  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.
