Model Status

This model has been built using the expressions found in the Conradie's 2010 paper "Restriction point control of the mammalian cell cycle via the cyclin E/Cdk2:p27 complex". This model was produced using the parameter values on page 360. This file is known to run on COR 0.9 and Open Cell.

Model Structure

Numerous top-down kinetic models have been constructed to describe the cell cycle. These models have typically been constructed, validated and analyzed using model species (molecular intermediates and proteins) and phenotypic observations, and therefore do not focus on the individual model processes (reaction steps). We have developed a method to: (a) quantify the importance of each of the reaction steps in a kinetic model for the positioning of a switch point [i.e. the restriction point (RP)]; (b) relate this control of reaction steps to their effects on molecular species, using sensitivity and co-control analysis; and thereby (c) go beyond a correlation towards a causal relationship between molecular species and effects. The method is generic and can be applied to responses of any type, but is most useful for the analysis of dynamic and emergent responses such as switch points in the cell cycle. The strength of the analysis is illustrated for an existing mammalian cell cycle model focusing on the RP [Novak B, Tyson J (2004) J Theor Biol230, 563-579]. The reactions in the model with the highest RP control were those involved in: (a) the interplay between retinoblastoma protein and E2F transcription factor; (b) those synthesizing the delayed response genes and cyclin D/Cdk4 in response to growth signals; (c) the E2F-dependent cyclin E/Cdk2 synthesis reaction; as well as (d) p27 formation reactions. Nine of the 23 intermediates were shown to have a good correlation between their concentration control and RP control. Sensitivity and co-control analysis indicated that the strongest control of the RP is mediated via the cyclin E/Cdk2:p27 complex concentration. Any perturbation of the RP could be related to a change in the concentration of this complex; apparent effects of other molecular species were indirect and always worked through cyclin E/Cdk2:p27, indicating a causal relationship between this complex and the positioning of the RP.

Restriction point control of the mammalian cell cycle via the cyclin E/Cdk2:p27 complex, Conradie R, Bruggeman F J, Ciliberto A, Csikasz-Nagy A, Novak B, Westerhoff H V, Snoep J L FEB Journal, 277, 357-367PubMed ID: 20015233

Time course of the mammalian cell division cycle. A time integration for 30 h is shown for six of the intermediates of the system. The G1-, S/G2- and M-phases for one cell cycle are indicated in the graph. The RP is also depicted in the G1-phase. It should be noted that, in contrast to other switch points in the model, the RP is not a hard coded event (i.e. it is not an explicit function, but rather an emergent property of the model), and it is empirically defined as the last time point where, upon cycloheximide (CHX) treatment, the cell would not finish the division cycle it started with. The CHX treatment was mimicked in the model by reducing the translation efficiency of the ribosomes [e or Eps(t)], a parameter found in all synthesis steps of the model, from 1.0 to 0.5. This definition was taken from the original publication in which the model was described [5].

Model Status

This CellML model runs in OpenCell and COR. It was created from equations [1] to [46]. The model parameters were taken from the Parameters document. The units have been checked and they are consistent. v_sw has been set to 0 in this particular model which means the model is de-coupled from the circadian clock. The CellML model runs to replicate the first part of figure 7B.

Model Structure

We propose an integrated computational model for the network of cyclin-dependent kinases (Cdks) that controls the dynamics of the mammalian cell cycle. The model contains four Cdk modules regulated by reversible phosphorylation, Cdk inhibitors, and protein synthesis or degradation. Growth factors (GFs) trigger the transition from a quiescent, stable steady state to self-sustained oscillations in the Cdk network. These oscillations correspond to the repetitive, transient activation of cyclin D/Cdk4-6 in G(1), cyclin E/Cdk2 at the G(1)/S transition, cyclin A/Cdk2 in S and at the S/G(2) transition, and cyclin B/Cdk1 at the G(2)/M transition. The model accounts for the following major properties of the mammalian cell cycle: (i) repetitive cell cycling in the presence of suprathreshold amounts of GF; (ii) control of cell-cycle progression by the balance between antagonistic effects of the tumor suppressor retinoblastoma protein (pRB) and the transcription factor E2F; and (iii) existence of a restriction point in G(1), beyond which completion of the cell cycle becomes independent of GF. The model also accounts for endoreplication. Incorporating the DNA replication checkpoint mediated by kinases ATR and Chk1 slows down the dynamics of the cell cycle without altering its oscillatory nature and leads to better separation of the S and M phases. The model for the mammalian cell cycle shows how the regulatory structure of the Cdk network results in its temporal self-organization, leading to the repetitive, sequential activation of the four Cdk modules that brings about the orderly progression along cell-cycle phases.

The original paper reference is cited below:

Temporal self-organization of the cyclin/Cdk network driving the mammalian cell cycle. Goldbeter A, Gerard C, 2009, Unite de Chronobiologie Theorique, 106(51), 21643-8. PubMed ID: 20007375

GF-induced oscillations in the Cdk network. (A) Below a sharp threshold in the concentration of GF, the Cdk network evolves to a stable steady state, whereas sustained oscillations occur above the threshold that corresponds to a bifurcation beyond which the steady state becomes unstable

Model Status

This CellML model runs in both OpenCell and COR to reproduce Figure 2 in the original published paper. The units have been checked and they are consistent.

Model Structure

ABSTRACT: The mechanism of circadian oscillations in the period protein (PER) in Drosophila is investigated by means of a theoretical model. Taking into account recent experimental observations, the model for the circadian clock is based on multiple phosphorylation of PER and on the negative feedback exerted by PER on the transcription of the period (per) gene. This minimal biochemical model provides a molecular basis for circadian oscillations of the limit cycle type. During oscillations, the peak in per mRNA precedes by several hours the peak in total PER protein. The results support the view that multiple PER phosphorylation introduces times delays which strengthen the capability of negative feedback to produce oscillations. The analysis shows that the rhythm only occurs in a range bounded by two critical values of the maximum rate of PER degradation. A similar result is obtained with respect to the rate of PER transport into the nucleus. The results suggest a tentative explanation for the altered period of per mutants, in terms of variations in the rate of PER degradation.

The original paper reference is cited below:

A Model for Circadian Oscillations in the Drosophila Period Protein (PER), Albert Goldbeter, 1995, Proceedings of the Royal Society of London, Series B, Biological Sciences, 261, 319-324. PubMed ID: 8587874

Schematic diagram of the model for circadian oscillations in PER protein and per mRNA.

This CellML model runs in OpenCell and COR. It is based on 3 different models: the a model is the goldbeter_1995 model and the b model is the leloup_gonze_goldbeter_1999b model. Both are available in the CellMl model repository. The c model comes from this paper and is based on equations 2.9, 2.10 and 2.11 in the published paper.

Model Structure

ABSTRACT: Circadian rhythms, which occur spontaneously with a period of about 24 h in a variety of organisms, allow their adaptation to the periodic variations of the environment. These rhythms are generated by a genetic regulatory network involving a negative feedback loop on transcription. Mathematical models based on the negative autoregulation of gene expression by the protein product of a clock gene account for the occurrence of self- sustained circadian oscillations. These models differ by their degree of complexity and, hence, by the number of variables considered. Some of these models can be considered as minimal because they contain a reduced number of biochemical processes and variables capable of producing sustained oscillations. In three of these minimal models, the period of the oscillations significantly changes with the rate of degradation of the clock protein. However, depending on the model considered, the period increases, decreases or passes through a maximum as a function of the protein degradation rate. We clarify the bases for these markedly different results by bringing to light the roles of (i) protein phosphorylation, which is required for protein degradation, and (ii) the velocity and degree of saturation of mRNA and protein degradation. Changes in the parameter values of the more complex of the minimal models can produce the period profiles observed in the other two models. The analysis allows us to reconcile the contradictory predictions for the dependence of the period on the clock protein degradation rate in three minimal models used to describe circadian rhythms.

The original paper reference is cited below:

Dependence of the period on the rate of protein degradation in minimal models for circadian oscillations , Claude Gerard, Didier Gonze, Albert Goldbeter, 2009, Phil. Trans. R. Soc. A , 367, 4665-4683. PubMed ID: 19884174

Schematic Diagram of Model A, the goldbeter_1995 model

Schematic Diagram of Model B, the leloup_1999 model

Schematic Diagram of Model C, the model based on equations 2.9, 2.10 and 2.11 in the published paper

Model Status

This CellML model runs in both PCEnv and COR to recreate the published results. The units have been checked and they are consistent. In this particular version of the model couples the Notch, Wnt and FGF signalling pathways together and so uses equations A17 and A18 in the appendix to replace equations A4 and A10 respectively.

Model Structure

ABSTRACT: The formation of somites in the course of vertebrate segmentation is governed by an oscillator known as the segmentation clock, which is characterized by a period ranging from 30 min to a few hours depending on the organism. This oscillator permits the synchronized activation of segmentation genes in successive cohorts of cells in the presomitic mesoderm in response to a periodic signal emitted by the segmentation clock, thereby defining the future segments. Recent microarray experiments [Dequeant, M.L., Glynn, E., Gaudenz, K., Wahl, M., Chen, J., Mushegian, A., Pourquie, O., 2006. A complex oscillating network of signaling genes underlies the mouse segmentation clock. Science 314, 1595-1598] indicate that the Notch, Wnt and Fibroblast Growth Factor (FGF) signaling pathways are involved in the mechanism of the segmentation clock. By means of computational modeling, we investigate the conditions in which sustained oscillations occur in these three signaling pathways. First we show that negative feedback mediated by the Lunatic Fringe protein on intracellular Notch activation can give rise to periodic behavior in the Notch pathway. We then show that negative feedback exerted by Axin2 on the degradation of beta-catenin through formation of the Axin2 destruction complex can produce oscillations in the Wnt pathway. Likewise, negative feedback on FGF signaling mediated by the phosphatase product of the gene MKP3/Dusp6 can produce oscillatory gene expression in the FGF pathway. Coupling the Wnt, Notch and FGF oscillators through common intermediates can lead to synchronized oscillations in the three signaling pathways or to complex periodic behavior, depending on the relative periods of oscillations in the three pathways. The phase relationships between cycling genes in the three pathways depend on the nature of the coupling between the pathways and on their relative autonomous periods. The model provides a framework for analyzing the dynamics of the segmentation clock in terms of a network of oscillating modules involving the Wnt, Notch and FGF signaling pathways.

The complete original paper reference is cited below:

Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways, Albert Goldbeter and Olivier Pourquie, 2008, Journal of Theoretical Biology, 252, 574-585. PubMed ID: 18308339

Schematic diagram representing the model for the segmentation clock based on negative feedback loops in the coupled FGF, Wnt and Notch signalling pathways.

Model Status

This CellML model runs in both OpenCell and COR to recreate the published results. The units have been checked and they are consistent. This particular version of the model uncouples the Notch, Wnt and FGF signalling pathways and uses equations A4 and A10 in the appendix rather than A17 and A18.

Model Structure

ABSTRACT: The formation of somites in the course of vertebrate segmentation is governed by an oscillator known as the segmentation clock, which is characterized by a period ranging from 30 min to a few hours depending on the organism. This oscillator permits the synchronized activation of segmentation genes in successive cohorts of cells in the presomitic mesoderm in response to a periodic signal emitted by the segmentation clock, thereby defining the future segments. Recent microarray experiments [Dequeant, M.L., Glynn, E., Gaudenz, K., Wahl, M., Chen, J., Mushegian, A., Pourquie, O., 2006. A complex oscillating network of signaling genes underlies the mouse segmentation clock. Science 314, 1595-1598] indicate that the Notch, Wnt and Fibroblast Growth Factor (FGF) signaling pathways are involved in the mechanism of the segmentation clock. By means of computational modeling, we investigate the conditions in which sustained oscillations occur in these three signaling pathways. First we show that negative feedback mediated by the Lunatic Fringe protein on intracellular Notch activation can give rise to periodic behavior in the Notch pathway. We then show that negative feedback exerted by Axin2 on the degradation of beta-catenin through formation of the Axin2 destruction complex can produce oscillations in the Wnt pathway. Likewise, negative feedback on FGF signaling mediated by the phosphatase product of the gene MKP3/Dusp6 can produce oscillatory gene expression in the FGF pathway. Coupling the Wnt, Notch and FGF oscillators through common intermediates can lead to synchronized oscillations in the three signaling pathways or to complex periodic behavior, depending on the relative periods of oscillations in the three pathways. The phase relationships between cycling genes in the three pathways depend on the nature of the coupling between the pathways and on their relative autonomous periods. The model provides a framework for analyzing the dynamics of the segmentation clock in terms of a network of oscillating modules involving the Wnt, Notch and FGF signaling pathways.

The complete original paper reference is cited below:

Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways, Albert Goldbeter and Olivier Pourquie, 2008, Journal of Theoretical Biology , 252, 574-585. (Full text and PDF versions of the article are available to subscribers on the journal website.) PubMed ID: 18308339

Schematic diagram representing the model for the segmentation clock based on negative feedback loops in the coupled FGF, Wnt and Notch signalling pathways.

Model Status

This CellML model runs in both OpenCell and COR to recreate the published results. The units have been checked and they are consistent. This particular version of the model describes the first control system with non-interacting oscillations.

Model Structure

ABSTRACT: THE demonstration of negative feedback control processes operating at the molecular level in cells is one of the most significant developments in modern biology. The phenomena of feedback inhibition and feedback repression, whereby enzymatic activities are controlled at the level of the enzyme and the gene, respectively, provide a firm experimental basis for the construction of dynamic models which represent the fundamental regulatory activity of cells. The behavior of these and other molecular control circuits thus constitutes the basis of cell physiology, and in effect provides the physiologist with his elementary units of function. The purpose of this paper is to illustrate the type of periodic behavior which can arise in model systems incorporating the essential control features of enzymatic regulatory processes, and to discuss the significance of oscillatory motion in relation to the organization of cellular processes in time.

The complete original paper reference is cited below:

Oscillatory behavior in enzymatic control processes, Brian C. Goodwin, 1965, Advances in Enzyme Regulation s, 3, 425-438. (An abstract and a PDF version of the article are available to subscribers on the journal website.) PubMed ID: 5861813

Schematic diagram of the first control system modelled in this study. Here Li is a genetic locus which produces messenger dbonudeic acid (mRNA) in quantities denoted by Xi. This mRNA then combines with ribosomes to form active protein-synthesizing aggregates (polysomes) designated by R, producing protein in quantity Yi. This protein assumed to be an enzyme then directs a metabolic transformation giving rise to a metabolic species M, which passes through a cellular pool, Pi. A fraction of the metabolite in the pool feeds back to the genetic locus where it serves to repress the activity of the gene, presumably in association with a macromolecule, the aporepresser.

Model Status

This CellML model runs in both OpenCell and COR to recreate the published results. The units have been checked and they are consistent. This particular version of the model describes the second control system with interactions between non-linear oscillations.

Model Structure

ABSTRACT: THE demonstration of negative feedback control processes operating at the molecular level in cells is one of the most significant developments in modern biology. The phenomena of feedback inhibition and feedback repression, whereby enzymatic activities are controlled at the level of the enzyme and the gene, respectively, provide a firm experimental basis for the construction of dynamic models which represent the fundamental regulatory activity of cells. The behavior of these and other molecular control circuits thus constitutes the basis of cell physiology, and in effect provides the physiologist with his elementary units of function. The purpose of this paper is to illustrate the type of periodic behavior which can arise in model systems incorporating the essential control features of enzymatic regulatory processes, and to discuss the significance of oscillatory motion in relation to the organization of cellular processes in time.

The complete original paper reference is cited below:

Oscillatory behavior in enzymatic control processes, Brian C. Goodwin, 1965, Advances in Enzyme Regulation , 3, 425-438. (An abstract and a PDF version of the article are available to subscribers on the journal website.) PubMed ID: 5861813

Schematic diagram of the second control system modelled in this study. It is to be expected from the complexity of intracellular processes that control circuits will interact in some manner. One type of interaction which seems very probable is for the repressor of one genetic locus to have a repressive effect on another locus. This situation can be represented schematically here.

Model Status

This CellML model runs in both OpenCell and COR, however it does not recreate the published results - the CellML model does not oscillate. The units have been checked and they are consistent. This particular version of the model describes the third control system of a circuit with limit cycles. We suspect the CellML model does not oscillate because it does not include time delays. These are mentioned in the main body of text in the published paper, but they are not descibed mathematically in the model equations.

Model Structure

ABSTRACT: THE demonstration of negative feedback control processes operating at the molecular level in cells is one of the most significant developments in modern biology. The phenomena of feedback inhibition and feedback repression, whereby enzymatic activities are controlled at the level of the enzyme and the gene, respectively, provide a firm experimental basis for the construction of dynamic models which represent the fundamental regulatory activity of cells. The behavior of these and other molecular control circuits thus constitutes the basis of cell physiology, and in effect provides the physiologist with his elementary units of function. The purpose of this paper is to illustrate the type of periodic behavior which can arise in model systems incorporating the essential control features of enzymatic regulatory processes, and to discuss the significance of oscillatory motion in relation to the organization of cellular processes in time.

The complete original paper reference is cited below:

Oscillatory behavior in enzymatic control processes, Brian C. Goodwin, 1965, Advances in Enzyme Regulation, 3, 425-438. (An abstract and a PDF version of the article are available to subscribers on the journal website.) PubMed ID: 5861813

Schematic diagram of the third control system modelled in this study: a control circuit with limit-cycle characteristics.

Model Status

This model has been built with the differential expressions in Leloup's 1998 paper for a circadian cycle in the case of continuous darkness on the system. This file is known to run in PCEnv and COR, and paramters can be changed to reproduce the diagrams in figure 2 and figure 4A-C (by changing v_sP, v_mP, and k_1). 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 by allowing the model to run till steady state to replicate figure2A. To replicate other figures, the model should be run untill they reach steady state (at least 48h)

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

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.

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

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.

Model Status

This CellML model runs in OpenCell to recreate the published results (figure 2A from the published paper, where KAC = 0.6nM). The model needs to be run for 160 hours with a step size of 0.1 hours, and MP, MB and MC are plotted. Ignore the first couple of oscillations as the model needs to stabilise. Please note that the model presented here is the extended version of the model which includes equations to define the role of REV-ERB-alpha in inhibiting the transcription of Bmal1. There are 19 equations and the parameters are listed as part of the Supplementary data set under Figure 8.

Model Structure

ABSTRACT: We present a computational model for the mammalian circadian clock based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, Clock, and Rev-Erb alpha genes. In agreement with experimental observations, the model can give rise to sustained circadian oscillations in continuous darkness, characterized by an antiphase relationship between Per/Cry/Rev-Erbalpha and Bmal1 mRNAs. Sustained oscillations correspond to the rhythms autonomously generated by suprachiasmatic nuclei. For other parameter values, damped oscillations can also be obtained in the model. These oscillations, which transform into sustained oscillations when coupled to a periodic signal, correspond to rhythms produced by peripheral tissues. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark cycles. Simulations show that the phase of the oscillations can then vary by several hours with relatively minor changes in parameter values. Such a lability of the phase could account for physiological disorders related to circadian rhythms in humans, such as advanced or delayed sleep phase syndrome, whereas the lack of entrainment by light-dark cycles can be related to the non-24h sleep-wake syndrome. The model uncovers the possible existence of multiple sources of oscillatory behavior. Thus, in conditions where the indirect negative autoregulation of Per and Cry expression is inoperative, the model indicates the possibility that sustained oscillations might still arise from the negative autoregulation of Bmal1 expression.

The original paper reference is cited below:

Toward a detailed computational model for the mammalian circadian clock, Jean-Christophe Leloup and Albert Goldbeter, 2003,PNAS, 100, 7051-7056. PubMed ID: 12775757

Model for circadian oscillations in mammals involving interlocked negative and positive regulations of Per, Cry, Bmal1, and Rev-Erb genes by their protein products.

Model Status

This CellML model runs in OpenCell to recreate the published results (figure 2C from the published paper). The model needs to be run for 160 hours with a step size of 0.1 hours, and MP, MB and MC are plotted. Ignore the first couple of oscillations as the model needs to stabilise. Also please note that in the version of the model presented here, parameter set 4 from the original paper has been used. The units have been checked and they are consistent.

Model Structure

ABSTRACT: We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to circadian rhythms autonomously generated by suprachiasmatic nuclei and by some peripheral tissues. By using different sets of parameter values producing circadian oscillations, we compare the effect of the various parameters and show that both the occurrence and the period of the oscillations are generally most sensitive to parameters related to synthesis or degradation of Bmal1 mRNA and BMAL1 protein. The mechanism of circadian oscillations relies on the formation of an inactive complex between PER and CRY and the activators CLOCK and BMAL1 that enhance Per and Cry expression. Bifurcation diagrams and computer simulations nevertheless indicate the possible existence of a second source of oscillatory behavior. Thus, sustained oscillations might arise from the sole negative autoregulation of Bmal1 expression. This second oscillatory mechanism may not be functional in physiological conditions, and its period need not necessarily be circadian. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark (LD) cycles. Long-term suppression of circadian oscillations by a single light pulse can occur in the model when a stable steady state coexists with a stable limit cycle. The phase of the oscillations upon entrainment in LD critically depends on the parameters that govern the level of CRY protein. Small changes in the parameters governing CRY levels can shift the peak in Per mRNA from the L to the D phase, or can prevent entrainment. The results are discussed in relation to physiological disorders of the sleep-wake cycle linked to perturbations of the human circadian clock, such as the familial advanced sleep phase syndrome or the non-24h sleep-wake syndrome.

Schematic diagram of the model for circadian oscillations in mammals including positive and negative feedback on the transcription of the Per, Cry and Bmal1 genes by their protein products. Per, Cry and Bmal1 are transcribed in the nucleus and are then transferred to the cytosol where their proteins are translated and the mRNAs are degraded. Light increases the rate of Per transcription.

The original paper reference is cited below:

Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms, Jean-Christophe Leloup and Albert Goldbeter, 2004, Journal of Theoretical Biology, 230, 541-562. PubMed ID: 15363675

Model Status

This CellML model runs in COR and OpenCell to reproduce the published results (figure 2a). The units are consistent. This particular version of the model has been translated from equations 1a-1j (Drosophila).

Model Structure

ABSTRACT: We examine theoretical models for circadian oscillations based on transcriptional regulation in Drosophila and Neurospora. For Drosophila, the molecular model is based on the negative feedback exerted on the expression of the per and tim genes by the complex formed between the PER and TIM proteins. For Neurospora, similarly, the model relies on the feedback exerted on the expression of the frq gene by its protein product FRQ. In both models, sustained rhythmic variations in protein and mRNA levels occur in continuous darkness, in the form of limit cycle oscillations. The effect of light on circadian rhythms is taken into account in the models by considering that it triggers degradation of the TIM protein in Drosophila, and frq transcription in Neurospora. When incorporating the control exerted by light at the molecular level, we show that the models can account for the entrainment of circadian rhythms by light-dark cycles and for the damping of the oscillations in constant light, though such damping occurs more readily in the Drosophila model. The models account for the phase shifts induced by light pulses and allow the construction of phase response curves. These compare well with experimental results obtained in Drosophila. The model for Drosophila shows that when applied at the appropriate phase, light pulses of appropriate duration and magnitude can permanently or transiently suppress circadian rhythmicity. We investigate the effects of the magnitude of light-induced changes on oscillatory behavior. Finally, we discuss the common and distinctive features of circadian oscillations in the two organisms.

The original paper reference is cited below:

Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora, Jean-Christophe Leloup, Didier Gonze, and Albert Goldbeter, 1999, Journal of Biological Rhythms, 14, 433-448. PubMed ID: 10643740

Scheme for the model for circadian oscillations in Drosophila involving negative regulation of gene expression by the PER-TIM protein complex. And beneath this is the scheme for the model for circadian rhythms in Neurospora. This model is based on negative feedback exerted by the protein FRQ on the transcription of the frq gene.

Model Status

This CellML model runs in COR and OpenCell to reproduce the published results (figure 2c). The units are consistent. This particular version of the model has been translated from equations 4a-4c (Neurospora).

Model Structure

ABSTRACT: We examine theoretical models for circadian oscillations based on transcriptional regulation in Drosophila and Neurospora. For Drosophila, the molecular model is based on the negative feedback exerted on the expression of the per and tim genes by the complex formed between the PER and TIM proteins. For Neurospora, similarly, the model relies on the feedback exerted on the expression of the frq gene by its protein product FRQ. In both models, sustained rhythmic variations in protein and mRNA levels occur in continuous darkness, in the form of limit cycle oscillations. The effect of light on circadian rhythms is taken into account in the models by considering that it triggers degradation of the TIM protein in Drosophila, and frq transcription in Neurospora. When incorporating the control exerted by light at the molecular level, we show that the models can account for the entrainment of circadian rhythms by light-dark cycles and for the damping of the oscillations in constant light, though such damping occurs more readily in the Drosophila model. The models account for the phase shifts induced by light pulses and allow the construction of phase response curves. These compare well with experimental results obtained in Drosophila. The model for Drosophila shows that when applied at the appropriate phase, light pulses of appropriate duration and magnitude can permanently or transiently suppress circadian rhythmicity. We investigate the effects of the magnitude of light-induced changes on oscillatory behavior. Finally, we discuss the common and distinctive features of circadian oscillations in the two organisms.

The original paper reference is cited below:

Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora, Jean-Christophe Leloup, Didier Gonze, and Albert Goldbeter, 1999, Journal of Biological Rhythms, 14, 433-448. PubMed ID: 10643740

Scheme for the model for circadian oscillations in Drosophila involving negative regulation of gene expression by the PER-TIM protein complex. And beneath this is the scheme for the model for circadian rhythms in Neurospora. This model is based on negative feedback exerted by the protein FRQ on the transcription of the frq gene.