合取布尔网络的稳定性结构
Stability Structures of Conjunctive Boolean Networks
摘要：
布尔网络是一个有限的动力系统，其变量从二进制集合中取值。每个变量的值更新规则是布尔函数，取决于所选的变量子集。布尔网络在基因调控网络的建模中得到了广泛的应用。本文主要研究一类特殊的布尔网络，称之为合取布尔网络。如果关联的值更新规则仅由AND运算组成，则布尔网络是合取的。已知有限动力系统的任何轨迹都将进入周期轨道。本文刻画了其基础图是强连通的合取布尔网络的所有周期轨道。特别地，我们建立了一个双射之间的一组周期轨道和一组二元项链的一定长度。我们进一步研究了周期轨道的稳定性。具体来说，我们通过改变状态的单个条目的值来扰动周期轨道中的状态。以扰动状态为初始条件的轨道将在有限的时间步长内进入另一个（可能是同一个）周期轨道。然后，我们提供了一个完整的表征所有这样的过渡从一个周期轨道到另一个。特别地，我们构造了一个有向图，其顶点是周期轨道，并且（有向）边表示轨道之间的过渡。我们称这样的有向图为合取布尔网络的稳定结构。
英文摘要：
A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter a periodic orbit. We characterize in this paper all periodic orbits of a conjunctive Boolean network whose underlying graph is strongly connected. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the conjunctive Boolean network.
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