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网络具备庞大的规模与复杂性，其微观层面的组分相互作用逻辑与宏观涌现动力学之间存在非平凡关联。本文通过形式化现有通路模块概念来应对该问题——通路模块定义为自动机网络动力学中（在无外部干扰条件下）受驱动节点子集扰动后必然发生的状态更新序列。我们提出一种从网络中自动提取通路模块的新型算法，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（微观动力学），无需计算网络动力学景观（宏观动力学）。具体而言，我们识别出能最大化通路长度且需组件间协同作用的复杂模块，由此提出动力学模块性的新视角：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子集扰动后必然跃迁至特定状态。因此，同一节点变量可根据其当前状态参与不同模块。网络动力学模块性的度量与复杂模块间的重叠度成反比，当复杂模块在网络动力学中完全可解耦时达到最大值。我们估算了包括果蝇体节极性网络在内的多个基因调控网络的动力学模块性，并探讨复杂模块识别与网络动力学模块性图景如何解释生物网络宏观动力学，例如揭示生物化学调控与信号网络中涌现计算（或集体行为）的（不同程度）可解耦基元。