Introduction 2+ 8 16 25 44 49 19 48 50 58 1 2+ 34 45 59 10 51 2+ 9 43 54 62 Figure 1. i i i  1 7 14 2+ 48 33 56 20 36 26 2+ 11 21 2+ 22 2+ 22 39 15 23 58 15 18 39 57 30 5 1 To understand better how signal transduction pathways function, to help find therapies and cures for immune system diseases, and to find better ways to control the immune system in transplant patients it would be useful to understand the NFAT and NFκB pathways in a quantitative fashion. Models and Mechanisms 1 4 22 TABLE 1. NFAT rate constants. Rate constant Value Source k 1 −1 Estimate k 2 −1 34 k 3 −1 Estimate k 4 −1 Estimate k 5 −1 52 k 6 −1 52 k 7 −1 Estimate k 8 −1 Estimate k 9 −1 Estimate k 10 −1 Estimate k 11 −1 −1 51 k 12 −1 10 k 13 −1 32 k 14 −1 34 k 15 −1 10 k 16 −1 −1 51 k 17 −1 62 k 18 −1 62 k 19 −3 −1 32 29 k 20 −1 32 29 k 21 −1 k 22 k 22 −1 Estimate TABLE 2. Concentration and volume parameters. Parameter Value Source Cell diameter 9000 nM 2 Nucleus diameter 6000 nM 2 n 3 Calculated c 3 Calculated 2+ 100 nM 2 2+ 1000 nM 2 Total calcineurin concentration 60.0 nM 17 Total NFκB concentration 52.9 nM 13 Total NFAT concentration 7.23 nM Estimate Total IκB concentration 55.0 nM 13 Number calcium ions required to activate each calcineurin molecule 3 29 Total PKCθ concentration 2000 nM Estimate Total PKCα concentration 1000 nM Estimate TABLE 3. PKCα membrane-binding parameters. Parameter Definition Value Source K Ca Dissociation constant 22.0 μM 38 h Calcium binding cooperativity 1.5 38 n Stoichiometric coefficient 2 mol/mol 38 K DAG Dissociation constant 10.2 nM 3 [DAG] Diacyl glycerol concentration 2000.0 nM 37 tr2a IκB constitutive mRNA synthesis −6 −1 25 tr2 IκB inducible mRNA synthesis −2 −1 −1 25 tr3 IκB mRNA degradation −4 −1 26 TABLE 4. NFκB rate constants. Rate constant Value Source k 5 −1 52 k 6 −1 52 k 19 −3 −1 32 29 k 20 −1 32 29 k 21 −1 k 22  k 22 −1 Estimate k 23 −1 −1 14 27 k 24 −1 14 k 25 −1 22 k 26 −1 −1 22 k 27 −1 k 2 k 2 14 k 28 −1 12 k 29 −1 k 3 k 2 14 k 30 −1 Estimate k 31 −1 Estimate k 32 −1 Estimate k 33 −1 Estimate k 34 −1 Estimate k 35 −1 56 k 36 −1 56 k 37 −1 56 k 38 −1 56 k 39 −1 56 k 40 −1 56 k 41 −1 60 k 42 −1 60 k 43 0.02 56 k 44 0.009 56 NFAT Model 1 i 40 46 31 * 22 2 i c 2 n 2+ k 1 k 22 Figure 2. i i  Using the law of mass action, this reaction scheme yields a system of 12 coupled first order differential equations describing the change in concentration with time for each of the 12 distinct chemical species (Appendix A). The parameters in the equations include the 22 rate constants and the number of calcium ions required to activate the phosphatase activity of each calcineurin molecule, designated by m in the figure. The time course for the concentrations of each of the chemical species from a given initial condition is determined by numerical solution of the system using a standard fourth order Runge–Kutta technique. 1 NFkB Model 1 55 56 28 IκB 61 20 6 26 22 2 2 n k 5 k 6 k 19 k 42 k 41 k 42 26 Results and Discussion 1 4 NFAT Model Results 1 2 i 5 i i 3 i i i i 2 Figure 3. 2  TABLE 5. NFAT resting steady-state concentrations. Species Concentration (nM) n 0.5219 c 0.1101 n 0.0505 c 0.0091 i n 0.2272 i c 9.4397 i n 0.0025 i c 0.0022 n 0.9477 c 0.0061 n 49.198 c 9.7108 n 2+ 100 c 2+ 100 3 3 n n i n 2 n 22 4 4 n i c n n n 4 n i c Figure 4. μ μ μ μ  4 n n 4 22 3 22 24 22 4 4 22 2 35 5 Figure 5. μ  NFκB Model Results 2 4 6 TABLE 6. NFκB resting steady-state concentrations. Species Concentration (nM) n 7.1779 c 0.0731 i n 0.2035 i c 0.0525 i n 0.2343 i c 0.1076 n 0.2376 c 52.961 n 0.0505 c 0.0091 n 49.198 c 9.7108 \documentclass[12pt]{minimal}
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\begin{document}$$\hbox{Ca}_{\rm c}^{2+}$$\end{document} 100 6 i 6 2 56 Figure 6. κ 2  6 2 22 3 7 22 Figure 7. Dependence of the percent active NFκB on the steady-state calcium concentration. For each data point the calcium ion concentration was fixed at the indicated value.  8 Figure 8. μ μ κ κ κ κ κ κ κ κ κ κ κ  8 2+ c 8 22 3 8 4 8 22 3 9 60 2+ 53 9 60 2 Figure 9. κ 2+ κ κ κ κ κ κ κ −1  9 k 41 −1 Sensitivity Analysis 10 \documentclass[12pt]{minimal}
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\begin{document}$$ \frac{\ln\frac{\left(\hbox{active transcription factor at} + 10\% \right)} {\left(\hbox{active transcription factor at} - 10\% \right)}} {\frac{\left(\hbox{rate constant times 1.1}\right)} {\left(\hbox{rate constant times 0.9 }\right)}} $$\end{document} Figure 10. (a) Sensitivity analysis for NFAT activation. (b) Sensitivity analysis for NFκB activation. In both cases, the rate constants were increased and decreased by 10%. The sensitivity of the steady state concentration of active transcription factor at 1.0 μM calcium to changes in the rate constants is shown.  10 k 9 k 10 k 13 k 14 k 15 k 16, k 19 k 20 k 9 k 10 k 13 k 14 k 15 k 16 4 41 k 11 k 12 k 19 k 20 k 5 k 6 5 10 k 30 k 41 k 42 k 23 k 24 k 25 k 26 k 27 k 28 k 43 k 44 k 41 k 42 9 k 43 k 44 k 30 k 29 k 23 k 24 k 25 k 26 k 27 k 28 Conclusions We have developed models for the calcium and PKCθ mediated activation and deactivation of the transcription factors NFAT and NFκB that incorporate experimentally determined reaction pathways and that simulate this action over physiological calcium concentrations. The model reproduces experimentally observed behaviors of both the NFAT and NFκB systems under a variety of conditions. 13 60 While the model is qualitatively accurate, simulation of the reporter gene levels observed by Dolmetsch and colleagues might be more closely approximated by the inclusion of additional features to the model. Several of these enhancements are obvious and consist of including elements both upstream and downstream of the system modeled. For example, binding of the transcription factors to the DNA might be added along with steps to describe the expression of the reporter genes observed in the experiment. Inclusion of these steps would require adding significant complexity to the model that might obstruct demonstration of the mechanisms governing activation of the calcium dependent transcription factors NFAT and NFκB. Thus, addition of these steps has been left for future work. in vitro in vivo 18 55 42 3 47 13