Mathematical problems of metabolic pathway organization from biochemical reactions

The problem of metabolic pathway building from biochemical reactions is considered. For the given pair “main substrate–main product” it is necessary to select the reactions performing this conversion and find all the attendant substrates and products. The approach is based on a system of stoichiometric equations for reaction rates. The mathematical problems are as follows: 1) a number of right-hand sides are initially unknown, 2) a part of the system with the known right-hand sides has less number of equations than variables, and 3) presence of irreversible reactions imposes restrictions on signs of the corresponding rates in the form of inequalities. The method is described enabling to solve these problems. It is found that many restrictions are represented as series of parallel hyperplanes in the space of variables. This fact makes it possible to eliminate many restrictions and to find out a big number of fixed variables the values of which, thus, are found before the complete solving of the whole problem. The search of unnecessary restrictions and fixed variables is of the form of specific iterations due to which the problem is substantially simplified. Examples are given which display effectiveness of the approach described.