goldbeter7

The SBML for this model was obtained from the BioModels database (BioModels ID: BIOMD0000000201) Biomodels notes: Reproduction of fig 4 using SBML odeSolver and xmgrace. JWS Online curation: This model was curated by reproducing the figures as described in the BioModels Notes. No additional changes were made.

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Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways.

  • Albert Goldbeter
  • Olivier Pourquié
J. Theor. Biol. 2008; 252 (3): 574-585
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.

Unit definitions have no effect on the numerical analysis of the model. It remains the responsibility of the modeler to ensure the internal numerical consistency of the model. If units are provided, however, the consistency of the model units will be checked.

Name Definition
1e-09 mole
60.0 second
0.016666666666666666 second^(-1.0)
16666666.666666666 mole^(-1.0) litre second^(-1.0)
1.6666666666666667e-11 mole litre^(-1.0) second^(-1.0)
1e-09 mole litre^(-1.0)
1.0 dimensionless
1.0 litre
Id Name Spatial dimensions Size
cytosol 3.0 1.0 litre
Id Name Initial quantity Compartment Fixed
A Axin2 protein 0.1 <substance_units>/litre cytosol
AK Axin2/Gsk3 destruction complex <assignment rule> <substance_units>/litre cytosol
B beta-catenin 0.1 <substance_units>/litre cytosol
BN nuclear beta-catenin 0.001 <substance_units>/litre cytosol
Bp phosph. beta-catenin 0.1 <substance_units>/litre cytosol
D Dsh protein 2.0 <substance_units>/litre cytosol
Dusp Dusp6 protein 0.1 <substance_units>/litre cytosol
ERKa active ERK 0.2 <substance_units>/litre cytosol
ERKi inactive ERK <assignment rule> <substance_units>/litre cytosol
ERKt ERK total 2.0 <substance_units>/litre cytosol
F Lunatic Fringe protein 0.001 <substance_units>/litre cytosol
Fgf Fgf 1.0 <substance_units>/litre cytosol
K Gsk3 3.0 <substance_units>/litre cytosol
Kt Kt 3.0 <substance_units>/litre cytosol
MAx Axin2 mRNA 0.1 <substance_units>/litre cytosol
MDusp Dusp6 mRNA 0.1 <substance_units>/litre cytosol
MF Lunatic fringe mRNA 0.1 <substance_units>/litre cytosol
N Notch protein 0.5 <substance_units>/litre cytosol
Na cytosolic NicD 0.2 <substance_units>/litre cytosol
Nan nuclear NicD 0.0 <substance_units>/litre cytosol
Rasa active Ras 0.5 <substance_units>/litre cytosol
Rasi inactive Ras <assignment rule> <substance_units>/litre cytosol
Rast Ras total 2.0 <substance_units>/litre cytosol
Xa active TF X 0.1 <substance_units>/litre cytosol
Xi inactive TF X <assignment rule> <substance_units>/litre cytosol
Xt X total 2.0 <substance_units>/litre cytosol

Initial assignments are expressions that are evaluated at time=0. It is not recommended to create initial assignments for all model entities. Restrict the use of initial assignments to cases where a value is expressed in terms of values or sizes of other model entities. Note that it is not permitted to have both an initial assignment and an assignment rule for a single model entity.

Definition
Id Name Objective coefficient Reaction Equation and Kinetic Law Flux bounds
AK_dissoc AK = A + K

theta * cytosol * (d1 * AK - a1 * A * K)
A_degradation A > ∅

theta * cytosol * vdAx * A / (KdAx + A)
A_translation ∅ > A

theta * cytosol * ksAx * MAx
BP_dephosphorylation Bp > B

theta * cytosol * VMP * Bp / (K2 + Bp)
B_degradation B > ∅

theta * cytosol * kd1 * B
B_phosphorylation B > Bp

theta * cytosol * VMK * KID / (KID + D) * B / (K1 + B) * AK / Kt
B_shuttling BN = B

theta * cytosol * (kt4 * BN - kt3 * B)
B_synth ∅ > B

theta * cytosol * vsB
Bp_degradation Bp > ∅

theta * cytosol * kd2 * Bp
Dusp_degradation Dusp > ∅

eta * cytosol * vdDusp * Dusp / (KdDusp + Dusp)
Dusp_translation ∅ > Dusp

eta * cytosol * ksDusp * MDusp
Erk_activation ∅ > ERKa

eta * cytosol * VMaErk * Rasa / Rast * ERKi / (KaErk + ERKi)
Erk_inactivation ERKa > ∅

eta * cytosol * kcDusp * Dusp * ERKa / (KdErk + ERKa)
F_degradation F > ∅

epsilon * cytosol * vdF * F / (KdF + F)
F_translation ∅ > F

epsilon * cytosol * ksF * MF
MAx_degradation MAx > ∅

theta * cytosol * vmd * MAx / (Kmd + MAx)
MAx_trans_BN ∅ > MAx

theta * cytosol * (vMB * pow(BN, n) / (pow(KaB, n) + pow(BN, n)))
MAx_trans_Xa ∅ > MAx

theta * cytosol * (vMXa * pow(Xa, m) / (pow(KaXa, m) + pow(Xa, m)))
MAx_trans_basal ∅ > MAx

theta * cytosol * v0
MDusp_degradation MDusp > ∅

eta * cytosol * VMdMDusp * MDusp / (KdMDusp + MDusp)
MDusp_transkription ∅ > MDusp

eta * cytosol * VMsMDusp * pow(Xa, q) / (pow(KaMDusp, q) + pow(Xa, q))
MF_degradation MF > ∅

epsilon * cytosol * vmF * MF / (KdMF + MF)
MF_transkription ∅ > MF

epsilon * cytosol * vsFK * pow(Nan, p) / (pow(KA, p) + pow(Nan, p))
N_activation Notch_activation N > Na

epsilon * cytosol * kc * N * pow(KIF, j) / (pow(KIF, j) + pow(F, j))
N_degradation N_degradation N > ∅

epsilon * cytosol * vdN * N / (KdN + N)
Na_degradation Na_degradation Na > ∅

epsilon * cytosol * VdNa * Na / (KdNa + Na)
Na_transport Na = Nan

epsilon * cytosol * (kt1 * Na - kt2 * Nan)
Nan_degradation Nan > ∅

epsilon * cytosol * VdNan * Nan / (KdNan + Nan)
Ras_activation ∅ > Rasa

eta * cytosol * VMaRas * pow(Fgf, r) / (pow(KaFgf, r) + pow(Fgf, r)) * Rasi / (KaRas + Rasi)
Ras_inactivation Rasa > ∅

eta * cytosol * VMdRas * Rasa / (KdRas + Rasa)
X_activation ∅ > Xa

eta * cytosol * VMaX * ERKa / ERKt * Xi / (KaX + Xi)
X_inactivation Xa > ∅

eta * cytosol * VMdX * Xa / (KdX + Xa)
n_synth Notch_synthesis ∅ > N

cytosol * epsilon * vsN

Global parameters

Id Value
K1 0.28 nanomolar
K2 0.03 nanomolar
KA 0.05 nanomolar
KID 0.5 nanomolar
KIF 0.5 nanomolar
KIG1 2.5 nanomolar
KaB 0.7 nanomolar
KaErk 0.05 nanomolar
KaFgf 0.5 nanomolar
KaMDusp 0.5 nanomolar
KaRas 0.103 nanomolar
KaX 0.05 nanomolar
KaXa 0.05 nanomolar
KdAx 0.63 nanomolar
KdDusp 0.5 nanomolar
KdErk 0.05 nanomolar
KdF 0.37 nanomolar
KdMDusp 0.5 nanomolar
KdMF 0.768 nanomolar
KdN 1.4 nanomolar
KdNa 0.001 nanomolar
KdNan 0.001 nanomolar
KdRas 0.1 nanomolar
KdX 0.05 nanomolar
Kmd 0.48 nanomolar
VMK 5.08 flux
VMP 1.0 flux
VMaErk 3.3 flux
VMaRas 4.968 flux
VMaX 1.6 flux
VMdMDusp 0.5 flux
VMdRas 0.41 flux
VMdX 0.5 flux
VMsMDusp 0.9 flux
VdNa 0.01 flux
VdNan 0.1 flux
a1 1.8 second_order_rate_constant
d1 0.1 first_order_rate_constant
epsilon 0.3 dimensionless
eta 0.3 dimensionless
j 2.0 dimensionless
kc 3.45 first_order_rate_constant
kcDusp 1.35 first_order_rate_constant
kd1 0.0 first_order_rate_constant
kd2 7.062 first_order_rate_constant
ksAx 0.02 first_order_rate_constant
ksDusp 0.5 first_order_rate_constant
ksF 0.3 first_order_rate_constant
kt1 0.1 first_order_rate_constant
kt2 0.1 first_order_rate_constant
kt3 0.7 first_order_rate_constant
kt4 1.5 first_order_rate_constant
m 2.0 dimensionless
n 2.0 dimensionless
p 2.0 dimensionless
q 2.0 dimensionless
r 2.0 dimensionless
theta 1.5 dimensionless
v0 0.06 flux
vMB 1.64 flux
vMXa 0.5 flux
vdAx 0.6 flux
vdDusp 2.0 flux
vdF 0.39 flux
vdN 2.82 flux
vmF 1.92 flux
vmd 0.8 flux
vsB 0.087 flux
vsF 3.0 flux
vsFK <assignment rule> flux
vsN 0.23 flux

Local parameters

Id Value Reaction

Assignment rules

Definition
Xi = Xt - Xa
ERKi = ERKt - ERKa
Rasi = Rast - Rasa
AK = Kt - K
vsFK = vsF * (KIG1 / (KIG1 + K))

Rate rules

Definition

Algebraic rules

Definition
Trigger Assignments