Modeling Molecular Interaction Networks with Nonlinear Ordinary Differential Equations
Modeling Molecular Interaction Networks with Nonlinear Ordinary Differential Equations
This chapter explains how cell biologists can make reliable connections between molecular interaction networks and cell behaviors, when intuition fails in all but the simplest cases. It proposes to make the connection by translating the reaction network into a set of nonlinear differential equations that describe how all the interacting species are changing with time. Differential equations define a vector field in the state space of the network. The vector field points to certain stable attractors, which can be correlated with long-term, stable behavior of the network and of the cell it governs. Transitions from one stable attractor to another represent the responses of the cell to specific perturbations (signals). A natural way to describe the signal-response properties of a regulatory network is in terms of a one-parameter bifurcation diagram, which efficiently displays the stable attractors (steady states and oscillators) and transitions between attractors as signal strength (the “parameter”) varies. These ideas are illustrated with simple examples of linear, hyperbolic, and sigmoidal signal-response curves; bistable switches based on positive feedback or mutual inhibition; and limit cycle oscillators based on substrate depletion, activator-inhibitor interactions, or time-delayed negative feedback.
Keywords: cell biology, molecular interaction networks, cell behavior, nonlinear differential equations
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