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 finally the death of the bacteria. On the other hand, the removal of SdpR also enhance the expression of SdpI and accelerate 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, it is believed that the positive loop is strong enough to outweigh the negative one, which guarantees this mechanism 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” locus itself is to kill their siblings rather than themselves to ensure the survival of themselves. Surly they can kill their siblings, but can they finally eliminate themselves, as we expects? 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 laboratory. Fortunately, we could resort to mathematical models to verify the validity of this locus theoretically. There are six independent variables in individual cells, and the 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:

Mark	Meaning
Imax	Mole number of free SdpI in cytoplasm.
Im	Mole number of SdpI in the cell membrane.
Cf	Mole number of free SdpC in cytoplasm.
Ci	Mole number of SdpC captured by SdpI.
Rf	Mole number of free SdpR in cytoplasm.
Ri	Mole number of SdpR captured by SdpI

To construct reasonable ordinary differential equation (ODE) model to describe and predict the operation of the suicide system, we followed the law of mass action, one basic law of chemistry and biology.
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 the process of 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 equation of the variables:


















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

Mark	Meaning
Imax	The maximal number of SdpI than can be fixed on the cell membrane.
k0	Constant describes the normal expression rate of SdpC
k1	Constant describes the self-repression effects of SdpC
k2	Constant describes the repression of SdpR on the expression of SdpC.
k3	Constant describes the rate of SdpI capturing SdpC
k4	Constant describes the normal expression rate of SdpR
k5	Constant describes the self-repression effects of SdpR
k6	Constant describes the rate of SdpI capturing SdpR
k7	Constant describes the normal expression rate of SdpI
k8	Constant describes the self-repression effects of SdpI
k9	Constant describes the repression of SdpR on the expression of SdpI
k10	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 k₀ 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 expression. Yet any modification on genome is notoriously time-consuming, which inhibited us from measuring them in person. We also looked up oceans of papers to confer their approximate ranges, but almost all papers are too fragmental to afford 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. 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:

1. k₀>>k₄≈k₇: k₀,k₄ and k₇ 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;
2. k₂>>k₉: 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;
3. k₁₀>>k₃,k₆：it is hard to predict the value of k₃ and k₈, yet we suppose both of them is much smaller than k₁₀ because SdpI is a kind of membrane protein inherently, and rarely exists as free protein
4. The primary values of all the six variables are very small or strictly zero. We expect it 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 description 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 account for the unexpected results: our model is too rough to include some assignable factor; or there are some implicit but objective limitation inside 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:

k0	k1	k2	k3	k4	k5	k6	k8	k8	k9	k10	Imax	Cf0	Rf0	If0	Im0	Ci0	Ri0
50	5	5	5	5	5	5	5	5	5	20	500	5	5	1	5	3	2

At the first glance this graph seemed fine. Initially the concentration of SdpC decreased slightly due to the capturing of SdpI and the repression of float SdpR, but gradually the positive feedback loop works, and C_f increases rapidly. But when we turned our attention to the curves of other parameters, things seemed not so perfect:


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 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:

k0	k1	k2	k3	k4	k5	k6	k8	k8	k9	k10	Imax	Cf0	Rf0	If0	Im0	Ci0	Ri0
100	5	5	5	5	5	5	20	5	5	20	500	5	5	1	5	3	2

In this group, we gave up one former assumption and set k₂ equal to k₉. We also gave positive values to I_m, C_i and R_i, which were considered to be zero at first. And by groups of stimulations we realized the value of k₂ does matter, as the derivative of C_f only increased slightly as k₂ 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, If 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 k₇. Here is another group of values and corresponding graph:

k0	k1	k2	k3	k4	k5	k6	k8	k8	k9	k10	Imax	Cf0	Rf0	If0	Im0	Ci0	Ri0
100	5	5	5	5	5	5	20	5	5	20	500	5	5	1	5	3	2

Although the derivative of Im is not seriously positive constantly, the three latter curves seemed much more reasonable. Hence, we extrapolated 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 k₇ also represses SdpC, and hence the copy number of SdpC must be larger. We kept all other parameters constant and gradually augmented k₀. The larger k₀, the more perfect the curve seemed, and here are the values table and graph where k₀ equals 400, 80 times larger than k₄.

k0	k1	k2	k3	k4	k5	k6	k8	k8	k9	k10	Imax	Cf0	Rf0	If0	Im0	Ci0	Ri0
400	5	5	5	5	5	5	20	5	5	20	500	5	5	1	5	3	2

Take the curve of C_f and R_f separately, the curves seemed more perfect:



In wild bacteria who are unable to produced SdpC, naturally k₀ equals zero. We expected C_f would decreased gradually and finally approximate zero, and here are the corresponding table and graph:

k0	k1	k2	k3	k4	k5	k6	k8	k8	k9	k10	Imax	Cf0	Rf0	If0	Im0	Ci0	Ri0
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 where we increased the primary concentration of SdpC and the normal expression rate of SdpI.

k0	k1	k2	k3	k4	k5	k6	k8	k8	k9	k10	Imax	Cf0	Rf0	If0	Im0	Ci0	Ri0
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 of singular cells indicate following results:

1. The character of free SdpC is most affected by k₀, if the copy number of SdpC is large enough, it is theoretically reasonable to commit suicide;
2. The influence of the value of Imax and k₂ is much limited;
3. The amount of free SdpI is always near zero;
4. SdpC will not increase limitlessly however we transform parameters;
5. To ensure the success of suicide, it is required k₀>>k₄>>k₇;
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 hand, if #4 engineered bacteria are not able to commit suicide, we can try to boost the expression of SdpI to adjust the bacteria.

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. Assume the diffusion of toxin among cells comply with diffusion law, that is, the diffusion rate is proportionate with the gradient of concentration. Further 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 bacterias, 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 locus.

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