| Citation |
|---|
| Leloup JC, Goldbeter A (1999) Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in Drosophila, J Theor Biol 198:445-459 http://www.sciencedirect.com/ |
| Description |
|---|
| A model of circadian oscillations. The parameters are set to ensure sustained oscillations. The cited reference also shows how chaos can occur in this model with different parameter values. |
| Rate constant | Reaction |
|---|---|
| k1 = 0.6 | C -> CN |
| k2 = 0.2 | CN -> C |
| k3 = 1.2 | P2 + T2 -> C |
| k4 = 0.6 | C -> P2 + T2 |
| kd = 0.01 | MP -> EmptySet |
| kd = 0.01 | P0 -> EmptySet |
| kd = 0.01 | P1 -> EmptySet |
| kd = 0.01 | P2 -> EmptySet |
| kd = 0.01 | T0 -> EmptySet |
| kd = 0.01 | T1 -> EmptySet |
| kdC = 0.01 | C -> EmptySet |
| kdN = 0.01 | CN -> EmptySet |
| KdP = 0.2 (hill khalf) | P2 -> EmptySet |
| kd + nudT/(KdT + T2[t]) | T2 -> EmptySet |
| kd + numT/(KmT + MT[t]) | MT -> EmptySet |
| (KIP^n*nusP)/(KIP^n + CN[t]^n) | EmptySet -> MP |
| (KIT^n*nusT)/(KIT^n + CN[t]^n) | EmptySet -> MT |
| KmP = 0.2 (hill khalf) | MP -> EmptySet |
| ksP = 0.9 | MP + EmptySet -> MP + P0 |
| ksT*MT[t] | EmptySet -> T0 |
| V1P/(K1P + P0[t]) | P0 -> P1 |
| V1T/(K1T + T0[t]) | T0 -> T1 |
| V2P/(K2P + P1[t]) | P1 -> P0 |
| V2T/(K2T + T1[t]) | T1 -> T0 |
| V3P/(K3P + P1[t]) | P1 -> P2 |
| V3T/(K3T + T1[t]) | T1 -> T2 |
| V4P/(K4P + P2[t]) | P2 -> P1 |
| V4T/(K4T + T2[t]) | T2 -> T1 |
| nudP = 2 (hill vmax) | P2 -> EmptySet |
| numP = 0.7 (hill vmax) | MP -> EmptySet |
| Variable | IC | ODE |
|---|---|---|
| C | 0.33 | C'[t] == -(k1*C[t]) - k4*C[t] - kdC*C[t] + k2*CN[t] + k3*P2[t]*T2[t] |
| CN | 1.74 | CN'[t] == k1*C[t] - k2*CN[t] - kdN*CN[t] |
| MP | 0.031 | MP'[t] == (KIP^n*nusP)/(KIP^n + CN[t]^n) - kd* MP[t] - (numP*MP[t])/(KmP + MP[t]) |
| MT | 0.031 | MT'[t] == (KIT^n*nusT)/(KIT^n + CN[t]^n) - MT[ t]*(kd + numT/(KmT + MT[t])) |
| P0 | 0.01 | P0'[t] == ksP*MP[t] - kd*P0[t] - (V1P*P0[t])/( K1P + P0[t]) + (V2P*P1[t])/(K2P + P1[t]) |
| P1 | 0.015 | P1'[t] == (V1P*P0[t])/(K1P + P0[t]) - kd*P1[t] - (V2P*P1[t])/(K2P + P1[t]) - (V3P*P1[t])/( K3P + P1[t]) + (V4P*P2[t])/(K4P + P2[t]) |
| P2 | 0.03 | P2'[t] == k4*C[t] + (V3P*P1[t])/(K3P + P1[t]) - kd*P2[t] - (V4P*P2[t])/(K4P + P2[t]) - (nudP*P2[t])/(KdP + P2[t]) - k3*P2[t]*T2[t] |
| T0 | 0.01 | T0'[t] == ksT*MT[t] - kd*T0[t] - (V1T*T0[t])/( K1T + T0[t]) + (V2T*T1[t])/(K2T + T1[t]) |
| T1 | 0.015 | T1'[t] == (V1T*T0[t])/(K1T + T0[t]) - kd*T1[t] - (V2T*T1[t])/(K2T + T1[t]) - (V3T*T1[t])/( K3T + T1[t]) + (V4T*T2[t])/(K4T + T2[t]) |
| T2 | 0.03 | T2'[t] == k4*C[t] + (V3T*T1[t])/(K3T + T1[t]) - k3*P2[t]*T2[t] - (V4T*T2[t])/(K4T + T2[t]) - T2[t]*(kd + nudT/(KdT + T2[t])) |
| Plot of Sustained Oscillations in Model |
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