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, 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.

in terms of chemical reactions, this process can be described as:

which can be simply described by a set of first order differential equations, given by:

in the equilibrium condition all equations are dependent only by the initial values of $[ComD]$ and $[AIP]$, which are give by the equations derived in the detection section of the modelling. According to London team the value of $[ComE^*]$ will eventually reach a limit concentration of $5x10^{-11}M$. Considering that the detection/signaling steps reach the equilibrium state rather rapidly, it is possible to consider the response step as the limiting factor, and the initial concentration

Diagnosis

According to London team the value of [ComE*] will eventually reach a limit concentration of 5x10^-11 M. Considering that the detection/signaling steps reach the equilibrium state rather rapidly, it is possible to consider the response step as the limiting factor, so it is possible to consider [ComE*]eq= 5x10^-11 M.

In our project, the main doubt and assumptions had to be made concerning the interaction of the proteins LasR and QteE, which has a repression mechanism still unknown. The first assumption made was that the LasR/QteE interaction has a 1:1 equivalence, which seems quite unnatural, but is the simplest aproximation to be made. Afterwards, we assumed that the interaction occur with the formation a complex o bonded LasR/QteE.

First Case

The set of equations describing our system for the first case are described bellow:

the constants alphaL, alphaq and alphaG are the kinetic for the production of LasR, qteE and GFP respectively. The constants yL, yq, yG are the degradation constant, which determinate the rate of degrading LasR, QteE and GFP respectively. The term "teta ([LasR] - [QteE]) is the Heaviside function (step function), being 1 when LasR > QteE and 0 when LasR < QteE, guaranteeing the production of GFP only when the qteE barrier is crossed. The equation describing the production o QteE was taken from the 2010-Kyoto team.

In equilibrium, the equations become:

Note that the equations describing the equilibrium of LasR and QteE can be substituted on the equation for the GFP equilibrium concentration. By this means, it is possible to model de amount of $GFP$ produced taking into account only the promoters [ComE*] and [IPTG], which are controllable in an experimental set-up. So, if all the constants are given, it is possible do infer the strength of the signal generated in terms of the $qteE$ barrier and the efficiency of the cleaving process the occur outside the cell.

Second Case

For the second case, we will consider the formation of a [QteE-LasR] complex. To take this into account it is necessary to add an intermediate step in the equations described in the previous section. This intermediate step is given by the following chemical reaction:

adding the one more differential equation to our system, which now becomes:

which in equilibrium will be:

The main difference that rises in this second case is the basal concentration of LasR, given that yC is different from 0, due to the dissociation of the LasR-qteE complex. In the case where the dissociation rate yC = 0, we recover the first case, where the value of [GFP]eq is dependent only in the difference between the values of [LasR] and [qteE].

Response

Since our diagnosis is based on the fluorescent feature of the GFP, it is important to evaluate whether or not our system produces enough GFP to be seen. This problems was already addressed by the 2013-USP-Brazil team, where they analysed the relative contrast generated by their reporter protein.

The main measure to be analysed is the relative contrast C(x) given by:

where I(x) is the luminous intensity of the signal in the position x and I_B(x) is the background light intensity.

The brightness generated by a fluorescent protein is measured in terms of the product of its Quantum Yield (QY)(the rate Phi between emitted and absorbed photons) and its Molar Extinction Coefficient(MEC) defined as the absorption force per mol per centimetre (M^-1 cm^-1).

The Beer-Lambert law states that the intensity I(x) suffers an exponential attenuation given by

where x(0) is the position of the source, x is the position of the observer and A is the Total Absorptivity of the medium with c(i) being the absorbing components and epsilon(i) their respective MECs. It is then assumed that the amount of light generated is proportional to the amount of background light available in the container, described by:

then, using the definition of the visual contrast C(x), we get:

assuming that the background light is homogeneous (i.e IB(x) = IB) we get a simple relation for the contrast:

It is possible then to define a visual threshold for the contrast, from which we can infer the minimum concentration of GFP needed to make the diagnosis visible, calculated from the relation:

where the values of Phi and epsilon can be extracted from the literature. For the Green Fluorescent Protein we have Phi = 0.79 and epsilon = 55 000 M^-1 cm^-1.

Estimating teh minimal GFP concentration

To estimate the value of [GFP]min, it is necessary to make a last assumption, concerning the value of A = A. Following the prescription given by the 2013-USP-Brazil team, we consider A = A, i.e homogeneous and constant. We will assume that half the light is absorbed by the surrounding air, meaning A = 0.5, this estimate is rather pessimistic, given that air opacity for visible light is around 10%. We also have to determine the distance of the observer, we consider that $1m$ is enough to give a good rough estimate. We then have:

which give us [GFP]min = 6.55 x 10^-5 M. In possession of this value, it is possible to construct a phase diagram, where we can determine whether or not our system will be visible in terms of the concentrations [IPTG] and [ComE*]. This conditions for the diagnosis to be possible are:

where the values of [LasR]eq and [QteE]eq can be calculated in terms of the initial values of [ComE*] and [IPTG].

Conclusion