Model I describes the in vivo bacterial two hybrid selection system:

The selection pathway can be illustrated by the following reaction sequence:

The reaction sequence starts with the expression of the Evobody-RpoZ translational fusion construct [(a)] on the one hand and the target protein-DNA binding domain translational fusion construct [(b)] on the other hand. Both resulting fusion proteins are degraded over time [(c)]. In the figure the degeneration is marked with a 0. Furthermore, together with the DNA binding site for the DNA binding domain (DNABD) both fusion proteins form the bacterial two hybrid complex [(d)]. When the complex is built the RNA polymerase gets recruited and transcribes the beta-lactamase. Just like the fusion proteins the beta-lactamase will be degraded after a while [(e)]. When the beta-lactamase and ampicillin meet in the periplasm the ampicillin will be deactivated and the cell survives.

Based on the reaction sequence the chemical reaction equations can be written down:

(I) k_exprP1 stands for the expression rate of plasmid 1, which holds the Evobody-RpoZ sequence [s^-1].
k_degEvo stands for the degeneration rate of the Evobody-RpoZ protein [s^-1].
(II) k_exprP2 stands for the expression rate of plasmid 2, which holds the target-DNA binding domain (DNABD) protein [s^-1].
k_degTarget stands for the degeneration of the target-DNABD protein [s^-1].
(III) k_ass stands for the association rate of the bacterial two hybrid complex [(s*M)^-1]. k_diss stands for the disassociation rate of the bacterial two hybrid complex [s^-1].
(IV) k_exprbla stands for the expression rate of beta-lactamase (bla) [s^-1].
(V) k_transpbla stands for the transport rate of beta-lactamase from cytoplasm to periplasm [s^-1].
(VI) k_degbla stands for the degeneration rate of the beta-lactamase [s^-1].
(VII) k_transpamp stands for the transport rate of ampicillin from medium to cytoplasm and back [s^-1].
(VIII) k_diff stands for the diffusion rate of beta-lactamase and ampicillin [(s*M)^-1].

Next the language of chemistry has to be translated into the language of mathematics. A system of ordinary differential equations is generated. As this system has to be solved numerically, we have chosen a notation the software package MATLAB® understands:

stand_expr stands for standard expression [M].
[Evobody] or Evobody stands for Evobody-RpoZ protein [M].
[Target] or Target stands for target-DNABD protein [M].
[DNA] stands for DNA [M].
[amp_Peri] stands for ampicillin in periplasm [M].
[amp_Medium] stands for ampicillin in medium [M].
[bla_Cyt] stands for beta-lactamase in cytoplasm [M].
[bla_Peri] stands for beta-lactamase in periplasm [M].
[Evobody-Target] stands for Evobody-Target protein complex [M].
[Target-DNA] stands for Target-DNA complex [M].
[Evobody-Target-DNA] stands for Evobody-Target-DNA complex [M].
k_{dissEvoTarget} stands for disassociation rate of the Evobody-Target complex [s^-1].
k_assEvoTarget stand for association rate of the Evobody-Target complex [(s*M)^-1].
k_{dissTargetDNA} stand for disassociation rate of the Evobody-Target complex [s^-1].
k_assTargetDNA stand for association rate of the Evobody-Target complex [(s*M)^-1].
k_degTarget stands for degeneration rate of the target protein [s^-1].
k_degEvo stands for degeneration rate of the Evobodies [s^-1].
k_exprblaETD stands for the expression rate of beta-lactamase with the bacterial two hybrid complex [s^-1].

The deactivation reaction of ampicillin through beta-lactamase is a diffusion-controlled reaction. An equation that considers the rate at which the reactants diffuse is needed [Atkins and De Paula, 2010]:

This equation describes the general diffusion of two molecules (figure 3). But in our case the enzyme beta-lactamase only reacts with ampicillin, if its active centrum is in close proximity to ampicillin (figure 4 B). When beta-lactamase and ampicillin meet correctly, ampicillin is deactivated as shown in figure 4, section B to D. Section E indicates a wrong encounter of beta-lactamase and ampicillin. That's why we decided to modify the equation according to [Bisswanger, 2004]:

R_min is defined as the distance in which the two reactant molecules, here ampicillin and beta-lactamase, react. This is the case, when they are in very close proximity. D is the sum of the diffusion coefficients of the two reactants in the solution. In our case the solution is the periplasm. The term diffusion describes the movement of a substance in a solution; the bigger the diffusion coefficient, the larger the diffusion velocity. The angle α restricts the reaction area of the enzyme to its active center. So a reaction between beta-lactamase and ampicillin only takes place, when ampicillin meets the active center of beta-lactamase.
Finally, the equation for the diffusion rate k_diff must be multiplied with the Avogadro constant to gain the unit [(s*M)^-1].

R_min was computated under the assumption that beta-lactamase and ampicillin react when they are in close vicinity and that the molecules are nearly spherical. Therefore, the minimal radius of a sphere that could contain the given mass of protein was estimated using the formula from [Erickson, 2009]:

for M in Dalton, R_min in nanometer. For beta-lactamase we obtained 2.0277 nm and for ampicillin 0.4649 nm.
To compute the diffusion coefficients we used the program WinHydroPro v1.00 PUB [Ortega, 2011]. We received 2.852*10^-9 for beta-lactamase and 1.237*10^-8 for ampicillin. The angle α were computed with the open-source program PyMOL [PyMOl, 2013]. α was determined to be 25°C. In the end we computed the reproduction rate in dependence of the ampicillin concentration in the periplasm in μg/ml. Therefore, we use the formula

where n is the ampicillin concentration in periplasm after 20 min. The time point 20 minutes were chosen, because the division rate of a E. coli cell is around 20 minutes.

A theoretically implemented model must always be validated with experimentally measured data. Therefore, we designed and conducted an amount of appropriate experiments. Firstly, we performed a the beta-lactamase PCA colorimetric assay, and, secondly, cultivation experiments. One part of the cultivation experiments was used for fitting the growth parameter r, and seperate experiments were used for validating our model.

To quantify the amount of active beta-lactamase in the periplasm of an E. coli bacterium and thus validate our mathematical program in this aspect, we performed an [osmotic shock] and a [beta-lactamase PCA colorimetric assay] [André Galarneau et. al, 2002]. The osmotic shock leads to the disruption of the periplasm and spreading of membrane proteins. To yield the maximum amount of active beta-lactamase an osmotic shock is used that does not use detergents that are harmful for the activity of beta-lactamase [Jalalirad, 2013]. With the beta-lactamase PCA colorimetric assay we determined the amount of beta-lactamase through the reduction of Nitrocefin. When Nitrocefin is split by beta-lactamase it changes his color from yellow to red like in figure 5. To measure how the beta-lactamase production changes over time we cultivated E. coli with a beta-lactamase plasmid for 6 hours and took probes every hour. Afterwards we did a lysis and a Nitrocefin assay. The results can be seen here:

In figure 5 it can be seen that the beta-lactamase concentration rises with proceeding cultivation. Also we measured the beta-lactamase concentration of the whole culture at every time point and computed with the use of the OD (figure 6) the beta-lactamase concentration for one cell. One cell comprises around 1.27848 * 10^-6 M beta-lactamase in the periplasm. We adapted our system accordingly.