# The Michaelis-Menten-Stueckelberg Theorem

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## Abstract

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## 1. Introduction

#### 1.1. Main Asymptotic Ideas in Chemical Kinetics

#### 1.2. The Structure of the Paper

#### 1.3. Main Results: One Asymptotic and Two Theorems

- The compounds are in fast equilibrium with the corresponding input reagents (QE);
- They exist in very small concentrations compared to other components (QSS).

- The activated complexes are in a quasi-equilibrium with the reactant molecules;
- Rates of the reactions are studied by studying the activated complexes at the saddle point of a potential energy surface.

## 2. QE and Preservation of Entropy Production

**MaxEnt**optimization problem:

**Theorem about preservation of entropy production.**

## 3. The Classics and the Classical Confusion

#### 3.1. The Asymptotic of Fast Reactions

#### 3.2. QSS and the Briggs-Haldane Asymptotic

- Enzymes (or catalysts, or radicals) participate in fast reactions and, hence, relax faster than substrates or stable components. This is obviously wrong for many QSS systems: For example, in the Michaelis-Menten system all reactions include enzyme together with substrate or product. There are no separate fast reactions for enzyme without substrate or product.
- Concentrations of intermediates are constant because in QSS we equate their time derivatives to zero. In general case, this is also wrong: We equate the kinetic expressions for some time derivatives to zero, indeed, but this just exploits the fact that the time derivatives of intermediates concentrations are small together with their values, but not obligatory zero. If we accept QSS then these derivatives are not zero as well: To prove this we can just differentiate the Michaelis-Menten formula (11) and find that [ES] in QSS is almost constant when $\left[S\right]\gg {K}_{M}$, this is an additional condition, different from the Briggs-Haldane condition $\left[E\right]+\left[AE\right]\ll \left[S\right]$ (for more details see [1,14,33] and a simple detailed case study [41]).

#### 3.3. The Michaelis and Menten Asymptotic

## 4. Chemical Kinetics and QE Approximation

#### 4.1. Stoichiometric Algebra and Kinetic Equations

#### 4.2. Formalism of QE Approximation for Chemical Kinetics

#### 4.2.1. QE Manifold

#### 4.2.2. QE in Traditional MM System

#### 4.2.3. Heterogeneous Catalytic Reaction

#### 4.2.4. Discussion of the QE procedure for Chemical Kinetics

## 5. General Kinetics with Fast Intermediates Present in Small Amount

#### 5.1. Stoichiometry of Complexes

#### 5.2. Stoichiometry of Compounds

#### 5.3. Energy, Entropy and Equilibria of Compounds

#### 5.4. Markov Kinetics of Compounds

#### 5.5. Thermodynamics and Kinetics of the Extended System

#### 5.6. QE Elimination of Compounds and the Complex Balance Condition

#### 5.7. The Big Michaelis-Menten-Stueckelberg Theorem

- Concentrations of the compounds ${B}_{\rho}$ are close to their quasiequilibrium values (65)$${\varsigma}_{j}=(1+\delta ){\varsigma}_{j}^{\mathrm{QE}}=(1+\delta ){\varsigma}_{j}^{*}(b,T)exp\left(\frac{{\sum}_{i}{\nu}_{ji}{\mu}_{i}(b,T)}{RT}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\delta \ll 1$$
- Concentrations of the compounds ${B}_{\rho}$ are much smaller than the concentrations of the components ${A}_{i}$: There is a small positive parameter $\epsilon \ll 1$, ${\varsigma}_{j}^{*}=\epsilon {\xi}_{j}^{*}$ and ${\xi}_{j}^{*}$ do not depend on ε;
- Kinetics of transitions between compounds ${B}_{i}\to {B}_{j}$ is linear (Markov) kinetics with reaction rate constants ${k}_{ji}=\frac{1}{\epsilon}{\kappa}_{ji}$.

## 6. General Kinetics and Thermodynamics

#### 6.1. General Formalism

**Warning**: This definition differs from the chemical potentials (22) by the factor $1/RT$: For constant volume the Massieu-Planck potential is $-F/T$ and we, in addition, divide it on R. On the other hand, we keep the same sign as for the chemical potentials, and this differs from the standard Legendre transform for S. (It is the Legendre transform for function $-S$).

- The energy of the Universe is constant.
- The entropy of the Universe tends to a maximum.

#### 6.2. Accordance Between Kinetics and Thermodynamics

#### 6.2.1. General Entropy Production Formula

#### 6.2.2. Detailed Balance

#### 6.2.3. Complex Balance

#### 6.2.4. G-Inequality

## 7. Linear Deformation of Entropy

#### 7.1. If Kinetics Does not Respect Thermodynamics then Deformation of Entropy May Help

#### 7.2. Entropy Deformation for Restoration of Detailed Balance

#### 7.3. Entropy Deformation for Restoration of Complex Balance

**The deficiency zero theorem.**

_{2}+2Pt with an arbitrarily small but positive constant in order to make the mechanism weakly reversible.

#### 7.4. Existence of Points of Detailed and Complex Balance

#### 7.5. The Detailed Balance is Needed More Often than the Complex Balance

## 8. Conclusions

- The detailed balance: The kinetic factors for mutually reverse reaction should coincide, ${\phi}^{+}={\phi}^{-}$. This identity is proven for systems with microreversibility (Section 7.5).
- The complex balance: The sum of the kinetic factors for all elementary reactions of the form ${\sum}_{i}{\alpha}_{i}{A}_{i}\to \dots $ is equal to the sum of the kinetic factors for all elementary reactions of the form $\dots \to {\sum}_{i}{\alpha}_{i}{A}_{i}$ (91). This identity is proven for all systems under the Michaelis-Menten-Stueckelberg assumptions about existence of intermediate compounds which are in fast equilibria with other components and are present in small amounts.

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**Figure 2.**A multichannel view on the complex transformation. The hidden reactions between compounds are included in an oval $\mathrm{S}$.

## Appendix

## 1. Quasiequilibrium Approximation

#### 1.1. Quasiequilibrium Manifold

**The slaving assumption.**

**The assumption of small fast-slow projection**.

**MaxEnt**optimization problem:

**Remark.**

#### 1.2. Preservation of Entropy Production

**Theorem about preservation of entropy production.**