Reachability Problems for Continuous Chemical Reaction Networks

Case, Adam; Lutz, Jack H.; Stull, Donald M.

doi:10.1007/978-3-319-41312-9_1

Cited by 7 publications

(9 citation statements)

References 15 publications

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“…A key concept in capturing rate-independent computation is the reachability relation (segmentreachability, Definition 2.3). Reference [8] showed that, given two states, deciding whether one is reachable from the other is solvable in polynomial time. This contrasts sharply with the hardness of the reachability problem for discrete CRNs, which is not even primitive recursive [14,21].…”

Section: Related Workmentioning

confidence: 99%

Rate-independent computation in continuous chemical reaction networks

Chen¹,

Doty²,

Soloveichik³

et al. 2014

Proceedings of the 5th Conference on Innovations in Theoretical Computer Science

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Understanding the algorithmic behaviors that are in principle realizable in a chemical system is necessary for a rigorous understanding of the design principles of biological regulatory networks. Further, advances in synthetic biology herald the time when we'll be able to rationally engineer complex chemical systems, and when idealized formal models will become blueprints for engineering.Coupled chemical interactions in a well-mixed solution are commonly formalized as chemical reaction networks (CRNs). However, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. Here we study the following problem: what functions : R → R can be computed by a CRN, in which the CRN eventually produces the correct amount of the "output" molecule, no matter the rate at which reactions proceed? This captures a previously unexplored, but very natural class of computations: for example, the reaction 1 + 2 → can be thought to compute the function = min( 1 , 2 ). Such a CRN is robust in the sense that it is correct whether its evolution is governed by the standard model of mass-action kinetics, alternatives such as Hill-function or Michaelis-Menten kinetics, or other arbitrary models of chemistry that respect the (fundamentally digital) stoichiometric constraints (what are the reactants and products?). We develop a formal definition of such computation using a novel notion of reachability, and prove that a function is computable in this manner if and only if it is continuous piecewise linear. CCS Concepts: • Theory of computation → Models of computation; • Computer systems organization → Analog computers.

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Section: Related Workmentioning

confidence: 99%

Rate-independent computation in continuous chemical reaction networks

Chen¹,

Doty²,

Soloveichik³

et al. 2014

Proceedings of the 5th Conference on Innovations in Theoretical Computer Science

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show abstract

“…It is natural to wonder about the computational complexity of determining whether or not we have c ⇒ * d for given continuous states c and d. It is shown in [19] that if c and d have only rational entries, then this problem can be solved in polynomial time. In contrast, the reachability problem for CRNs using the usual reachability relation for states of Section 2 is much harder, cf.…”

Section: Rate-independent Computation With Continuous Chemical Reactimentioning

confidence: 99%

Computing with chemical reaction networks: a tutorial

Brijder

2019

Nat Comput

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Chemical reaction networks (CRNs) model the behavior of chemical reactions in well-mixed solutions and they can be designed to perform computations. In this tutorial we give an overview of various computational models for CRNs. Moreover, we discuss a method to implement arbitrary (abstract) CRNs in a test tube using DNA. Finally, we discuss relationships between CRNs and other models of computation.

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“…The mathematical model of discrete state chemical reaction networks is equivalent to an important model of theoretical computer science, namely, the so-called vector addition systems with states (VASS) or equivalently Petri nets [20,21]. Hence the discrete chemical reaction network reachability problem is equivalent to the extensively studied problem of vector addition system (VAS) reachability.…”

Section: Complexitymentioning

confidence: 99%

“…The VAS reachability problem is known to be decidable [22][23][24][25], and for the space complexity we have EXSPACE lower bound [26]. Unfortunately, contrary to the proven polynomial time complexity of reachability of rate independent continuous state chemical reaction networks [21], in the case of discrete state reaction networks it is not known whether there exists an algorithm of primitive-recursive time complexity deciding this problem [27].…”

Section: Complexitymentioning

confidence: 99%

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

Szlobodnyik

Szederkényi

2019

Complexity

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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.

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Reachability Problems for Continuous Chemical Reaction Networks

Cited by 7 publications

References 15 publications

Rate-independent computation in continuous chemical reaction networks

Rate-independent computation in continuous chemical reaction networks

Computing with chemical reaction networks: a tutorial

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

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