On Projection-Based Model Reduction of Biochemical Networks-- Part I: The Deterministic Case

This paper addresses the problem of model reduction for dynamical system models that describe biochemical reaction networks. Inherent in such models are properties such as stability, positivity and network structure. Ideally these properties should be preserved by model reduction procedures, although traditional projection based approaches struggle to do this. We propose a projection based model reduction algorithm which uses generalised block diagonal Gramians to preserve structure and positivity. Two algorithms are presented, one provides more accurate reduced order models, the second provides easier to simulate reduced order models. The results are illustrated through numerical examples.Comment: Submitted to 53rd IEEE CD

On Projection-Based Model Reduction of Biochemical Networks-- Part II: The Stochastic Case

In this paper, we consider the problem of model order reduction of stochastic biochemical networks. In particular, we reduce the order of (the number of equations in) the Linear Noise Approximation of the Chemical Master Equation, which is often used to describe biochemical networks. In contrast to other biochemical network reduction methods, the presented one is projection-based. Projection-based methods are powerful tools, but the cost of their use is the loss of physical interpretation of the nodes in the network. In order alleviate this drawback, we employ structured projectors, which means that some nodes in the network will keep their physical interpretation. For many models in engineering, finding structured projectors is not always feasible; however, in the context of biochemical networks it is much more likely as the networks are often (almost) monotonic. To summarise, the method can serve as a trade-off between approximation quality and physical interpretation, which is illustrated on numerical examples.Comment: Submitted to the 53rd CD

On the Mathematical Structure of Balanced Chemical Reaction Networks Governed by Mass Action Kinetics

Motivated by recent progress on the interplay between graph theory, dynamics, and systems theory, we revisit the analysis of chemical reaction networks described by mass action kinetics. For reaction networks possessing a thermodynamic equilibrium we derive a compact formulation exhibiting at the same time the structure of the complex graph and the stoichiometry of the network, and which admits a direct thermodynamical interpretation. This formulation allows us to easily characterize the set of positive equilibria and their stability properties. Furthermore, we develop a framework for interconnection of chemical reaction networks, and we discuss how the formulation leads to a new approach for model reduction

Symbolic Methods for Biological Networks D2.1 Report on Scalable Methods for Tropical Solutions (T1.2): ANR-DFG SYMBIONT Project ANR-17-CE40-0036

Tropical geometry can be used to find the order of time scales of variables in chemical reaction networks and search for model reductions [SGF+15]. In this report, we consider the problem of solving tropical equilibration problems in ODE systems of the BioModels model repository. We are interested in the existence of solutions both in R and Z. We present three methods and study their scalability to solve complete equilibration problems. The first two methods, a naive polyhedral method using PtCut [Lüd20c], and a Satisfiability Modulo Theories (SMT) method recently introduced in [Lüd20a] using the SMT solver CVC4 [BCD+11], compute the set of solutions over real numbers. The SMT approach is significantly faster than the polyhedral approach, by up to two orders of magnitude. Furthermore, this method provides an anytime algorithm, thus offering a way to compute parts of the solution when the polyhedral approach is infeasible. The third method, the Constraint Programming (CP) method presented in [SFR14] and implemented in Biocham-4, computes integer equilibrations. The CP approach presents similar performance as the SMT method, mostly below two minutes computation time for the polynomial and rational fractional ODE systems in this benchmark. This method also reveals that 30% of the models that can be equilibrated over the reals have in fact no integer solution. These evaluation results show the scalability of the SMT and CP solvers for solving both real and integer tropical equilibration problems on real-size problems