called activogravity length to characterize the importance of gravitational forces relative to internal active contraction in driving the bending of the active plate. When the lateral plate dimension is larger than the activogravity length (about 100 micron), gravitational forces become important. Such gravitaxis behaviors at multicellular scales may play significant roles in the morphogenesis and in the up-down symmetry broken during tissue development.