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Team:HIT-Harbin/project/model
From 2012.igem.org
Staphylococcus aureus AgrA, the transcriptional component of a quorum sensing system and global regulator of virulence that upregulates secreted virulence factors and down-regulates cell wall-associated proteins, can bind in both the P2 and P3 promoter regions of the agr locus.
The structure of AgrA, described by online software PDB (Protein Data Bank), has ten β strands arranged into three antiparallel β sheets and a small α helix. The sheets are arranged roughly parallel to each other in an elongated β-β-β sandwich. A hydrophobic five-stranded β sheet (sheet 2: β3-β7) is at the center of the domain with two smaller amphipathic β sheets (sheet 1: β1-β2 and sheet 3: β8-β10) positioned on either side.
AgrC belongs to the histidine protein kinase (HPK) family and in particular to the HPK10 subfamily of QS peptide HPKs, which are predicted to consist of six or seven N-terminal transmembrane segments and a C-terminal cytoplasmic kinase domain.
The 3D structures of HPK10 kinases have not been determined, but topology modeling using different prediction methods has indicated that AgrC has either five or six transmembrane segments. We have tried many different tools for modeling the structure of AgrC.
Fig.2 shows the predicted topology of AgrC using the MEMSAT3 program, which indicates six transmembrane helices (18-24 amino acid sequence domains), three extracellular loops, with high probability of N-terminal in the cytoplasmic by TMHMM program.
The Mathematical Model
The detecting system can sense the existence of S.aureus by responding to the AIPs secreted only from S.aureus, accelerating the production of Lux I, which releases signal protein AHL into the environment (Fig. 3). We have constructed a set of ordinary differential equations to mathematically analyze this novel genetic circuit. Considering that the system is quite complicated, we make several reasonable assumptions to simplify it.
(1) In order to simulate the detecting part separately, we neglect the mechanism of AIPs production by S.aureus, watching the AIPs dynamic variation directly.
(2) The population consists of a number (n) of cells with a cytoplasmic volume (v), and is located in a medium with activating AIPs concentration of P, neglecting inhibiting AIPs.
(3) All reactions are modeled by mass action principles, except transcription which obeys saturation kinetics, and all variables are explained in Table 1.
(4) There is no delay in synthesis of either substance, subjecting to degradation all the time.
We set up 5 ODEs (ordinary differential equations) in our model to describe the detecting circuit, the interaction between AIP and AgrC, the AgrA phosphorylation process, the Lux I synthesis in response to phosphorylated AgrA binding to the promoter. The parameters are defined when necessary and all described in Table 2.
P is a special variable. In this model, we cannot predict the AIP’s variation, for the reason of the killing system having a negative effect of S.aureus population, which determines the concentration of AIP directly. The following figure shows all the substances levels in response to the concentration of AIP Fig. 4. It can be concluded that the phosphorylated AgrA (Api) and unphosphorylated AgrA would be at stable levels after a certain concentration of AIP. Besides, Lux I is on the increase via adding AIP in the whole environment, which is identical to the next part.
In this section, we analyze the substances level with time passing, by control AIP be at a high level (P=0.8). As shown in Fig. 5, Lux I would stay at a stable level when AIP be controlled as a constant value, which is the most important output of this model. So we can make bold prediction that the whole system would be at stable state, including detecting and killing parts.
The variation of unphosphorylated AgrA (A) shown in the figure is corresponding to biological mechanism (the green line). Until the complex between AgrC and AIP (Ccp) reaches threshold value, level of unphosphorylated AgrA would not stop increase at top speed.
As far as possible, parameter values are based on present publications on the agr system and on biological plausibility. The degradation rates for all components were equally set.
In summary, the protein structure prediction model of AgrA and topology analysis of membrane protein AgrC are conforming to the expected function, which also need further validation in the lab. The mathematical model of this system is part of a complete gene circuit. But under the reasonable assumption, we conclude some valuable information from the output of ODEs, of which the most important is all the substances would stay at a stable level when AIP be controlled as a constant value.
Reference
[1] Erik G., Patric N., Stefan K., Stanffan A., Characterizing the Dynamics of the Quorum-Sensing System in Staphylococcus aureus. J Mol Microbiol Biotechnol 2004; 8: 232-242.
[2] Sidote D.J., Barbieri C.M., Wu T., Stock A.M., Structure of the Staphylococcus aureus AgrA LytTR Domain Bound to DNA Reveals a Beta Fold with a Novel Mode of Binding. Structure 2008; 16: 727-735.
[3] Rasmus O.J., Klaus W., Simon R.C., Weng C.C., Paul W., Differential Recognition of Staphylococcus aureus Quorum-Sensing Signals Depends on Both Extracellular Loops 1 and 2 of the Transmembrane Sensor AgrC. J. Mol. Biol. 2008; 381: 300– 309.
It is can be saw that we have divide this model into two parts-detecting and killing. And in this part, we will depict the second part. According to the schematic, we have the reaction process as follows:
Assumptions
We model the dynamics of the synthetic Lysostaphin by accounting for the key reactions during the functioning of this system. In writing down the kinetic rate expression for these reactions, we make the following assumptions:
1. All the components are assumed to decay with first-order kinetics.
2. For constitutively expressed genes, the mRNA production rate is assumed to be constant. The synthesis rate of a protein is assumed to be proportional to the concentration of the corresponding mRNA.
3. In order to simplify, we set the initial value of extracellular AHLs constant.
4. The flux of AHL across the cell membrane is proportional to the concentration difference between the intracellular and extracellular space.
5. Actually, we do not have sufficient data for full parameters. However, we have information to estimate their value or use value that someone else has estimated yet.
6. Lysostaphin gene expression follows Michaelis-Menten-type kinetics and other reactions follow mass action kinetics. There is no crosstalk between different AHL signals.
ODEs
The state variables and parameters are described in detail in Table 1 and 2. Based on the listed reactions (in Table3), we write a series of ordinary differential equations(ODEs) to describe this part.
. Transcription of luxR gene = v1 – dMM （F1） . Interaction between intracellular AHL and LuxR-protein = a(A’-A) – dAA - kPAR + dPP （F2） = kRM – dRR – kPAR + dPP (F3) = kP*R*A – dP*P (F4) . Product Lysostaphin-protein by induced = – dNN (F5) = kLN – dLL (F6) There I should explain to you is in the equation F5, what the binding to promoter PluxI is the dimerization of P. So we use P2 rather than P. Table 1. State variable of the full model Variable Description A [AHL] in the target cell A’ [AHL] in the medium M [luxR-mRNA] N [lys-mRNA] R [LuxR-protein] P [LuxR-AHL] L [Lysostaphin-protein] Table 2. Kinetic Parameters of the full model Parameter Description Base value v1 Transcription rate for luxR-mRNA 3 dM LuxR-mRNA decay rate constant 2 c Maximal rate of lysostaphin gene transcription 10 α Sensitivity of lysostaphin gene transcription to AHL 1 η AHL diffusion rate constant across the cell membrane 10 dA AHL intracellular decay rate constant 2 kP AHL/LuxR binding rate constant 20 dP AHL/LuxR dissociation rate constant 10 kR LuxR synthesis rate constant 10 dR LuxR decay rate constant 2 kL Lysostaphin synthesis rate constant 10 dL Lysostaphin decay rate constant 2 Parameter values The base parameter setting of the model is listed in table 2. Several parameter values are directly taken from the literature or derived from literature data. For other parameters where we lack quantitative information, we use educated guesses that are biologically feasible and able to simulate our experimental findings. Then we set these values to our ODEs and solve by Matlab. Here is our function: ---matlab 代碼的鏈接--- And we write code as follows in Command Window of Matlab: [t,y] = ode45(@mysolve,[0 5],[1;0;0;0;5;0]); plot(t,y(:,4),t,y(:,6)) plot(y(:,4),y(:,6)) Thus, we get figures as follows: Fig 2. All the variables change with time. A’=5; M=1; LuxR=5. In order to complete this picture, we need presume some conditions: The time of t=0 is that extracellular AHL diffuse to the target cell Amount of mRNA is less than that of protein. So M=1 but A’=5、LuxR=5. What we care most is the amount of P and L(that is Lysostaphin). Thus we use Fig3 to spell out the relation between P and L. Fig 2. [P] and [L] change with time. In fig2 and fig3 , we can conclude: Binding of AHL and LuxR are fast. So P can reach the peak in a short time. Until the amount of P reach about 3.2, Lysostaphin increase quickly. So the threshold to induce promoter luxI in our model maybe 3.2. The moment AHL diffuse to intracellular, LuxR decrease sharp cause the binding to AHL has a high affinity. But when the extracellular AHL change, what’s the influence to the amount of Lysostaphin? With the purpose to solve the problem, we set all ODEs equal 0 that is steady-state. Then we write M.file to solve this problem in Matlab. Fig 4. In steady state, the relation of extracellular AHL and Lysostaphin As we can see in this picture, we can see when extracellular AHL less than 5, Lysostaphin change quickly along with extracellular AHL; but when extracellular AHL more than 5, the change of Lysostaphin become slow even steady. Then we can think that if we want to increase Lysostaphin significant, changing extracellular AHL in the region 0 to 5 may be needed. References Frederick K., Balagaddé et al (2008) A synthetic Escherichia coli predator-prey system. Molecular System Biology 4; Article number 187. Sally James, Staffan Kjelleberg, Patric Nilsson al.(2000) Luminescence Control in the Marine Bacterium Vibrio ficheri: An Analysis of the Dynamics of lux Regulation. J.Mol.Biol.(2000)296,1127-1137
