Introduction

In every problem concerning modelling, the main objective is always to reduce the system as much as possible. By this means, is possible to understand the fundamental features and behaviours of said system. However, when studying biological systems, this approach may not be possible due to the complexity of the interactions and the large amount of parameters necessary to describe each component. With our problem is not different, for it lies on the scope of the biological sciences. Taking this complexity into account, we chose to model our system in the simplest way possible: by means of simple ordinary differential equations, i.e chemical kinetics.

Chemical kinetics usually consists of first order differential equations of the form:

describing the evolutions of concentrations in time, in terms of initial conditions and some parameters $k$ that describing the strength of the dynamics. Each equation corresponds to one chemical reaction occurring along the system evolution. Usually, one seeks to find the equilibrium solution, i.e

which is considered to be the true state of the system after a long period of time. In our case, the chemical reactions are those that regulates the binding/unbinding of promoters, the enzyme activity, the expression of the reporter gene, and so on. So, each part of our genetic circuit will be represented by a equation, which is usually coupled with the equations describing other parts.

Detection and Signaling

The first problem we have to address is the efficiency of the detection module and it's components. Our system consists in an AIP linked with a surface protein, which is cleaved in the absence of Cystatin C, triggering the phosphorylation of the ComE protein. This problem was already studied by the 2010-Imperial College London team[\textbf{Citar London IC}], where they analysed the cleaving rate of AIP as well as the minimum amount of AIP needed for the ComE phosphorylation process to occur.

Detection

The first problem we should consider is the production of free AIP in the medium given the proper activation stimuli, which in our case is the absence of Cystatin C. This problem is simply a enzyme/substrate interaction, where as a first approximation, the effective enzyme concentration can be assumed to be

from which we considered a 1:1 ration in the Cystatin/Cathepsin interaction. The cleaving process obey the reaction:

which can be modelled by a simple set of first order differential equations, given by:

where

is the kinetic constant that regulates the amount of free Cathepsin in the solution, taking into account the unbinding of the enzyme-substrate complex (yES), as well as taking into account the releasing of Cathepsin after the cleavage (Kcleav). We considered that no new surface proteins are generated, given that we have a negative diagnosis, this is the worst case scenario.

The problem is a bit more complex than what is presented here, given to the diffusion nature of this process, but in equilibrium it is wise to assume that AIP concentrations are steady an homogeneous in the equilibrium state. Having the dynamics of $[AIP]$ described, the next step is to understand the phosphorylation process of the ComE protein.

Signaling

Our signaling system consists in the phosphorylation of the ComE protein, altering it to a functional state, where it acts as a promoter for the production of LasR. The chemical pathway involving ComE has 4 steps:

• The binding of AIP to the ComD receptor.
• The phosphorilation of the AIP-ComD complex.
• The biding of the ComE protein to the AIP-ComD* complex
• The releasing of ComE* in the medium.