Chen Kuan-Wei, Liao Kang-Ling, Shih Chih-Wen
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan.
Department of Biology, The University of North Carolina at Chapel Hill, Chapel Hill, NC, USA.
J Math Biol. 2018 Jan;76(1-2):97-150. doi: 10.1007/s00285-017-1138-1. Epub 2017 May 25.
Somitogenesis is the process for the development of somites in vertebrate embryos. This process is timely regulated by synchronous oscillatory expression of the segmentation clock genes. Mathematical models expressed by delay equations or ODEs have been proposed to depict the kinetics of these genes in interacting cells. Through mathematical analysis, we investigate the parameter regimes for synchronous oscillations and oscillation-arrested in an ODE model and a model with transcriptional and translational delays, both with Michaelis-Menten type degradations. Comparisons between these regimes for the two models are made. The delay model has larger capacity to accommodate synchronous oscillations. Based on the analysis and numerical computations extended from the analysis, we explore how the periods and amplitudes of the oscillations vary with the degradation rates, synthesis rates, and coupling strength. For typical parameter values, the period and amplitude increase as some synthesis rate or the coupling strength increases in the ODE model. Such variational properties of oscillations depend also on the magnitudes of time delays in delay model. We also illustrate the difference between the dynamics in systems modeled with linear degradation and the ones in systems with Michaelis-Menten type reactions for the degradation. The chief concerns are the connections between the dynamics in these models and the mechanism for the segmentation clocks, and the pertinence of mathematical modeling on somitogenesis in zebrafish.
体节发生是脊椎动物胚胎中体节发育的过程。这一过程由分割时钟基因的同步振荡表达进行适时调控。已提出用延迟方程或常微分方程表示的数学模型来描述这些基因在相互作用细胞中的动力学。通过数学分析，我们研究了一个常微分方程模型以及一个具有转录和翻译延迟（二者均为米氏型降解）的模型中同步振荡和振荡停止的参数范围。对这两个模型的这些范围进行了比较。延迟模型容纳同步振荡的能力更强。基于该分析以及从分析扩展而来的数值计算，我们探究了振荡的周期和幅度如何随降解速率、合成速率以及耦合强度而变化。对于典型参数值，在常微分方程模型中，周期和幅度会随着某些合成速率或耦合强度的增加而增大。振荡的这种变化特性在延迟模型中还取决于时间延迟的大小。我们还阐述了用线性降解建模的系统与具有米氏型降解反应的系统在动力学方面的差异。主要关注点在于这些模型中的动力学与分割时钟机制之间的联系，以及数学建模对斑马鱼体节发生的相关性。
