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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta_{LuxR}} , 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)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[LuxR]}{dt} = k_1 + k_3[A] - k2[LuxR][AHL]- \delta_{LuxR}[LuxR]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[AHL]}{dt} = k_3[A] - k2[LuxR][AHL]- \delta_{AHL}[AHL]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[A]}{dt} = -k_3[A] + k2[LuxR][AHL]- k_4[A][P] + k_5[AP]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[P]}{dt} = -k_4[A][P] + k_5[AP]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[AP]}{dt} = k_4[A][P] - k_5[AP]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[AHL]}{dt} = k_3[A] - k_2[LuxR][AHL]- \delta_{AHL}[AHL]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[A]}{dt} = -k_3[A] + k_2[LuxR][AHL]- k_4[A][P] + k_5[AP]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[P]}{dt} = -k_4[A][P] + k_5[AP]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[AP]}{dt} = k_4[A][P] - k_5[AP]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d[GFP]}{dt} = k_6[AP]\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) - \delta_{GFP}[GFP]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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)}
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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \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)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle K_E \ }
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] | ||
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta_{GFP} } | 0.029 hrs ^{-1} | degradation rate of GFP | Literature ~~ give reference |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta_{LuxR} } | x [units] | degradation rate of LuxR | |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta_{AHL} } | 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.