IGEM:IMPERIAL/2007/Dry Lab/Modelling/ID
Contents
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. It functions by exploiting the inherent AHL production employed by the quorum-sensing bacteria, in the formation of such structures.
~~Insert diagram illustrating this phenomenon
Our project attempts to improve where previous methods of biofilm detection have proven ineffective: first and foremost, by focussing on the sensitivity of the system, to low levels of AHL production (bacterial chatter).
In doing so, a complete 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, how can the system performance be tailored, by exploiting the remaining state variables (e.g. varying initial [LuxR] and/or [pLux]).
The system performance here revolves most importantly around AHL sensitivity; however, the effect on, maximal output of fluorescent reporter protein and/or response time is, likewise, of great importance.
Our approach, involves the proposal of two simple constructs, varying with respect to the manner in which LuxR is introduced into the system:
- Construct 1 - represented by T9002, incorporates constitutive expression of LuxR by pTET.
- Construct 2 - simpler in nature, lacks pTET; LuxR is introduced in purified form here.
~~Here explain briefly why Construct 2 was selected, i.e. we were concerned with the time the system would take to reach steady-state (that is before energy-dependence was considered) - due to almost negligible [math] \delta_{LuxR}[/math], etc .
Selection of model structure
At reasonably high molecular concentrations of the state variables, a continuous model can be adopted, which is represented by a system of ordinary differential equations.
It is for this reason that our approach to modelling the system follows a deterministic, continuous approximation. In developing this model, we were interested in the behaviour at steady-state, that is when the system has equilibrated and the concentrations of the state variables remain constant.
In adopting this approach, we perform the following assumptions:
Assumptions
- Ignore spatial information of the system; we ignore molecular dynamics of the system - this is a kinetic model.
- Keep track of total number of molecules of each type - by tracking the concentrations of these state variables (as a continuous variable)
- System is homogeneous - well-stirred, so that the molecules of each type are spread uniformly throughout the spatial domain. In doing so, we assume thermal equilibrium
- Volume of the spatial domain remains constant
The system kinetics are determined by the following six coupled-ODEs.
Establishing a representative model
- 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 be 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]\frac{d[LuxR]}{dt} = k_1 + k_3[A] - k2[LuxR][AHL]- \delta_{LuxR}[LuxR][/math]
[math]\frac{d[AHL]}{dt} = k_3[A] - k2[LuxR][AHL]- \delta_{AHL}[AHL][/math]
[math]\frac{d[A]}{dt} = -k_3[A] + k2[LuxR][AHL]- k_4[A][P] + k_5[AP][/math]
[math]\frac{d[P]}{dt} = -k_4[A][P] + k_5[AP][/math]
[math]\frac{d[AP]}{dt} = k_4[A][P] - k_5[AP][/math]
[math]\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]\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]\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[AP][/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]
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]\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]K_E \ [/math] the half-saturation coefficient
State Variables
Create similar table for state variables, as for parameter table below
Model Parameters
Populate parameters table
Parameter | Value | Description | Comment (literature, derived?) |
---|---|---|---|
k_{1} | x [units] | max. transcription rate of constitutive promoter (pTET) | Estimate |
k_{2} | |||
k_{3} | |||
k_{4} | |||
k_{5} | |||
k_{6} | x [units] | ||
[math] \delta_{GFP} [/math] | 0.029 hrs ^{-1} | degradation rate of GFP | Literature ~~ give reference |
[math] \delta_{LuxR} [/math] | x [units] | degradation rate of LuxR | |
[math] \delta_{AHL} [/math] | x [units] | degradation rate of AHL |
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.