Team:Manchester/dspb
Antibiofilm Mechanism Modelling
Aims: To estimate the efficiency of our engineered bacteria at reducing formation of pathogenic bacterial colonies.
We set out to model the amount of dispersin B our engineered bacteria might be capable of producing in our catheter coating. Dispersin B degrades biofilm which is produced by pathogenic bacteria and helps to keep them alive. Our dispersin B will be secreted from our system in response to a molecule known as autoinducer-2 (AI-2) which is produced by pathogenic E. coli. Thus in our model we needed to model bacterial population, AI-2 production, dspB production and the relationship between dispersin B, biofilm and cell survival.
We were successful in creating the necessary equations to model our system, however we did not gain an accurate estimate of our engineered bacteria's ability to reduce the growth of pathogenic bacteria due to the presence of too many unknown variables. However after identifying these variables we created lab protocols that future times could use to characterise these variables and consolidate our quorum-biofilm model which could be used by future teams.
Our assumptions:
- Diffusion of dispersin B throughout the catheter was not a rate limiting step and thus the intracellular concentration, which we caculated, was the same as the extracellular conntration
- The production rate of the Lsr-operon (LsrABCD) is the same as the production rate of Lsr-DspB in our engineered bacteria.
- The concentration of our engineered abcteria remained constant throughout the model.
Variables
| Symbol | Description | Initial Value |
|---|---|---|
| t | Time | 0 minutes |
| [OP] | Lsr-operon concentration | 0 M |
| [REG] | Lsr Regulator Cocnentration | 0 M |
| [R] | LsrR Concentration | 0 M |
| [Ap] | Intracellular Phosphorylated AI-2 concentration | 0 M |
| [Aout] | Extracellular AI-2 Concentration | Estimate/model this uM |
Equations
Equation 1
$$ \frac{d\left[OP\right]}{dt\ }\ =\ \frac{k_{op}}{1+\left(\frac{\left[R\right]}{k_{1}}\right)^{nOP}+\left(\frac{\left[REG\right]}{K_{4}}\right)^{nOP}}\ -\ k_{deg}\left[OP\right] $$Equation 1 describes the rate of change of protein concentration produced by the Lsr promoter. Specifically this equation refers to the LsrABCD operon however in our model we assumed this concentration would be the same for our recombinant protein.
Equation 2
$$\frac{d\left[REG\right]}{dt}=\ \frac{k_{op}}{1+\left(\frac{\left[R\right]}{k_{1}}\right)^{nOP}+\left(\frac{\left[REG\right]}{K_{4}}\right)^{nOP}}\ -\ k_{deg}\left[REG\right]-k_{5}\left[Ap\right]\left[REG\right] $$Equation 2 is describing the rate of change of a hypothetical Lsr-regulator molecule suggested to exist in the source literature. [2]
Equation 3
$$ \frac{d\left[R\right]}{dt}=\frac{k_{r}}{1+\left(\frac{\left[R\right]}{k_{2}}\right)^{nR}}\ -\ k_{deg}\left[R\right]\ -\ k_{3}\left[Ap\right]\left[R\right] $$Equation 3 describes the rate of synthesis of LsrR
Equation 4
$$ \frac{d\left[Ap\right]}{dt}\ =\ k_{f}\left[A_{out}\right]+k_{imp}\left[OP\right]\left[A_{out}\right]-k_{3}\left[Ap\right]\left[R\right]-k_{5}\left[Ap\right]\left[REG\right] $$This equation describes the concentration of phosphorylated AI-2 inside the cell. This is contributed to by flux of AI-2 through the LsrABCD channel complex and alternative pathways. Ap is depleted by AI-2 binding to the LsrR protein and the theoretical Lsr-regulator molecule.
Equation 5
$$\frac{d\left[A_{out}\right]}{dt}\ =\ k_{AI2}\cdot X-k_{f}\left[A_{out}\right]-k_{imp}\cdot\left[OP\right]\cdot \left[A_{out}\right]$$Equation 5 gives us the rate of decrease of the exrracellular AI-2 concentration, this assumes that we have estimated an initial concentration of AI-2 which is not very realistic. In our model we added equations to describe bacterial concentration and AI-2 production, further details can be found here:https://2021.igem.org/Team:Manchester/Model
Equation 6
$$\frac{d\left[ApG\right]}{dt}\ =\ k_{5}\left[Ap\right]\left[REG\right]\ -\ k_{deg}\left[ApG\right]$$This equation gives us the rate of change of the Lsr-regulator/AI-2 complex.
Equation 7
$$ \frac{d\left[ApR\right]}{dt}\ =\ k_{3}\left[Ap\right]\left[R\right]-k_{deg}\left[ApR\right]$$This equation gives us the rate of change of the LsrR/AI-2 complex.
Equation 8
$$ \frac{d\left(X\right)}{dt}=X\cdot\left(1-\frac{X}{N_{max}}\right)\cdot\left(1-\frac{N_{min}}{X}\right)^{c}\cdot\left(r-B_{exposed}\cdot k_{dying}\right) $$Constants r and c are adjusted from: https://doi.org/10.1016/j.fm.2004.01.007 . This equation describes the growth of pathogenic bacteria. With the second term being negative and refereing to the fraction of bacteria which are exposed and not covered by biofilm (Bexposed), and thus more likely to die at rate (kdying).
Equation 9
$$ \frac{d\left(B_{exposed}\right)}{dt}\ =\ k_{break}\cdot\left[OP\right]\cdot\left(1-B_{exposed}\right)-k_{biofilm}\cdot B_{exposed} $$This equation describes the rate at which biofilm is broken down/created and thus the fraction of exposed bacteria (exposed refers to bacteria that are not covered by biofilm). This of course relies on the concentration of dispersin B and the rate at which bacteria produce biofilm.
We input these equations with the following parameter values:
Parameters
| Parameter | Description | Value |
|---|---|---|
| kop | Lsr-synthesis rate | 7 uM-1 min-1 [2] |
| kr | LsrR synthesis rate | 2 min-1 [2] |
| k1 | Repression Coefficient (Lsr-operon) | 0.2 uM [2] |
| k2 | Repression coefficient (LsrR) | 0.1 uM [2] |
| k3 | AI-2/LsrR binding rate | 0.5 uM-1 min-1 [2] |
| k4 | Repression coefficient (Lsr-regulator) | 65 uM [2] |
| k5 | REG/AI-2 interaction | 0.0001 uM-1 min-1 [2] |
| kf | AI-2 fluc through alternative pahways | 0.01 uM-1 min-1 [2] |
| kimp | AI-2 import via LasrABCD | 0.01 uM-1 min-1 [2] |
| nOP | Cooperativity coefficient (Lsr-operon) | 4 [2] |
| knR | class="model-th"Cooperativity coefficient (LsrR) | 4 [2] |
| kdeg | Protein decay rate | 0.02 min-1 [2] |
| kbreak* | Rate at which dispersin B dissovles biofilm | Range from 0.0005 to 0.08 uM-1 |
| kAI2* | AI-2 Production Rate | 0.0006 uM min-1 |
| kbiofilm* | Rate at which pathogenic bacteria produce biofilm | Mean value chosen to be 3 (no units) |
| r | Logistic function constant, divided paper’s value by 60 to change time units to mins | 0.038 min-1[5] |
| c | Logistic Function Constant | 0.74 [5] |
| kdying* | Rate of pathogenic bacteria population decrease | 0.076 min-1 |
*Estimated Uncharacterised Paramters
The unkoown parameters are:
- kbreak: The rate at which dispersin B breaks down biofilm.
- kbiofilm: Tha rate at which pathogenic bacteria produce biofilm.
- kdying: The rate at which pathogenic bacteria die when exposed outside of the biofilm.
- kAI2:The rate of AI-2 production by pathogenic bacteria
For each of the unknown parameters, we have chosen the mean estimated values in a way that produced the oscillating behavior of the bacteria and DspB protein, i.e. our system is on the edge of working efficiently. To analyse the influence of each parameter on the calculation results, these parameters are selected from a distribution of values that are evenly spaced from an arbitrarily-set range around their mean estimated values. When performing the calculations, we have kept all other parameters at their average values while varying one of the four estimated parameters. The results from this can be seen in the following four graphs:
These results suggest that kbreak, kbiofilm and kdying all have a large effect on the bacterial population however kAI2 has little effect. To fully consolidate this model as a potential tool for future teams to use in designing their own antibiofilm mechanisms, these unknown parameters must be experimentally characterized. For a team to use this model in their project they can do this by adapting our model with a synthesis model of their own promoter, altering their bacterial growth parameters to suit their environment, altering the kbreak value to suit their antibiofilm molecule and integrating the rest of our equations into their adapted model.
Overall, we have produced a system of equations that has the potential to be used by future teams in the design statge of their project that could help determine the feasibility of chosen antibiofilm mechanisms. We would encourage other teams to build upon our work to further expand this model and it's uses.
References
- Li J, Attila C, Wang L, Wood TK, Valdes JJ, Bentley WE. Quorum sensing in Escherichia coli is signaled by AI-2/LsrR: effects on small RNA and biofilm architecture. J Bacteriol. 2007 Aug;189(16):6011-20. doi: 10.1128/JB.00014-07. Epub 2007 Jun 8. PMID: 17557827; PMCID: PMC1952038.
- https://doi.org/10.1371/journal.pcbi.1002172
- https://doi.org/10.3389/fmicb.2019.01835
- https://doi.org/10.1038/ismej.2015.89
- https://doi.org/10.1016/j.fm.2004.01.007
- DspB assay: https://link.springer.com/article/10.1134/S0026261715040062#citeas