Spatiotemporal Dynamics and Pattern Formations of an Activator-Substrate Model with Double Saturation Terms

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.

Spatiotemporal Patterns in a General Networked Activator-Substrate Model

For the purpose of understanding the spatiotemporal pattern formation in the random networked system, a general activator-substrate model with network structure is introduced. Firstly, we investigate the boundedness of the non-constant steady state of the elliptic system of the continuous media system. It is found that the non-constant steady state admits their upper and lower bounds with certain conditions. Then, one investigates some properties and non-existence of the non-constant steady state with the no-flux boundary conditions. The main results show that the diffusion rate of activator should greater than the diffusion rate of substrate. Otherwise, there might be no pattern formation of the system. Afterwards, a general random networked activator-substrate model is made public. The conditions of the stability, the Hopf bifurcation, the Turing instability and a co-dimensional-two Turing-Hopf bifurcation are yield by the method of stability analysis and bifurcation theorem. Finally, we choose a suitable sub-system of the general activator-substrate model to verify the theoretical results, and full numerical simulations are well verified these results. Especially, an interesting finding is that the stability of the positive equilibrium will switch from unstable to stable one with the change of the connection probability of the nodes, this is different from the pattern formation in the continuous media systems.

A Thermodynamics-Based Nonlocal Bar-Elastic Substrate Model with Inclusion of Surface-Energy Effect

This paper presents a bar-elastic substrate model to investigate the axial responses of nanowire-elastic substrate systems considering the effects of nonlocality and surface energy. The thermodynamics-based strain gradient model is adopted to capture the nonlocality of the bar-bulk material while the Gurtin-Murdoch surface theory is utilized to consider the surface energy. To characterize the bar-surrounding substrate interaction, the Winkler foundation model is employed. In a direct manner, system compatibility conditions are obtained while within the framework of the virtual displacement principle, the system equilibrium condition and the corresponding natural boundary conditions are consistently obtained. Three numerical simulations are conducted to investigate the characteristics and behaviors of the nanowire-elastic substrate system: the first is conducted to reveal the capability of the proposed model to eliminate the paradoxical behavior inherent to the Eringen nonlocal differential model; the second is employed to characterize responses of the nanowire-elastic substrate system; and the third is aimed at demonstrating the dependence of the system effective Young’s modulus on several system parameters.

Cortical cell stiffness is independent of substrate mechanics

Cortical stiffness is an important cellular property that changes during migration, adhesion, and growth. Previous atomic force microscopy (AFM) indentation measurements of cells cultured on deformable substrates suggested that cells adapt their stiffness to that of their surroundings. Here we show that the force applied by AFM onto cells results in a significant deformation of the underlying substrate if it is softer than the cells. This ‘soft substrate effect’ leads to an underestimation of a cell’s elastic modulus when analyzing data using a standard Hertz model, as confirmed by finite element modelling (FEM) and AFM measurements of calibrated polyacrylamide beads, microglial cells, and fibroblasts. To account for this substrate deformation, we developed the ‘composite cell-substrate model’ (CoCS model). Correcting for the substrate indentation revealed that cortical cell stiffness is largely independent of substrate mechanics, which has significant implications for our interpretation of many physiological and pathological processes.

Cell focal adhesion clustering leads to decreased and homogenized basal strains

The subendothelial matrix of the artery is a complex mechanical environment where endothelial cells respond to and affect changes upon the underlying substrate. Our recent work has demonstrated that endothelial cell strain heterogeneity increases on a more heterogeneous underlying subendothelial matrix, and these cells display increased focal adhesion presence on stiffer substrate areas. However, the impact of these grouped focal adhesions on endothelial cell strains has not been explored. Here, we use finite element modeling to investigate the effects of micro-scale stiffness heterogeneities and focal adhesion location and stiffness on endothelial cell strains. Shear stress applied to the apical cell layer demonstrated a minimal effect on cell strain values while substrate stretch had a greater effect on cell strain in the cell-substrate model. The addition of focal adhesions into the computational model (cell-FA-substrate model) predicted a decrease and homogenization of the cell strains. For simulations including focal adhesions, stiffer and more distributed adhesions caused increased and more heterogeneous endothelial cell strains. Overall, our data indicate that cells may group focal adhesions to minimize and homogenize their basal strains.