Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA 24061-0477, USA.
Bull Math Biol. 2010 Aug;72(6):1425-47. doi: 10.1007/s11538-010-9501-z. Epub 2010 Jan 20.
For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.
对于许多生物网络来说，网络的拓扑结构限制了它的动态特性。特别是，反馈回路起着至关重要的作用。本文的结果定量地描述了（无符号）反馈回路对一类基因调控网络离散模型动态的约束。通过仅使用与（AND）或或（OR）运算符，可以得到合取（conjunctive）布尔网络和析取（disjunctive）布尔网络，它们构成了一个由 canalyzing 函数组成的网络子类，用于描述许多已发表的基因调控机制。在反馈回路的研究中，通常将布线图分解为每个都是强连通的连接组件。结果表明，对于具有强连通布线图的合取布尔网络，反馈回路结构完全决定了网络的长期动态。建立了一个给定长度的极限环的精确数量的公式，并确定了可以出现哪些极限环长度。对于一般的布线图，情况要复杂得多，因为一个强连通组件中的反馈回路会影响其他组件中的反馈回路。本文通过强连通组件的偏序集的性质，提供了给定长度的极限环数量的一个精确下界和一个上界。
