/* * A Model for Circadian Rhythms in Drosophila Incorporating the * Formation of a Complex between the PER and TIM Proteins * * Model Status * * This model has been built with the differential expressions * in Leloup's 1998 paper for a circadian cycle in the case of * a light-dark (12:12 LD) cycle. This file is known to run in * PCEnv and COR to reproduce the diagrams in figure 4D-F (taking * into account light-induced TIM degration via an oscillating * v_dT). The initial conditions (for M_P, M_T, C_N, C, P_0, P_1, * P_2, T_0, T_1, T_2) have been set after allowing the model to * run till steady state to replicate figure4D-F. Note the rescaling * of the publication diagrams for figure4E and F * * Model Structure * * Abstract: The authors present a model for circadian oscillations * of the Period (PER) and Timeless (TIM) proteins in Drosophila. * The model for the circadian clock is based on multiple phosphorylation * of PER and TIM and on the negative feedback exerted by a nuclear * PER-TIM complex on the transcription of the perand tim genes. * Periodic behavior occurs in a large domain of parameter space * in the form of limit cycle oscillations. These sustained oscillations * occur in conditions corresponding to continuous darkness or * to entrainment by light-dark cycles and are in good agreement * with experimental observations on the temporal variations of * PER and TIM and of per and tim mRNAs. Birhythmicity (coexistence * of two periodic regimes) and aperiodic oscillations (chaos) * occur in a restricted range of parameter values. The results * are compared to the predictions of a model based on the sole * regulation by PER. Both the formation of a complex between PER * and TIM and protein phosphorylation are found to favor oscillatory * behavior. Determining how the period depends on several key * parameters allows us to test possible molecular explanations * proposed for the altered period in the perl and pers mutants. * The extended model further allows the construction of phase-response * curves based on the light-induced triggering of TIM degradation. * These curves, established as a function of both the duration * and magnitude of the effect of a light pulse, match the phase-response * curves obtained experimentally in the wild type and pers mutant * of Drosophila. * * A Model for Circadian Rhythms in Drosophila Incorporating the * Formation of a Complex between the PER and TIM Proteins, Leloup * JC, Goldbeter A, 1998, Journal of Biological Rhythms, 13, 70-87 * PubMed ID: 9486845 * * Model Diagram * * [[Image file: leloup_1998.png]] * * Scheme of the model for circadian oscillations in Drosophila * involving negative regulation of gene expression by PER and * TIM. per (MP) and tim (MT) mRNAs are synthesized in the nucleus * and transferred into the cytosol, where they accumulate at the * maximum rates vsP and vsT, respectively. There they are degraded * enzymatically at the maximum rates, vmP and vmT, with the Michaelis * constants, KmP and KmT. The rates of synthesis of the PER and * TIM proteins, respectively proportional to MP and MT, are characterized * by the apparent first-order rate constants ksP and ksT. Parameters * ViP (ViT) and KiP (KiT) (i = 1, . . . 4) denote the maximum * rate and Michaelis constant of the kinase(s) and phosphatase(s) * involved in the reversible phosphorylation of P0 (T0) into P1 * (T1) and P1 (T1) into P2 (T2), respectively. The fully phosphorylated * forms (P2 and T2) are degraded by enzymes of maximum rate vdP * and vdT and of Michaelis constants KdP and KdT and reversibly * form a complex C (association and dissociation are characterized * by the rate constants k3 and k4), which is transported into * the nucleus at a rate characterized by the apparent first-order * rate constant k1. Transport of the nuclear form of the PER-TIM * complex (CN) into the cytosol is characterized by the apparent * first-order rate constant k2. The negative feedback exerted * by the nuclear PER-TIM complex on per and tim transcription * is described by an equation of the Hill type (see first terms * in Equations 1a and 1e) in which n denotes the degree of cooperativity * and KIP and KIT are the threshold constants for repression. */ import nsrunit; unit conversion on; unit hour=3600 second^1; unit per_hour=2.7777778E-4 second^(-1); unit per_litre=1E3 meter^(-3); // unit nanomolar predefined unit nanomolar_hour=2.7777778E-10 meter^(-3)*second^(-1)*mole^1; unit per_nanomolar_hour=2.7777778E2 meter^3*second^(-1)*mole^(-1); math main { realDomain time hour; time.min=0; extern time.max; extern time.delta; real v_sP nanomolar_hour; v_sP=0.8; real v_mP nanomolar_hour; v_mP=0.8; real K_IP nanomolar; K_IP=1; real K_mP nanomolar; K_mP=0.2; real v_sT nanomolar_hour; v_sT=1; real v_mT nanomolar_hour; v_mT=0.7; real K_IT nanomolar; K_IT=1; real K_mT nanomolar; K_mT=0.2; real k_d per_hour; k_d=0.01; // Var below replaced by constant in model eqns to satisfy unit correction // real n dimensionless; // n=4; real k_1 per_hour; k_1=1.2; real k_2 per_hour; k_2=0.2; real k_dN per_hour; k_dN=0.01; real C(time) nanomolar; when(time=time.min) C=0.1790; real M_P(time) nanomolar; when(time=time.min) M_P=0.09107; real M_T(time) nanomolar; when(time=time.min) M_T=1.427; real C_N(time) nanomolar; when(time=time.min) C_N=1.203; real k_3 per_nanomolar_hour; k_3=1.2; real k_4 per_hour; k_4=0.6; real k_dC per_hour; k_dC=0.01; real P_0(time) nanomolar; when(time=time.min) P_0=0.02324; real P_1(time) nanomolar; when(time=time.min) P_1=0.02210; real P_2(time) nanomolar; when(time=time.min) P_2=0.01251; real T_0(time) nanomolar; when(time=time.min) T_0=0.5420; real T_1(time) nanomolar; when(time=time.min) T_1=0.8000; real T_2(time) nanomolar; when(time=time.min) T_2=4.733; real V_1P nanomolar_hour; V_1P=8; real V_2P nanomolar_hour; V_2P=1; real V_3P nanomolar_hour; V_3P=8; real V_4P nanomolar_hour; V_4P=1; real K_1P nanomolar; K_1P=2; real K_2P nanomolar; K_2P=2; real K_3P nanomolar; K_3P=2; real K_4P nanomolar; K_4P=2; real K_dP nanomolar; K_dP=0.2; real v_dP nanomolar_hour; v_dP=2; real k_sP per_hour; k_sP=0.9; real V_1T nanomolar_hour; V_1T=8; real V_2T nanomolar_hour; V_2T=1; real V_3T nanomolar_hour; V_3T=8; real V_4T nanomolar_hour; V_4T=1; real K_1T nanomolar; K_1T=2; real K_2T nanomolar; K_2T=2; real K_3T nanomolar; K_3T=2; real K_4T nanomolar; K_4T=2; real K_dT nanomolar; K_dT=0.2; real v_dT(time) nanomolar_hour; real k_sT per_hour; k_sT=0.9; real was_PI dimensionless; was_PI=3.141592653589793; real v_dT_dark nanomolar_hour; v_dT_dark=2; real v_dT_light nanomolar_hour; v_dT_light=4; real P_t(time) nanomolar; real T_t(time) nanomolar; // // M_P:time=(v_sP*(K_IP^4/(K_IP^4+C_N^4))-v_mP*(M_P/(K_mP+M_P))-k_d*M_P); M_T:time=(v_sT*(K_IT^4/(K_IT^4+C_N^4))-v_mT*(M_T/(K_mT+M_T))-k_d*M_T); C_N:time=(k_1*C-k_2*C_N-k_dN*C_N); // C:time=(k_3*P_2*T_2-k_4*C-k_1*C+k_2*C_N-k_dC*C); // P_0:time=(k_sP*M_P-V_1P*(P_0/(K_1P+P_0))+V_2P*(P_1/(K_2P+P_1))-k_d*P_0); P_1:time=(V_1P*(P_0/(K_1P+P_0))-V_2P*(P_1/(K_2P+P_1))-V_3P*(P_1/(K_3P+P_1))+V_4P*(P_2/(K_4P+P_2))-k_d*P_1); P_2:time=(V_3P*(P_1/(K_3P+P_1))-V_4P*(P_2/(K_4P+P_2))-k_3*P_2*T_2+k_4*C-v_dP*(P_2/(K_dP+P_2))-k_d*P_2); // T_0:time=(k_sT*M_T-V_1T*(T_0/(K_1T+T_0))+V_2T*(T_1/(K_2T+T_1))-k_d*T_0); T_1:time=(V_1T*(T_0/(K_1T+T_0))-V_2T*(T_1/(K_2T+T_1))-V_3T*(T_1/(K_3T+T_1))+V_4T*(T_2/(K_4T+T_2))-k_d*T_1); T_2:time=(V_3T*(T_1/(K_3T+T_1))-V_4T*(T_2/(K_4T+T_2))-k_3*P_2*T_2+k_4*C-v_dT*(T_2/(K_dT+T_2))-k_d*T_2); // v_dT=(if (sin(was_PI*time/(12 hour))<=0) v_dT_dark else v_dT_light); // P_t=(P_0+P_1+P_2+C+C_N); // T_t=(T_0+T_1+T_2+C+C_N); }