Model : BIOMD0000000012

Creator
Nicolas, Le Novère, EMBL-EBI, lenov@ebi.ac.uk
Vijayalakshmi, Chelliah, EMBL-EBI, viji@ebi.ac.uk
Lukas, Endler, EMBL-EBI, lukas@ebi.ac.uk
Nick, Juty, EMBL-EBI, juty@ebi.ac.uk
Bruce, Shapiro, Jet Propulsion Laboratory, bshapiro@caltech.edu
Created: 2009-01-20 14:03
Modified: 2013-07-10 10:59

id BIOMD0000000012 _000001
nameElowitz2000 - Repressilator
time
substance
extent
volume
area
length
L2V3 layout-V1 render-V1

BQM_IS
MODEL6615351360
BQM_IS
BIOMD0000000012
BQM_IS_DESCRIBED_BY
10659856
BQB_HAS_TAXON
562
BQB_IS_VERSION_OF
GO:0040029
Elowitz2000 - Repressilator

This model describes the deterministic version of the repressilator system.

The authors of this model (see reference) use three transcriptional repressor systems that are not part of any natural biological clock to build an oscillating network that they called the repressilator. The model system was induced in Escherichia coli.

In this system, LacI (variable X is the mRNA, variable PX is the protein) inhibits the tetracycline-resistance transposon tetR (Y, PY describe mRNA and protein). Protein tetR inhibits the gene Cl from phage Lambda (Z, PZ: mRNA, protein),and protein Cl inhibits lacI expression. With the appropriate parameter values this system oscillates.

This model is described in the article:

Elowitz MB, Leibler S.
Nature. 2000 Jan; 403(6767):335-338

Abstract:

Networks of interacting biomolecules carry out many essential functions in living cells, but the 'design principles' underlying the functioning of such intracellular networks remain poorly understood, despite intensive efforts including quantitative analysis of relatively simple systems. Here we present a complementary approach to this problem: the design and construction of a synthetic network to implement a particular function. We used three transcriptional repressor systems that are not part of any natural biological clock to build an oscillating network, termed the repressilator, in Escherichia coli. The network periodically induces the synthesis of green fluorescent protein as a readout of its state in individual cells. The resulting oscillations, with typical periods of hours, are slower than the cell-division cycle, so the state of the oscillator has to be transmitted from generation to generation. This artificial clock displays noisy behaviour, possibly because of stochastic fluctuations of its components. Such 'rational network design may lead both to the engineering of new cellular behaviours and to an improved understanding of naturally occurring networks.

The model is based upon the equations in Box 1 of the paper; however, these equations as printed are dimensionless, and the correct dimensions have been returned to the equations, and the parameters set to reproduce Figure 1C (left).

The original model was generated by B.E. Shapiro using Cellerator version 1.0 update 2.1127 using Mathematica 4.2 for Mac OS X (June 4, 2002), November 27, 2002 12:15:32, using (PowerMac,PowerPC, Mac OS X,MacOSX,Darwin).

Nicolas Le Novere provided a corrected version generated by SBMLeditor on Sun Aug 20 00:44:05 BST 2006. This removed the EmptySet species. Ran fine on COPASI 4.0 build 18.

Bruce Shapiro revised the model with SBMLeditor on 23 October 2006 20:39 PST. This defines default units and correct reactions. The original Cellerator reactions while being mathematically correct did not accurately reflect the intent of the authors. The original notes were mostly removed because they were mostly incorrect in the revised version. Tested with MathSBML 2.6.0.

Nicolas Le Novere changed the volume to 1 cubic micrometre, to allow for stochastic simulation.

Changed by Lukas Endler to use the average livetime of mRNA instead of its halflife and a corrected value of alpha and alpha0.

Moreover, the equations used in this model were clarified, cf. below.

The equations given in box 1 of the original publication are rescaled in three respects (lowercase letters denote the rescaled, uppercase letters the unscaled number of molecules per cell):

  • the time is rescaled to the average mRNA lifetime, t_ave: τ = t/t_ave
  • the mRNA concentration is rescaled to the translation efficiency eff: m = M/eff
  • the protein concentration is rescaled to Km: p = P/Km

α in the equations should be in units of rescaled proteins per promotor and cell, and β is the ratio of the protein to the mRNA decay rates or the ratio of the mRNA to the protein halflife.

In this version of the model α and β are calculated correspondingly to the article, while p and m where just replaced by P/Km resp. M/eff and all equations multiplied by 1/t_ave . Also, to make the equations easier to read, commonly used variables derived from the parameters given in the article by simple rules were introduced.

The parameters given in the article were:

promotor strength (repressed) ( tps_repr ): 5*10 -4 transcripts/(promotor*s)
promotor strength (full) ( tps_active ): 0.5 transcripts/(promotor*s)
mRNA half life, τ 1/2,mRNA : 2 min
protein half life, τ 1/2,prot : 10 min
K M : 40 monomers/cell
Hill coefficient n: 2

From these the following constants can be derived:

average mRNA lifetime ( t_ave ): τ 1/2,mRNA /ln(2) = 2.89 min
mRNA decay rate ( kd_mRNA ): ln(2)/ τ 1/2,mRNA = 0.347 min -1
protein decay rate ( kd_prot ): ln(2)/ τ 1/2,prot
transcription rate ( a_tr ): tps_active*60 = 29.97 transcripts/min
transcription rate (repressed) ( a0_tr ): tps_repr*60 = 0.03 transcripts/min
translation rate ( k_tl ): eff*kd_mRNA = 6.93 proteins/(mRNA*min)
α : a_tr*eff*τ 1/2,prot /(ln(2)*K M ) = 216.4 proteins/(promotor*cell*Km)
α 0 : a0_tr*eff*τ 1/2,prot /(ln(2)*K M ) = 0.2164 proteins/(promotor*cell*Km)
β : k_dp/k_dm = 0.2

Annotation by the Kinetic Simulation Algorithm Ontology (KiSAO):

To reproduce the simulations run published by the authors, the model has to be simulated with any of two different approaches. First, one could use a deterministic method ( KISAO_0000035 ) with continuous variables ( KISAO_0000018 ). One sample algorithm to use is the CVODE solver ( KISAO_0000019 ). Second, one could simulate the system using Gillespie's direct method ( KISAO_0000029 ), which is a stochastic method ( KISAO_0000036 ) supporting adaptive timesteps ( KISAO_0000041 ) and using discrete variables ( KISAO_0000016 ).

To the extent possible under law, all copyright and related or neighbouring rights to this encoded model have been dedicated to the public domain worldwide. Please refer to CC0 Public Domain Dedication for more information.


UnitDefinitions [3] name units sbo cvterm
volume metaid_0000029 cubic microns 10 15 litre
substance metaid_1000000 item item
time metaid_0000030 minute 60 s

Compartments [1] name size constant spatial dimensions units derived units sbo cvterm
cell _000002 1.0 3 10 15 litre

Species [6] name compartment hasOnlySubstanceUnits boundaryCondition constant initialAmount initialConcentration conversionFactor units substanceUnits derivedUnits sbo cvterm
PX PX LacI protein cell 0.0 item
BQB_IS
P03023
PY PY TetR protein cell 0.0 item
BQB_IS
P04483
PZ PZ cI protein cell 0.0 item
BQB_IS
P03034
X _905769 LacI mRNA cell 0.0 item
BQB_IS_VERSION_OF
CHEBI:33699; C00046
BQB_ENCODES
P03023
Y _905781 TetR mRNA cell 20.0 item
BQB_IS_VERSION_OF
CHEBI:33699; C00046
BQB_ENCODES
P04483
Z _905802 cI mRNA cell 0.0 item
BQB_IS_VERSION_OF
CHEBI:33699; C00046
BQB_ENCODES
P03034

Parameters [16] name constant value unit derived unit sbo cvterm
beta metaid_0000022 beta 0.2 None
alpha0 metaid_0000023 alpha0 0.2164 None
alpha metaid_0000024 alpha 216.404 None
eff metaid_0000025 translation efficiency 20.0 None
n metaid_0000026 n 2.0 None
KM metaid_0000027 KM 40.0 None
tau_mRNA metaid_0000028 mRNA half life 2.0 None
tau_prot metaid_0000128 protein half life 10.0 None
t_ave metaid_0000032 average mRNA life time tau_mRNA 2 None
kd_mRNA metaid_0000132 kd_mRNA 2 tau_mRNA None
kd_prot metaid_0000133 kd_prot 2 tau_prot None
k_tl metaid_0000233 k_tl eff t_ave None
a_tr metaid_0900235 a_tr ps_a ps_0 60 None
ps_a metaid_0800235 tps_active 0.5 None
ps_0 metaid_0500235 tps_repr 0.0005 None
a0_tr metaid_0000234 a0_tr ps_0 60 None

Rules [9]   assignment name derived units sbo cvterm
t_ave metaid_0500035 = tau_mRNA 2 None
beta metaid_0240045 = tau_mRNA tau_prot None
k_tl metaid_0400235 = eff t_ave None
a_tr metaid_1000237 = ps_a ps_0 60 None
a0_tr metaid_0100236 = ps_0 60 None
kd_prot metaid_0010335 = 2 tau_prot None
kd_mRNA metaid_0020435 = 2 tau_mRNA None
alpha metaid_0230035 = a_tr eff tau_prot 2 KM None
alpha0 metaid_0240035 = a0_tr eff tau_prot 2 KM None

Reactions [12] name equation modifiers kinetic law derived units sbo cvterm
Reaction1 _905823 degradation of LacI transcripts
X ➞ kd_mRNA X item
BQB_IS_VERSION_OF
GO:0006402
Reaction2 _905842 degradation of TetR transcripts
Y ➞ kd_mRNA Y item
BQB_IS_VERSION_OF
GO:0006402
Reaction3 _905862 degradation of CI transcripts
Z ➞ kd_mRNA Z item
BQB_IS_VERSION_OF
GO:0006402
Reaction4 _905882 translation of LacI
➞ PX X k_tl X item
BQB_IS_VERSION_OF
GO:0006412
Reaction5 _905903 translation of TetR
➞ PY Y k_tl Y item
BQB_IS_VERSION_OF
GO:0006412
Reaction6 _905923 translation of CI
➞ PZ Z k_tl Z item
BQB_IS_VERSION_OF
GO:0006412
Reaction7 _905943 degradation of LacI
PX ➞ kd_prot PX item
BQB_IS_VERSION_OF
GO:0030163
Reaction8 _905962 degradation of TetR
PY ➞ kd_prot PY item
BQB_IS_VERSION_OF
GO:0030163
Reaction9 _905982 degradation of CI
PZ ➞ kd_prot PZ item
BQB_IS_VERSION_OF
GO:0030163
Reaction10 _906002 transcription of LacI
➞ X PZ a0_tr a_tr KM n KM n PZ n 1 item
BQB_IS_VERSION_OF
GO:0006351
Reaction11 _906022 transcription of TetR
➞ Y PX a0_tr a_tr KM n KM n PX n 1 item
BQB_IS_VERSION_OF
GO:0006351
Reaction12 _906042 transcription of CI
➞ Z PY a0_tr a_tr KM n KM n PY n 1 item
BQB_IS_VERSION_OF
GO:0006351