Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.
J Theor Biol. 2010 Oct 21;266(4):641-56. doi: 10.1016/j.jtbi.2010.07.022. Epub 2010 Jul 24.
Prior work on the dynamics of Boolean networks, including analysis of the state space attractors and the basin of attraction of each attractor, has mainly focused on synchronous update of the nodes' states. Although the simplicity of synchronous updating makes it very attractive, it fails to take into account the variety of time scales associated with different types of biological processes. Several different asynchronous update methods have been proposed to overcome this limitation, but there have not been any systematic comparisons of the dynamic behaviors displayed by the same system under different update methods. Here we fill this gap by combining theoretical analysis such as solution of scalar equations and Markov chain techniques, as well as numerical simulations to carry out a thorough comparative study on the dynamic behavior of a previously proposed Boolean model of a signal transduction network in plants. Prior evidence suggests that this network admits oscillations, but it is not known whether these oscillations are sustained. We perform an attractor analysis of this system using synchronous and three different asynchronous updating schemes both in the case of the unperturbed (wild-type) and perturbed (node-disrupted) systems. This analysis reveals that while the wild-type system possesses an update-independent fixed point, any oscillations eventually disappear unless strict constraints regarding the timing of certain processes and the initial state of the system are satisfied. Interestingly, in the case of disruption of a particular node all models lead to an extended attractor. Overall, our work provides a roadmap on how Boolean network modeling can be used as a predictive tool to uncover the dynamic patterns of a biological system under various internal and environmental perturbations.
先前关于布尔网络动态的研究，包括对状态空间吸引子和每个吸引子的吸引域的分析，主要集中在节点状态的同步更新上。虽然同步更新的简单性使其非常有吸引力，但它没有考虑到与不同类型的生物过程相关的各种时间尺度。已经提出了几种不同的异步更新方法来克服这一限制，但对于同一系统在不同更新方法下显示的动态行为，还没有进行任何系统的比较。在这里，我们通过结合理论分析（如标量方程的求解和马尔可夫链技术）以及数值模拟来填补这一空白，对先前提出的植物信号转导网络布尔模型的动态行为进行了全面的比较研究。先前的证据表明，这个网络允许出现振荡，但还不清楚这些振荡是否持续。我们使用同步和三种不同的异步更新方案对这个系统进行了吸引子分析，包括未受干扰（野生型）和受干扰（节点中断）的系统。这项分析表明，虽然野生型系统具有与更新无关的固定点，但除非满足某些过程的时间和系统初始状态的严格约束，否则任何振荡最终都会消失。有趣的是，在特定节点中断的情况下，所有模型都导致了扩展的吸引子。总的来说，我们的工作提供了一个路线图，说明布尔网络建模如何可以作为一种预测工具，在各种内部和环境干扰下揭示生物系统的动态模式。
