Given the large size and complexity of most biochemical regulation and signaling networks, there is a non-trivial relationship between the micro-level logic of component interactions and the observed macro-dynamics. Here we address this issue by formalizing the existing concept of pathway modules, which are sequences of state updates that are guaranteed to occur (barring outside interference) in the dynamics of automata networks after the perturbation of a subset of driver nodes. We present a novel algorithm to automatically extract pathway modules from networks and we characterize the interactions that may take place between modules. This methodology uses only the causal logic of individual node variables (micro-dynamics) without the need to compute the dynamical landscape of the networks (macro-dynamics). Specifically, we identify complex modules, which maximize pathway length and require synergy between their components. This allows us to propose a new take on dynamical modularity that partitions complex networks into causal pathways of variables that are guaranteed to transition to specific states given a perturbation to a set of driver nodes. Thus, the same node variable can take part in distinct modules depending on the state it takes. Our measure of dynamical modularity of a network is then inversely proportional to the overlap among complex modules and maximal when complex modules are completely decouplable from one another in the network dynamics. We estimate dynamical modularity for several genetic regulatory networks, including the Drosophila melanogaster segment-polarity network. We discuss how identifying complex modules and the dynamical modularity portrait of networks explains the macro-dynamics of biological networks, such as uncovering the (more or less) decouplable building blocks of emergent computation (or collective behavior) in biochemical regulation and signaling.

翻译：鉴于大多数生物化学调控与信号传导网络规模庞大且结构复杂，其微观层面的组件交互逻辑与宏观可观测动态之间存在非平凡关联。针对该问题，本文通过形式化已有的通路模块概念进行探讨——这类模块指代自动机网络动态中，在扰动特定驱动节点子集后（若无外界干扰）必然发生的状态更新序列。我们提出一种新型算法，可自动从网络中提取通路模块，并刻画模块间可能存在的相互作用关系。该方法仅利用单个节点变量的因果逻辑（微观动态），无需计算网络的动态景观（宏观动态）。具体而言，我们识别出能最大化通路长度且需要组件间协同作用的复合模块，由此提出动态模块性的新诠释：将复杂网络划分为变量因果通路，这些通路在给定驱动节点扰动后会确保进入特定状态。因此，同一节点变量可能根据其当前状态参与不同模块。网络的动态模块性度量与复合模块间的重叠程度呈反比，当复合模块在网络动态中完全解耦时达到最大值。我们估算了多个基因调控网络的动态模块性，包括黑腹果蝇体节极性网络。研究探讨了识别复合模块及网络动态模块性图谱如何揭示生物网络的宏观动态本质——例如揭示生物化学调控与信号传导中涌现计算（或集体行为）的（不同程度）可解耦构成单元。