Computational complexity of atomic chemical reaction networks

Doty, David; Zhu, Shaopeng

doi:10.1007/s11047-018-9687-9

“…The entry of can be interpreted as the total number of atoms in compound . In [ 78 ], a RN is called primitive atomic if each reaction preserves the total number of atoms. Thus a RN is primitive atomic if and only if it is conservative, cf.…”

Section: Realizations Of Reaction Networkmentioning

confidence: 99%

What makes a reaction network “chemical”?

Müller

¹

,

Flamm

²

,

Stadler

³

2022

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Background Reaction networks (RNs) comprise a set X of species and a set $$\mathscr {R}$$ R of reactions $$Y\rightarrow Y'$$ Y → Y ′ , each converting a multiset of educts $$Y\subseteq X$$ Y ⊆ X into a multiset $$Y'\subseteq X$$ Y ′ ⊆ X of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix $$\mathbf {S}\in \mathbb {R}^{X\times \mathscr {R}}$$ S ∈ R X × R have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for $$\mathbf {S}$$ S that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively. Results Chemically plausible RNs allow neither a perpetuum mobile, i.e., a “futile cycle” of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix $$\mathbf {S}$$ S . The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and—in a stricter sense—futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. It is also sufficient for the existence of a realization in terms of sum formulas, obeying conservation of “atoms”. In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless $$\mathbf {S}$$ S implies that they are “obligatory isomers”. In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN. Conclusions “Chemical” RNs are directed hypergraphs with a stoichiometric matrix $$\mathbf {S}$$ S whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain $$\mathbf {S}$$ S by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesting avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes.

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“…The entry m x of m can be interpreted as the total number of atoms in compound x ∈ X. In [71], a RN is called primitive atomic if each reaction preserves the total number of atoms. Thus a RN is primitive atomic if and only if it is conservative, cf.…”

Section: Conservation Of Atoms and Moietiesmentioning

confidence: 99%

What makes a reaction network "chemical"?

Müller¹,

Flamm²,

Stadler³

2022

Preprint

0

View full text Add to dashboard Cite

Background: Reaction networks (RNs) comprise a set X of species and a set R of reactions Y → Y , each converting a multiset of educts Y ⊆ X into a multiset Y ⊆ X of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix S ∈ R X×R have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for S that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively.Results: Chemically plausible RNs allow neither a perpetuum mobile, i.e., a "futile cycle" of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix S. The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and -in a stricter sense -futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. Furthermore, it is sufficient for the existence of a realization in terms of sum formulas, obeying conservation of "atoms". In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless S implies that they are "obligatory isomers". In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN.Conclusions: "Chemical" RNs are directed hypergraphs with a stoichiometric matrix S whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain S by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesing avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes.

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“…On more theoretical constructions, important progress has been made on Population Protocols [1,5,10], where anonymous agents with only a constant amount of memory available randomly interact with each other and are able to compute functions that do not seem possible at first, like electing leaders. Population protocols are a typical example of a passive model, and are related to chemical reaction networks [9]. Additionally, cellular automata use very simple update rules that give rise to interesting patterns [2,11].…”

Section: Related Workmentioning

confidence: 99%

On the Convergence of Network Systems

Kipouridis¹,

Tsichlas²

2019

Preprint

0

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The apparent disconnection between the microscopic and the macroscopic is a major issue in the understanding of complex systems. To this extent, we study the convergence of repeatedly applying local rules on a network, and touch on the expressive power of this model. We look at network systems and study their behavior when different types of local rules are applied on them. For a very general class of local rules, we prove convergence and provide a certain member of this class that, when applied on a graph, efficiently computes its k-core and its (k − 1)-crust giving hints on the expressive power of such a model. Furthermore, we provide guarantees on the speed of convergence for an important subclass of the aforementioned class. We also study more general rules, and show that they do not converge. Our counterexamples resolve an open question of (Zhang, Wang, Wang, Zhou, KDD -2009) as well, concerning whether a certain process converges. Finally, we show the universality of our network system, by providing a local rule under which it is Turing-Complete.

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Computational complexity of atomic chemical reaction networks

Cited by 5 publications

References 39 publications

What makes a reaction network “chemical”?

What makes a reaction network “chemical”?

What makes a reaction network "chemical"?

On the Convergence of Network Systems

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