# Team:USTC CHINA/Modeling/KillSwitch

### From 2013.igem.org

# Introduction

To be more user-friendly, 4# circuit contains a reporting system. After melting in water, the spores germinate and express blue pigment protein to report the optimal using time.

Meanwhile, 4# circuit could also ensure biosafety. Since other circuits do not contain self-killing device, 4# engineered bacterial would kill all the bacterial after being used.

# Designing of the suicide system

We designed a circuit of killing switch based on its endogenous genetic system.

In *B.subtilis*, when stationary phase begins, the environmental pressure increases and nutrition becomes limited, forcing *B.subtilis* to produce spores. Then the community will be divided into two different parts. One is trying to kill the other one for adequate nutrient, delaying the production of spores and obtaining competitive advantages. Killing is mediated by the exported toxic protein SdpC. SdpI will appear on the membrane surface to avoid itself from being damaged. As a membrane protein, SdpI could bind free SdpC and autopressor SdpR, removing SdpR’s inhibition against I and R, producing more SdpI to offset SdpC and finally guaranteeing the subgroup alive, thereby delaying the spores production.

We transfered SdpC, which is fused with promoter SdpI/R into high copy plasmids, to damage the balance of the system and kill whole colony. When SdpC appears, SdpI on the membrane will bind free SdpC and adsorb SdpR to cease its inhibition against SdpI P/R, trying to produce more SdpI. At the same time, it will activate the promoter SdpR/I in our circuit and generate more SdpC. The system would fall into an infinite loop, and according to our modeling, the amount of SdpC can increase beyond the accommodation of SdpI. Thus, the cells with protection mechanism will finally collapse due to too much SdpC. All above forms the killing device. We also designed a test circuit, which contains promotor grac and sdpABC solely, to check the toxicity of SdpC.

There are both positive and negative feedback loops in this process. On the one hand, SdpI is unable to sequestrate the autorepressor, SdpR, until it captures the toxin, SdpC. The accumulation of SdpC will thus facilitate SdpI to capture more SdpR and thereby relieve the repression of SdpR, stimulating the expression of itself. This is the positive feedback loop, which leads to the increasing accumulation of SdpC and the death of the colony. On the other hand, the removal of SdpR also enhances the expression of SdpI and accelerates the sequestration of SdpC, which forms a negative feedback loop whose effects contradict the positive feedback loop. However, since the copy number of SdpC is much higher, we believe that the positive loop is strong enough to outweigh the negative one, which guarantees this circuit will finally leads to collapse instead of equilibrium.

# The ODE model of singular cells

There is no denying fact that the essential goal of engineered bacterias who carry this so called “suicide” circuit itself is to kill their siblings rather than themselves and ensure the survival of themselves. Surely they can kill their siblings, but can they finally eliminate themselves, as we expect? The trivial experiment protocol and huge uncertainty had put off our experiment, and as expected, we failed to achieve the construction of complete reporter system in our limited laboratory. Fortunately, we could resort to mathematical models to verify the validity of this circuit theoretically. There are six independent variables in individual cells, and theoretically if the initial conditions are fixed, all of them will be the univariate functions of time. The following table illustrates the mark and meaning of each variable:Taken as a statement about kinetics, the law states that the rate of an elementary reaction (a reaction that proceeds through only one transition state, which is one mechanistic step) is proportional to the product of the concentrations of the participating molecules. In modern chemistry this is derived using statistical mechanics. Despite the complicated chemical reactions involved in transcription and translation, it is common and logically sound to view the expression of one particular gene as an elementary reaction and assume the repression effects of the protein itself encodes and the repressor are both linear.

According to the law of mass action, we got six independent differential equations of the variables:

The following table explain the constants in the above ODE group:

Mark | Meaning |

I_{max} |
Mole number of free SdpI in cytoplasm. |

I_{m} |
Mole number of SdpI in the cell membrane. |

C_{f} |
Mole number of free SdpC in cytoplasm. |

C_{i} |
Mole number of SdpC captured by SdpI. |

R_{f} |
Mole number of free SdpR in cytoplasm. |

R_{i} |
Mole number of SdpR captured by SdpI |

Mark | Meaning |

I_{max} |
The maximal number of SdpI than can be fixed on the cell membrane. |

k_{0} |
Constant describes the normal expression rate of SdpC |

k_{1} |
Constant describes the self-repression effects of SdpC |

k_{2} |
Constant describes the repression of SdpR on the expression of SdpC. |

k_{3} |
Constant describes the rate of SdpI capturing SdpC |

k_{4} |
Constant describes the normal expression rate of SdpR |

k_{5} |
Constant describes the self-repression effects of SdpR |

k_{6} |
Constant describes the rate of SdpI capturing SdpR |

k_{7} |
Constant describes the normal expression rate of SdpI |

k_{8} |
Constant describes the self-repression effects of SdpI |

k_{9} |
Constant describes the repression of SdpR on the expression of SdpI |

k_{10} |
Constant describes the rate of SdpI binding to the cell membrane |

## Discussions on the constants

All the constants given above is steady and theoretically measurable when all the conditions are constant. For example, we could measure k_{0} by constructing a new engineered bacteria, which contains the gene encoding SdpC and marker gene alone, and observing the influence of the concentration of SdpC on its own expression. Yet any modification on genome is notoriously time-consuming, which inhibited us from measuring them in our tiny laboratory. We also looked up oceans of papers to confer their approximate ranges, but almost all papers are too fragmental to provide any valid information. Therefore, we decided to assume all these constant according to our limited information and make a qualitative analysis instead of quantifiable analysis, while in fact the latter one is impossible. All units and dimensions were temporarily ignored. In other words, our model aims at justifying the validity of this suicide mechanism rather than predicting the exact time or any other parameters of the system.
Despite the fact that we have hardly any accurate data on these constants, there are some limitations that we extrapolated from known information before we further explore this model:

- k
_{0}>>k_{4}≈k_{7}: k_{0},k_{4}and k_{7}represent the normal expression rate of SdpC, SdpR and SdpC separately, and the copy number of SdpC is much larger than that of SdpR and SdpI, whereas the value of the latter two is approximately equal; - k
_{2}>>k_{9}: the existence of free SdpR represses the expression of both SdpI and SdpC, and similarly, since the copy number of SdpC is much higher, we expected the repression effect was stronger accordingly; - k
_{10}>>k_{3},k_{6}：it is hard to predict the value of k_{3}and k_{8}, but we supposed both of them were much smaller than k_{10}because SdpI is a membrane protein inherently, and rarely exists as free protein; - The primary values of all the six variables are very small or strictly zero. We expect this as the most logical initial status. If the primary value of any variable is relatively large, the suicide mechanism may not run normally.

## Stimulation and discussion

Simple and rough as the above model is, it does theoretically sound. To test the validity of this model, we first tried to get analytic solution of the ODE set. If this analytic solution exists, we could further investigate the interaction among those variables, and draw some phase planes to get accurate and mathematically perfect descriptions of this model. Unfortunately but expectedly, the existence of analytic solution was negated by MATLAB, and we had to assume groups of values for these constants in advance and analyze the arithmetic solutions instead. These arithmetic solutions not only justified this mechanism is effective enough to commit cell suicide but also indicated some unexpected, or even weird results that beyond our wildest imagination. There are two possibility accounting for the unexpected results: our model is too rough to include some assignable factor; or there are some implicit but objective limitation inside this model, which may be substantiate by later experiments or papers.

When we explored the arithmetic solutions of this ODE set, we received nearly one hundred warnings from MATLAB and for many times our most powerful computer ran out of its 8GB memory, but sometimes we can receive the solution within seconds. We had adjusted our parameters for several times before we got our first solution. Here is the values of parameters for this group, and the graph of arithmetic solutions is also given:

k_{0} |
k_{1} |
k_{2} |
k_{3} |
k_{4} |
k_{5} |
k_{6} |
k_{8} |
k_{8} |
k_{9} |
k_{10} |
I_{max} |
C_{f0} |
R_{f0} |
I_{f0} |
I_{m0} |
C_{i0} |
R_{i0} |

50 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 20 | 500 | 5 | 5 | 1 | 5 | 3 | 2 |

_{f}increases rapidly. But when we turned our attention to the curves of other parameters, things seemed not so platonic:

The curve of I

_{m}, C

_{i}and R

_{i}contradicted our common sense severely. First, I

_{m}>C

_{i}>R

_{i}is expected to be tenable all the time, which precludes the intersects among the three curves; Second, there is no mechanism in this system that could decrease their concentration, and all of them are expected to be increasing function; Third and most serious, never will them be negative, as they represent the concentration of real substances. Then we adjusted the parameters slightly for several times. To eliminate those absurd curves, we reconsidered some assumptions. Here we listed another representative group of parament values and relative graph:

k_{0} |
k_{1} |
k_{2} |
k_{3} |
k_{4} |
k_{5} |
k_{6} |
k_{8} |
k_{8} |
k_{9} |
k_{10} |
I_{max} |
C_{f0} |
R_{f0} |
I_{f0} |
I_{m0} |
C_{i0} |
R_{i0} |

100 | 5 | 5 | 5 | 5 | 5 | 5 | 20 | 5 | 5 | 20 | 500 | 5 | 5 | 1 | 5 | 3 | 2 |

k_{0} |
k_{1} |
k_{2} |
k_{3} |
k_{4} |
k_{5} |
k_{6} |
k_{8} |
k_{8} |
k_{9} |
k_{10} |
I_{max} |
C_{f0} |
R_{f0} |
I_{f0} |
I_{m0} |
C_{i0} |
R_{i0} |

100 | 5 | 5 | 5 | 5 | 5 | 5 | 20 | 5 | 5 | 20 | 500 | 5 | 5 | 1 | 5 | 3 | 2 |

_{2}equal to k

_{9}. We also gave positive values to I

_{m}, C

_{i}and R

_{i}, which were considered zero at first. And by groups of stimulations we realized the value of k

_{2}does matter, as the derivative of C

_{f}only increased slightly as k

_{2}lowers, and the positive values failed to avoid the weird phenomenon in the latter three curves.

We also found that however we adjusted the primary value of I

_{f}and other parameters, I

_{f}dropped into approximately zero extremely rapidly at the initial stage and remained balanced, which might account for why the derivatives of the latter curves were abnormally negative. Thus we modified another assumption and increased k

_{7}. Here is another group of values and the corresponding graph:

Although the derivative of I_{m} is not seriously positive constantly, the three latter curves seemed much more reasonable. Hence, we extrapolated that although SdpI and SdpR share the same promoter, the expression of SdpI must much faster than SdpR to ensure successful “suicide”. Additionally, the increase of k_{7} also represses SdpC, which requires the copy number of SdpC must be larger.
We kept all other parameters constant and gradually augmented k_{0}. The larger k_{0}, the more perfect the curve seemed, and here are the values table and graph where k_{0} equals 400, 80 times larger than k_{4}.

k_{0} |
k_{1} |
k_{2} |
k_{3} |
k_{4} |
k_{5} |
k_{6} |
k_{8} |
k_{8} |
k_{9} |
k_{10} |
I_{max} |
C_{f0} |
R_{f0} |
I_{f0} |
I_{m0} |
C_{i0} |
R_{i0} |

400 | 5 | 5 | 5 | 5 | 5 | 5 | 20 | 5 | 5 | 20 | 500 | 5 | 5 | 1 | 5 | 3 | 2 |

_{f}and R

_{f}separately, the curves seemed more perfect:

In wild bacteria who are unable to produced SdpC, naturally k

_{0}equals zero. We expected C

_{f}would decreased gradually and finally approximate zero, and here are the corresponding table and graph:

k_{0} |
k_{1} |
k_{2} |
k_{3} |
k_{4} |
k_{5} |
k_{6} |
k_{8} |
k_{8} |
k_{9} |
k_{10} |
I_{max} |
C_{f0} |
R_{f0} |
I_{f0} |
I_{m0} |
C_{i0} |
R_{i0} |

0 | 5 | 5 | 5 | 5 | 5 | 5 | 30 | 5 | 5 | 20 | 500 | 8 | 5 | 1 | 5 | 3 | 2 |

Wired but not surprising, there were intersects among the latter three curve, and C

_{f}decreases continually when it is negative. We continued to try groups of these parameters, and this is the best one in which we increased the primary concentration of SdpC and the normal expression rate of SdpI.

k_{0} |
k_{1} |
k_{2} |
k_{3} |
k_{4} |
k_{5} |
k_{6} |
k_{8} |
k_{8} |
k_{9} |
k_{10} |
I_{max} |
C_{f0} |
R_{f0} |
I_{f0} |
I_{m0} |
C_{i0} |
R_{i0} |

0 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 20 | 500 | 8 | 5 | 1 | 5 | 3 | 2 |

The curve of C

_{f}and R

_{f}alone:

In spite of minimal abnormal phenomenon (C

_{f}was negative in later stage), this graph roughly testified that in wild bacterial the concentration of float SdpC will drop to nearly zero quickly. In sum, the ODE model on singular cell indicates following results:

- The concentration of free SdpC is most affected by k
_{0}, if the copy number of SdpC is large enough, it is theoretically reasonable to commit suicide; - The influence of the value of I
_{max}and k_{2}is much limited; - The amount of free SdpI is always near zero;
- SdpC will not increase limitlessly however we transform parameters;
- To ensure successful suicide, it is required k
_{0}>>k_{4}>>k_{7};

The last conclusion was our biggest windfall, and we have verified the validity of this suicide mechanism in math. On the one hand, if further experiments proven #4 engineered bacteria will kill both siblings and themselves, it is highly like that the expression rate SdpI is much larger than SdpR even if they share the same promoter; on the other, if #4 engineered bacteria are not able to commit suicide, we can try to boost the expression of SdpI to adjust the kill switch.

# Discussion on colonies

In reality, the engineered bacteria aims at killing its siblings instead of itself, and at first almost all toxin SdpC will be secreted outside the bacteria. We assume the diffusion of toxin among cells comply with diffusion law, that is, the diffusion rate is proportionate with the gradient of concentration. Further we assume the death concentration of SdpC is same to all bacteria expect those who contain this locus, the average life expectancy is bacteria will hinge on the rate and distribution of engineered bacteria, and the distribution of life expectancy of bacteria is similar to that of average free path of gas molecules. As long the coefficient of diffusion is large enough, any engineered bacteria, no matter how few, is adequate to devastate the whole colony. Alike to the average free path of thin gas, the average suicide time of the whole reporter system is inversely proportional with the square root of the rate of engineer bacteria containing this circuit.# References

Parallel pathways of repression and antirepression governing the transition to stationary phase in *Bacillus subtilis*
AV Banse, A Chastanet, L Rahn-Lee…,PNAS ,2008