Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA.
Bull Math Biol. 2012 Feb;74(2):422-33. doi: 10.1007/s11538-011-9692-y. Epub 2011 Dec 3.
We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are not significantly better than functions of sufficient depth for many applications of the modeling and reverse engineering of biological networks.
我们引入了函数的嵌套 canalyzing 深度，用于衡量其保持嵌套 canalyzing 结构的程度。我们描述了具有给定深度的函数的结构，并计算了变量的预期活性和敏感度。这种分析量化了 canalyzation 如何导致布尔网络中更高的稳定性。它推广了嵌套 canalyzing 函数（NCF）的概念，NCF 正是具有最大深度的函数。NCF 已被提议作为基因调控网络模型，但它们的结构通常过于严格，并且极其稀疏。我们发现，随着 canalyzing 深度的增加，函数对输入扰动的敏感度会降低，但稳定性的收益会迅速减少。此外，我们还表明，随着深度的增加，使用这些函数的网络的动态很快接近临界状态，这表明真实网络具有一定程度的 canalyzing 深度，并且对于生物网络的建模和反向工程的许多应用，NCF 并不比具有足够深度的函数显著更好。
