Dual Chemical Reaction Networks and Implications for Lyapunov-Based Structural Stability

Abstract

Given a class of (bio)Chemical Reaction Networks (CRNs) identified by a stoichiometric matrix S, we define as dual reaction network, CRN * , the class of (bio)Chemical Reaction Networks identified by the transpose stoichiometric matrix S . We consider both the dynamical systems describing the time evolution of the species concentrations and of the reaction rates. First, based on the analysis of the Jacobian matrix, we show that the structural (i.e., parameter-independent) local stability properties are equivalent for a CRN and its dual CRN * . We also assess the structural global stability properties of the two dual networks, analysing both concentration and rate representations. We prove that the existence of a polyhedral (or piecewise-linear) Lyapunov function in concentrations for a CRN is equivalent to the existence of a piecewise-linear in rates Lyapunov function for the dual CRN * ; in fact, if V is a polyhedral Lyapunov function for a CRN, the dual polyhedral function V * is a piecewise-linear in rates Lyapunov function for the dual network. We finally show how duality can be exploited to gain additional insight into biochemical reaction networks.

Structural polyhedral stability of a biochemical network is equivalent to finiteness of the associated generalised Petri net

“…, θ n } 0 is a diagonal matrix of positive time constants. Proposition 2: Structural polyhedral stability, guaranteed when Procedure 1 stops in finite time, implies the structural stability of system (11) for any arbitrary diagonal Θ 0.…”

“…As shown in [8], [9], this is equivalent to applying the Procedure 1 to the dual system ẇ(t) = C D(t)B w(t). In view of duality properties [11], [14], [31], a PLF exists for the primal system if and only if it exists for its dual.…”

“…Structural analysis investigates how several systems often encountered in nature enjoy important properties in view of their interconnection structure, regardless of parameter values. Here, we consider structural stability and stabilisation of biochemical networks [12], [18], [23], adopting piecewiselinear Lyapunov functions [8], [9], [10], which -along with the complementary piecewise-linear-in-rate Lyapunov functions [1], [2], [3], [9] that can be seen as their dual [11] -have proven effective in the stability analysis of chemical reaction networks. A recent contribution [4] shows that this type of functions can be very useful to detect, more in general, nonoscillatory behaviours.…”

Graphical characterizations of robust stability in biological interaction networks

“…Examples include the theory of complex balance [6], [7], [8], and the theory of monotone BINs [9]. More recently, stability certificates have been constructed via Robust Lyapunov Functions (RLFs) in reaction [10,11,12], and concentration coordinates [13,14,15,12,16]. Except for a small subclass of BINs (see §III.C), such methods mainly utilize computational algorithms to construct RLFs via either iterative algorithms or linear programs.…”