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# Mathematica code for the map kinase cascade rate equations

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 Huang and Ferrell Mathematica code for the MAP kinase cascade rate equations. First we define the concentrations and rate constants. For the most part, the nomenclature here is exactly what I used in the paper in Fig. 1 and Eqs 1-35, except that to try to minimize confusion, I have called MAPKK mek and MAPKKK mos; these are the relevant MAPKK and MAPKKK in the Xenopus oocyte system, and using the names mek and mos rather than MAPKK and MAPKKK makes it easier to tell the enzymes apart at a glance. Concentrations are in micromolar units. mostot=.003; e1tot=.0003; e2tot=.0003; mektot=1.2; mekpasetot=.0003; mapktot=1.2; mapkpasetot=.12; a1=1000; a2=1000; a3=1000; a4=1000; a5=1000; a6=1000; a7=1000; a8=1000; a9=1000; a10=1000; d1=150; d2=150; d3=150; d4=150; d5=150; d6=150; d7=150; d8=150; d9=150; d10=150; k1=150; k2=150; k3=150; k4=150; k5=150; k6=150; k7=150; k8=150; k9=150; k10=150; Next we solve the rate equations. For the choice of the rate constants used here, t = 5000 is more than long enough to reach steady state. NDSolve[{ mos'[t]==-a1*(mostot-mosstar[t]-mose1[t]-mosstare2[t]-mekmosstar[t]-mekstarmosstar[t])*(e1tot-mose1[t])+d1*mose1[t]+k2*mosstare2[t], mose1'[t]==a1*(mostot-mosstar[t]-mose1[t]-mosstare2[t]-mekmosstar[t]-mekstarmosstar[t])*(e1tot-mose1[t])-(d1+k1)mose1[t], mosstar'[t]==-a2*mosstar[t]*(e2tot-mosstare2[t])+d2*mosstare2[t]+k1*mose1[t]+(k3+d3)*mekmosstar[t]-a3*mosstar[t]*(mektot-mekstar[t]-mekstarstar[t]-mekstarmekpase[t]-mekstarstarmekpase[t]-mekmosstar[t]-mekstarmosstar[t]-mapkmekstarstar[t]-mapkstarmekstarstar[t])+(k5+d5)*mekstarmosstar[t]-a5*mekstar[t]*mosstar[t], mosstare2'[t]==a2*mosstar[t](e2tot-mosstare2[t])-(d2+k2)*mosstare2[t], mos[0]==mostot, mose1[0]==0, mosstar[0]==0, mosstare2[0]==0, mek'[t]==-a3*(mektot-mekstar[t]-mekstarstar[t]-mekstarmekpase[t]-mekstarstarmekpase[t]-mekmosstar[t]-mekstarmosstar[t]-mapkmekstarstar[t]-mapkstarmekstarstar[t])*mosstar[t]+d3*mekmosstar[t]+k4*mekstarmekpase[t], mekmosstar'[t]==a3*(mektot-mekstar[t]-mekstarstar[t]-mekstarmekpase[t]-mekstarstarmekpase[t]-mekmosstar[t]-mekstarmosstar[t]-mapkmekstarstar[t]-mapkstarmekstarstar[t])*mosstar[t]-(d3+k3)*mekmosstar[t], mekstar'[t]==-a4*mekstar[t]*(mekpasetot- mekstarmekpase[t]-mekstarstarmekpase[t])+ d4*mekstarmekpase[t]+k3*mekmosstar[t]+ k6*mekstarstarmekpase[t]+d5*mekstarmosstar[t]-a5*mekstar[t]*mosstar[t], mekstarmekpase'[t]==a4*mekstar[t]*(mekpasetot- mekstarmekpase[t]-mekstarstarmekpase[t])-(d4+ k4)*mekstarmekpase[t], mekstarmosstar'[t]==a5*mekstar[t]*mosstar[t]- (d5+k5)*mekstarmosstar[t], mekstarstar'[t]==k5*mekstarmosstar[t]- a6*mekstarstar[t]*(mekpasetot-mekstarmekpase[t]- mekstarstarmekpase[t])+d6*mekstarstarmekpase[t]-a7*mekstarstar[t]*(mapktot-mapkstar[t]-mapkstarstar[t]-mapkstarmapkpase[t]-mapkstarstarmapkpase[t]-mapkmekstarstar[t]-mapkstarmekstarstar[t])+(d7+k7)*mapkmekstarstar[t]+(d9+k9)*mapkstarmekstarstar[t]-a9*mapkstar[t]*mekstarstar[t], mekstarstarmekpase'[t]==a6*mekstarstar[t]* (mekpasetot-mekstarmekpase[t]- mekstarstarmekpase[t])-(d6+k6)*mekstarstarmekpase[t], mek[0]==mektot, mekmosstar[0]==0, mekstar[0]==0, mekstarmekpase[0]==0, mekstarmosstar[0]==0, mekstarstar[0]==0, mekstarstarmekpase[0]==0, mapk'[t]==-a7*(mapktot-mapkstar[t]-mapkstarstar[t]-mapkstarmapkpase[t]-mapkstarstarmapkpase[t]-mapkmekstarstar[t]-mapkstarmekstarstar[t])*mekstarstar[t]+d7*mapkmekstarstar[t]+k8*mapkstarmapkpase[t], mapkmekstarstar'[t]==a7*(mapktot-mapkstar[t]-mapkstarstar[t]-mapkstarmapkpase[t]-mapkstarstarmapkpase[t]-mapkmekstarstar[t]-mapkstarmekstarstar[t])*mekstarstar[t]-(d7+k7)mapkmekstarstar[t], mapkstar'[t]==k7*mapkmekstarstar[t]-a8*mapkstar[t]*(mapkpasetot-mapkstarmapkpase[t]-mapkstarstarmapkpase[t])+d8*mapkstarmapkpase[t]-a9*mapkstar[t]*mekstarstar[t]+d9*mapkstarmekstarstar[t]+k10*mapkstarstarmapkpase[t], mapkstarmekstarstar'[t]==a9*mapkstar[t]*mekstarstar[t]- (d9+k9)*mapkstarmekstarstar[t], mapkstarstar'[t]==-a10*mapkstarstar[t]*(mapkpasetot-mapkstarmapkpase[t]-mapkstarstarmapkpase[t])+d10*mapkstarstarmapkpase[t]+k9*mapkstarmekstarstar[t], mapkstarmapkpase'[t]==a8*mapkstar[t]*(mapkpasetot-mapkstarmapkpase[t]-mapkstarstarmapkpase[t])-(d8+k8)*mapkstarmapkpase[t], mapkstarstarmapkpase'[t]==a10*mapkstarstar[t]*(mapkpasetot-mapkstarmapkpase[t]-mapkstarstarmapkpase[t])-(d10+k10)*mapkstarstarmapkpase[t], mapk[0]==mapktot, mapkmekstarstar[0]==0, mapkstar[0]==0, mapkstarmekstarstar[0]==0, mapkstarstar[0]==0, mapkstarmapkpase[0]==0, mapkstarstarmapkpase[0]==0 }, {mos,mose1,mosstar,mosstare2,mek,mekmosstar,mekstar,mekstarmosstar,mekstarmekpase,mekstarstar,mekstarstarmekpase,mapk,mapkmekstarstar,mapkstar,mapkstarmekstarstar,mapkstarstar,mapkstarmapkpase,mapkstarstarmapkpase}, {t,0,5000},MaxSteps->2000] The results can be tabulated or plotted. For extensive calculations, the NDSolve expression can be looped. This is the set of equations used for most of the calculations in the PNAS paper. This is a place to start, but the equations and constants should be modified in light of some recent findings: 1. Probably both mekstar (MAPKK-P) and mekstarstar (MAPKK-PP) catalyze mapk phosphorylation (Alessi et al. EMBO J. 13:1610-1619 [1994]). Exactly how much of the phosphorylation is catalyzed by which species is as yet unclear. The relative activities of mono- and bisphosphorylated Mek have not been measured directly. Mutational analysis suggests that the bisphosphorylated form is much more active than the monophosphorylated form, but studies of phosphatase-treated Mek suggest that both forms are similar in activity. 2. Natalie Ahn has now estimated some Km and Vmax values for the phosphorylation of MAPK by MEK (Mansour et al. [1996] Biochemistry 35:15529-15536). I recommend that you pick a, d, and k values consistent with Natalie’s measurements. 3. There are really 4 MAPK species, not three--MAPK, MAPK-Thr-P, MAPK-Tyr-P, and MAPK-PP. All 4 are formed to some extent. 4. In the oocyte we now know that the MAPK phosphatases are constitutively active and have a rough idea of how rapid each dephosphorylation step is. The apparent zero-order rate constants for dephosphorylation are: MAPK-Tyr-P -> MAPK, 0.10 min-1 MAPK-PP -> MAPK-Tyr-P, 0.08 min-1 MAPK-PP -> MAPK-Thr-P, 0.08 min-1 MAPK-Thr-P -> MAPK, 0.45 min-1 [Sohaskey and Ferrell, Molecular Biology of the Cell 10:3729-3743 (1999)] 5. The concentration of MAPK in oocytes is a bit lower than we originally assumed--it’s about 0.3 uM. **************************************************** James Ferrell, M.D., Ph.D. Professor of Molecular Pharmacology Professor of Biochemistry Stanford University School of Medicine CCSR, 269 Campus Drive Stanford CA 94305-5174 Phone: 650 725-0765 Fax: 650 725-2253 Email: james.ferrell@stanford.edu

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