# Model I

## Selection system

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:

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

(I)

k

(II)

(III)

(IV)

(V)

(VI)

(VII)

(VIII)

*k*stands for the expression rate of plasmid 1, which holds the Evobody-RpoZ sequence [_{exprP1}*s*^{-1}].k

_{degEvo}stands for the degeneration rate of the Evobody-RpoZ protein [*s*^{-1}].(II)

*k*stands for the expression rate of plasmid 2, which holds the target-DNA binding domain (DNABD) protein [_{exprP2}*s*^{-1}].*k*stands for the degeneration of the target-DNABD protein [_{degTarget}*s*^{-1}].(III)

*k*stands for the association rate of the bacterial two hybrid complex [(_{ass}*s*M*)^{-1}].*k*stands for the disassociation rate of the bacterial two hybrid complex [_{diss}*s*^{-1}].(IV)

*k*stands for the expression rate of beta-lactamase (bla) [_{exprbla}*s*^{-1}].(V)

*k*stands for the transport rate of beta-lactamase from cytoplasm to periplasm [_{transpbla}*s*^{-1}].(VI)

*k*stands for the degeneration rate of the beta-lactamase [_{degbla}*s*^{-1}].(VII)

*k*stands for the transport rate of ampicillin from medium to cytoplasm and back [_{transpamp}*s*^{-1}].(VIII)

*k*stands for the diffusion rate of beta-lactamase and ampicillin [(_{diff}*s*M*)^{-1}].### Differential equations

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

[Evobody] or Evobody stands for Evobody-RpoZ protein [M].

[Target] or Target stands for target-DNABD protein [M].

[DNA] stands for DNA [M].

[amp

[amp

[bla

[bla

[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].

_{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*stands for disassociation rate of the Evobody-Target complex [_{dissEvoTarget}*s*^{-1}].*k*stand for association rate of the Evobody-Target complex [(_{assEvoTarget}*s*M*)^{-1}].*k*stand for disassociation rate of the Evobody-Target complex [_{dissTargetDNA}*s*^{-1}].*k*stand for association rate of the Evobody-Target complex [(_{assTargetDNA}*s*M*)^{-1}].*k*stands for degeneration rate of the target protein [_{degTarget}*s*^{-1}].*k*stands for degeneration rate of the Evobodies [_{degEvo}*s*^{-1}].*k*stands for the expression rate of beta-lactamase with the bacterial two hybrid complex [_{exprblaETD}*s*^{-1}].### Diffusion

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 2008):

*R*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

_{min}*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*must be multiplied with the Avogadro constant

_{diff}*N*to gain the unit [(

_{A}*s*M*)

^{-1}].

*R*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):

_{min}
for V in nm

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

^{3}, 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.### Experimental validation

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.

#### Beta-lactamase PCA colorimetric assay

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 * 10

^{8}cells. One cell comprises around 1.27848 * 10^{-6}M beta-lactamase in the periplasm. We adapted our system accordingly.### References

- Atkins, Peter W.; Paula, Julio de (2010): Physical chemistry. 9. ed. Oxford u.a.: Oxford Univ. Press.
- Bisswanger, Hans (2008): Enzyme kinetics. Principles and methods. 2., rev. and updated ed. Weinheim: Wiley-VCH. Online accessible at http://site.ebrary.com/lib/alltitles/docDetail.action?docID=10301967.
- Erickson, Harold P. (2009): Size and shape of protein molecules at the nanometer level determined by sedimentation, gel filtration, and electron microscopy. In: Biological procedures online 11, S. 32–51. DOI: 10.1007/s12575-009-9008-x.
- Galarneau, André; Primeau, Martin; Trudeau, Louis-Eric; Michnick, Stephen W. (2002): |[beta]|-Lactamase protein fragment complementation assays as in vivo and in vitro sensors of protein|[ndash]|protein interactions. In: Nature Biotechnology 20 (6), S. 619–622. DOI: 10.1038/nbt0602-619.
- Jalalirad, Reza (2013): Selective and efficient extraction of recombinant proteins from the periplasm of Escherichia coli using low concentrations of chemicals. In: Journal of industrial microbiology & biotechnology 40 (10), S. 1117–1129. DOI: 10.1007/s10295-013-1307-1.