Model of our circuit

The tumor sensing circuit is composed of several proteins interacting with small molecules (AHL and lactate) and DNA (at the transcription factors binding sites). To establish a model describing the behavior of our circuit, we first had to understand the way these interactions are happening inside of the cell. BLABLA references DAVID. Here is a detailed overview of the tumor sensing circuit:

Simplification of the lactate sensing

Let us first focus on the lactate sensing part of the circuit. In the cell, two proteins are produced:

• LldP: a transmembrane protein enabling the transport of extracellular lactate into E. coli.
• LldR: a transcription factor, repressing the expression of the hybride transcription factor when unbound to lactate, and deactivated when bound to it.

To model precisely the regulation of the hybrid promoter by lactate, it would be necessary to take into account all the following points:

• How the intracellular lactate concentration behaves in regards to the expression level of LldP and the extracellular lactate concentration
• What is the binding constant between LldR and the lactate
• What is the binding dynamics of Lldr to the operon, and how it affects the transcription rate downstream

In an effort to simplify our model to reduce it to the most meaningful parameters, and because it has already extensively been studied and characterized by previous iGEM teams, we have chosen not to take into account the complexity of the lactate sensing pathway and rather use a phenomenological model to describe its influence. We rely on the characterization of the lactate sensor using several expression regulation sequences done by the ETH 2015 iGEM team. We consider therefore that lactate sensing follows a Hill function as following:

\[\begin{aligned} P_{\text{Lac}} &\simeq \frac{\left(\frac{[\text{Lac}]}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}}{1 + \left(\frac{[\text{Lac}]}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}}\end{aligned}\]

As a result, the schematics of the circuit can be simplified this way:

Quorum Sensing sensor modelization

The sensing of the bacterial cell density is done via a quorum sensing circuit. The principles behind quorum sensing is that each bacteria produces via the expression of the enzyme LuxI a basal amount of a small chemical (here AHL) that diffuses in the environment and into neighboring cells. When AHL is in sufficient quantity, it binds to the intracellular LuxR and induces the production of more LuxI, which in turn results in the production of more AHL. This positive feedback loop results in the activation of the operon containing the luxI gene when the cell density reaches a critical threshold.

Concerning LuxR

LuxR is under a constitutive promoter of strength \( a_{\text{luxR}} \) and its degradation rate is \( d_{\text{luxR}} \). AHL binds and stabilizes LuxR; LuxR-AHL molecules can only act as transcription factors when they form a tetramer (2*AHL+2*LuxR). Since we are modeling the steady state, the following simplifications apply:

• We consider that the total amount of LuxR present in the cell is constant, and only depends on its constitutive expression and degradation rate.

• We consider the global binding equilibrium between LuxR and AHL without taking into account the intermediary dimers.

We can therefore write the following equations:

\[\begin{aligned} [\text{LuxR}]_0 &= \frac{a_{\text{LuxR}}}{d_{\text{LuxR}}} & \text{steady state concentration} \\ [\text{LuxR-AHL}] &= K_{LuxRAHL} [\text{LuxR}]^2 [\text{AHL}]^2 & \text{rapid binding equilibrium} \\ [\text{LuxR}] &= [\text{LuxR}]_0 - 2 [\text{LuxR-AHL}] & \text{mass conservation}\end{aligned}\]

Hybrid Lux-Lac promoter

The expression of the main operon containing the luxI, bfr and azurin genes is regulated by a hybrid promoter activated by the quorum sensing and repressed by the lactate sensing (the repression being released in presence of lactate). This hybrid promoter should behave as a AND-gate: mathematically, this corresponds to multiplying the Hill functions describing their behavior.

Along with the lactate concentration, the intracellular levels of LuxR-AHL complexes affect LuxI expression. With \( a_{\text{LuxI}} \) being the maximal production rate of LuxI, \( d_{\text{LuxI}} \) the degradation rate, \( k_{\text{LuxI}} \) the leakiness of the promoter and \( P_{\text{Lux-Lac}} \) the combined effect of the \( P_{\text{Lux}} \) and \( P_{\text{Lac}} \) regulating sequences behavior, the ODE governing the production of \( [\text{LuxI}] \) can be written as following:

\[\frac{\mathrm{d} [\text{luxI}]}{\mathrm{d} t} = a_{\text{LuxI}} (k_{\text{LuxI}} + (1 - k_{\text{LuxI}}) P_{\text{Lux-Lac}}) - d_{\text{LuxI}} [\text{luxI}]\]

where

\[\begin{aligned} P_{\text{Lux-Lac}} &= P_{\text{Lux}} \, P_{\text{Lac}} \\ \\ P_{\text{Lux}} &= \frac{\left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR-AHL}}} \right)^{n_{\text{LuxR}}}}{1 + \left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR-AHL}}} \right)^{n_{\text{LuxR}}}}\end{aligned}\]

Solving the above at steady state, we get:

\[\begin{aligned} \frac{\mathrm{d} [\text{luxI}]}{\mathrm{d} t} &= 0 \\ [\text{luxI}] &= \frac{a_{\text{luxI}}}{d_{\text{luxI}}} (k_{\text{luxI}} + (1 - k_{\text{luxI}}) P_{\text{Lux-Lac}})\end{aligned}\]

Production of AHL

AHL is produced intracellularly by LuxI and diffuses then freely through the membrane [1]. Modeling the production of AHL quite straightforward: it is proportional to the amount of LuxI present intracellularly. To describe the production per unit of volume though, we have to take into account the bacteria cell density present locally and take it as a dilution coefficient (for instance, if the cells occupy locally half of the volume, then the intracellularly produced AHL would be instantly diluted two times as it diffuses into the surrounding environment).

\[\begin{aligned} P_{\text{AHL}} &= d_{\text{cell}} a_{\text{AHL}} [\text{luxI}] \end{aligned}\]