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

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# 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 δLuxR, 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)

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

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

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

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

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

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

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

$\frac{d[LuxR]}{dt} = k_1\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) + k_3[A] - k_2[LuxR][AHL]- \delta_{LuxR}[LuxR]$
$\frac{d[AHL]}{dt} = k_3[A] - k_2[LuxR][AHL]- \delta_{AHL}[AHL]$

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

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

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

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

$\frac{d[E]}{dt} = -\alpha_{1}k_1\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg) - \alpha_{2}k_6[AP]\bigg(\frac{[E]^n}{K_E^n + [E]^n}\bigg)$

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

### State Variables

Create similar table for state variables, as for parameter table below

### Model Parameters

Populate parameters table

Parameter Value Description Comment (literature, derived?)
k1 x [units] max. transcription rate of constitutive promoter (pTET) Estimate
k2
k3
k4
k5
k6 x [units]
δGFP 0.029 hrs -1 degradation rate of GFP Literature ~~ give reference
δLuxR x [units] degradation rate of LuxR
δAHL 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.