From 2008.igem.org

ODEs used in modeling

In this section, we will have a short discussion on the ODEs used for Deterministic Modeling in our iGem project.

For the modeling exercise, we use mainly simple linear ODEs to describe the system we have at hand. These ODEs are used to describe processes such as constitutive transcription/ translation etc. Nonlinear equations are used for more complex processes such as regulated transcription. The choice of these equations are in hope that they can describe the system to a relative degree of accuracy and yet be simple enough to be solved. This is very much an engineering compromise between a rigorous description and getting an answer that is good enough which is required.

Our systems are based on the assumptions that Lumped parameter models are sufficient to describe them.
Lumped parameter models are those in which spatial variations are ignored and the dependent variables are uniform throughout the whole system. The only variations are those with respect to time leading to mathematical description in terms of the ODEs used.

The following equations show the break down of the different equations that will be used in this modeling exercise. By understanding this section, it would make the understanding of the system of ODEs used in our iGEm modeling exercise.

Constant synthesis & Linear Synthesis

Simple ode to describe constant synthesis
Gives an explicit analytical solution
Unique solution once a IC is posed

Linear Degradation

Rate of degradation is proportional to how much of the molecule is present
Gives an explicit analytical solution
Constant half life

Simple Forward Reaction

[C] : Complex
kc : Rate constant of complex formation

This equation ignores the fact that dissociation of the complex occurs. We can do so if the dissociation is much slower than the formation.

Single solvable equation for the unknown C
Simple, unique solution available with I.C

Phosphorylation and Dephosphorylation

Assumptions:

Linear kinetic rate laws apply only if XT is much less than the Michaelis constants of both kinase and phosphotase

XT :	total cost of X protein in phosphorylated and unphosphorylated form
S :	protein kinase concentration
k2 :	accounts for protein phosphotase

Modeled after simple linear kinetics
Gives a hyperbolic signal response curve when X plotted vs S

Regulated Transcription

[P]:	Protein Formed
µ:	Repression, µ=0; Activation, µ=1
K:	Hill Constant Value of input that gives 50% response
n:	Hill coefficient Slope of signal-response curve at this input signal
d:	degradation of protein
k1:	basal gene expression
k:	signal-dependent gene expression
a:	correlation between k1 and k, 0<a<1

This ODE attempts to capture characteristics of the mRNA dynamics
For our modeling, all our detection systems activates some form of transcription. Therefore µ=1 in all cases for our modeling exercise.

ODE system used in model

The system can be viewed as two parts. The first part comprises of lactose induced production of colicin E7 and the immunity protein. The second part comprises of a detection mechanism that produces the lysis protein upon the detection of both Iron ions and Ai-2 ( Autoinducer 2).

Lactose controlled production of E7 + Imm

Variables

LacI =	A
Lactose =	B
E7 =	C

LacI production

Transcription of LacI gene mRNA

[MA]:	LacI mRNA concentration
k1A:	kinetic constant of transcription
U:	system input
d1A:	degradation constant for mRNA

Translation of LacI Protein

[PA]:	Protein concentration
k2A:	kinetic constant of translation
d2A:	degradation constant for Protein

Complex formation between LacI and Lactose

[ComAB] :	Complex of LacI and Lactose
K3AB :	Rate constant of complex formation
[PA]o :	Protein concentration of LacI at the time of Lactose addition
[B]o :	Initial concentration of Lactose

E7 production

Transcription of E7 gene mRNA

[MC]:	E7 mRNA Concentration
k1C:	kinetic constant of transcription
a :	constitutive portion , 0<a<1
KC:	Hill constant
nC:	Hill coefficient
d1C:	degradation constant for mRNA

Translation of E7 Protein

[PC] :	Protein concentration
k2C :	kinetic constant of translation
d2C :	degradation constant for Protein

Variables of Iron and Ai2 controlled production of Lysis