Rational positive systems for reaction networks

The purpose of the lecture associated with this paper is to present problems, concepts, and theorems of control and system theory for a subclass of the rational positive systems of which examples have been published as models of biochemical cell reaction networks. The recent advances in knowledge for the genome of plants, animals, and humans now lead to increased interest in cell biology. Knowledge is needed on how a cell as a functional unit operates biochemically and how the reaction network is influenced by the genome via the enzymes. In principle it is possible to model the complete biochemical reaction network of a cell though this program has so far been carried out only for small compartments of such networks. Mathematical analysis for such reaction networks then leads to a system of ordinary differential equations or of partial differential equations. Often the ordinary differential equations are of polynomial or of rational form. The number of reactions in a cell can be as high as 15.000 (about half the number of estimated genomes) and the number of chemical compounds as high as 20.000. A detailed mathematical analysis of amathematical model of the complete cell reaction network may therefore not be possible in the short run. Hence there is an interest to develop procedures to obtain from high-order mathematical models approximations in the form of low-order mathematical models. The formulation of approximate models requires understanding of the dynamics of the system, in particular of its algebraic and graph-theoretic structure and of its rate functions. It is the aim of the author to contribute to this research effort. In this lecture attention is restricted to mathematical models for biochemical cell reaction networks in the form of rational positive systems. These systems are called positive because the state vector represents masses or concentrations of chemical compounds and the external input vectors represent inputs into the network of externally available chemical compounds and of enzymes produced by the nucleus of the cell. The dynamics of the system is often modelled as a polynomial map but in this lecture attention it is restricted to rational maps (each component equals a quotient of two polynomials). Such a dynamics arises for example in the model of Michealis- Menten kinetics due to a singular perturbation of a bilinear system. The mathematical model of the glycolysis of Trypanosoma brucei is phrazed almost entirely in terms of a rational positive system and this model is regarded as realistic, see [2]. A book on biochemical reaction networks is that of R. Heinrich and S. Schuster, see [1].The subclass of rational positive systems considered in this lecture is specific due to the conditions imposed by the modeling of biochemical cell reaction networks. It is precisely because of these physically determined conditions that the subclass merits further study. The properties of such systems differ to a minor extent from those of polynomial systems considered. The graph-theoretic and the algebraic structure of rational positive systems make the analysis interesting. A book on mathematical control and system theory is [3] and a paper on polynomial positive systems is [4]. The main topics of the lecture are: - The mathematical framework of rational positive systems for biochemical reaction networks. - The system theoretic results on the interconnection and decomposition of rational positive systems, on the realization problem, and the dissipation and conservation properties. - The formulation of control problems for biochemical reaction networks and preliminary concepts and results for these problems

Complex and detailed balancing of chemical reaction networks revisited

The characterization of the notions of complex and detailed balancing for mass action kinetics chemical reaction networks is revisited from the perspective of algebraic graph theory, in particular Kirchhoff's Matrix Tree theorem for directed weighted graphs. This yields an elucidation of previously obtained results, in particular with respect to the Wegscheider conditions, and a new necessary and sufficient condition for complex balancing, which can be verified constructively.Comment: arXiv admin note: substantial text overlap with arXiv:1502.0224

System theory of rational positive systems for cell reaction networks

Biochemical reaction networks are in realistic cases best modeled as rational positive systems. Rational positive systems for biochemical cell reaction networks are defined as dynamic systems which are rational in the state but linear in the inputs. An academic example is provided. The positive orthant is positively or forward invariant for the differential equation of the system. Results are presented for the realizability of an input-output relation as a rational positive systems and for the form of state-space isomorphism

Intermediates, Catalysts, Persistence, and Boundary Steady States

For dynamical systems arising from chemical reaction networks, persistence is the property that each species concentration remains positively bounded away from zero, as long as species concentrations were all positive in the beginning. We describe two graphical procedures for simplifying reaction networks without breaking known necessary or sufficient conditions for persistence, by iteratively removing so-called intermediates and catalysts from the network. The procedures are easy to apply and, in many cases, lead to highly simplified network structures, such as monomolecular networks. For specific classes of reaction networks, we show that these conditions for persistence are equivalent to one another. Furthermore, they can also be characterized by easily checkable strong connectivity properties of a related graph. In particular, this is the case for (conservative) monomolecular networks, as well as cascades of a large class of post-translational modification systems (of which the MAPK cascade and the $n$-site futile cycle are prominent examples). Since one of the aforementioned sufficient conditions for persistence precludes the existence of boundary steady states, our method also provides a graphical tool to check for that.Comment: The main result was made more general through a slightly different approach. Accepted for publication in the Journal of Mathematical Biolog

A Linear Programming Approach to Weak Reversibility and Linear Conjugacy of Chemical Reaction Networks

15 páginas, 2 figuras.-- The final publication is available at www.springerlink.comA numerically effective procedure for determining weakly reversible chemical reaction networks that are linearly conjugate to a known reaction network is proposed in this paper. The method is based on translating the structural and algebraic characteristics of weak reversibility to logical statements and solving the obtained set of linear (in)equalities in the framework of mixed integer linear programming. The unknowns in the problem are the reaction rate coefficients and the parameters of the linear conjugacy transformation. The efficacy of the approach is shown through numerical examples.Matthew D. Johnston and David Siegel acknowledge the support of D. Siegel’s Natural Sciences and Engineering Research Council of Canada Discovery Grant. Gàbor Szederkényi acknowledges the support of the Hungarian National Research Fund through grant no. OTKA K-83440 as well as the support of project CAFE (Computer Aided Process for Food Engineering) FP7-KBBE-2007-1 (Grant no: 212754).Peer reviewe