SBML Model Report
Model name: “Yildirim2003 Lac Operon”
2LATEX
May 6, 2016
1 General Overview
This is a document in SBML Level 2 Version 4 format. This model was created by the following
three authors: Harish Dharuri1 , Adam Halasz2 and Vijayalakshmi Chelliah3 at June 21st 2006
at 2:47 p. m. and last time modified at February twelveth 2014 at 4:36 p. m. Table 1 gives an
overview of the quantities of all components of this model.
Table 1: Number of components in this model, which are described in the following sections.
Element
compartment types
species types
events
reactions
global parameters
rules
Quantity
0
0
0
16
23
0
Element
compartments
species
constraints
function definitions
unit definitions
initial assignments
Quantity
1
9
0
0
2
0
Model Notes
This a model from the article:
Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data.
Yildirim N, Mackey MC Biophys. J. 2003 12719218 ,
Abstract:
1 California
Institute of Technology, [email protected]
Engineering, [email protected]
3 EMBL-EBI, [email protected]
2 Penn
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1
A mathematical model for the regulation of induction in the lac operon in Escherichia coli is
presented. This model takes into account the dynamics of the permease facilitating the internalization of external lactose; internal lactose; beta-galactosidase, which is involved in the
conversion of lactose to allolactose, glucose and galactose; the allolactose interactions with the
lac repressor; and mRNA. The final model consists of five nonlinear differential delay equations
with delays due to the transcription and translation process. We have paid particular attention
to the estimation of the parameters in the model. We have tested our model against two sets of
beta-galactosidase activity versus time data, as well as a set of data on beta-galactosidase activity
during periodic phosphate feeding. In all three cases we find excellent agreement between the
data and the model predictions. Analytical and numerical studies also indicate that for physiologically realistic values of the external lactose and the bacterial growth rate, a regime exists
where there may be bistable steady-state behavior, and that this corresponds to a cusp bifurcation
in the model dynamics.
The model reproduces the time profile of beta-galactosidase activity as shown in Fig 3 of the
paper. The delay functions for transcription (M) and translation (B and P) have been implemented by introducing intermediates ( I1, I2 and I3) in the reaction scheme which then give
their respective products (I1-> M, I2 ->B and I3 ->P) after an appropriate length of time. The
steady state values, attained upon simulation of model equations, for Allolactose (A), mRNA
(M), beta-galactosidase (B), Lactose (L), and Permease (P) match with those predicted by the
paper. The model was successfully tested on Jarnac, MathSBML and COPASI
This model originates from BioModels Database: A Database of Annotated Published Models
(http://www.ebi.ac.uk/biomodels/). It is copyright (c) 2005-2010 The BioModels.net Team.
For more information see the terms of use .
To cite BioModels Database, please use: Li C, Donizelli M, Rodriguez N, Dharuri H, Endler L,
Chelliah V, Li L, He E, Henry A, Stefan MI, Snoep JL, Hucka M, Le Novre N, Laibe C (2010)
BioModels Database: An enhanced, curated and annotated resource for published quantitative
kinetic models. BMC Syst Biol., 4:92.
2 Unit Definitions
This is an overview of five unit definitions of which three are predefined by SBML and not
mentioned in the model.
2.1 Unit time
Name minute
Definition 60 s
2.2 Unit substance
Name millimoles
Definition mmol
2
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2.3 Unit volume
Notes Litre is the predefined SBML unit for volume.
Definition l
2.4 Unit area
Notes Square metre is the predefined SBML unit for area since SBML Level 2 Version 1.
Definition m2
2.5 Unit length
Notes Metre is the predefined SBML unit for length since SBML Level 2 Version 1.
Definition m
3 Compartment
This model contains one compartment.
Id
Name
Table 2: Properties of all compartments.
SBO
Spatial
Size Unit Constant
Dimensions
cell
3
1
litre
3.1 Compartment cell
Outside
2
3
This is a three dimensional compartment with a constant size of one litre.
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3
4
4 Species
This model contains nine species. The boundary condition of one of these species is set to true so that this species’ amount cannot be
changed by any reaction. Section 7 provides further details and the derived rates of change of each species.
Produced by SBML2LATEX
Id
Name
M
B
A
L
P
I1
I2
I3
Le
mRNA
Betagalactosidase
allolactose
lactose internal
permease
PartialmRNA
PartialBetagalactosidase
PartialPermease
External Lactose
Table 3: Properties of each species.
Compartment
cell
cell
cell
cell
cell
cell
cell
cell
cell
Derived Unit
mmol · l−1
mmol · l−1
mmol · l−1
mmol · l−1
mmol · l−1
mmol · l−1
mmol · l−1
mmol · l−1
mmol · l−1
Constant
Boundary
Condition
2
–
2
–
2
–
2
–
2
–
2
–
2
–
2
–
2
3
2
–
2
–
2
–
2
–
2
–
2
–
2
–
2
–
2
3
5 Parameters
This model contains 23 global parameters.
Id
mu
gamma M
gamma B
gamma A
gamma 0
K
alpha M
tau B
alpha A
K L1
alpha B
KA
beta A
tau M
KL
gamma L
gamma P
alpha L
alpha P
tau P
beta L1
K Le
K1
Name
Table 4: Properties of each parameter.
SBO
Value
Unit
0.023
0.411
8.33 · 10−4
0.520
7.25 · 10−7
7200.000
9.97 · 10−4
2.000
17600.000
1.810
0.017
1.950
21500.000
0.100
0.970
0.000
0.650
2880.000
10.000
0.830
2650.000
0.260
25200.000
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Constant
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
5
6
6 Reactions
This model contains 16 reactions. All reactions are listed in the following table and are subsequently described in detail. If a reaction is
affected by a modifier, the identifier of this species is written above the reaction arrow.
No¯
Table 5: Overview of all reactions
Reaction Equation
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Id
Name
1
2
3
r m1
r m2
r m3 i1
Basal mRNA Synthesis
mRNA Degradation
allolactose controlled mRNA synthesis
4
r i1
5
6
r b1
r b2 i2
allolactose controlled partial mRNAsynthesis
Beta galactosidase Degradation
Beta galactosidase synthesis
7
8
r i2
r a1
Partial Beta galactosidase synthesis
Basal Allolactose Degradation
9
r a2
Betagalactosidase mediated AllolactoseDegradation
10
11
r a3 l1
r l2
Beta galactosidase reaction
lactose degradation
12
r l3
Lactose transport out
13
14
15
r l4
r p1
r p2 i3
Lactose transport in
permease degradation
permease synthesis
16
r i3
partial permease synthesis
0/ −→ M
M −→ 0/
I1 −→ M
A
0/ −
→ I1
B −→ 0/
I2 −→ B
M
0/ −
→ I2
A −→ 0/
B
A−
→ 0/
B
L−
→A
L −→ 0/
P
L−
→ 0/
P, L e
0/ −−−→ L
P −→ 0/
I3 −→ P
M
0/ −
→ I3
SBO
6.1 Reaction r m1
This is an irreversible reaction of no reactant forming one product.
Name Basal mRNA Synthesis
Reaction equation
0/ −→ M
(1)
Product
Table 6: Properties of each product.
Id Name
SBO
M
mRNA
Kinetic Law
Derived unit contains undeclared units
v1 = vol (cell) · gamma 0
(2)
6.2 Reaction r m2
This is an irreversible reaction of one reactant forming no product.
Name mRNA Degradation
Reaction equation
M −→ 0/
(3)
Reactant
Table 7: Properties of each reactant.
Id Name
SBO
M
mRNA
Kinetic Law
Derived unit contains undeclared units
v2 = vol (cell) · [M] · (gamma M + mu)
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(4)
7
6.3 Reaction r m3 i1
This is an irreversible reaction of one reactant forming one product.
Name allolactose controlled mRNA synthesis
Reaction equation
I1 −→ M
(5)
Reactant
Table 8: Properties of each reactant.
Id Name
SBO
PartialmRNA
I1
Product
Table 9: Properties of each product.
Id Name
SBO
M
mRNA
Kinetic Law
Derived unit contains undeclared units
v3 =
vol (cell) · [I1]
tau M
(6)
6.4 Reaction r i1
This is an irreversible reaction of no reactant forming one product influenced by one modifier.
Name allolactose controlled partial mRNA synthesis
Reaction equation
A
0/ −
→ I1
Modifier
8
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(7)
Table 10: Properties of each modifier.
Id Name
SBO
allolactose
A
Product
Table 11: Properties of each product.
Id Name
SBO
PartialmRNA
I1
Kinetic Law
Derived unit contains undeclared units
v4 = vol (cell) · alpha M ·
K 1 · exp (mu · tau M · −2) · [A]2 + 1
K + K 1 · exp (−2 · mu · tau M) · [A]2
(8)
6.5 Reaction r b1
This is an irreversible reaction of one reactant forming no product.
Name Beta galactosidase Degradation
Reaction equation
B −→ 0/
(9)
Reactant
Table 12: Properties of each reactant.
Id Name
SBO
B
Betagalactosidase
Kinetic Law
Derived unit contains undeclared units
v5 = vol (cell) · [B] · (gamma B + mu)
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(10)
9
6.6 Reaction r b2 i2
This is an irreversible reaction of one reactant forming one product.
Name Beta galactosidase synthesis
Reaction equation
I2 −→ B
(11)
Reactant
Table 13: Properties of each reactant.
Id Name
SBO
PartialBetagalactosidase
I2
Product
Table 14: Properties of each product.
Id Name
SBO
B
Betagalactosidase
Kinetic Law
Derived unit contains undeclared units
v6 =
vol (cell) · [I2]
tau B
(12)
6.7 Reaction r i2
This is an irreversible reaction of no reactant forming one product influenced by one modifier.
Name Partial Beta galactosidase synthesis
Reaction equation
M
0/ −
→ I2
Modifier
10
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(13)
Table 15: Properties of each modifier.
Id Name
SBO
M
mRNA
Product
Table 16: Properties of each product.
Id Name
SBO
I2
PartialBetagalactosidase
Kinetic Law
Derived unit contains undeclared units
v7 = vol (cell) · alpha B · [M] · exp ((mu · tau B))
(14)
6.8 Reaction r a1
This is an irreversible reaction of one reactant forming no product.
Name Basal Allolactose Degradation
Reaction equation
A −→ 0/
(15)
Reactant
Table 17: Properties of each reactant.
Id Name
SBO
A
allolactose
Kinetic Law
Derived unit contains undeclared units
v8 = vol (cell) · [A] · (gamma A + mu)
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(16)
11
6.9 Reaction r a2
This is an irreversible reaction of one reactant forming no product influenced by one modifier.
Name Betagalactosidase mediated Allolactose Degradation
Reaction equation
B
A−
→ 0/
(17)
Reactant
Table 18: Properties of each reactant.
Id Name
SBO
A
allolactose
Modifier
Table 19: Properties of each modifier.
Id Name
SBO
Betagalactosidase
B
Kinetic Law
Derived unit contains undeclared units
v9 = vol (cell) · beta A · [B] ·
[A]
K A + [A]
(18)
6.10 Reaction r a3 l1
This is an irreversible reaction of one reactant forming one product influenced by one modifier.
Name Beta galactosidase reaction
Reaction equation
B
L−
→A
Reactant
12
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(19)
Table 20: Properties of each reactant.
Id Name
SBO
lactose internal
L
Modifier
Table 21: Properties of each modifier.
Id Name
SBO
B
Betagalactosidase
Product
Table 22: Properties of each product.
Id Name
SBO
allolactose
A
Kinetic Law
Derived unit contains undeclared units
v10 = vol (cell) · alpha A · [B] ·
[L]
K L + [L]
(20)
6.11 Reaction r l2
This is an irreversible reaction of one reactant forming no product.
Name lactose degradation
Reaction equation
L −→ 0/
(21)
Reactant
Table 23: Properties of each reactant.
Id Name
SBO
L
lactose internal
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13
Kinetic Law
Derived unit contains undeclared units
v11 = vol (cell) · [L] · (gamma L + mu)
(22)
6.12 Reaction r l3
This is an irreversible reaction of one reactant forming no product influenced by one modifier.
Name Lactose transport out
Reaction equation
P
L−
→ 0/
(23)
Reactant
Table 24: Properties of each reactant.
Id Name
SBO
lactose internal
L
Modifier
Table 25: Properties of each modifier.
Id Name
SBO
P
permease
Kinetic Law
Derived unit contains undeclared units
v12 = vol (cell) · beta L1 · [P] ·
[L]
K L1 + [L]
(24)
6.13 Reaction r l4
This is an irreversible reaction of no reactant forming one product influenced by two modifiers.
Name Lactose transport in
14
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Reaction equation
P, L e
0/ −−−→ L
(25)
Modifiers
Table 26: Properties of each modifier.
Id
Name
SBO
permease
External Lactose
P
Le
Product
Table 27: Properties of each product.
Id Name
SBO
lactose internal
L
Kinetic Law
Derived unit contains undeclared units
v13 = vol (cell) · alpha L · [P] ·
[L e]
K Le + [L e]
(26)
6.14 Reaction r p1
This is an irreversible reaction of one reactant forming no product.
Name permease degradation
Reaction equation
P −→ 0/
(27)
Reactant
Table 28: Properties of each reactant.
Id Name
SBO
P
permease
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15
Kinetic Law
Derived unit contains undeclared units
v14 = vol (cell) · [P] · (gamma P + mu)
(28)
6.15 Reaction r p2 i3
This is an irreversible reaction of one reactant forming one product.
Name permease synthesis
Reaction equation
I3 −→ P
(29)
Reactant
Table 29: Properties of each reactant.
Id Name
SBO
PartialPermease
I3
Product
Table 30: Properties of each product.
Id Name
SBO
P
permease
Kinetic Law
Derived unit contains undeclared units
v15 =
vol (cell) · [I3]
tau B + tau P
(30)
6.16 Reaction r i3
This is an irreversible reaction of no reactant forming one product influenced by one modifier.
Name partial permease synthesis
16
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Reaction equation
M
0/ −
→ I3
(31)
Modifier
Table 31: Properties of each modifier.
Id Name
SBO
M
mRNA
Product
Table 32: Properties of each product.
Id Name
SBO
I3
PartialPermease
Kinetic Law
Derived unit contains undeclared units
v16 = vol (cell) · alpha P · [M] · exp (−1 · mu · (tau B + tau P))
(32)
7 Derived Rate Equations
When interpreted as an ordinary differential equation framework, this model implies the following set of equations for the rates of change of each species.
Identifiers for kinetic laws highlighted in gray cannot be verified to evaluate to units of
SBML substance per time. As a result, some SBML interpreters may not be able to verify the
consistency of the units on quantities in the model. Please check if
• parameters without an unit definition are involved or
• volume correction is necessary because the hasOnlySubstanceUnits flag may be set to
false and spacialDimensions> 0 for certain species.
7.1 Species M
Name mRNA
Initial concentration 6.26 · 10−4 mmol · l−1
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17
This species takes part in five reactions (as a reactant in r m2 and as a product in r m1, r m3 i1
and as a modifier in r i2, r i3).
d
M = v1 + v3 − v2
dt
(33)
7.2 Species B
Name Betagalactosidase
Initial concentration 0 mmol · l−1
This species takes part in four reactions (as a reactant in r b1 and as a product in r b2 i2 and
as a modifier in r a2, r a3 l1).
d
B = v6 − v5
dt
(34)
7.3 Species A
Name allolactose
Initial concentration 0.038 mmol · l−1
This species takes part in four reactions (as a reactant in r a1, r a2 and as a product in r a3 l1
and as a modifier in r i1).
d
A = v10 − v8 − v9
dt
(35)
7.4 Species L
Name lactose internal
Initial concentration 0.372 mmol · l−1
This species takes part in four reactions (as a reactant in r a3 l1, r l2, r l3 and as a product
in r l4).
d
L = v13 − v10 − v11 − v12
dt
(36)
7.5 Species P
Name permease
Initial concentration 0.0149 mmol · l−1
This species takes part in four reactions (as a reactant in r p1 and as a product in r p2 i3 and
as a modifier in r l3, r l4).
d
P = v15 − v14
dt
18
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(37)
7.6 Species I1
Name PartialmRNA
Initial concentration 0 mmol · l−1
This species takes part in two reactions (as a reactant in r m3 i1 and as a product in r i1).
d
I1 = v4 − v3
dt
(38)
7.7 Species I2
Name PartialBetagalactosidase
Initial concentration 0 mmol · l−1
This species takes part in two reactions (as a reactant in r b2 i2 and as a product in r i2).
d
I2 = v7 − v6
dt
(39)
7.8 Species I3
Name PartialPermease
Initial concentration 0 mmol · l−1
This species takes part in two reactions (as a reactant in r p2 i3 and as a product in r i3).
d
I3 = v16 − v15
dt
(40)
7.9 Species L e
Name External Lactose
Initial concentration 0.08 mmol · l−1
This species takes part in one reaction (as a modifier in r l4), which does not influence its rate
of change because this constant species is on the boundary of the reaction system:
d
Le=0
dt
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(41)
19
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a
a
SBML2LATEX was developed by Andreas Dräger , Hannes Planatscher , Dieudonné M Wouamba , Adrian
Schrödera , Michael Huckab , Lukas Endlerc , Martin Golebiewskid and Andreas Zella . Please see http:
//www.ra.cs.uni-tuebingen.de/software/SBML2LaTeX for more information.
a Center
for Bioinformatics Tübingen (ZBIT), Germany
Institute of Technology, Beckman Institute BNMC, Pasadena, United States
c European Bioinformatics Institute, Wellcome Trust Genome Campus, Hinxton, United Kingdom
d EML Research gGmbH, Heidelberg, Germany
b California
20
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