# Why Alpha System Should Work – a deterministic model of alpha system

When modelling our system, we began with the simplest method known – deterministic rate equations. Moreover, from the design it was clear that the most important benchmark for the signal within the system would be the concentration of AHL as a function of time, so we began by modelling this part of our system. It took only a simple derivation (see [1]) to obtain these equations which characterize this part of the system:

*d*[

*mRNA*

_{LuxI}])/(

*dt*) = (

*v*

_{B}+

*v*

_{A}

*k*

_{A}[

*AHL*]

^{2})/(1 +

*k*

_{A}[

*AHL*]

^{2}) −

*γ*

_{mRNALuxI}[

*mRNA*

_{LuxI}] +

*GateI*

*d*[

*LuxI*])/(

*dt*) =

*α*

_{LuxI}[

*mRNA*

_{LuxI}] −

*γ*

_{LuxI}[

*LuxI*]

*d*[

*AHL*])/(

*dt*) =

*α*

_{AHL}[

*AHL*] −

*γ*

_{AHL}[

*AHL*]

(For a glossary see [1]).

We began to analyze this system by attempting to simplify it, by assuming a steady state solution wherever possible. Using this method (see [2]) we managed to obtain this equation:

*d*[

*AHL*])/(

*dt*) = (

*v*

_{B}+

*v*

_{A}

*k*

_{A}[

*AHL*]

^{2})/(1 +

*k*

_{A}[

*AHL*]

^{2}) −

*γ*

_{AHL}[

*AHL*] +

*GateI*

It is clear from the goals of our system, that we want to have some sort of bi-stability in the result, when the term Gate I is small (see [2]). The answer to whether this condition is met, would obviously depend on the constants of the system for which we could not find a reliable source, but using a simple geometric analysis of the phase space (see [3]), we were able to produce a graph showing for which values of (v_{1},v_{2}) we could configure the system (by changing the IPTG concentration and the OD) to show bi-stability: