Demystification of Entangled Mass Action Law

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Demystiﬁcation of Entangled Mass Action Law

A. N. Kirdina,b,∗, S. V. Stasenkoa

aLobachevsky University, Nizhni Novgorod, Russia

bInstitute for Computational Modelling, Russian Academy of Sciences, Siberian Branch,

Krasnoyarsk, Russia

Abstract

Recently, Gorban (2021) analysed some kinetic paradoxes of the transition

state theory and proposed its revision that gave the “entangled mass action

law”, in which new reactions were generated as an addition to the reaction

mechanism under consideration. These paradoxes arose due to the assump-

tion of quasiequilibrium between reactants and transition states.

In this paper, we provided a brief introduction to this theory, demon-

strating how the entangled mass action law equations can be derived in the

framework of the standard quasi steady state approximation in combination

with the quasiequilibrium generalized mass action law for an auxiliary re-

action network including reactants and intermediates. We also proved the

basic physical property (positivity) for these new equations, which was not

obvious in the original approach.

Keywords: transition state; quasiequilibrium; quasi steady state; entangled

mass action law; generalised mass action law

1. Introduction

Mass Action Law (MAL) exists in two basic forms:

∗Corresponding author

Email addresses: kirdinalexandergp@gmail.com (A. N. Kirdin),

stasenko@neuro.nnov.ru (S. V. Stasenko)

Preprint submitted to Elsevier

arXiv:2210.00787v3 [physics.chem-ph] 28 Mar 2023

1. Equilibrium (or static) MAL that describes chemical equilibria by the

systems of algebraic equations;

2. Dynamic (or kinetic) MAL invented for description of chemical dynam-

ics.

Both versions were proposed by Guldberg and Waage in a series of papers in

1864–1879.

In physical chemistry, the static MAL was developed to the most gen-

eral thermodynamic form for perfect and non-perfect systems by Gibbs [1].

The physical justiﬁcation of the dynamic MAL was provided in 1935 simul-

taneously by Eyring, Polanyi and Evans. They introduced transition states

(activated complexes) as universal intermediates in chemical reactions (we

refer to the analytic review [2] for the basic notions and further references).

The static MAL was used in Transitions State Theory (TST) to describe

the quasiequilibrium between the reactants and the transition state. Re-

cently it was demonstrated that the assumption about quasiequilibrium be-

tween reactants and transition states leads to some paradoxes in modelling

of multistage or reversible reactions [3,4]. A new kinetic framework for the

TST was proposed.

Surprisingly, these new models led to the same (generalised) MAL expres-

sions but with the entanglement eﬀect: the MAL rate of some elementary

reactions are also included in the reaction rates of other reactions. These

results created a mystery: if we tried to abandon MAL quasiequilibrium as-

sumption then we came to the same MAL with the shuﬄed reaction rates.

This puzzle must be solved. Some technical questions also remain open. For

example, positivity of the quasi steady state concentrations of the interme-

diates should be proven. Explicit formulas for entropy production are also

very desirable.

In this work, we obtained the following results:

1. We demonstrated how the new entangled MAL equations may be de-

rived in the framework of the standard quasi steady state assumption

combined with the quasiequilibrium generalised mass action law.

2. We proved a basic physical property (positivity) of the new entangled

MAL equations, which was not obvious in the original work [4]

In Sec. 2we brieﬂy describe the kinetic paradoxes in the TST. In Sec. 3

the fundamentals of entangled MAL are presented with the Positivity The-

orem and a simple (perhaps, the simplest) example. Sec. 4demystiﬁes the

entangled MAL theory and represents a complex reaction as a network of the

ﬁrst order transitions between intermediates and generalised MAL transitions

between the complexes of the reactants and the corresponding intermediates.

2. Quasiequilibrium paradox in transition state theory

The most prominent approach to justifying Mass Action Law (MAL) was

provided by Eyring, Polanyi and Evans [2]. They introduced transition states

(activated complexes) as universal intermediates in chemical reactions. The

basic textbook scheme is (1

A+B[A−B]→Products .(1)

Here we use the notation [A−B] for the transition state or activated com-

plex. The key assumption was that the activated complexes are in quasi-

equilibrium with the reactants. Therefore, the quasiequilibrium concentra-

tions of the activated complex can be estimated using thermodynamics, and

the overall reaction rate is the product of this concentration and the reaction

rate constant for the [A−B]→Products transition.

An additional assumption is the low concentration of the activated com-

plex compared to the concentrations of the reactants. Without this hypoth-

esis, the MAL formulas cannot be produced [5].

Thus, the problem of estimating the reaction rate was divided into two

tasks:

•Thermodynamic equilibration A+B[A−B];

•Dynamic evaluation of transition rate [A−B]→Products .

This nice picture hides several problems. First of all, the reaction can be

reversible. Consider, just for simplicity, products C+D(2):

A+B[A−B]C+D. (2)

Microreversibility and detailed balance require that the reverse reaction in (2)

follows the same route as the initial reaction. According to the quasiequilib-

rium assumption, the reaction [A−B]C+Dshould be also in quasiequilir-

ium. Simple algebra demonstrates that these two quasiequilibrium assump-

tions imply complete equilibrium and reaction vanishes [3,4]. Two solution

to this paradox were proposed:

1. Consider the model with two intermediates and transition between

them:

A+B[A−B][C−D]C+D.

Asymptotic assumptions about two quasiequilibria, A+B[A−B]

and [C−D]C+D, and smallness of the [A−B] and [C−D]

concentrations lead to classical MAL in very wide conditions [5,6].

2. Abandon the quasiequilibrium hypothesis but keep the assumption that

the concentration of the active complex is much smaller than that of the

reactants and prove this assumption when possible [3]. This assump-

tion violates the polynomial MAL and leads to more complex rational

reaction rate dependencies.

Both approaches have a long history. Combination of quasiequilibrium

and small concentration assumptions for intermediate compounds was used

by Michaelis and Menten in 2013 [7]. Stueckelberg in 1952 [8] used the same

two assumptions for analysis of the Boltzmann equation beyond microre-

versibility and proved general semidetailed balance that is known now also

as cyclic balance or complex balance. The quasiequilibrium condition in en-

zyme kinetics was abolished by [9]. (For more modern analysis we refer to

the work by [10].) They assumed only the smallness of intermediate concen-

trations and obtained a nonpolynomial reaction rate, which was called the

Michelis–Menten kinetics. The same formula was proposed by [3] for general

transition state kinetics.

This approach may give correct answers but has some logical issues: We

aim to justify MAL for general (non-linear) kinetics. The transition state

theory uses thermodynamic deﬁnition of quasiequilibrium (the static MAL)

and simple ﬁrst order Markov kinetics for transition of activated complex.

The result is the MAL kinetics for nonlinear reactions of arbitrary complexity.

But if we would like to apply the Briggs–Haldane approach then we must

assume dynamic MAL for all elementary transitions from scratch, before

justifying.

The assumption about small concentrations of the intermediates was used

explicitly in enzyme kinetics [7,9], in gas kinetics [8], and in kinetics of

heterogeneous catalytic reactions [11]. In TST, it is used usually implicitly

and, therefore, needs further clariﬁcation. Let us take the basic example from

the popular textbook [12] (Section “Transition State Theory”): A+B

C→P, where Aand Bare the reactants, Cis the activated complex and Pis

the product. The fast quasiequilibrium assumption gives [C] = K[A][B]. The

TST produces an estimate of the reaction rate constant kfor the transition

C→P. After that, we have to exclude [C] from the material balance

equations using the quasiequilibrium assumption.

The material balance gives

d[P]

dt =−dM

dt =kK[A][B],(3)

where M= [A] + [B] + [C] Notice that ∆ = [B]−[A] does not change in the

reaction. The quasiequilibrium assumption provides the quadratic equation

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摘要：

DemysticationofEntangledMassActionLawA.N.Kirdina,b,,S.V.StasenkoaaLobachevskyUniversity,NizhniNovgorod,RussiabInstituteforComputationalModelling,RussianAcademyofSciences,SiberianBranch,Krasnoyarsk,RussiaAbstractRecently,Gorban(2021)analysedsomekineticparadoxesofthetransitionstatetheoryandproposedi...

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