Mathematical modeling and experimental research are relatively complementary to each other. Biology have gradually became a qualitative field of research. In our study, we made respectively three models to quantify the peptide accumulation with S. aureus proliferation under the four types of Agr system, to predict the gene expression pattern of the four types of Agr system, and to simulate the gene expression of AimR-AimP system. To help readers understand our modeling theory, we also wrote attentions for all the modeling process. At last, we expect to improve mathematical modeling to make it more convenient to use.

Introduction:

In order to quantify the peptide concentration accumulation with S. aureus proliferation under different conditions, we regarded the each S. aureus as isolated unit. This prediction will be incorporated into the equation to calculate（simulate） the total protein amount at the community level.

Figure 1. The agr-Quorum Sensing System of S.aureus

The Agr-QS system is a quorum-sensing system in S. aureus and typical for Gram-positive bacteria. As shown in figure 1, when the transcription factor from AgrA phosphorylation binds with P2 promoter, agrB and agrD were continuously promoted to express AgrB and AgrD respectively[2]. In the Agr system, the protein of interest production process is briefly shown as below：

Modeling:

Our study is based on the quorum-sensing system of Gram-positive bacteria. In order to model the biological law of S. aureus, the prioritized task is to quantify the peptide accumulation with S. aureus proliferation under the four types of Agr system.

Assumption:

1. P2 promoter is always in large quantity and that its binding to the transcription factor from AgrA phosphorylation happens on a faster time scale to promote the genes downstream to express.

2. The product never binds with the free enzyme.

3. The conversion between enzyme-substrate and enzyme-product is much faster than that of association and dissociation events.

We tried to use Michaelis-Mentin kinetics equation[3] to predict the peptide production rate. We made several modifications on the equation to accommodate the transcriptional regulatory mode of the P2 promoter.

Attention:

1. [P2:F], [mRNA] and [P] represent the concentrations of P2 promoter and the transcription factor from AgrA phosphorylation, mRNA and product protein respectively.

2. The rate K is not just a simple constant and is given as the Hill function in the equations.

The remaining symbols are defined in Table 1.

Table 1 Definition of symbols used in the Michaelis-Mentin kinetics

Analysis and Results:

Using the model we established with MATLAB, we quantified the peptide concentration accumulation with bacterial proliferation under different type of agr systems as shown in Figure 2. As it’s shown, the higher of the bacterial flora quantity, the higher expression levels of protein of interest. After a period of time, the peptide concentration was in a platform period.

Figure 2. the peptide concentration accumulation with bacterial proliferation

The advantage of our model with Michaelis-Mentin kinetics equation is that we have not had to directly measure data for all of our enzymes, which is a difficult process. The results of our forecasting model matches well with previous relative researches. In addition, we can make the parameters of the equation much closer to the reality with experiment data.

Reference:
[1].https://2008.igem.org/Team:Cambridge/Modeling
[2] James, E.H., A.M. Edwards, and S. Wigneshweraraj, Transcriptional downregulation of agr expression in Staphylococcus aureus during growth in human serum can be overcome by constitutively active mutant forms of the sensor kinase AgrC. FEMS Microbiol Lett, 2013. 349(2): p. 153-62
[3]. https://2015.igem.org/Team:Oxford/Modeling

Introduction:

In order to predict the gene expression pattern of the four types of Agr system under different concentration of AIP, we regarded the each S. aureus as isolated unit. We would like to have the receiver part of the Agr-QS system in form of a biobrick. It becomes useful to think about inputs and outputs to and from such a biobrick. We shall use the concentration of AIP as a universal signal to "connect" biobricks to each other. The function of AIP to receiver was shown as following in detail.

Figure 3. the gene expression model of agr system.

As shown in figure 3 above, once AIP enters into the Agr receiver system, the AgrC was phosphorylated and changed its structure at the same time. Immediately, the AgrA was phosphorylated and then promoted the target gene to express, thus resulting in fluorescence. [4]

Modeling:

In our study, we divided the Agr system into Agr sender system and Agr receiver system. In model 1, we quantified the peptide concentration accumulation of Agr sender system. In model 2, we want to quantify the GFP relative concentration expressed by the target gene under different concentration of AIP.

Assumption:

1. phosphorylation and enzyme complex formation rates must be significantly higher than protein expression rates.

2. The concentration of [CP] is in a quasi-steady state.

The equations of this model after simplified read [5]:

Attention:

1. [P] denote the concentration of AIP, [A] denote the concentration of AgrA and similarly for the other proteins, [CP] denote the obvious enzyme complex and [A+] be AgrA phosphorylated.

2. AIP exists (or is only a chemical player) in the extracellular environment whereas cell enzymes (such as AgrA and AgrC) exist only within a cell.

3. The maximum promoter expression rate was 1.

The remaining symbols are defined in Table 2.

Table 2 Definition of symbols used in the recursion equation

Analysis and Results:

The gene expression of different type of Agr systems under different concentration of AIP was shown in figure 2. When the concentration of AIP was small, the extracellular signal molecule concentration was always at a fairly low level, and when the concentration of AIP is achieved at 0.02umol/l, the signal molecules began to increase rapidly, and we found that the signal growth trend was delayed but not stopped, and the growth rate of the curves were different due to different type.

Figure 4. the gene expression of different type of Agr systems under different concentration of AIP

Throughout the plots we observe step-like behavior - this is typical for quorum-sensing systems that react sensitively to local cell densities. The step-like shape of the curve agrees with previously published results [6].

Reference:
[4]. Marchand N, Collins C H. Synthetic Quorum Sensing and Cell–Cell Communication in Gram-Positive Bacillus megaterium[J]. Acs Synthetic Biology, 2016, 5(7):597.
[5].https://2008.igem.org/Team:Cambridge/Modeling
[6].[Novick et al. 1995] Cell density control of staphylococcal virulence mediated by an octapeptide pheromone

Introduction:

This models, which can predict the gene expression of AimR-AimP systems, is practical for many research. In this model, the genetic circuit describes the biochemical reactions taking place inside the cell (Figure5). The intracellular model describes the fluctuations in concentrations of each substance by modelling each chemical reaction.

Figure 5.the gene expression model of AimR-AimP system[7]

This QS system of AimR-AimP remains to be further studied for potential application. The QS response is quite tightly regulated as expression occurs only once the local bacterial density passes a fixed threshold value, similar unlike a switch. This system is encoded

by three phage genes: aimP, which produces the peptide; aimR, the intracellular peptide receptor; and aimX, a negative regulator of lysogeny. In the sender system, aimP produces the peptide and the concentration of peptide changes along with time. While in the receiver system, the peptide binds with aimX which inhibites the target gene expression. The mechanism can briefly described in the following:

Modeling:

In this model, we want to quantify the gene expression of the complete AimR-AimP system including the sender and receiver. Let us now consider the complete AimR-AimP system as illustrated in the figure 5. For the rate equation involving [P], we need to add both contributions from the sender and receiver systems and only take a single decay. In the lab，one can "simply" (Biologists clearly agree with this statement) put the sender and receiver to recreate the AimR-AimP system with biobricks.

Assumption:

1. Phosphorylation and enzyme complex formation rates must be significantly higher than protein expression rates.

The equations of this model read[8-9]:

Attention:

1. [P] denote the concentration of peptide of AimR-AimP system , [aimR] and [aimX] respectively denote the concentration of aimR and aimX, [CP] denote the peptide of AimR-AimP system and aimR complex.

2. The remaining symbols are defined in Table 2..

Table 3 Definition of symbols used in the recursion equation

Analysis and Results:

As it’s shown, in the early stages of the experiment, extracellular signal molecules increased with time, but soon the signal molecules increased to a bottleneck. Into the late experiment, the signal molecules continue to decrease, and ultimately reach a fairly low level, making the signal molecular curve to form a first increase after the trend.

Figure 6. the gene expression of different type of AimR-AimP systems

The tendency of this curve above matches well with our experimental data. And this crest-like shape of the curve agrees with previously published results[7].

Reference:
[7] Erez Z, Steinberger-Levy I, Shamir M, et al. Communication between viruses guides lysis-lysogeny decisions[J]. Nature, 2017, 541(7638):488.
[8].https://2008.igem.org/Team:Cambridge/Modeling
[9].James, E.H., A.M. Edwards, and S. Wigneshweraraj, Transcriptional downregulation of agr expression in Staphylococcus aureus during growth in human serum can be overcome by constitutively active mutant forms of the sensor kinase AgrC. FEMS Microbiol Lett, 2013. 349(2): p. 153-62