Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop
Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop
复制标题
具有非均匀衰减率的分段仿射基因网络模型的周期解：负反馈环的情况
作者：
Etienne Farcot;J. Gouzé
本文研究了一类基因调控网络模型方程的周期解。与以前的绝大多数研究不同，它没有假设所有的衰变率都是相同的。为了处理这种更一般的情况，我们依赖于这些系统的单调性。在另一种假设下，证明了单调凹算子的经典不动点定理可以应用于这些系统。所需的假设在几何术语中表示为所谓焦点上的对准条件。作为应用，我们证明了3维或更多维负反馈环系统稳定周期轨道的存在唯一性，以及2维系统唯一稳定平衡点的存在唯一性。这扩展了Snoussi定理，该定理仅证明了这些轨道的存在。
This paper concerns periodic solutions of a class of equations that model gene regulatory networks. Unlike the vast majority of previous studies, it is not assumed that all decay rates are identical. To handle this more general situation, we rely on monotonicity properties of these systems. Under an alternative assumption, it is shown that a classical fixed point theorem for monotone, concave operators can be applied to these systems. The required assumption is expressed in geometrical terms as an alignment condition on so-called focal points. As an application, we show the existence and uniqueness of a stable periodic orbit for negative feedback loop systems in dimension 3 or more, and of a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only.