A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins
Ethan
Choi
Auckland Bioengineering Institute, University of Auckland
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
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.
This component stores information related to activities in the nucleus
per mRNA synthesized in the nucleus and transferred into the cytosol
tim mRNA synthesized in the nucleus and transferred into the cytosol
nuclear form of the PER_TIM complex
$\frac{d \mathrm{M\_P}}{d \mathrm{time}}=\mathrm{v\_sP}\frac{\mathrm{K\_IP}^{n}}{\mathrm{K\_IP}^{n}+\mathrm{C\_N}^{n}}-\mathrm{v\_mP}\frac{\mathrm{M\_P}}{\mathrm{K\_mP}+\mathrm{M\_P}}-\mathrm{k\_d}\mathrm{M\_P}\frac{d \mathrm{M\_T}}{d \mathrm{time}}=\mathrm{v\_sT}\frac{\mathrm{K\_IT}^{n}}{\mathrm{K\_IT}^{n}+\mathrm{C\_N}^{n}}-\mathrm{v\_mT}\frac{\mathrm{M\_T}}{\mathrm{K\_mT}+\mathrm{M\_T}}-\mathrm{k\_d}\mathrm{M\_T}\frac{d \mathrm{C\_N}}{d \mathrm{time}}=\mathrm{k\_1}C-\mathrm{k\_2}\mathrm{C\_N}-\mathrm{k\_dN}\mathrm{C\_N}$
This component stores information related to activities in the cytosol
PER-TIM Complex reversibly formed in the cytosol and transported into the nucleus
$\frac{d C}{d \mathrm{time}}=\mathrm{k\_3}\mathrm{P\_2}\mathrm{T\_2}-\mathrm{k\_4}C-\mathrm{k\_1}C+\mathrm{k\_2}\mathrm{C\_N}-\mathrm{k\_dC}C$
This component stores information for the phosphorylation of PER in the cytosol
synthesized unphosphorylated PER protein
monophosphorylated PER protein
biphosphorylated PER protein
$\frac{d \mathrm{P\_0}}{d \mathrm{time}}=\mathrm{k\_sP}\mathrm{M\_P}-\mathrm{V\_1P}\frac{\mathrm{P\_0}}{\mathrm{K\_1P}+\mathrm{P\_0}}+\mathrm{V\_2P}\frac{\mathrm{P\_1}}{\mathrm{K\_2P}+\mathrm{P\_1}}-\mathrm{k\_d}\mathrm{P\_0}\frac{d \mathrm{P\_1}}{d \mathrm{time}}=\mathrm{V\_1P}\frac{\mathrm{P\_0}}{\mathrm{K\_1P}+\mathrm{P\_0}}-\mathrm{V\_2P}\frac{\mathrm{P\_1}}{\mathrm{K\_2P}+\mathrm{P\_1}}-\mathrm{V\_3P}\frac{\mathrm{P\_1}}{\mathrm{K\_3P}+\mathrm{P\_1}}+\mathrm{V\_4P}\frac{\mathrm{P\_2}}{\mathrm{K\_4P}+\mathrm{P\_2}}-\mathrm{k\_d}\mathrm{P\_1}\frac{d \mathrm{P\_2}}{d \mathrm{time}}=\mathrm{V\_3P}\frac{\mathrm{P\_1}}{\mathrm{K\_3P}+\mathrm{P\_1}}-\mathrm{V\_4P}\frac{\mathrm{P\_2}}{\mathrm{K\_4P}+\mathrm{P\_2}}-\mathrm{k\_3}\mathrm{P\_2}\mathrm{T\_2}+\mathrm{k\_4}C-\mathrm{v\_dP}\frac{\mathrm{P\_2}}{\mathrm{K\_dP}+\mathrm{P\_2}}-\mathrm{k\_d}\mathrm{P\_2}$
This component stores information for the phosphorylation of TIM in the cytosol
synthesized unphosphorylated TIM protein
monophosphorylated TIM protein
biphosphorylated TIM protein
$\frac{d \mathrm{T\_0}}{d \mathrm{time}}=\mathrm{k\_sT}\mathrm{M\_T}-\mathrm{V\_1T}\frac{\mathrm{T\_0}}{\mathrm{K\_1T}+\mathrm{T\_0}}+\mathrm{V\_2T}\frac{\mathrm{T\_1}}{\mathrm{K\_2T}+\mathrm{T\_1}}-\mathrm{k\_d}\mathrm{T\_0}\frac{d \mathrm{T\_1}}{d \mathrm{time}}=\mathrm{V\_1T}\frac{\mathrm{T\_0}}{\mathrm{K\_1T}+\mathrm{T\_0}}-\mathrm{V\_2T}\frac{\mathrm{T\_1}}{\mathrm{K\_2T}+\mathrm{T\_1}}-\mathrm{V\_3T}\frac{\mathrm{T\_1}}{\mathrm{K\_3T}+\mathrm{T\_1}}+\mathrm{V\_4T}\frac{\mathrm{T\_2}}{\mathrm{K\_4T}+\mathrm{T\_2}}-\mathrm{k\_d}\mathrm{T\_1}\frac{d \mathrm{T\_2}}{d \mathrm{time}}=\mathrm{V\_3T}\frac{\mathrm{T\_1}}{\mathrm{K\_3T}+\mathrm{T\_1}}-\mathrm{V\_4T}\frac{\mathrm{T\_2}}{\mathrm{K\_4T}+\mathrm{T\_2}}-\mathrm{k\_3}\mathrm{P\_2}\mathrm{T\_2}+\mathrm{k\_4}C-\mathrm{v\_dT}\frac{\mathrm{T\_2}}{\mathrm{K\_dT}+\mathrm{T\_2}}-\mathrm{k\_d}\mathrm{T\_2}$
This component creates the v_dT oscillation to account for the light-induced TIM degration in the 12:12 LD cycle
maximum rate enzyme degration of the fully phosphorylated TIM (affected by exposure to light)
$\mathrm{v\_dT}=\begin{cases}\mathrm{v\_dT\_dark} & \text{if $\sin \left(\frac{\mathrm{PI}\mathrm{time}}{12}\right)\le 0$}\\ \mathrm{v\_dT\_light} & \text{otherwise}\end{cases}$
This component calculates the total quantity of PER in the entire model
The total (nonconserved) quantities of the PER proteins
$\mathrm{P\_t}=\mathrm{P\_0}+\mathrm{P\_1}+\mathrm{P\_2}+C+\mathrm{C\_N}$
This component calculates the total quantity of TIM in the entire model
The total (nonconserved) quantities of the TIM proteins
$\mathrm{T\_t}=\mathrm{T\_0}+\mathrm{T\_1}+\mathrm{T\_2}+C+\mathrm{C\_N}$
A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins (Light-Dark Cycle Model)
Choi
Ethan
mcho099@aucklanduni.ac.nz
The University of Auckland
Auckland Bioengineering Institute
2009-12-10
A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins
This is the CellML description of Leloup and Goldbeter's 1998 mathematical model for circadian oscillations of the Period and Timeless proteins in Drosophila
Ethan Choi
Drosophila
keyword
Circadian Rhythms
biochemical oscillations
phase-response curve
Drosophila
PER
TIM
model
9486845
Leloup
Jean-Christophe
Goldbeter
Albert
A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins
1998-02
Journal of Biological Rhythms
13
70
87