Nardini John T, Bortz D M
Department of Applied Mathematics, University of Colorado, Boulder 80309-0526, United States.
SIAM J Appl Math. 2018;78(3):1712-1736. doi: 10.1137/16M1108546. Epub 2018 Jun 19.
Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.
近期的生物学研究试图了解生化信号通路，如丝裂原活化蛋白激酶（MAPK）家族，如何在伤口愈合过程中影响细胞群体的迁移。费希尔方程由于其简单的性质和已知的行波解，已被广泛用于模拟实验性伤口愈合检测。然而，这个具有时间和空间自变量的偏微分方程无法解释生化活性对伤口愈合的影响。为此，我们推导了一个具有时间、空间和生化信号通路活性水平自变量的结构化费希尔方程，并证明了该方程存在自相似行波解。我们展示了这些方法也适用于一般的结构化反应扩散方程和趋化方程。我们还考虑了一个基于MAPK激活的具有不同表型的更复杂模型，并通过数值研究各种生化活性的时间模式如何导致群体迁移速率的增加和降低。
