IGEM:IMPERIAL/2007/Dry Lab/Modelling/ID

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(Model 1: Steady-state is attained; limitless energy supply ''(link here to derivation)'')
(Selection of model structure)
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*determinants for system behaviour? - external influences, internal structure...
*determinants for system behaviour? - external influences, internal structure...
*assign system variables
*assign system variables
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 +
====Assumptions====
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*Ignore spatial information of the system; we ignore molecular dynamics of the system.
 +
*Keep track of number of molecules of each type - concentrations of these state variables.
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*Thus assume that the system is homogeneous - well-stirred, so that the molecules of each type are spread uniformly throughout spatial domain.  *assume thermal equilibrium
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*assume constant volume of spatial domain
==Our models==
==Our models==

Revision as of 20:50, 16 October 2007

Contents

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
  1. is the level of description macro- or microscopic
  2. choice of a deterministic or stochastic (!) approach
  3. use of discrete or continuous variables
  4. choice of steady-state, temporal, or spatio-temporal description
  • determinants for system behaviour? - external influences, internal structure...
  • assign system variables

Assumptions

  • Ignore spatial information of the system; we ignore molecular dynamics of the system.
  • Keep track of number of molecules of each type - concentrations of these state variables.
  • Thus assume that the system is homogeneous - well-stirred, so that the molecules of each type are spread uniformly throughout spatial domain. *assume thermal equilibrium
  • assume constant volume of spatial domain

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 initial approach assumed that energy would in unlimited supply, and that our system would eventually reach steady-state (Model 1). Experimentation suggested otherwise; our system needed to be amended. This lead to the development of model 2, an energy-dependent network, where the dependence on energy assumed Hill-like dynamics:

Model 1: Steady-state is attained; limitless energy supply (link here to derivation)

\frac{d[LuxR]}{dt} = k_1 + k_3[A] - k2[LuxR][AHL]- \delta_{LuxR}[LuxR]


\frac{d[AHL]}{dt} = k_3[A] - k2[LuxR][AHL]- \delta_{AHL}[AHL]

\frac{d[A]}{dt} = -k_3[A] + k2[LuxR][AHL]- k_4[A][pLux] + k_5[AP]

\frac{d[P]}{dt} = -k_4[A][P] + k_5[P]

\frac{d[AP]}{dt} = k_4[A][pLux] - k_5[AP]

\frac{d[GFP]}{dt} = k_6[AP] - \delta_{GFP}[GFP]

Model 2: Equations developed through steady-state analysis; however due to limited energy supply, we operate in the transient regime

\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]
\frac{d[AHL]}{dt} = k_3[A] - k_2[LuxR][AHL]- \delta_{AHL}[AHL]

\frac{d[A]}{dt} = -k_3[A] + k_2[LuxR][AHL]- k_4[A][P] + k_5[AP]

\frac{d[P]}{dt} = -k_4[A][P] + k_5[P]

\frac{d[AP]}{dt} = k_4[A][P] - k_5[AP]

\frac{d[GFP]}{dt} = k_6[AP]\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) - \delta_{GFP}[GFP]

\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)


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
\alpha_i \ represents the energy consumption due to gene transcription. It is a function of gene length.
n is the positive co-operativity coefficient (Hill-coefficient)
K_E \ the half-saturation coefficient



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.

Generalities of the Model

Simulations

Sensitivity Analysis

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