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Team:ULB-Brussels/Project/Mathematical
From 2009.igem.org
Contents |
Detailed Quorum Sensing system with integrated Toggle Switch
Sensibility to the initial condition
In the previous section, we have seen that the production process can start with only a few concentration of IPTG inside the system. This characteristics can be a problem from a practical point of view. If we look at the situation from a commercial or industrial point of view it is obvious that it is mandatory to implement a control system in order to avoid a non desired production of glue. In this section we present a theoretical improvement of the actual system which can solve this problem.
An improvement proposal
A natural way to improve our system is to use the well known toggle switch system in order
to reduce the sensitiveness of our system. The toggle switch system is described in [15].
The schema for this new model is shown on Figure 19. The only difference with the model in
the Figure 16 is the presence of a negative regulation of the c2 repressor on the c1 repressor.
The equation 18 is then modified as follows:
Let focus on the two first equations of the system. Assuming that we start from an initial state
in which [L] (t = 0) = 0, the two first equations reduce to (see the corresponding diagram on
Figure 20):
This is similar to the dynamical system described in [15]. The difference is that in the first equation the initial value of ¯ appears. If we choose the parameters α1 and α2 so that the system is initially in the bistability region, we can put the system in a state for which [c2] dominates the global dynamics. In that way, we are sure that the system will stay in its initial configuration.
We can see that in this case, we have a strong connection between the initial concentration of [c2] and the concentration of IPTG we have to add in order to switch to the production mode. It means that the system will not start to produce glue if a too small amount of inductor is added inside the system (by accident for example).
Dynamics of the system without inductor
When IPTG is lacking (β= 0), we find the same behaviour as in the previous model ( figure 21), the results from the previous linear stability analysis remain valid. On the figure 21(a) we see that as previously we need to put a minimal value of [c2] to stay in the initial configuration without inductor. However, it can be seen on the figure 21(b) that we recover the same influence of the initial value of the [c2] concentration on the amplitude of the initial perturbation.
Dynamics of the system with inductor
In this case we focus on the correlation which has been established between the minimal amount of IPTG which is needed to start the glue production and the initial value of the [c2] concentration. The bifurcation diagram is shown on Figure 22. If we compare with the analog diagram of the previous model ( Figure 18(a)), we see that in this new model, there is an obvious correlation between the initial value of [c2] and the quantity of IPTG which has to be added in order to produce the glue. We also notice that the needed IPTG values are higher. On Figure 23 we observe the time evolution of the different concentrations when the glue production is activated. About the glue production we have the same behaviour as for the model without toggle switch. The main difference lies in the dynamics of [c1]: it increases rapidly to a very high value. This is due to the very high initial concentration of [IPTG] which is added in this case. When [c2] varies we observe the same type of influence on the time evolution as for the case without toggle switch. We have to notice that, like in the previous model, when the glue production process starts, the final amount of glue which is produced is not influenced by the value of IPTG concentration or [c2] concentration.
Discussion and conclusion
In this part, we described our biological model from a dynamical point of view. Our aim was to identify the function of the main biobricks components present in the model. We showed that the system is able to switch from a stage of zero glue production to a stage where a steady state of glue production is reached. In our last model, this transition is entirely regulated by the quantity of IPTG inductor added in the system. Lacking such an inductor we observed that the system reaches a steady state for which there is no glue. But we saw that if the initial concentration [c2] is too low, the glue production can start without inductor. In the second configuration of our model, the glue production can start for a very small quantity of IPTG. This can be a problem from a practical point of view. Because of this high sensitiveness, the glue production could start just by accident. In order to increase the robustness of our system in regard to the IPTG concentration, it is useful to improve the current system by adding a toggle switch system between the c1 repressor and the c2 repressor biobricks. Indeed, with this last improvement, there is a minimal quantity of IPTG which is needed to start the glue production, This minimal quantity is strongly correlated to the concentration of [c2] initially found in the system. Moreover an increase in the initial quantity of [c2] leads to a diminution of the disturbance amplitude in the glue concentration if there is no IPTG. In the presence of inductor, the decrease in the initial concentration [c2] leads to an increase in the growth rate of the glue production.
From a more theoretical point of view, we can also address the following question: for this global system, which genes should we choose in order to obtain an optimal equilibrium between robustness and glue production? To answer that we studied the behaviour of our theoretical model with integrated toggle switch when the parameters values (such as activation concentration, degradation rates which are intrinsic to the genes) are modified. Firstly, an increase in the [c2] degradation rate (γ2) leads to an increase in the glue production rate when IPTG is added. Secondly, when we lower the value of the kLr and kLi constant, an increase in the glue production rate is also observed (see figure 24(b)). Thirdly, a decrease in the k1 constant value leads to a crash of glue production even in presence of IPTG. As we said in the previous section, the total amount of produced glue depends also on the intrinsic characteristics of the biobricks: a decrease in the γparE coefficient leads to a lower quantity of glue when the stationary state is reached (see figure 24(a)), α7 has also a similar influence.
