Many animal cells that crawl on extracellular substrates exhibit durotaxis, i.e., directed migration toward stiffer substrate regions. This has implications in several biological processes including tissue development and tumor progression. Here, we introduce a phenomenological model for single-cell durotaxis that incorporates both elastic deformation-mediated cell-substrate interactions and the stochasticity of cell migration. Our model is motivated by a key observation in an early demonstration of durotaxis: a single, contractile cell at a sharp interface between a softer and a stiffer region of an elastic substrate reorients and migrates toward the stiffer region. We model migrating cells as self-propelling, persistently motile agents that exert contractile traction forces on their elastic substrate. The resulting substrate deformations induce elastic interactions with mechanical boundaries, captured by an elastic potential. The dynamics is determined by two crucial parameters: the strength of the cellular traction-induced boundary elastic interaction (A), and the persistence of cell motility (Pe). Elastic forces and torques resulting from the potential orient cells perpendicular (parallel) to the boundary and accumulate (deplete) them at the clamped (free) boundary. Thus, a clamped boundary induces an attractive potential that drives durotaxis, while a free boundary induces a repulsive potential that prevents antidurotaxis. By quantifying the steady-state position and orientation probability densities, we show how the extent of accumulation (depletion) depends on the strength of the elastic potential and motility. We compare and contrast crawling cells with biological microswimmers and other synthetic active particles, where accumulation at confining boundaries is well known. We define metrics quantifying boundary accumulation and durotaxis, and present a phase diagram that identifies three possible regimes: durotaxis, and adurotaxis with and without motility-induced accumulation at the boundary. Overall, our model predicts how durotaxis depends on cell contractility and motility, successfully explains some previous observations, and provides testable predictions to guide future experiments.

中文翻译：

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弹性相互作用与持续的细胞运动竞争以驱动 durotaxis


许多在细胞外基质上爬行的动物细胞表现出双轮性，即向更坚硬的基质区域定向迁移。这对包括组织发育和肿瘤进展在内的几个生物过程有影响。在这里，我们介绍了一个单细胞 durotaxis 的现象学模型，该模型结合了弹性变形介导的细胞-基质相互作用和细胞迁移的随机性。我们的模型受到 durotaxis 早期演示中的关键观察的启发：弹性基材较软区域和较硬区域之间尖锐界面处的单个收缩单元重新定向并向较硬区域迁移。我们将迁移细胞建模为自我推进、持续运动的试剂，在其弹性基材上施加收缩牵引力。由此产生的基体变形会引起与机械边界的弹性相互作用，由弹性势捕获。动力学由两个关键参数决定：细胞牵引诱导的边界弹性相互作用的强度 （A） 和细胞运动的持久性 （Pe）。由电位定向单元垂直（平行）于边界产生的弹性力和扭矩，并在夹紧（自由）边界处累积（耗尽）它们。因此，钳制边界会感应出驱动双轴的吸引电位，而自由边界会感应出排斥电位，从而阻止反双轴。通过量化稳态位置和取向概率密度，我们展示了积累（消耗）的程度如何取决于弹性电位和运动的强度。 我们将爬行细胞与生物微游泳器和其他合成活性粒子进行比较和对比，其中在限制边界的积累是众所周知的。我们定义了量化边界积累和反作用的指标，并提出了一个相图，确定了三种可能的机制：反作用和趋性，在边界处有和没有运动诱导的积累。总体而言，我们的模型预测了 durotaxis 如何依赖于细胞收缩性和运动性，成功地解释了以前的一些观察结果，并提供了可测试的预测来指导未来的实验。