Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin–Voigt model of linear viscoelasticity to represent the stress–strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the ECM, while no patterns are observed when the Kelvin–Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress–strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.

生物组织中模式形成的机械和机械化学模型已用于研究各种生物医学系统，特别是发育生物学，并描述细胞与其局部环境之间的物理相互作用。这些模型的原始形式由细胞密度平衡方程、细胞外基质 (ECM) 密度平衡方程以及描述细胞-ECM 系统机械平衡的力平衡方程组成。在细胞-ECM系统可以被视为各向同性线性粘弹性材料的假设下，通常使用线性粘弹性的Kelvin-Voigt模型定义力平衡方程来表示ECM的应力-应变关系。然而，由于 ECM 成分的多方面生物物理性质，该模型无法有效捕获流变方面的内容，因此，根据图案形成过程和所考虑的生物组织类型，线性的其他本构模型粘弹性可能更适合。在本文中，我们系统地评估了生物组织图案形成力学模型中 ECM 不同应力-应变本构方程的图案形成潜力。通过线性稳定性分析获得的结果以及由此得出的色散关系支持这样的观点，即类流体本构模型（例如麦克斯韦模型和杰弗里模型）具有比类固体模型（例如开尔文模型）高得多的图案形成潜力。 –Voigt 模型和标准线性实体模型。 数值模拟结果证实了这一点，数值模拟结果表明，在其他条件相同的情况下，使用麦克斯韦模型表示 ECM 的应力应变关系时会出现空间模式，而当使用开尔文模型表示 ECM 的应力应变关系时则不会观察到任何模式。 –采用Voigt模型。我们的研究结果表明，需要进一步的实证工作来获取不同生物组织中 ECM 组件的机械性能的详细定量信息，以便为 ECM 提供具有应力应变本构方程的图案形成的机械和机械化学模型，从而提供更忠实地代表底层组织流变学。