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CALTECH CS 191A - Appendix

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PNAS(99)673.supp2.pdfTable 1. Decomposition of the core deterministic model of Fig. 1 into detailed reaction stepsSupporting information for Gonze et al. (January 15, 2002) Proc. Natl. Acad. Sci. USA,10.1073/pnas.022628299.AppendixKinetic Equations of the Deterministic Model. In the model schematized in Fig. 6, thetemporal variation of the concentrations of mRNA (MP) and the various forms of clockprotein, cytosolic (P0, P1, P2) or nuclear (PN), is governed by the following system of kineticequations (1, 2):dMPdt=vsKInKIn+PNn−vmMPKm+MPdP0dt=ksMP−v1P0K1+P0+v2P1K2+P1dP1dt=v1P0K1+P0- v2P1K2+P1−v3P1K3+P1+v4P2K4+P2[1]dP2dt=v3P1K3+P1- v4P2K4+P2−vdP2Kd+P2−k1P2+k2PNdPNdt=k1P2−k2PNThe results shown in Fig. 2A (see article) have been obtained by numerical integration of Eqs.1 for the following parameter values:KI = 2 nM, n = 4, vs=0.5 nMh−1, vm=0.3 nMh−1, Km=0.2 nM, ks=2.0 h−1,v1=6.0 nMh−1, K1=1.5 nM , v2=3.0 nMh−1, K2=2.0 nM, v3=6.0 nMh−1, K3=1.5 nM,v4=3.0 nMh−1,K4=2.0 nM, vd=1.5 nMh−1, Kd=0.1 nM, k1=2.0 h−1, k2=1.0 h−1.Decomposition of the Deterministic Model into Elementary Reaction Steps. To performstochastic simulations of the circadian clock mechanism, the deterministic model schematizedin Fig. 6, governed by the five kinetic equations (Eqs. 1), is decomposed into a detailedreaction system consisting of 30 elementary steps. These steps are listed in Table 1 with theprobability of their occurrence, denoted wi (i = 1,... 30). Each wi is the product of a rateconstant times the number(s) of molecules involved in the reaction step. Because eachenzymatic reaction is decomposed fully into elementary steps, enzyme-substrate complexesare considered explicitly. The detailed reaction system thus contains 22 variables instead of 5in the deterministic model. In Table 1, the central column shows the reaction steps involvingthe indicated molecular species, with the rate constant indicated above the arrow. In the rightcolumn, showing the probability of occurrence of the various reaction steps, italicized capitalsdenote the numbers of molecules of the corresponding species involved in the particularreaction step.Steps 1-8 pertain to the formation and dissociation of the various complexes between the genepromoter and nuclear protein PN. G denotes the unliganded promoter of the gene, and GPN,GPN2, GPN3, and GPN4 denote the complexes formed by the gene promoter with 1, 2, 3, or 4PN molecules. Step 9 relates to the active state of the promoter leading to expression of thegene and synthesis of mRNA (MP). In the case considered we assume that only the complexbetween the promoter and four molecules of PN is inactive. Steps 10-12 pertain to thedegradation of MP by enzyme Em through formation of the complex Cm. Step 13 relates tosynthesis of unphosphorylatyed clock protein (P0) at a rate proportional to the number ofmRNA molecules. Steps 14-16 refer to the phosphorylation of P0 into P1 by kinase E1 throughformation of complex C1. Steps 17-19 refer to the dephosphorylation of P1 into P0 byphosphatase E2 through formation of complex C2. Steps 20-25 pertain to the correspondingphosphorylation of P1 into P2 and dephosphorylation of P2 into P1. Steps 26-28 relate to thedegradation of the phosphorylated form P2 by enzyme Ed through formation of complex Cd.Steps 29 and 30 refer to entry of P2 into and exit of PN from the nucleus, respectively.Parameter Values for Stochastic Simulations. Stochastic simulations of the detailedreaction system consisting of the 30 reaction steps listed in Table 1 have been carried out bymeans of the algorithm proposed by Gillespie (3, 4), in which in a random, infinitesimal timeinterval computed by the method, one of the i reactions occurs with a probability proportionalto wi (i = 1,... 30). Parameter values used for stochastic simulations are listed in Table 2.Remarks. In the column listing the probability of occurrence of the various reaction steps inTable 1, kinetic constants related to bimolecular reactions are scaled by Ω (3, 4). Whenvarying Ω to modify the numbers of molecules involved in the circadian oscillatorymechanism, we wish to keep the number of gene promoters (G) equal to unity withoutaltering the relative weights of the different probabilities wi so as to keep dynamic behaviorconsistent with that predicted by the corresponding deterministic model governed by Eqs. 1.The numbers of enzyme molecules and the kinetic constants related to the steps involving Gtherefore are multiplied by Ω in Table 2 listing the parameter values.The maximum value of ai (i = 1, ..4) considered in Figs. 2 and 3 ranges from 103 to 5 × 104molecule-1 h-1 for Ω ranging from 10 to 500 (see Table 2). For a nuclear volume of 10-13 liters,for which a concentration of 1nM corresponds to 60 molecules per nucleus, these values of aicorrespond to values of the bimolecular rate constant ranging from 1.5 × 1010 to 7.5 × 1011M-1 s-1. Such values are larger than the diffusion limit of 108–109 M-1s-1 usually considered forbimolecular rate constants. However, values of up to 1010 M-1 s-1 (5, 6) or even higher values(7) characterize the binding of a repressor to the gene promoter because of a “facilitateddiffusion” process mediated by encounter of the protein with the DNA molecule followedeither by sliding (6-9) or direct intersegment transfer of the protein on DNA (6). The values ofbimolecular rate constants ai considered in a previous report (10) were bounded by the“classical” diffusion limit; this may explain the lack of robustness reported by the authors,because at lower values of ai the oscillations are more affected by molecular noise (seearticle).In steps 1-8 in Table 1, parameters aj and dj (j = 1,…, n; with n = 1, 2, 3, or 4) are chosen suchthat the dissociation constant Ki = di/ai (with KIn=Kjj=1n∏ , where KI denotes the inhibitionconstant in the nondeveloped, deterministic model governed by Eqs. 1) decreases as thenumber of molecules of PN bound to the promoter increases (see Table 2); these conditionsenhance the cooperativity of the repression process.1. Goldbeter, A. (1995) Proc. R. Soc. London Ser. B 261, 319-324.2. Goldbeter, A. (1996) Biochemical Oscillations and Cellular Rhythms: The MolecularBases of Periodic and Chaotic Behavior (Cambridge Univ. Press, Cambridge, U.K.).3. Gillespie, D. T. (1976) J. Comp. Physiol. 22, 403-434.4. Gillespie, D. T. (1977) J. Phys. Chem. 81, 2340-2361.5. Riggs, A. D., Bourgeois, S. & Cohn, M. (1970) J. Mol.


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