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

Model Development for Infector Detector

Formulation of the problem

As described earlier, Infector Detector (ID) is a simple biological detector, which serves to expose bacterial biofilm, by exploiting the inherent AHL production employed by the quorum-sensing bacteria, in the formation of such structures. Our project serves to improve on previous failures and relatively ineffective current methods of biofilm detection, first and foremost, by focussing on the sensitivity of the system, to low levels of AHL production (bacterial chatter).
In doing so, a rigorous investigation of the level of sensitivity to [AHL] needs to be performed - in other words, what is the minimal [AHL] for appreciable expression of reporter protein. Furthermore, establish a functional range for AHL detection. How does increased [AHL] impact on maximal output of reporter protein?
Also, can the system performance be tailored, by exploiting the remaining state variables (e.g. varying initial [LuxR] and/or [pLuxR]). The system performance here revolves most importantly around AHL sensitivity; however, the effect on, maximal output of fluorescent reporter protein and/or response time affected is of much importance too.

Our initial approach involved deterministic methods leading to a steady-state of the reacting species. For this purpose, LuxR being constitutively expressed by pTET. However,

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.