Abstract
Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. 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. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.
Highlights
Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components
Whereas network motifs are typically modeled using ordinary differential equations (ODEs) with variables reacting to one another instantaneously, we explore network motif modeling using delay differential equations (DDEs), which have derivatives depending explicitly on the value of variables at times in the past
We thoroughly examine the most common network motifs[2,6] with explicit delays and present an approachable, step-by-step view of the mathematical analysis in order to make such delay equations easy to use for biologists and others
Summary
Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. The time scales and delays are explicit (Fig. 1), better capturing dynamics where intermediate steps are not fast, as compared to ODE models, which may fail to capture real delays, hide their effects by oversimplification, or require additional variables and parameters to predict complex phenomena[17,23,32] Such delay models have been productive in a variety of biological scenarios, such as processes with many intermediate but unimportant steps[33,34] and agestructured populations[34,35]. With DDEs, multiple steps (“cascades”) within a network can be rigorously simplified into a single step with delay (see below), an approach which has been explored in a variety of biological contexts[18,33,34,40,41,42] This makes interpretation of the phenomenology simpler than with ODEs and reduces the number of equations and parameters in the model[18,23].
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