Our model


Part modeling was not an easy task. Let me explain. I am a novice in the field of mathematical engineering and that made more than seven years since I was not made of Biology.

Our project is based on the feedback cycle of chemiotaxie in Escherichia coli. To develop our model, it was first essential to know the ins and out. To do so, the study of narratives on the general biology were paramount; it was only later that I got interested in the chemiotaxie.

Subsequently, we made contact with a brilliant modeler of this field from the University of Aix-Marseille: Ms. Elysabeth Remi. Together, we tried to synthesize the system to be modeled and some approximations which we considered natural at first.

Model System

The scheme is simple. Each cell of Escherichia coli has an amount of CheA in its cytoplasm. This one will phosphorylate CusR which in turn will catalyze the formation of ppGpp (via RelA) and the SerA. The formation of ppGpp will cause failure of cell division. SerA is a protein that will lead to the formation of Serine via SerC and SerB. This intracellular Serine will migrate to the outside of the cell. However, the histidine kinase CheA is sensitive to the gradient of Serine outside, thanks to chemoreceptor. Moreover, since the outer homogeneous media is considered, all cells will receive the same gradient outside Serine. This is the start of cell synchronization. Increasing Serine will create a decrease in the phosphorylation of CheA, involving a decrease CusR phosphorylated. Thus, the rate will decrease ppGpp allow cell division and of all cells at the same time. After their split, the pattern will repeat itself.

Simplifying assumptions

In order to simplify our model and lack of experimental data, we overlook some things.

  1. First, we assume that the formation of Serine is done directly by SerA. In reality, training SerC and SerB are much faster than those of SerA and Serine.
  2. The concentration of ppGpp being by fluorescence, we do not get that of RelA. We will take into account that the formation of ppGpp by CusR phosphorylated.
  3. CheA phosphorylates naturally CheY and not CusR. We have considered the kinetic constants of CheY for CusR.
  4. We do not know the constants of CheA deactivation by the gradient of Serine. We've replaced by those of the gradient Aspartate which are easily found in the literature.
  5. The threshold of ppGpp for cell division is not known, we took a concentration we know realistic: 1 μM.
  6. 6. Some kinetics are expected to follow a law of Michaelis-Menten. However, the system is rather slow and concentrations vary only slightly around the concentrations of half-maximal speed. Thus, we assume that they follow linear laws.
  7. We didn't have the time to test the model experimentally so we took an arbitrary time scale.

Differential equations

Here are the differential equations modeling our system.

Initial values

Some constants have been found experimentally or guessed from existing data.

Resolution of the system

Thanks to our sponsor, MathWorks, and his gift of licenses, we made our computing and simulation in Matlab.

Then, we implemented a Runge-Kutta 4 to solve our system. We made our program so that latency between each phase of the cycle. This aspect allows, among other things, to account for the time between the formation of a protein and its interaction with the rest of the cell; for example, the time required for the Serine out after its formation.

Now, let's see the result of our work.

1. First Case

We can see that our system is oscillating. It is indeed the expected result since after cell division, the system must be reset (with the exception of the outer Serine). But the system needs time to qualibrer (here, from beginning to the first cell division).

2. Second Case

We will now see some pathological cases. First, we will observe if all our parameters are multiplied by 5 and here is the result.

CheA concentration reaches zero once. However, its derivative is not zero, the system can still go. Then the next cycle, its derivative is also canceled and the system stabilizes.

3. Third Case

Michaelis-Menten constants is divided by 20.

We can even observe more. These constants act as timer. Here, they are very small and the system does not boot even. In next case, we observed that if it is big, the system is perfectly calibrated. This is very understandable. If they are too small, the Michaelis-Menten equations react as constants. Then, some parts of the system will no longer interact with others. For example, the concentration of CheA no longer influences that of CusR.

4. Fourth Case

In this case, we took Michaelis-Menten constant and multiplied by only two parameters. It is observed however that the system is still oscillating and the system is not far from the initial case. So we do have a range of data sets allowing us to stick a little better with reality. This range can also enable us to overcome the errors due to our assumptions that may be too simplistic.