IGEM:IMPERIAL/2007/Dry Lab/Modelling/ID: Difference between revisions
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==Our models== | ==Our models== | ||
* '''Introduction''' | |||
We can condition the system in various manners, but for the purposes of our project, Infector Detector, we will seek a formulation which is valid for both constructs considered. Our system can thus be modellied by the following Dynamical System: | |||
<br><br> | |||
<br><br> | |||
The system of equations for the two constructs varies strictly with respect to the value of the parameter k1. Construct 1 possesses a non-zero k1 rate constant, whereas for construct 2, a zero value is assumed. | |||
===Model 1: Steady-state is attained; limitless energy supply === | ===Model 1: Steady-state is attained; limitless energy supply === | ||
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===Model 2: Equations developed through steady-state analysis; however due to limited energy supply, we operate in the transient regime=== | ===Model 2: Equations developed through steady-state analysis; however due to limited energy supply, we operate in the transient regime=== | ||
<math>\frac{d[LuxR]}{dt} = k_1\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) + k_3[A] - k_2[LuxR][AHL]- \delta_{LuxR}[LuxR]</math> | |||
<br> | |||
<math>\frac{d[AHL]}{dt} = k_3[A] - k_2[LuxR][AHL]- \delta_{AHL}[AHL]</math> | |||
<math>\frac{d[A]}{dt} = -k_3[A] + k_2[LuxR][AHL]- k_4[A][P] + k_5[AP]</math> | |||
<math>\frac{d[P]}{dt} = -k_4[A][P] + k_5[P]</math> | |||
<math>\frac{d[AP]}{dt} = k_4[A][P] - k_5[AP]</math> | |||
<math>\frac{d[GFP]}{dt} = k_6[AP]\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) - \delta_{GFP}[GFP]</math> | |||
<math>\frac{d[E]}{dt} = -\alpha_{1}k_1\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) - \alpha_{2}k_6[AP]\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg)</math> | |||
<br> | |||
where:<br><br> | |||
[A] represents the concentration of AHL-LuxR complex<br> | |||
[P] represents the concentration of pLux promoters<br> | |||
[AP] represents the concentration of A-Promoter complex<br> | |||
k1, k2, k3, k4, k5, k6 are the rate constants associated with the relevant forward and backward reactions<br> | |||
<math>\alpha_i \ </math> represents the energy consumption due to gene transcription. It is a function of gene length.<br> | |||
n is the positive co-operativity coefficient (Hill-coefficient)<br> | |||
<math>K_E \ </math> the half-saturation coefficient | |||
==Generalities of the Model== | ==Generalities of the Model== |
Revision as of 17:17, 16 October 2007
Model Development for Infector Detector
Formulation of the problem
- Questions to be answered with the approach
- Verbal statement of background
- What does the problem entail?
- Hypotheses employed
Selection of model structure
- Present general type of model
- is the level of description macro- or microscopic
- choice of a deterministic or stochastic (!) approach
- use of discrete or continuous variables
- choice of steady-state, temporal, or spatio-temporal description
- determinants for system behaviour? - external influences, internal structure...
- assign system variables
Our models
- Introduction
We can condition the system in various manners, but for the purposes of our project, Infector Detector, we will seek a formulation which is valid for both constructs considered. Our system can thus be modellied by the following Dynamical System:
The system of equations for the two constructs varies strictly with respect to the value of the parameter k1. Construct 1 possesses a non-zero k1 rate constant, whereas for construct 2, a zero value is assumed.
Model 1: Steady-state is attained; limitless energy supply
Model 2: Equations developed through steady-state analysis; however due to limited energy supply, we operate in the transient regime
[math]\displaystyle{ \frac{d[LuxR]}{dt} = k_1\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) + k_3[A] - k_2[LuxR][AHL]- \delta_{LuxR}[LuxR] }[/math]
[math]\displaystyle{ \frac{d[AHL]}{dt} = k_3[A] - k_2[LuxR][AHL]- \delta_{AHL}[AHL] }[/math]
[math]\displaystyle{ \frac{d[A]}{dt} = -k_3[A] + k_2[LuxR][AHL]- k_4[A][P] + k_5[AP] }[/math]
[math]\displaystyle{ \frac{d[P]}{dt} = -k_4[A][P] + k_5[P] }[/math]
[math]\displaystyle{ \frac{d[AP]}{dt} = k_4[A][P] - k_5[AP] }[/math]
[math]\displaystyle{ \frac{d[GFP]}{dt} = k_6[AP]\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) - \delta_{GFP}[GFP] }[/math]
[math]\displaystyle{ \frac{d[E]}{dt} = -\alpha_{1}k_1\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) - \alpha_{2}k_6[AP]\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) }[/math]
where:
[A] represents the concentration of AHL-LuxR complex
[P] represents the concentration of pLux promoters
[AP] represents the concentration of A-Promoter complex
k1, k2, k3, k4, k5, k6 are the rate constants associated with the relevant forward and backward reactions
[math]\displaystyle{ \alpha_i \ }[/math] represents the energy consumption due to gene transcription. It is a function of gene length.
n is the positive co-operativity coefficient (Hill-coefficient)
[math]\displaystyle{ K_E \ }[/math] the half-saturation coefficient