Model

Background

For the part of fengycin inhibition, we here designed a pathway to simulate the producing of Authoinducing peptide (AIP) through the translation of AgrD gene, and the activation of the toxin production through the activation of the P3 promoter for RNAIII gene.

Design

For the inhibition effect of fengycin, the AgrC would be activated and phosphorylates the AgrA to initiate the transcription and translation of RNA III gene, which when the AgrC is bound by fengycin, the following process would not be initiated.

Aims of our model

Predict the impacts of different concentration of Fengycin on the expression of RNAIII gene.

Modeling process

In the calculations below, the concentration of the receptor would be represented by [R], the concentration of the fengycin would be represented by [F], and the concentration of AIP would be represented by [A]. The concentration of the gene AgrD is also included in the modeling process, represented by [AgrD]. We assume that all AgrC receptors are phosphorylated by ATP.

For approaching the result, different states would be identified. The states included in the model are listed below:

Then, we apply the concept of ordinary differential equations (ODE) for each state.

The ODE of the free receptor (AgrC) can be written as:

Here, the symbols k_F- and k_F+ stand for the rate of association and dissociation between AgrC and Fengycin, and the symbols k_A-and k_A+ stand for the rate of association and dissociation between AgrC and AIP.

The ODE of the free AIP and the fengycin in cytoplasm can be respectively written as:

Here, the symbol β stands for the rate of transcription and translation (assumes that the degradation of the mRNA from AgrD transcription has little effect) of the AgrD gene, and the k_{degradation a} and k_{degradation f} respectively stands for the degradation rate of AIP and Fengycin.

The ODE of the status of binding between AgrC and AIP/AgrC and Fengycin are respectively:

The ODE of the receptor binding with AIP can be written as:

Within these states, only the state [RA] would represent the activated receptors. The auto-transduction pathway involves the process of activating of AgrA through the ATP binded to the receptor. We applied the Michaels-Mentos formula to predict the concentration of phosphorylated AgrA (represented by [AA]):

which k_cat the rate of turnover to phosphorylated product, and K_m represented the maximum catalytic rate of [AgrC]. Since we understand the [AgrC] is equal to the number of [RA], then the formula can be substituted as

The expression of RNA III required the activator of corresponding P3 promoter (, which would be achieved by the binding of phosphorylated on the promoter, which the bound fraction can be expressed in the formula of (using Hill coefficient):

Follow the bounding rate, the transcription rate can summarized in:

Model testing and discussion

We use R to stimulate our results of the ODE constructed.

As presented above, the concentrations of free AgrC, AIPs and fengycin all drop to their lowest points suddenly after the simulation starts. Reversely, the concentration of the AgrC bound with the fengycin suddenly reaches its peak in the same time with the sudden drop with the concentrations of AgrC, AIPs and the fengycin. The curve of AgrC bounded with the AIP appears to present a logarithmic increasing curve and reaches its stationary phase after the 7th minutes. Corresponding to the logarithmic increasing of RA, the increasing of phosphorylated AgrA presents a linear increasing trend. Most importantly, the expression of RNA III presents a logarithmic decreasing trend, which suggest the effects of inhibition of the quorem sensing, thus suggests the effects of introducing fengycin in decreasing biofilms formation.

Reference

[1]Bridge, L., King, J., Hill, S., & Owen, M. (2010). Mathematical modelling of signalling in a two-ligand G-protein coupled receptor system: Agonist–antagonist competition. Mathematical Biosciences, 223(2), 115-132. https://doi.org/10.1016/j.mbs.2009.11.005 [2]Jabbari, S., King, J. R., & Williams, P. (2010). A mathematical investigation of the effects of inhibitor therapy on three putative phosphorylation cascades governing the two-component system of the agr operon. Mathematical Biosciences, 225(2), 115-131. https://doi.org/10.1016/j.mbs.2010.03.001 [3]Srivastava, S. K., Rajasree, K., Fasim, A., Arakere, G., & Gopal, B. (2014). Influence of the AgrC-AgrA Complex on the Response Time of Staphylococcus aureus Quorum Sensing. [4]Wang, B., Zhao, A., Novick, R., & Muir, T. (2014). Activation and inhibition of the receptor histidine kinase AgrC occurs through opposite helical transduction motions. Molecular Cell, 53(6), 929-940. https://doi.org/10.1016/j.molcel.2014.02.029 [5]Wong, P., Gladney, S., & Keasling, J. D. (1997). Mathematic model of the lac operon: Inducer exclusion, catabolite repression, and diauxic growth on glucose and lactose. Biotechnol, 132-143.

Parameters

Figure 2. parameters