A Mathematical model captures the essential dynamics of the system in the form of mathematical equations and helps to study and analyze the biological system before stepping into lab work. All of the chemical reactions in the system can essentially be written in the form of differential equations that capture the biological processes in the cell. These equations can then be simulated and the dynamics can be analyzed, in order to understand how a particular network is going to behave inside the cell.
The two molecular processes that are central to the functioning of a cell are transcription and translation. The cell consists of DNA, which contains all the genetic information of the cell. It contains information for synthesis of various proteins required for normal functioning of the cell. The process of transcription leads to creation of mRNA from DNA which contains the information for protein synthesis. This mRNA is then translated into protein with the help of ribosomes and tRNA.
Thus, the entire set of reactions happening inside a cell leading to the expression of a gene can be broken down into the following –
Here, the major processes occurring are as follows -
mRNA
mRNA is being produced from the plasmid DNA that has been introduced into the cell via transformation, by the process of transcription.
The mRNA produced is also being degraded, because it has a certain half life (just like radioactive elements decay, all chemicals have a half life, and so do DNA and RNA!).
Protein
Protein is produced from the mRNA transcript by the process of translation.
Protein is also degraded since it has a half life, similar to the mRNA.
What needs to be noted before we start to write a model for this is that this is a very simplistic model that makes use of several assumptions and simplifications. This is because biological systems are extremely complex, and at a single instant of time, there are several hundred reactions happening. Thus, we need to simplify and lump certain intermediate reactions, in order to have some quantitative estimate of how our system will behave.
Some of the assumptions made here are -
mRNA is made directly from DNA, and all the other components and processes in between, such as pulling of RNA polymerase (RNAp) by the TATA box in the promoter, binding of RNAp to the promoter and initiation of transcription are sufficiently fast, so that the parameters can be lumped and variables (such as RNAp and Promoter) can be ignored.
Degradation of molecules such as mRNA and protein is spontaneous, and is not triggered or accelerated by certain components (such as ssrA).
The rates of production and degradation are constant.
mRNA is not degraded, damaged, or consumed in any way during translation or production (transcription).
Total DNA inside a cell is constant.
Mass action kinetics is valid for the reactions occurring above
