# Team:Aix-Marseille/Modeling

### From 2014.igem.org

# Our model

## Introduction

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.

- 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*. - 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. *CheA*phosphorylates naturally*CheY*and not*CusR*. We have considered the kinetic constants of*CheY*for*CusR*.- 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. - The threshold of
*ppGpp*for cell division is not known, we took a concentration we know realistic: 1 μM. - 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.
- 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.