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Quantitative evolutionary stability analysis and design guidelines for kill switches

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1. Concentration dynamics of kill switches

This chapter summarizes the first working phase of the iGEM Marburg 2016 modeling work: The continuous ODE model presented previously is used with the biologically relevant parameters to compute the concentrations' dynamics \(\vec{X}(t)\) for all times. This shows that the designs are comparable in terms of the toxin expression levels. The latter three designs are comparable in terms of the number of model parameters.

At first, the lab environment is simulated in order to evaluate the toxin expression during the kill switch's OFF state. After that, the organism is released into wild life conditions – no inducers present – and the toxin expression of the activated kill switch is compared to the OFF state. From there, a toxin threshold is found.

This sections serves as verification that the non-mutated kill switch designs work as intended.

(a) BNU China 2014

The BNU China 2014 kill switch (find design here) is simulated and the results are shown in figure 1. One observes a significant increase of the toxin concentration once the organism is released into wild life conditions which lacks an inducer. The lethal toxin threshold is defined as based on the simulation results. This is the mean of the kill switch's OFF and ON state. It is consistently drawn as dashed red line in the following.

Figure 1. Simulated BNU China 2014 kill switch dynamics. All involved concentrations are simulated. The first subplot shows the externally set inducer – IPTG – concentration. The last subplot shows the resulting toxin expression and the lethal toxin threshold \(\theta\). All subplots in-between correspond to the concentrations of intermediate substances.

Interestingly, one can observe a noticeable delay between the inducer turned off and the toxin turning on. This is due to the signal traveling through the network at finite speed.

The next kill switch designs represent different generic kill switch topologies and abstract from specific biological substances to substances that behave just like their biological counterparts. This enables to focus on the bare kill switch design (see algorithm description for further details here).

(b) Parallel (OR)

This design utilizes two redundant toxin regulating branches (find design here). Being connected by an OR gate, this provides a backup mechanism in case one branch is disabled through mutations. The result can be found in figure 2.

Figure 2. The modeled Parallel (OR) kill switch with biologically correct parameters. The first two subplots show the two abstract inducers’ concentrations \(I_a\) and \(I_b\). The second inducer, \(I_b\), is kept on after release to resemble a destroyed kill switch through mutations. The blue curve in the last subplot shows the toxin expression with both branches working properly while the dashed green line with branch b deactivated. Since both concentrations are well above the threshold \(\theta\), the kill switch is in its ON state in both cases.

(c) Serial

The Serial kill switch design is simulated (find design here). Figure 3 shows the kill switch typical behavior.

Figure 3. The modeled Serial kill switch with inducer in the upper subplot and the toxin in the lower subplot. The toxin threshold is exceeded such that the kill switch is in its ON state after the inducer having been turned off.

Since this design introduces a longer serial cascade than the BNU China 2014 design, the delay between inducer deactivation and toxin expression is longer.

(d) Parallel (AND)

A parallel design that connects both branches with an AND gate is studied (find design here). Figure 4 shows that both branches need to be functional to trigger the kill switch's ON state.

Figure 4. Simulated Parallel (AND) kill switch. The subplots are equivalent to the one shown in figure 2. Artificially deactivating a branch leads to a kill switch remaining in its OFF state. A single branch does not regulate the toxin.

As evident from figure 4, the deactivation of one of the two branches leads to the kill switch being in the OFF state permanently - thus, the branches are indeed linked by an AND gate.

2. Evolutionary stability

For different incubation times and mutation rates, the escape probability of the BNU China 2014 kill switch design is studied. Figure 5 shows the results from which two intuitive results are confirmed:

The higher the mutation rate, the larger the escape probability: As the likelihood for a mutation to occur increases, the probability of the linear cascade to be destroyed through one of these mutations increases as well.
Longer incubation times lead to higher escape probabilities: This is due to more possible attempts for mutations to destroy the kills switch.

Figure 5. Evolutionary stability of BNU China 2014 kill switch. The escape probabilities for different mutation rates \(\eta\) are shown. Longer incubation times and mutation rates lead to larger escape probabilities.

(b) Time scaling by mutation rate

Scaling the incubation time with the mutation rate lets the escape probability curves collapse onto one master curve. This can be explained: Hitting a certain threshold takes more incubation time when a low mutation rate is applied but it takes less incubation generations when working with a high mutation rate. The master curve for the BNU China 2014 kill switch can be found in figure 6.

Figure 6. Evolutionary stability master curve of BNU China 2014 kill switch. The mutation rate scales the incubation time axis so that the curves collapse onto a master curve for a unified time scale.

Therefore, the mutation rate is arbitrary for quantifying the evolutionary stability of a kill switch design. Computations have been performed for \(\eta\in\{0.4081875 \cdot 10^{-3}, 0.816375 \cdot 10^{-3}, 1.63275 \cdot 10^{-3}\}\) but only \(\eta=1.63275 \cdot 10^{-3}\) is shown in the following. Other mutation rates are equivalent and the results show the same behavior.

(c) Design guidelines for a kill switch

We are providing design guidelines for a kill switch that is robust against evolutionary stability and show quantitative evidence for our suggestion. A kill switch design can be understood as a network with the toxin being a node that is controlled through regulation pathways.

The BNU China 2014 kill switch design uses a single regulation pathway to control its toxin MazF. From intuition it is clear that a kill switch with multiple parallel regulation pathways for its toxin – with only one required to control the toxin expression – should be less prone to being destroyed by mutations. Let a mutation destroy one of the parallel regulation pathways: There are still other pathways that back up the destroyed one and ensure a working kill switch.

We utilize three comparable kill switches that differ in their network topology. Figure 7 shows the escape probabilities for the Parallel (OR), Serial and Parallel (AND) designs. Parallel (OR) – implementing a topology that allows destroyed branches to be backed up – shows by far the smallest escape probability. Where a single mutation is capable of destroying the Parallel (AND) and Serial design, a destroyed pathway can be compensated by the functional regulation pathway for the Parallel (OR) design.

Figure 7. For \(\eta=1.63275\cdot 10^{-3}\), the escape probabilities of the three kill switch designs are shown against the incubation time. The Parallel (OR) design has the smallest escape probabilities. The explanation can be found in the text.

Concluding, we suggest to implement a redundant toxin regulation for kill switches. This significantly increases the evolutionary stability and decreases the escape probability of genetically modified organisms into the environment. Biotechnological applications ranging from small to industrial scale can be made safer with having this in mind.

However, introducing additional topology such as multiple redundant toxin regulations increases a kill switches complexity. There are more points of contact for mutations to destroy the topology. Translating this in the model we use, more complex kill switches have more ODE parameters that can be mutated. Therefore, the escape probability is expected to somehow scale with a kill switch's complexity.

This remaining question of a trade-off between kill switch complexity and desired network topology to find a robust kill switch is studied in the last chapter.

(d) Comparison between BNU China 2014 and Parallel (OR) kill switch

We compare the escape probabilities of the BNU China 2014 design and the hypothetical Parallel (OR) kill switch. The latter introduces on the one hand redundant toxin regulation but on the other hand an increased complexity. That manifests in almost double the number of ODE parameters in comparison to the BNU China 2014 design.

Figure 8 shows that the stability-optimized Parallel (OR) kill switch is more robust against evolution and mutations than the BNU China 2014 topology. The gain in evolutionary stability through the additional regulation units overcompensates the introduced complexity. Finally, this leads to a more robust kill switch when using a redundant toxin regulation.

Figure 8. Comparison between BNU China 2014 and Parallel (OR) kill switch. The latter one shows better robustness in terms of escape probabilities.

This result has various applications in real life. Redundant toxin regulation not only ensures projects in Synthetic Biology to be safer but also enables longer incubation times until a certain threshold in escape probability is reached. Organisms can therefore be longer incubated without becoming evolutionary instable. This simplifies industrial applications and increases overall efficiency in production processes.

The code framework is available upon request (Martin Lellep, mailto:martin.lellep@physik.uni-marburg.de). We provide possible outlooks on the following website.

Team:Marburg/Modeling/Results