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 (see opening in figure 4). 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 N_A 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 V in nm³, M in Dalton and 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 was computed with the open-source program PyMOL [PyMOl, 2013]. α was determined to be 25°. In the end we computed the reproduction rate in dependence of the ampicillin concentration in the periplasm, adapting the formula of Potsdam's:

where n is the ampicillin concentration in periplasm. The time point 20 minutes was chosen, because the division rate of an 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 beta-lactamase PCA colorimetric assay, and, secondly, a cultivation experiment. The cultivation experiment was used to measure the beta-lactamase concentration with the Nitrocefin assay over time. The experiments were essential to adapt model I.

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 its 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. 1 OD is converted to 8 * 10⁸ cells. One cell comprises around 1.27848 * 10^-6 M beta-lactamase in the periplasm. We adapted our system accordingly.