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

Model Development for Infector Detector

Formulation of the problem

Questions to be answered with the approach

• Level of sensitivity of system to [AHL] - what is the minimal [AHL] for appreciable expression of reporter protein

1. Effect of range of [AHL] upon system - does continued increase in [AHL] increase output?
2. If saturating behaviour is exhibited, is there a point at which greater [AHL] actually supresses output

• Can sensitivity to AHL, be tailored by varying initial [LuxR]
• Can sensitivity to AHL, be tailored by introducing increased [pLux] - in other words increasing [DNA]
• Description of response times - for construct 1 and contruct 2

• 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)

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

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

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

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

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

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.