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Computational complexity of atomic chemical reaction networks

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Abstract

Chemical reaction network has been a model of interest to both theoretical and applied computer scientists, and there has been concern about its physical-realisticity which calls for study on the atomic property of chemical reaction networks. Informally, a chemical reaction network is “atomic” if each reaction may be interpreted as the rearrangement of indivisible units of matter. There are several reasonable definitions formalizing this idea. We investigate the computational complexity of deciding whether a given network is atomic according to each of these definitions. Primitive atomic, which requires each reaction to preserve the total number of atoms, is shown to be equivalent to mass conservation. Since it is known that it can be decided in polynomial time whether a given chemical reaction network is mass-conserving (Mayr and Weihmann, in: International conference on applications and theory of petri nets and concurrency, Springer, New York, 2014), the equivalence we show gives an efficient algorithm to decide primitive atomicity. Subset atomic further requires all atoms be species, so intuitively this type of network is endowed with a “better” property than primitive atomic (i.e. mass conserving) ones in the sense that the atoms are not just abstract indivisible units, but also actual participants of reactions. We show that deciding if a network is subset atomic is in \({\mathsf{NP}}\), and “whether a network is subset atomic with respect to a given atom set” is strongly \({\mathsf{NP}}\)-\({\mathsf {complete}}\). Reachably atomic, studied by Adleman et al. (On the mathematics of the law of mass action, Springer, Dordrecht, 2014. https://doi.org/10.1007/978-94-017-9041-3_1), and Gopalkrishnan (2016), further requires that each species has a sequence of reactions splitting it into its constituent atoms. Using a combinatorial argument, we show that there is a polynomial-time algorithm to decide whether a given network is reachably atomic, improving upon the result of Adleman et al. that the problem is decidable. We show that the reachability problem for reachably atomic networks is \({\mathsf {PSPACE}}\)-\({\mathsf {complete}}\). Finally, we demonstrate equivalence relationships between our definitions and some cases of an existing definition of atomicity due to Gnacadja (J Math Chem 49(10):2137, 2011).

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Notes

  1. This usage of the term “atomic” is different from its usage in traditional areas like operating system or syntactic analysis, where an “atomic” execution is an uninterruptable unit of operation (Silberschatz et al. 2013).

  2. Note that by the argument above, reversible networks are weakly reversible and hence consistent, which establishes the equivalence between self-replicability and criticality. One may further observe from Deshpande and Gopalkrishnan (2013) that this equivalence translates to the equivalence between the mass conserving property and the reachability property above.

  3. There is typically a positive real-valued rate constant associated to each reaction, but we ignore reaction rates in this paper and consequently simplify the definition.

  4. For example, for \(V = \{v_1,v_2,\ldots , v_5\}\), \(C = (v_1\vee v_3\vee v_4)\wedge (v_3\vee v_2\vee v_5)\), \(v_{11}=v_1, v_{12}=v_3, v_{13}=v_4, \ldots , v_{23} = v_5\). Correspondingly, \(S_{11}=S_1,X_{11}=X_1, \cdots , S_{23}=S_5, X_{23}=X_5\).

  5. To continue the example in the previous footnote, the set of reactions shall be:

    $$\begin{aligned} &3P+2F+T\rightarrow {} S_1+ S_3 + S_4\\ & 3P+2F+T\rightarrow {} S_3 + S_2 + S_5\\ & 3Q+2F+T\rightarrow {} X_1 + X_3 + X_4\\ & 3Q+2F+T\rightarrow {} X_3 + X_2 + X_5\\ & S_i + Q\rightarrow {} X_i + P\ (i = 1,2,\ldots , 5). \end{aligned}$$

  6. We point out that the set of atoms \(M\ne \{S\in \varLambda \mid \exists ({{\mathbf {r}}},{\mathbf {p}})\in R\, {{\text { s.t. }}}\,(\{1S\}={{\mathbf {r}}})\wedge (\Vert {\mathbf {p}}\Vert \ge 2)\}\), so we have to test the \(\Vert {\mathbf {d}}_S\Vert \ge 2\) condition in later steps. This is because it might be the case that the only reaction \(({{\mathbf {r}}},{\mathbf {p}})\) with \({{\mathbf {r}}}=\{1S\}\) turns out to be an isomerization reaction. A counter example would be:

    $$\begin{aligned} A\rightarrow & {} B\\ B\rightarrow & {} 2C \end{aligned}$$

    By our definition \(M = \{S\in \varLambda \mid \exists ({{\mathbf {r}}},{\mathbf {p}})\in R\ {{\text { s.t. }}}\,\{1S\}={{\mathbf {r}}}\}\), we shall correctly identify \(M=\{A,B\}\), yet the added condition \(\Vert {\mathbf {p}}\Vert \ge 2\) would make \(M=\{B\}\), a mis-identification.

  7. That is, if (\(\forall S\in M'\))(\(\forall ({{\mathbf {r}}},{\mathbf {p}})\in R\)) (\({{\mathbf {r}}}=\{1S\}\Rightarrow [{\mathbf {p}}]\cap M'\ne \emptyset\)), then for each species S in \(M'\) there will be \(S'\in M'\) s.t. \(\Vert S\Vert _1>\Vert S'\Vert _1\), contradicting the finiteness of \(M'\).

  8. We organize the construction procedure as an algorithm to make the description more concise.

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Acknowledgements

The authors are thankful to Manoj Gopalkrishnan, Gilles Gnacadja, Javier Esparza, Sergei Chubanov, Matthew Cook, and anonymous reviewers for their insights and useful discussion.

Funding

Funding was provided by NSF (Grant No. 1619343).

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Authors and Affiliations

  1. 2069 Academic Surge, University of California, Davis, Davis, CA 95616, USA

    David Doty

  2. University of Maryland, College Park, College Park, MD 20740, USA

    Shaopeng Zhu

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  1. David Doty
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  2. Shaopeng Zhu
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Correspondence to Shaopeng Zhu.

Additional information

This paper is based on Doty and Zhu (2018), which omitted many detailed proofs to various theorems and lemmas, such as the poly-time decidability of Reachably-Atomic. These proofs have been included in the current version of the paper. We have also corrected some errors and typos that appeared in the conference version.

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Doty, D., Zhu, S. Computational complexity of atomic chemical reaction networks. Nat Comput 17, 677–691 (2018). https://doi.org/10.1007/s11047-018-9687-9

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