Conservation Laws in Biochemical Reaction Networks

Abstract

We study the existence of linear and non-linear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The non-linear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks.Here, using Darboux theory of integrability we provide necessary structural (i.e. parameter independent) conditions on a reaction network to guarantee the existence of non-linear conservation laws of certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that a non-linear first integrals (similar to Lotka-Volterra system) only exists under the same structural condition. We finally show that the existence of such a first integral generally implies that the system is persistent and has stable steady states. We illustrate our results by examples.

Sensitivity analysis methods in the biomedical sciences

“…After completing the initial mathematical model, model analysis is performed [86]. This includes a variety of techniques such as practical identifiability [111,120,134], stability [102] and bifurcation analysis [105], as well as SA, which is the main topic of this paper. Within the mathematical modelling workflow, it is possible to perform model analysis even without evaluating the model by using, for instance, structural identifiability techniques [177,106].…”

Biological networks with singular Jacobians: their origins and adaptation criteria

“…While it was relatively easy to detect conserved relation in this 3x3 system, more effort is necessary to do this in higher dimensional cases and the not all systems are amenable to existing techniques (see e.g. Mahdi et al [15] and discussion therein).…”

“…However, this requires knowing the system structure prior to any modifications. For systems with linear conservation laws, it may be easy to spot the cause of the singularity, but as can be seen in Mahdi et al [15], identifying non-linear conservation laws within the system can be a strenuous and time consuming process. Furthermore, if the system is known to have several conservation laws or built-in assumptions affecting various nodes, removing the singularities may take longer than utilizing the general conditions for adaptation presented in this work.…”

On the Liouville integrability of Edelstein’s reaction system in R3

“…Concerning the cases (iv) and (v), we discard the first case because by insertion of (iv) into (iii), we get c 1 < 0, which is not possible. We study the second case, for which we have m = 2n by insertion of (iii) and (v) into (9).…”

“…Edelstein's system has one such first integral, namely H = y + z. With some exceptions the non-linear first integrals have rarely been considered, see [1,10,9] for some general considerations and [8,6,7,12] for specific examples. Non-linear first integral are often useful for studying the dynamics of the system, see for example [12,9].…”