The mathematical modeling of physical phenomena and complex biology is a real challenge for mathematicians. In the face of the mass arrival of biological data (particularly in genetics), it becomes important to provide models allowing to exploit these to help understand the phenomena that come into play.
In this article, we expose the problem of genetic regulation and propose a model using hybrid systems.
It is the group of molecular regulators that work together with each other and other substances within the cell to run the gene expression levels of proteins and mRNA.
The taking into account of mathematical modeling in several important competitions through tests like TIPE for ENSI competitions, the modeling test at Aggregation certainly poses a lot of problems Preparation theses. In this text, we are going to develop a few remarks that are the consequence of our participation in several of the juries mentioned above. Now that the mainstream press is echoing issues related to the place of mathematics in general education, we believe that we should not act as we do it too often (and very often with very conservative arguments) but ourselves offer solutions. First, you have to know what we are talking about. And experience shows that it is not easy! It is enough to look at the texts proposed in several contests to see that the modeling process, which begins with the establishment of the model, is completely obeyed. It is replaced by an exercise in classical mathematics on an agreed system of equations.
How to build models in this framework? Is it reasonable to think of modeling the functioning of a cell, taking into account the fact that the historical dimension is one of the characteristics of the living? This remark is made, we will propose in the rest of this article a current example of a modeling problem in molecular biology. This example, although historic, is still current. The concept of a genetic network in this part, we will try to briefly describe what is meant by a genetic regulatory network. To do this, we will first familiarize ourselves with certain basic concepts of cell biology, in particular enzymatic mechanics and genetic control.
Cell biology is a very recent science since the discovery of the DNA molecule dates only from 1868, and it was not until the middle of the twentieth century. Even as we have understood that this molecule is indeed genetic material. This science seeks, in fact, to explain the various mechanisms which unfold on the molecular scale and which, being linked in a complex way to each other, constitute the activity. In other words, it tries to understand how a cell (that is to say an elementary entity of organic life) does to feed itself, divide, and organize the multitude of tasks necessary for its survival? We now know that each cell has a genetic background that carries a tremendous amount of information. It is this mass of information that is used to direct the metabolism of the cell, just like a factory in which a command center directs machinery. Basically, genes are translated or expressed to produce the proteins that perform cellular functions; this is the central dogma of cell biology. However, even if we gradually decode the human genome, we are still unable to describe with precision the entire complex links that exist between cellular metabolism and ‘genetic information.
The basic building blocks of organic matter are proteins. These are large molecules made up of a sequence of more or less long amino acids. This sequencing has the effect of giving them a complex spatial form that determines its function. Some of these molecules, enzymes, have the ability to accelerate a certain type of biochemical reaction. The presence of a catalyst is fundamental because it is she who determines whether the reaction takes place or not. So we see that the enzymes are direct players in cell life: by their presence or their absence, they contribute to carrying out such or such task.
To better understand this phenomenon, let’s take the example of food cells of bacteria. To survive, the first task of the cell is to consistently produce energy. To do this, it must degrade the glucose molecules that it has and recover the energy that results from this degradation. However, the bacteria may be lacking in glucose (if by example, the outside environment is lacking). It can then turn to another source of energy, lactose. This sugar is more difficult to exploit because it requires special treatment: it must be allowed to enter the cell, and then turn it into glucose. These two operations require the presence of two respective enzymes, Permease, and β-galactosidase. If the cell wants to survive in the event of glucose deficiency, it will, therefore, have to start producing these two enzymes quickly in order to adapt to their new living conditions.
The various models which have been proposed in an attempt to give a mathematical approach to genetic networks fall essentially into two categories, systems discrete and continuous systems. After a brief description of these two models, we will present a hybrid model, which describes the continuous evolution of networks while taking into account the discreet aspect that characterizes them. In the Boolean idealization of genetic networks, we model a gene (or any other biological stimulus) by a Boolean variable. A gene is, therefore, either on (transcribed) or off (not transcribed). We then represent the positive or negative influences of one gene on the others by Boolean functions. The advantage of such a model is, of course, its simplicity of calculation. We can thus simulate relatively large networks and answer, for example, general questions such as the number of cell types.
Living systems are incredibly complex and heterogeneous. We must, therefore, isolate subsystems (at the risk of giving too much space to certain entities such as genes) and try to make a mathematical model of them. Traditionally, the problem has been the lack of necessary measures to calibrate the models. However, a major change is underway with the arrival of tools from the Human Genome program. These tools (from bioinformatics, robotic manipulation techniques, biochips, etc.) provide us with a very large amount of new data. A natural step in biostatistics is underway to interpret this data, followed by the “dynamic” modeling step, which should lead to the appearance of more realistic models.