To summarize everything mentioned in the Wild Cell and Modified Cell tabs, the complete equation system is presented with all the final modifications. Aditionally, simulations are carried out to visualize the impact that the aiiA enzyme had over the QS.
Based on the law of mass action that states that the rate of a chemical reaction is proportional to the product of the concentrations of the reactants, the whole modified ecosystem is represented in the next chemical reaction network
To be able to visualize the numbers obtained from modeling, we simulated the behavior described before, on the MathWorks® platform MATLAB. Thanks to these simulations we were able to compare the graphs between the variables concentrations inside a Wild Erwinia amylovora and a Modified one. The results were the following:
EamI gene behavior
In the graphic of the wild cell (graph 1) we observed that the EamI gene expression is not significant until the cell density (p) reaches a critical value, around 1800 in this particular scenario. Considering a time horizon of 300 and a range of values of cell density from 0 to 5000 we obtained an increasing EamI concentration from zero to a stationary value of 136.4.
In contrast, in the graphic of the modified cell (graph 2), we observed that in the same horizon of time and with the same range of cell density, there is not enough time and density to reach quorum sensing, sustaining the hypothesis that the aiiA will help to inhibit the virulence gene expression. The EamI protein will reach a concentration equal to 0.9567, 99.30% less than the observed in the wild cell.
In comparisson with the previous behavior, the Complex concentration in the wild cell also tends to reach a stationary value at .1893 (as seen in graph 3), remarkably lower than the observed in the EamI concnetration. Altough it does not seem very relevant, from an initial condition of zero to its stationary state, it does represent an important increase in the P concentration.
Now, in the modified cell (graph 4) this does not happen and instead we observed a slowly increasing behavior that tends to 1.7257x10-5, we can even say that it relatively stays at zero. Under the same contitions of time horizon and cell density range, we again found an inhibited behavior in the modified cell.
In both cases, the Complex will always depend on the EamR protein concentration (RT) because the total number of EamR monomers will determinate the amount of finite complexes that can be formed. In this case the RT parameter was fixed at 0.5, avoiding that the any Complex concentrations exceeded this value.
Intracellular AHL behavior
In the both graphics we observed that the intracellular AHL concentration is influenced by the value of the cell density parameter. But, that the level of influence also is affected by the presence or absence of aiiA gene expression.
In the left graphic, representing the wild cell model, we identified in this particular scenario (with a cell density range from 0 to 5000) that there is not until the cell density reached a value near to 1800 that the Intracellular AHL concentration increased considerably from zero to 0.7276 . Notice that there is not a stationary value showed in the graphic.
In the right graphic, representing the cell model considering the presence of aiiA gene, the increment of Intracellular AHL concentration is not significant, as it went from 0.00067 to 0.0017 reaching a value 99.76% smaller than the one obtained without the influence of the aiiA.
Extracellular AHL behavior
The wild cell extracellular AHL behavior observed in graph 7 tends to reach a value of .5457. As seen in the Complex behavior, the logistic tendency was observed but at what seems to be a low value. Then again taking under consideration that it started at zero, its growth is considerable big enough.
Unlike the wild cell, the modified cell reaches an even lower value of 0.0013. In spite of what it seems to be a growing behavior, with the same conditions of time and cell density range, the amount of Ae was significantly delayed.
In this scenario, the D parameter has a lot of impact in the Ae(t) behavior. This happens because at a lower extracellular AHL loss value, the more AHL molecules are available in the media to enter the cells and a lower cell density could trigger the QS. In the other hand a high fixed value of D will represent a high rate of AHL loss and more cell density will be needed for reaching the QS.
Behavior through time
In the previous simulations, the cell density was compared against each variable to visualize the individual critical values of cell population that would trigger QS. In this section it is the time who is analyzed against the four variables at once. In graph 9 At first sight it looks like the Complex, intracellular and extracellular AHL concentrations are not changing through time, since the blue line, which represent the EamI concentration, presents a big increase around the 130 x-axis value. It is not until making a zoom to the image that we can appreciate the logistic growth not only in the EamI concentration, but also in the rest of the other three variables.
In the modified cell (graph 10) we identified two main differences with the wild cell model. First, that in the same horizon of time the variables have not yet reached a logistic growth. And second, the final concentration values are considerably smaller than the ones obtained in the wild cell, in fact all of them reduced in almost a 99%.
After analyzing the contrast between the wild and modified cell we concluded that under the same conditions (time and cell density) the modified cell will have a significant inhibition in reaching the critical value for the QS activation and therefore the virulence expression, due to the aiiA capacity to delay the growing behavior of all variables (EamI, AHL-EamR complex, intracellular and extracellular AHL). Nevertheless, if the horizon of time and the range of cell density are expanded strategically (as did in graph 11), the modified cell, in fact, can reach the QS. In this particular scenario, the QS was not completely eradicated but remarkably delayed. The modified cell needs around four times the time and cell density of the wild cell to be able to reach the critical value that triggers the QS.
Noticing that a basal state value of the EamI protein was taken under consideration which influenced in the synthesis of EamI, it did not matter that this value was as small as we wanted, the concentration of EamI in the media always existed. This is the reason why a predisposition of reaching QS inevitably remains, but always depending on the intrinsic characteristics of Erwinia amylovora.
Apart from the numerical results, our experimental results concluded the same thing, the aiiA enzyme partially inhibits, or more properly said, delays the QS gene expression and therefore the virulence expression too. In the Demonstrate section, an experiment was carried out on apple tree leaves, where they were inoculated with a modified Erwinia amylovora and with wild-type cells. Differences in disease development (both the severity and onset time) were seen between the wild-type E. amylovora and the one transformed with the aiiA BioBrick (BBa_K2471000). When incorporating the aiiA BioBrick into E. amylovora, leaf necrosis was inhibited in comparison with the wild-type one.
It is important to conclude that the activity of only one enzyme will not inhibit the virulence by itself, reason why the project revolves around the interaction of the three enzymes: aiiA, epsE, yhjH, to counteract the virulence in an integral way.
Modified CellGraph 11. Quorm sensing conditions for a Modified cell.
If you are interested in learning and understanding a little bit more of our model simulations, you can check-out our MATLAB codes with just a click.
Through the model we described the E. amylovora's behavior changes caused by the aiiA enzyme insertion. Nevertheless, the project is based on the combination of three enzymes to reach our goal and inhibit the E. amylovora's virulence factors. We are aware that there is more to do, so with a little help from our friends from IONIS Paris, the yhjH enzyme model is now a future project. They helped us to 3D model the yhjH enzyme (shown below) and to take certain conditions under consideration to its proper activity. Thanks to this edition model, the unregulated gene inhibition over a regulated one can be taken as a basis, and together with the 3D simulation, a second model could be achieved.
Barnard, A. and Salmond, G. (2006) Quorum sensing in Erwinia species. Analytical and Bioanalytical Chemistry, 387(2), pp.415-423.
Frederick K. Balagaddé et al. (2008) A synthetic Escherichia coli predator–prey ecosystem, EMBO, 187, pp. 1-26.
Ingalls, B. (2013) Modeling of Chemical Reaction Networks & Gene Regulatory Networks. From Mathematical Modeling in Systems Biology(pp. 21-314 ). England: MIT press.
James, S. et al. (2000) Luminescence Control in the Marine Bacterium Vibrio fischeri: An Analysis of the Dynamics of lux Regulation., JMB, 296, pp. 1127-1137.
Koczan JM, McGrath MJ, Zhao Y, Sundin GW (2009) Contribution of Erwinia amylovora exopolysaccharides amylovoran and levan to biofilm formation: implications in pathogenicity. Phytopathology 99:1237-1244
Tzvia Iljon, Jenna Stirling and Robert J.Smith (2012) A mathematical model describing an outbreak of fire blight. En Understanding the Dynamics of Emerging and Re-Emergign Infectious Diseases Using Mathematical Models(91-104). University of Ottawa, Canada: Transworld Research Network.
Venturi, V., Venuti, C., Devescovi, G., Lucchese, C., Friscina, A., Degrassi, G., Aguilar, C. and Mazzucchi, U. (2017) The plant pathogen Erwinia amylovora produces acyl-homoserine lactone signal molecules in vitro and in planta.