
Contents
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6.1 Introduction 6.1 Introduction
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6.2 Basic Building Blocks 6.2 Basic Building Blocks
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6.2.1 From a Wiring Diagram to a Set of ODEs 6.2.1 From a Wiring Diagram to a Set of ODEs
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6.2.2 Constant Synthesis 6.2.2 Constant Synthesis
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6.2.3 Linear Degradation 6.2.3 Linear Degradation
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6.2.4 Autocatalytic Production 6.2.4 Autocatalytic Production
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6.2.5 Dimerization 6.2.5 Dimerization
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6.2.6 Michaelis-Menten Kinetics 6.2.6 Michaelis-Menten Kinetics
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6.3 Simple Networks and Signal-Response Curves 6.3 Simple Networks and Signal-Response Curves
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6.3.1 Synthesis and Degradation 6.3.1 Synthesis and Degradation
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6.3.2 Phosphorylation and Dephosphorylation 6.3.2 Phosphorylation and Dephosphorylation
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6.4 Networks with Feedback 6.4 Networks with Feedback
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6.4.1 What Is Feedback? 6.4.1 What Is Feedback?
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6.4.2 Negative Feedback 6.4.2 Negative Feedback
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6.4.3 Phase Planes, Vector Fields, and Nullclines 6.4.3 Phase Planes, Vector Fields, and Nullclines
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6.4.4 Positive Feedback 6.4.4 Positive Feedback
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6.4.5 Mutual Antagonism 6.4.5 Mutual Antagonism
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6.5 Networks That Oscillate 6.5 Networks That Oscillate
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6.5.1 Activator-Inhibitor 6.5.1 Activator-Inhibitor
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6.5.2 Substrate-Depletion 6.5.2 Substrate-Depletion
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6.5.3 Delayed Negative Feedback 6.5.3 Delayed Negative Feedback
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6.6 A Multiple-Feedback Network: p53 and Mdm2 6.6 A Multiple-Feedback Network: p53 and Mdm2
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6.7 Conclusions 6.7 Conclusions
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6. Modeling Molecular Interaction Networks with Nonlinear Ordinary Differential Equations
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Published:March 2006
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Abstract
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.
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