Team:Tsinghua/Model
The Quorum Sensing model
Outline
Quorum Sensing plays an important role in biofilms, as it enables bacteria to act as a whole rather than individuals. In our project, quorum sensing in the Pseudomonas aeruginosa (below as P. aeruginosa) biofilm is vital as our modified E. coli relies on the signaling molecules to perform the functions we desired. To describe the process, we adapt existed models for quorum sensing in P. aeruginosa, and utilize it to obtain values of concentrations of substances we wanted with respect to time. On the other hand, since the mechanisms in our modified E. coli is rather genuine and only few data are available for documentation, we made several assumptions to describe the NO producing processes in our E.coli.
In general, we hope to apply the model as a way to observe trends of the concentrations of substances before actually performing experiments. In addition, the model might give us insights about the mechanisms and help us decide the optimal amount of E. coli in order to obtain the desired NO amount.
Model Assumptions and Parameters
The model consists of the two regulatory systems of P. aeruginosa, the las system and the rhl system. The variables involved are mainly the responsible components, complexes and external autoinducers. Below is a schematic diagram showing the gene regulation systems in P. aeruginosa.
Firstly, we assume that there is no shortage of substrate for autoinducer production, thus there is noneed to explicitly model the biosynthesis of 3-oxo-C12-HSL and C4-HSL by LasI and RhlI.
Secondly, we assume that production of LasR, RasL, and RhlR follows Michaelis–Menten kinetics. This enables us to focus on the main mechanism and omit the influence of mRNA producing them.
Thirdly, we assume that the production of 3-oxo-C12-HSL and C4-HSL also follow Michaelis–Menten kinetics, and that the autoinducers freely diffuse across the cell membrane.
Finally, we assume the E.coli only interreact with the biofilm via BHL quorum sensing, that is, the E.coli is independent from activities of the biofilm and function individually as a function of BHL.
Below are the variables and parameters we use in the model:
The overall model is governed by 8 ODE equations describing the intercellular activities of P. aeruginosa, and external concentration of autoinducers are also taken in account.
Model Design
The model mainly splits into three parts: proteins, dimers and autoinducers. Each part described using ODEs with assumptions made above.
The below 3 equations describe the rate of concentration of proteins in the cell, with the main contribution term being the Michaelis–Menten kinetics production term, and other describing bindings and dissociations of complexes, following the assumptions made above:
With the same assumptions, rate of concentration of autoinducers can also be described as the follow 2 equations:
At last, for the dimers, the rates are simply described as production via law of mass at certain rate and dissociation at certain rate:
We now turn towards the density of the biofilm to the sample space, that is, how the size of the biofilm would affect the behavior of the quorum sensing. By introducing the local volume fraction, ρ , one could describe the concentration of external autoinducers with respect to the fraction. As assumed above, the external autoinducers could diffuse into the cell membrane freely via a given conductance and decay with a rate. One thing to notice is that transportation of autoinducers in P. aeruginosa consists of two ways: passive diffusion and active pumping through the cell membrane. Yet in our model we solely consider the diffusion process.
We further assume the cell density is uniform with respect to the dimensions. This helps us simplify the model and focus more on the variations of the concentrations. In addition, we assume that the extracellular space is well-mixed, and hence we would not need to consider different area respectively.
With assumptions made, the concentrations of external autoinducers could be described with ODEs as follow:
Since the variables have altered, the original equations needed to be rewrite with the volume fraction term, namely the variable [3-oxo-C12-HSL] and [C4-HSL]:
Model Experiment
We apply MATLAB odesolver to solve the ODE systems and produce plots for analysis.
From the above result, we can see that the concentration of C4-HSl converges to steady state within a period of time. This provides solid basis for our E. coli to perform the function as it implies that a stable amount of C4-HSl could be detected, thus a non-decreasing amount of NO could be produced.
It is also worth notice that in the rhl system, the 3-oxo-C12-HSL can delay activation of the rhl system by binding to RhlR. Thus, production of C4-HSL might be affected. We experiment this complex action by changing the binding rate of RhlR and 3-oxo-C12-HSL from 0.1 to 0.7.
which could be seen does not have much impact on the concentration of C4-HSL along time. So such effect could be neglected.
On the other hand, by varying the value of local volume fraction, we could see that the larger the biofilm is, the higher the concentration of the autoinducers. This is quite intuitive as the production of autoinducers is correlated to the number of cells present.
The E.coli Function
From the above model and experiment, we can specify the concentration of C4-HSl given parameters and initial conditions, which enables us to model the mechanism of the E. coli function as now the input is clarified. In our model, we describe the E. coli as a function mapping the concentration of C4-HSl to the concentration of NO.
Since the NOS part is rather genuine and only few related experiments are performed before, it becomes difficult for us to determine a kinetic model with precise parameters for the E.coli function. This makes us turn into observing the activation of GFP against the concentration of C4-HSl in the E. coli. By assuming that the activation rate of NOS is the same as GFP, we can obtain an approximation of concentration of NOS given concentration of C4-HSl, and one can estimate the NO generated by the E. coli. An equation describing the process states follow:
in which 0《a《1 is to be determined via data fitting or experiment, and the exponential term represents the activation of NOS in the E.coli. It should be notice that the exponential term in fact should be described with Michaelis–Menten kinetics as follows However, in our project we focus more on the early stage of NO release by the E.coli. Therefore, exponential function would be a great candidate to estimate the output.
Discussion
From the model and experiments we performed, it could be seen that given enough substrate for the P. aeruginosa biofilm to grow, there is enough C4-HSL to activate our modified E. coli to produce amount of NO needed to induce dispersal of the P. aeruginosa biofilm. This provides numerical evidences and data for the experiments and further solidifies our idea. The E. coli function we proposed could provide a rough estimation of amount of NO produced at a specific time, which could also be used to estimate amount of E.coli we needed to produce a specific amount of NO as the function is bijective, which ensures a unique and existed solution for the given NO.
On the other hand, there are still some defections in the model. One of them are the local volume fraction parameter. In the above model the volume fraction is simply a constant which does not reveal actual scenarios very much. One improvement could be replacing the parameter with time-dependent growth function, either exponential or logistic, as:
and:
Where:
We believe this would produce more valid and realistic simulation to the interactions happening in the biofilm.
Discussion
From the model and experiments we performed, it could be seen that given enough substrate for the P. aeruginosa biofilm to grow, there is enough C4-HSL to activate our modified E. coli to produce amount of NO needed to induce dispersal of the P. aeruginosa biofilm. This provides numerical evidences and data for the experiments and further solidifies our idea. The E. coli function we proposed could provide a rough estimation of amount of NO produced at a specific time, which could also be used to estimate amount of E.coli we needed to produce a specific amount of NO as the function is bijective, which ensures a unique and existed solution for the given NO.
On the other hand, there are still some defections in the model. One of them are the local volume fraction parameter. In the above model the volume fraction is simply a constant which does not reveal actual scenarios very much. One improvement could be replacing the parameter with time-dependent growth function, either exponential or logistic, as:
and:
Where:
We believe this would produce more valid and realistic simulation to the interactions happening in the biofilm.
Where:
We believe this would produce more valid and realistic simulation to the interactions happening in the biofilm.
Reference:
[1]1. Fagerlind, M. G. et al. The role of regulators in the expression of quorum-sensing signals in Pseudomonas aeruginosa. Journal of molecular microbiology and biotechnology 6, 88-100 (2003).
[2]2. Klapper I, Dockery J. Mathematical description of microbial biofilms. (2010). SIAM Rev 52:000–000
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