Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, China.
Math Biosci. 2010 Nov;228(1):100-9. doi: 10.1016/j.mbs.2010.08.012. Epub 2010 Sep 9.
Many genetic regulatory networks (GRNs) have the capacity to reach different stable states. This capacity is defined as multistability which is an important regulation mechanism. Multiple time delays and multivariable regulation functions are usually inevitable in such GRNs. In this paper, multistability of GRNs is analyzed by applying the control theory and mathematical tools. This study is to provide a theoretical tool to facilitate the design of synthetic gene circuit with multistability in the perspective of control theory. By transforming such GRNs into a new and uniform mathematical formulation, we put forward a general sector-like regulation function that is capable of quantifying the regulation effects in a more precise way. By resorting to up-to-date techniques, a novel Lyapunov-Krasovskii functional (LKF) is introduced for achieving delay dependence to ensure less conservatism. New conditions are then proposed to ensure the multistability of a GRN in the form of linear matrix inequalities (LMIs) that are dependent on the delays. Our multistability conditions are applicable to several frequently used regulation functions especially the multivariable ones. Two examples are employed to illustrate the applicability and usefulness of the developed theoretical results.
许多遗传调控网络（GRNs）具有达到不同稳定状态的能力。这种能力被定义为多稳定性，这是一种重要的调控机制。在这样的 GRNs 中，通常不可避免地存在多个时滞和多变量调节函数。本文应用控制理论和数学工具分析了 GRNs 的多稳定性。本研究旨在为合成具有多稳定性的基因电路的设计提供一种理论工具，从控制理论的角度来看。通过将这些 GRNs 转化为一个新的和统一的数学公式，我们提出了一个通用的扇形调节函数，能够以更精确的方式量化调节效果。借助最新的技术，引入了一种新的 Lyapunov-Krasovskii 泛函（LKF），以实现时滞相关，确保更小的保守性。然后提出了新的条件，以确保 GRN 的多稳定性，其形式为线性矩阵不等式（LMIs），这些条件依赖于时滞。我们的多稳定性条件适用于几种常用的调节函数，特别是多变量调节函数。通过两个例子来说明所提出的理论结果的适用性和有用性。
