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 shows 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

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

Regulated Transcription

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 production

• Transcription of LacI gene mRNA

• Translation of LacI Protein

Complex formation between LacI and Lactose

E7 production

• Transcription of E7 gene mRNA

• Translation of E7 Protein

Variables of Iron and Ai2 controlled production of Lysis

Variables

Phosphorylation of Ai-2

LsrR production

• Transcription of LsrR gene mRNA

• Translation of LsrR protein

Complex formation between Ai-2-phos and LsrR

The hard part here is to determine what is the initial concentration of the protein LsrR and Ai-2-phos. A way out of this mess is to simplify the equation even more.

AND GATE odes

• Transcription of SupD gene mRNA

• Transcription of T7ptag gene mRNA

• Complex formation between tRNA and mRNA

• Translation of t7 protein

Lysis production

• Transcription of Lysis gene mRNA

• Translation of Lysis protein