Developing efficient computational methods to assess the impact of external interventions on the dynamics of a network model is an important problem in systems biology. This paper focuses on quantifying the global changes that result from the application of an intervention to produce a desired effect, which we define as the total effect of the intervention. The type of mathematical models that we will consider are discrete dynamical systems which include the widely used Boolean networks and their generalizations. The potential interventions can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system. We use a class of regulatory rules called nested canalizing functions that frequently appear in published models and were inspired by the concept of canalization in evolutionary biology. In this paper, we provide a polynomial normal form based on the canalizing properties of regulatory functions. Using this polynomial normal form, we give a set of formulas for counting the maximum number of transitions that will change in the state space upon an edge deletion in the wiring diagram. These formulas rely on the canalizing structure of the target function since the number of changed transitions depends on the canalizing layer that includes the input to be deleted. We also present computations on random networks to compare the exact number of changes with the upper bounds provided by our formulas. Finally, we provide statistics on the sharpness of these upper bounds in random networks.

开发有效的计算方法以评估外部干预对网络模型动力学的影响是系统生物学中的重要问题。本文着重于量化因采用干预措施产生预期效果而导致的总体变化，我们将其定义为总效果的干预。我们将考虑的数学模型类型是离散的动力学系统，其中包括广泛使用的布尔网络及其推广。潜在的干预措施可以由一组节点和边缘来表示，这些节点和边缘可以被操纵以对系统产生期望的效果。我们使用一类称为嵌套渠化功能的监管规则，该规则经常出现在已发布的模型中，并受到进化生物学中渠化概念的启发。在本文中，我们基于调节函数的渠化性质提供了多项式范式。使用该多项式范式，我们给出了一组公式，用于计算在接线图中删除边后状态空间中将变化的最大跃迁数。这些公式依赖于目标函数的渠化结构，因为更改的过渡次数取决于包含要删除的输入的渠化层。我们还介绍了随机网络上的计算，以将更改的确切数量与我们的公式提供的上限进行比较。最后，我们提供了有关随机网络中这些上限的清晰度的统计信息。