Transcription factors control production of other transcription factors 6, 7, kinases the activation of other kinases 1, 3, 4, cells the growth of other cells 8, 9, and species the population size of other species 10. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.īiological regulation consists of complex networks of dynamical interactions 1, 2, 3, 4, 5. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically.
However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components.
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