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Modelling of cell-free bisphenol A degradation

In our project we want to construct a cell-free biosensor that can detect bisphenol A (BPA). To be more specific we want to fuse proteins to silicia beads that degrade the BPA and detect the NAD⁺ that is set free in the reaction by a NAD⁺-dependent ligase (also attached to the beads) in combination with molecular beacons.

First reaction

The first reaction is the degradation of BPA by the Cytochrome P450 monooxygenase system (the components of this system are cytochrome P450 (<partinfo>K123000</partinfo>), ferredoxin (<partinfo>K123001</partinfo>) and ferredoxin reductase (<partinfo>K525499</partinfo>)). The velocity of the BPA degradation can be calculated by Michaelis Menten enzyme kinetics:

The velocity V₁ was calculated in dependency of the enzyme concentration and the k_cat-value for the monooxygenase system of [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1317416/ 3.9 min^-1].

Second reaction

In the second reaction the NAD⁺-dependent DNA-ligase LigA from E.coli (<partinfo>K525710</partinfo>) is used to transfer the molecular beacons from the closed state b_cs to the open state b_os. It is described by the following reaction scheme:

The rate of a reaction with two substrates is calculated by the following equation (assuming initial product concentration of 0):

where K_iA is the binding affinity of the inhibitor to substrate A, K_mA and K_mB the Michaelis-Menten constants for substrate A respectively B in the specific reaction. So the amount of the molecular beacon open state is described by the equation

For this we assume, that the molecular amount of NAD⁺ is equal to the molecular amount of degraded BPA. The required electrones for the degradation of one BPA molecule, more precisely for the activation of the oxygen in the cytochrome P450 monooxygenase system, are released by the oxidation of one molecule of the cofactor NADH.

The velocity V₂ is calculated by multiplicating the k_cat-value of 0.0212 s^-1 found in the [http://onlinelibrary.wiley.com/doi/10.1046/j.1432-1327.2000.01377.x/pdf literature] with the used concentration of NAD⁺. For K_mNAD⁺ and K_mbos we use the values we found in the literature ([http://www.sciencedirect.com/science/article/pii/S0003269702003020 Chen et al., 2002]):

Regarding K_iNAD⁺ we considered AMP and NMN as possible inhibitors ([http://www.sciencedirect.com/science/article/pii/S0003269702003020 Chenet al., 2002]) for the reaction. The literature provides half maximal inhibitory concentrations (IC₅₀-values) of 3.1 mM for AMP and 9.5 for NMN µM. We assume that AMP has no significant influence in our model, as the IC₅₀-value is quite high. The IC₅₀-value of NMN on the other hand is in closer range to concentrations that migth be achieved during the performance of our test, so we do consider it in our calculations. To determine K_iNAD⁺ we used the formula

yielding an value of 0.45 µM for NMN.

Implementation of the equations for the first and the second reaction in matlab:

Figure 2 shows the simulation of our model for a timespan of 120 minutes.

Fig. 2: Simulation of extracellular degradation of BPA in combination with the activation of the molecular beacon.

Modelling of intracellular bisphenol A degradation

The modelling was done with the software [http://www.berkeleymadonna.com/ Berkeley Madonna] using the [http://en.wikipedia.org/wiki/Runge–Kutta_methods#Common_fourth-order_Runge.E2.80.93Kutta_method common fourth-order Runge-Kutta] method to solve the equations. The model was fitted to the measured data by the function "curve fit" in Berkeley Madonna to calculate the parameters, constants etc.

To model the BPA degradation by E. coli carrying BioBricks for BPA degradation (<partinfo>K123000</partinfo> and <partinfo>K123001</partinfo>) the cell growth has to be described first. The observed growth of E. coli on (our) LB medium was [http://en.wikipedia.org/wiki/Diauxie diauxic] with two different growth phases. Cell growth is a [http://en.wikipedia.org/wiki/First_order_kinetics#First-order_reactions first-order reaction] and is mathematically described as

with the specific growth rate µ and the cell count X. The specific growth rate is dependent on the concentration of the growth limiting substrate (e.g. glucose) and can be described as

with the substrate concentration S, the Monod constant K_S and the maximal specific growth rate µ_max ([http://www.annualreviews.org/doi/abs/10.1146/annurev.mi.03.100149.002103 Monod, 1949]). Because LB medium is a complex medium we cannot measure the substrate concentration so we have to assume an imaginary substrate concentration. Due to the diauxic growth two different substrates with different Monod constants and consumption rates are necessary to model the cell growth. The amount of a substrate S can be modelled as follows

with the specific substrate consumption rate per cell q_S. The whole model for the diauxic growth of E. coli on LB medium with two not measurable (imaginary) substrates looks like:

The specific BPA degradation rate per cell q_D is modelled with an equation like eq. (3). In the beginning of the cultivations, when E. coli growths on the "good" imaginary substrate S₁, no BPA degradation is observed. When this substrate is consumed, the BPA degradation starts. The model for this diauxic behavior is as follows:

Fig. 1 shows a comparison between modelled and measured data for cultivations with BPA degrading E. coli. In Tab. 1 the parameters for the model are given obtained by curve fitting the model to the data.

Fig. 1: Comparison between modelled (lines) and measured (dots) data for cultivations of E. coli KRX carrying BPA degrading BioBricks. The BioBricks K525512 (polycistronic bisdAB genes behind medium strong promoter, shown in black) and K525517 (fusion protein between BisdA and BisdB, expressed with medium strong promoter, shown in red) were cultivated at least five times in E. coli KRX in LB + Amp + BPA medium at 30 °C, using 300 mL shaking flasks without baffles with silicon plugs. The BPA concentration (closed dots) and the cell density (open dots) is plotted against the cultivation time.

The specific BPA degradation rate per cell q_D is about 50 % higher when using the fusion protein compared to the polycistronic bisdAB gene. This results in an average 9 hours faster, complete BPA degradation by E. coli carrying <partinfo>K525517</partinfo> compared to <partinfo>K525512</partinfo> as observed during our cultivations. The fusion protein between BisdA and BisdB improves the BPA degradation by E. coli.

References

Sasaki M, Akahira A, Oshiman K, Tsuchido T, Matsumura Y (2005b) Purification of Cytochrome P450 and Ferredoxin, Involved in Bisphenol A Degradation, from Sphingomonas sp. Strain AO1, [http://aem.asm.org/cgi/content/abstract/71/12/8024 Appl Environ Microbiol 71(12):8024-8030].

Georlette D, Jonsson Z, Van Petegem F, Chessa J-P, Van Beeumen J, Huebscher U, Gerday C (2000) A DNA ligase from psychrophile Pseudoalteromonas haloplanktis gives insigth into the adaption of proteins to low temperatures, [http://onlinelibrary.wiley.com/doi/10.1046/j.1432-1327.2000.01377.x/pdf Eur.J.Biochem 267:3502-3512].

Chen X, Hentz N, Hubbard F, Meier T, Sittampalam S, Zhao G (2002) Development of a fluorescence resonance energy transfer assay for measuring the activity of Streptococcus pneumoniae DNA ligase, an enzyme essential for DNA replication, repair and recombination, [http://www.sciencedirect.com/science/article/pii/S0003269702003020 Anal Biochem 309:232-240].

Monod J (1949) The growth of bacterial cultures, Annu Rev Microbiol [http://www.annualreviews.org/doi/abs/10.1146/annurev.mi.03.100149.002103 3:371-394].