On the Mathematics of the Law of Mass Action

Abstract

In 1864, Waage and Guldberg formulated the "law of mass action." Since that time, chemists, chemical engineers, physicists and mathematicians have amassed a great deal of knowledge on the topic. In our view, sufficient understanding has been acquired to warrant a formal mathematical consolidation. A major goal of this consolidation is to solidify the mathematical foundations of mass action chemistry -to provide precise definitions, elucidate what can now be proved, and indicate what is only conjectured. In addition, we believe that the law of mass action is of intrinsic mathematical interest and should be made available in a form that might transcend its application to chemistry alone. We present the law of mass action in the context of a dynamical theory of sets of binomials over the complex numbers.

Computational Complexity of Atomic Chemical Reaction Networks

“…The most directly related previous work is that of Adleman, Gopalkrishnan, Huang, Moisset, and Reishus [1] and of Gnacadja [21], which we now discuss in conjunction with our results.…”

“…Adleman, Gopalkrishnan, Huang, Moisset, and Reishus [1] defined reachably atomic chemical reaction networks and proved the global attractor conjecture holds for such networks. Gnacadja [21], attacking similar goals, defined a notion of atomicity called "species decomposition" and showed a similar result.…”

“…Gnacadja [21], attacking similar goals, defined a notion of atomicity called "species decomposition" and showed a similar result. We establish links between our definitions and those of both [21] and [1] in Section C. We discuss related complexity issues in Sections 6 and C. In particular, Adleman et al [1] showed that it is decidable whether a given network is reachably atomic. This is not obvious since the condition of a species being separable into its constituent atoms via reactions appears to require an unbounded search.…”

“…Furthermore, at least two inequivalent definitions exist in the literature [1,21]. It is not the goal of this paper to identify a single correct definition.…”

“…We show that deciding if a network is subset atomic is in NP, and "whether a network is subset atomic with respect to a given atom set" is strongly NP-complete. Reachably atomic, studied by Adleman, Gopalkrishnan et al [1, 23], 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.…”