Gene

Biological network

Quantum mechanics

Engineering

Biology

Differential equation

Control engineering

Applied mathematics

Biochemistry

Computer science

Gene regulatory network

Computational biology

Physics

Nonlinear system

Mathematics

Network motif

Feed forward

Mathematical analysis

Delay differential equation

Ordinary differential equation

Combinatorics

Dynamical systems theory

Biological system

Gene expression

Nature Communications

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.

Nonlinear delay differential equations and their application to modeling biological network motifs