"Modeling Network Motifs as Linear Dynamical Systems"
The dynamic behavior of a system of interacting components is determined by the characteristics of the individual components, as well as the way in which they interact. From the system designer's point of view, there are two "handles" for adjusting dynamics: component parameters and connectivity. The idea that connectivity influences the range of possible behaviors of a system is a natural extension of the structure-function relation that is pervasive in biology. A certain network structure may favor particular dynamic behaviors, while precluding other behaviors. We present a method for characterizing possible dynamic behaviors of a system of interacting components when there is only information about connectivity. We model network motifs as linear systems, sampling the model parameters with random values. Using this Monte Carlo approach, we characterize a given connectivity in terms stability and oscillation. We find that certain motifs are inherently stable in the face of variation of parameter values. Other motifs are inherently oscillatory. These findings suggest a functional explanation for the observation that certain connectivities are frequently observed in real networks. A prevalence of stable sub-structures seems plausible. Among the most stable 3-node and 4-node motifs in our simulations are the feed-forward loop, the bi-fan, and the bi-parallel, identified by Milo, et al. in transcriptional networks, neurons, food webs, electronic circuits, and the World Wide Web (1). Additionally, linear chains, single-input, and multi-input motifs were identified by Lee, et al. in the S. cerevisiae transcriptional network (2). These motifs share a common structural feature: feedbacks are absent. Modeled as a linear system, motifs lacking feedbacks exhibit stable, non-oscillatory behavior. The most oscillatory 3-node motif in our simulations is the simple loop. In addition to the corresponding 4-node loop, we identified thirteen additional 4-nodestructures that are unstable oscillators, ten of which contained one or more feed-forward loops as a subcomponent of the 4-node structure. When the feed-forward loop is considered as an autonomous unit, it behaves a stable non-oscillator. Connected to a fourth node, no longer technically a FFL in many cases, the new motif may exhibit drastically different behavior. Of course, there are ways of extending the feed-forward loop such that the original behavior is preserved. Thus, the dynamic behavior of this motif depends on the context in which it is found and where the boundary of the motif is drawn. The question arises whether 3 and 4-node motifs are self contained functional units, or parts of larger functional structure. This modeling exercise suggests that consideration of motifs as autonomous functional units can lead to certain conclusions about their behavior. When considered part of a larger system, even if that new system is achieved by the addition of a single node, the original behavior may no longer be plausible. REFERENCES 1. R. Milo, et al. Network Motifs: Simple Building Blocks of Complex Networks. Science 298, 824 (2002). 2. T. Lee, et al. Transcriptional Regulatory Networks in Saccharomyces cerevisiae. Science 298, 799 (2002). 3. S. Mangan and U. Alon. Structure and function of the feed-forward loop network motif. PNAS 100. 11980-11985 (2003)