Paul Elijah, Pogudin Gleb, Qin William, Laubenbacher Reinhard
California Institute of Technology, Pasadena, CA, USA.
Courant Institute of Mathematical Sciences, New York University, New York, NY, USA.
Complexity. 2020;2020. doi: 10.1155/2020/3687961. Epub 2020 Jan 20.
Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. In this paper, we investigate the dynamics of a random Boolean network with such properties using analytical methods and simulations. From our simulations, we observe that Boolean networks with higher canalizing depth have generally fewer attractors, the attractors are smaller, and the basins are larger, with implications for the stability and robustness of the models. These properties are relevant to many biological applications. Moreover, our results show that, from the standpoint of the attractor structure, high canalizing depth, compared to relatively small positive canalizing depth, has a very modest impact on dynamics. Motivated by these observations, we conduct mathematical study of the attractor structure of a random Boolean network of canalizing depth one (i.e., the smallest positive depth). For every positive integer , we give an explicit formula for the limit of the expected number of attractors of length in an -state random Boolean network as goes to infinity.
布尔网络是计算生物学中一种流行的建模框架，用于捕捉分子网络（如基因调控网络）的动态变化。据观察，许多已发表的此类网络模型是由驱动具有某些所谓“渠道化”属性动态变化的调控规则定义的。在本文中，我们使用分析方法和模拟研究具有此类属性的随机布尔网络的动态变化。从我们的模拟中可以观察到，具有更高渠道化深度的布尔网络通常吸引子更少，吸引子更小，而盆地更大，这对模型的稳定性和鲁棒性具有影响。这些属性与许多生物学应用相关。此外，我们的结果表明，从吸引子结构的角度来看，与相对较小的正渠道化深度相比，高渠道化深度对动态变化的影响非常有限。受这些观察结果的启发，我们对渠道化深度为1（即最小正深度）的随机布尔网络的吸引子结构进行了数学研究。对于每个正整数 ，我们给出了一个明确的公式，用于计算当 趋于无穷大时， 状态随机布尔网络中长度为 的吸引子的预期数量的极限。
