Box 2. Primer on modeling NF-B pathways using differential equations

The core of our original model of NF-B signaling is depicted below as a set of linked biochemical reactions. The diagram omits reactions (e.g. dissociation, reactions involving IB and IB) that are present in the full model but are not essential to oscillatory behavior. Complexes are denoted by ':' and generic sources and sinks for synthesis and degradation are denoted by '.' Rate parameters are shown above their respective reactions, named according to the convention of the original model (Hoffmann et al, 2002). The input into the model is a step increase in IKK, which is a surrogate for TNF stimulation. This allows the first reaction, IKK binding to IB–NF-B complex (a7), to proceed. The steps of phosphorylation, ubiquitination, and proteosomal degradation of IB within this complex are lumped into a single reaction whose products are free IKK and free NF-B (r4). NF-B enters the nucleus, denoted by the suffix 'n' (k1). This leads to synthesis of IB mRNA transcript, denoted by the suffix 't' (tr2). The half-life of the transcript is determined by tr3. Translation leads to synthesis of new IB (tr1), whose half-life is determined by deg1. IB can enter (tp1) and leave (tp2) the nucleus, and in the nucleus, IB is also denoted with the suffix 'n.' Nuclear IB and NF-B associate (a4), and together are exported to the cytoplasm (k2). In all, these steps form a negative feedback loop (also described in Box 1), whose overall sequence is shown by the blue arrow. Mass action kinetics are used to convert these biochemical reactions into a system of ordinary differential equations. For example, the equation for the time rate of change of cytoplasmic IB–NF-B complex is given by

where the terms show increases in the amount of complex due to association of IB and NF-B (a4) and export of nuclear complex (k2), and decreases in the amount of complex due to association with IKK (a7). Equations are written in this way for each chemical species. In the full version of the original model, similar reactions govern the behavior of IB and IB, resulting in additional differential equations. In this model formulation, the parameters are biochemical rates of association, dissociation, catalysis, transport, synthesis, and degradation. Thus, their values may be quantitatively measured or constrained by biochemical experiments. The procedure we used is summarized in the main text. Finally, to run the model, the initial concentrations of each species must be specified. (Running the model means to numerically solve the differential equations, e.g. with Mathematica's NDSolve function, to determine time courses of the concentrations of each species.) We initialized the model with a biologically plausible total level of NF-B (0.1 M) with all other concentrations set to zero. The basal state of the cell (non-stimulated) is simulated by running the model starting from this initial state until it reaches steady state. At steady state, NF-B is found in the cytoplasm and nucleus, as well as free or complexed with IB, but is predominantly found complexed in the cytoplasm in accordance with experimental observations. Following a step increase in IKK, the model can be further run to simulate the effects of TNF stimulation.