Reachability Analysis of Low‐Order Discrete State Reaction Networks Obeying Conservation Laws

Abstract

In this paper we study the reachability problem of sub- and superconservative discrete state chemical reaction networks (d-CRNs). It is known that a subconservative network has bounded reachable state space, while that of a superconservative one is unbounded. The reachability problem of superconservative reaction networks is traced back to the reachability of subconservative ones. We consider network structures composed of reactions having at most one input and one output species beyond the possible catalyzers. We give a proof that, assuming all the reactions are charged in the initial and target states, the reachability problems of sub- and superconservative reaction networks are equivalent to the existence of nonnegative integer solution of the corresponding d-CRN state equations. Using this result, the reachability problem is reformulated as an Integer Linear Programming (ILP) feasibility problem. Therefore, the number of feasible trajectories satisfying the reachability relation can be counted in polynomial time in the number of species and in the distance of initial and target states, assuming fixed number of reactions in the system.

Polynomial Time Reachability Analysis in Discrete State Chemical Reaction Networks Obeying Conservation Laws

“…We assumed that the reaction networks are sub-or superconservative with additional constraints on their structure. We used the result that the reachability relation is equivalent to the non-negative integer solution of the d-CRN state equation in the considered sub-classes of reaction network structures [24]. We proved that in the studied sub-classes of d-CRNs, the stoichiometric matrix is guaranteed to be totally unimodular.…”

“…In the general case, the d-CRN reachability problem can be formulated as an integer programming (IP) decision problem [23]. For specific sub-classes of d-CRN structures, it was proven that the reachability relation can be decided with relaxed time complexity using the d-CRN state equation as the constraint set of the problem [23,24]. The aim of this paper is to provide theoretically guaranteed bounds on the computational complexity of the d-CRN reachability problem for specific sub-classes of d-CRN topologies.…”

“…We prove that for sub-classes of sub-and superconservative reaction network structures the NP-hard problem of d-CRN reachability can be reformulated as linear programming problem of polynomial complexity. In [24] necessary and sufficient conditions are obtained for which the d-CRN reachability relation is equivalent to the nonnegative integer solution of the d-CRN state equation. Using this result, the authors expressed the reachability problem as integer decision problem for which well-decoupled time complexity can be obtained by means of the Lenstra and Barvinok algorithms.…”

“…[24] Let us consider a subconservative or superconserva-tive d-CRN N = (S, C, R) with stoichiometric matrix Γ ∈ {−1, 0, 1} n×l and Γ − ∈ {0, 1} n×l and C = S ∪ {∅}. Let us assume that for each r ∈ R, n i=1 [y + ] i ≤ 1 and n i=1 [y − ] i = 1.…”