Phase transitions of biological phenotypes by means of a prototypical PDE model

IF 0.3 Q4 MATHEMATICS
C. Mascia, P. Moschetta, C. Simeoni
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引用次数: 1

Abstract

Abstract The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-parabolic PDE model composed of two equations, one for the representative of the epithelial phenotype, and the second describing the mesenchymal phenotype. The system involves two positive constants, the relaxation time and a measure of invasiveness, moreover an essential feature is the presence of a nonlinear reaction function, typically assumed to be S-shaped. An identity characterizing the speed of propagation of the fronts is proven, together with numerical evidence of the existence of traveling waves. The latter is obtained by discretizing the system by means of an implicit-explicit finite difference scheme, then the algorithm is validated by checking the capability of the so-called LeVeque–Yee formula to reproduce the value of the speed furnished by the above cited identity. Once such justification has been achieved, we concentrate on numerical experiments relative to Riemann initial data connecting two stable stationary states of the underlying ODE model. In particular, we detect an explicit transition threshold separating regression regimes from invasive ones, which depends on critical values of the invasiveness parameter. Finally, we perform an extensive sensitivity analysis with respect to the system parameters, exhibiting a subtle dependence for those close to the threshold values, and we postulate some conjectures on the propagating fronts.
通过典型PDE模型研究生物表型的相变
基本的研究是在所谓的上皮-间质转化及其反向间质-上皮转化的框架中繁殖锋的存在和(数值)可观察性,这在肿瘤的发展中起着至关重要的作用。为此,我们提出了一个简化的一维双曲抛物型PDE模型,由两个方程组成,一个代表上皮表型,第二个描述间质表型。该系统涉及两个正常数,松弛时间和入侵度量,此外一个基本特征是非线性反应函数的存在,通常被认为是s形的。证明了锋面传播速度的恒等式,并给出了行波存在的数值证据。后者是通过隐式-显式有限差分格式对系统进行离散得到的,然后通过检查所谓的LeVeque-Yee公式再现上述恒等式所提供的速度值的能力来验证算法。一旦实现了这样的证明,我们就集中精力进行与连接底层ODE模型的两个稳定稳态的黎曼初始数据相关的数值实验。特别是,我们检测到一个明确的过渡阈值,将回归机制与侵入性机制分开,这取决于侵入性参数的临界值。最后,我们对系统参数进行了广泛的敏感性分析,显示出对接近阈值的那些参数的微妙依赖,并且我们在传播前沿假设了一些猜想。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
3
审稿时长
16 weeks
期刊介绍: Communications in Applied and Industrial Mathematics (CAIM) is one of the official journals of the Italian Society for Applied and Industrial Mathematics (SIMAI). Providing immediate open access to original, unpublished high quality contributions, CAIM is devoted to timely report on ongoing original research work, new interdisciplinary subjects, and new developments. The journal focuses on the applications of mathematics to the solution of problems in industry, technology, environment, cultural heritage, and natural sciences, with a special emphasis on new and interesting mathematical ideas relevant to these fields of application . Encouraging novel cross-disciplinary approaches to mathematical research, CAIM aims to provide an ideal platform for scientists who cooperate in different fields including pure and applied mathematics, computer science, engineering, physics, chemistry, biology, medicine and to link scientist with professionals active in industry, research centres, academia or in the public sector. Coverage includes research articles describing new analytical or numerical methods, descriptions of modelling approaches, simulations for more accurate predictions or experimental observations of complex phenomena, verification/validation of numerical and experimental methods; invited or submitted reviews and perspectives concerning mathematical techniques in relation to applications, and and fields in which new problems have arisen for which mathematical models and techniques are not yet available.
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