# Thermodynamic validity criterion for the irreversible

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arXiv:2006.06476v1 [physics.bio-ph] 11 Jun 2020

Thermodynamic validity criterion for the irreversible Michaelis-Menten equation

Val´erie Voorsluijs1,2*, Francesco Avanzini1**, Massimiliano Esposito1***

1 Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg. 2 Luxembourg Centre for Systems Biomedicine, University of Luxembourg, L-4365 Esch-sur-Alzette, Luxembourg.

* [email protected] ** [email protected] *** [email protected]

Abstract

Enzyme kinetics is very often characterised by the irreversible Michaelis-Menten (MM) equation. However, in open chemical reaction networks such as metabolic pathways, this approach can lead to significant kinetic and thermodynamic inconsistencies. Based on recent developments in nonequilibrium chemical thermodynamics, we present a validity criterion solely expressed in terms of the equilibrium constant of the enzyme-catalysed reaction. When satisfied, it guarantees the ability of the irreversible MM equation to generate kinetic and thermodynamic data that are quantitatively reliable for reasonable ranges of concentrations. Our validity criterion is thus a precious tool to ensure reliable kinetic and thermodynamic modelling of pathways. We also show that it correctly identifies the so-called irreversible enzymatic reactions in glycolysis and Krebs cycle.

February 23, 2022

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Author summary

Living systems need to continuously feed on nutrients present in their surrounding to gather the energy necessary to their survival. That energy is extracted by a complex network of chemical reactions which constitutes metabolism. Enzymes are proteins that speed up and regulate the reactions without being consumed or produced by them. Most enzymes are characterised by only two kinetic parameters which are measured experimentally and reported in tables. By doing so, one implicitly assumes enzymes follow a kinetic mechanism called the irreversible Michaelis-Menten (MM) scheme. While this assumption usually holds in vitro, it is by no means ensured to apply to living systems. Furthermore, the irreversible MM scheme is a priori inconsistent with thermodynamics, the theory describing energy conversion in systems ranging from car engines to molecular motors. In this paper, we propose a way to easily test if the MM equation holds and explain how to make the irreversible MM scheme consistent with thermodynamics. Beside providing precious information about which enzymes need to be characterised beyond the irreversible MM scheme, our work paves the way for more realistic thermodynamic considerations in biochemistry to understand the efficiency with which living systems process energy.

Introduction

Enzymes are ubiquitous proteins catalysing biochemical reactions with very high efficiency. They are involved in a broad array of physiological processes, ranging from metabolism to cell signalling, and usually display a high selectivity with respect to their substrate. Enzyme kinetics can be subject to regulation by activators and/or inhibitors, and the fluxes along the biochemical pathways can hence be modulated in the course of time and, for example, adapt to the metabolic demand. Variations in kinetic rates can also indicate the presence of a disease or be induced by a pharmacological treatment. Appropriate mathematical descriptions of enzyme kinetics are thus crucial to improve the current understanding of physiology in health and disease.

The analysis of complex reactions networks such as biochemical pathways usually relies on reduced mathematical models incorporating a smaller number of variables

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February 23, 2022

and/or parameters [1, 2]. Simplifications can be achieved by nondimensionalisation and scaling of the governing equations [3], and/or by making assumptions regarding the kinetics. More precisely, when the variables of the system evolve on very different timescales, the dynamics is dominated by the behaviour of the slow variables and the fast variables, which are driven towards a pre-equilibrium or a steady state, can be adiabatically eliminated [4]. This idea is at the core of the equilibrium and quasi-steady-state approximations [1, 2]. However, this elimination can lead to thermodynamic inconsistencies in the model [5] and these aspects should be taken into account [6].

In the context of enzyme kinetics, the Michaelis-Menten (MM) equation is extensively used to characterise the rate of enzyme-catalysed reactions [7, 8]. This equation not only relies on the steady-state approximation, but additionally assumes an irreversible mechanism. While some enzymes such as invertases, phosphatases and peptidases can indeed be considered as “one-way” catalysts, where the backward transformation is negligible, most enzymatic reactions are reversible [8, 9]. If the reaction product has not accumulated in sufficiently large quantities to make the reverse reaction significant, as is the case at the beginning of the reaction, the irreversible MM equation can be applied to reversible processes without jeopardising the relevance of kinetic results [9]. For example, enzyme characterisation typically takes place in this regime in order to avoid the effects of the reverse reaction [8]. However, how can we ascertain that the MM equation provides a good description of the product formation rate in reaction pathways [10]? In this case, each reaction can be assimilated to an open system where the substrate and product are continuously injected and removed, which contrasts with the conditions of enzyme characterisation, usually proceeding in closed environments (i.e. without exchange of matter).

In order to provide a more robust framework for enzyme modelling, it thus appears crucial to derive explicit conditions of validity for the irreversible MM equation. Different theoretical approaches have been adopted to this end [11–13], but mainly focussed on kinetic and conformational aspects of catalysis. So far, the thermodynamic validity of the MM equation has been sidestepped, probably because of the theoretical issues raising from the irreversible character of the reaction [14]. In this paper, we address that gap by deriving a nonequilibrium thermodynamic criterion indicating

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February 23, 2022

whether the irreversible MM equation is a valid model for a given single-substrate enzymatic reaction. The novelty of our approach is twofold. We not only provide modellers with an explicit criterion indicating when the irreversible MM rate is thermodynamically consistent. We also show that nonequilibrium thermodynamics can be applied to irreversible reactions.

This paper is organised as follows. In Methods, we describe and compare the irreversible and reversible MM equations before introducing the entropy production rate (EPR) of reactive systems. This quantity is used to derive our validity criterion, which is one of the key results of this paper. The validity of our approach is examined in two ways. On the one hand, we test our criterion by applying it to well-characterised biochemical reactions. On the other hand, we quantify the error made on the EPR when it is estimated via the irreversible flux. We finally summarise our results and discuss the relevance of our approach to less well characterised pathways.

Methods

The Michaelis-Menten equations

Irreversible case The most common mechanism representing an enzyme-catalysed reaction of the type S → P follows the kinetic scheme:

E + S −−k−1− ES −k→2 E + P,

(1)

k−1

where substrate S binds to enzyme E to form a complex ES that further releases the

intact enzyme and the reaction product P. This mechanism, which neglects the reverse

reaction and regulation processes such as product inhibition, was used to derive the

irreversible MM equation:

Jirr = kcat [E]0 [S] ,

(2)

[S] + KM

where Jirr is the reaction rate, [S] is the concentration of the substrate, [E]0 is the total concentration of the enzyme ([E]0 = [E] + [ES]), kcat is the catalytic constant or turnover number, and KM is the Michaelis constant [8]. As ES is initially absent, [E]0

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also corresponds to the initial concentration of E. For the two-step mechanism shown in (1), kcat = k2 and KM = k−1k+1 k2 , but Eq (2) also applies to more complex kinetic schemes involving multiple steps. The catalytic efficiency of an enzyme, defined as kcat/KM , and the limiting rate, V = kcat [E]0, are also widely used in the literature to characterise enzymes.

The derivation of Eq (2) relies on the assumption that the formation of the complex is fast and the enzyme rapidly saturated, leading to a quasi-steady state for the complex (i.e. dt [ES] = 0) [1, 2, 15]. This condition is fulfilled if the substrate is in large excess with respect to the enzyme or if KM is relatively low compared to the substrate concentration. A deeper timescale analysis of the system provides a refined condition guaranteeing the validity of the quasi-steady-state approximation [1]:

[E]0

1

1,

(3)

[S]0 + KM 1 + (k−1/k2) + (k1 [S]0 /k2)

where [S]0 is the initial substrate concentration.

Reversible case Using the quasi-steady-state approximation for ES, a rate expression can be derived similarly for the reversible scheme

E + S −−k−1− ES −−k−2− E + P,

(4)

k−1

k−2

and reads

Jrev

=

[E]0

(ks

[S]

−

kp

[P]) ,

(5)

1 + K[SM]s + K[PM]p

where [P] is the product concentration, ks = k−k11+k2k2 , kp = kk− −11+k−k22 , KMs = k−1k+1 k2 , KMp = k−k1−+2k2 and the other notations are the same as in Eq (2). Although feasible and based on experimental procedures similar to the handling of irreversible enzymes,

the characterisation of reversible enzymatic processes is less straightforward, since it

requires measurements for backward and forward reactions [9, 16], and tends to be

avoided. Using the irreversible MM equation is thus usually more convenient and we

now examine the thermodynamic conditions under which this approximation can be

made.

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Nonequilibrium thermodynamics of reversible processes

To investigate the validity of the irreversible MM equation, we base our analysis on the EPR, Σ˙ . As a consequence of the second law of thermodynamics, the EPR of a system is always non-negative and is equal to zero at equilibrium. In this Section, we introduce key thermodynamic quantitites involved in the computation of EPR for elementary processes and the reversible MM scheme, before extending this formalism to the irreversible MM scheme, as further detailed in Results. A more systematic description of the nonequilibrium thermodynamics of chemical reaction networks can be found in refs [17] and [6].

Elementary processes

In the absence of mass and heat transport, the EPR associated with a set of N chemical

reactions is given by Σ˙ = N Jρ ATρ ≥ 0, (6)

ρ=1

where T is the absolute temperature while Jρ and Aρ are the net rate and affinity of

reaction ρ, respectively. In nonequilibrium thermodynamics, the latter factor is often

referred to as the force acting on the system while the first factor is interpreted as the

response or the flux of the system. The equilibrium state is characterised by zero forces

and hence zero fluxes.

For elementary processes, the net rate is Jρ = J+ρ − J−ρ and, according to the law

of mass action, the reaction rates associated with the forward (+ρ) and backward (−ρ)

reactions are given by

J±ρ = k±ρ

Z

νσ±ρ σ

,

(7)

σ

where Zσ is the concentration of species σ and νσ±ρ is the stoichiometric coefficient of species σ in reaction ±ρ. The forward and backward reaction currents become equal at

equilibrium (this is the principle of detailed balance), so the net reaction rate becomes

zero. It follows that

k+ρ

eq Sρ

= Zσ σ ,

(8)

k−ρ σ

where Sρσ = νσ−ρ − νσ+ρ is the net stoichiometric coefficient of species σ in reaction ρ and superscript “eq” denotes equilibrium conditions. The right-hand side of Eq (8) is the

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equilibrium constant, Kρ, related to the standard Gibbs free energy of reaction by

∆ρG◦

k+ρ

Kρ = exp −

=,

(9)

RT

k−ρ

where R is the gas constant. On the other hand, the affinity of reaction ρ is defined by [18]

Aρ = − Sρσµσ,

(10)

σ

where µσ is the chemical potential of species σ. Under the hypothesis of local

equilibrium, i.e. state variables such as temperature and pressure relax to equilibrium on

a much faster timescale than reaction rates so the expressions for the thermodynamic

potentials derived at equilibrium still hold locally out-of-equilibrium [18], the chemical

potential µσ is given by

µσ = µ◦σ + RT ln Zσ,

(11)

where µ◦σ denotes the standard chemical potential of species σ. Standard conditions correspond to atmospheric pressure p◦ = 1 bar and molar concentrations Zσ◦ = 1 mol L−1. Finally, standard chemical potentials are directly related to the standard Gibbs free energy of reaction

∆ρG◦ = Sρσµ◦σ.

σ

(12)

Combining equations (7), (9), (10), (11) and (12) yields

k+ρ

Z νσ+ρ

σ

Aρ = RT ln

σ −ρ = RT ln J+ρ

k−ρ Zσνσ

J−ρ

σ

(13)

and the EPR, which can then be rewritten as

Σ˙ = (J+ρ − J−ρ) R ln J+ρ

(14)

ρ J−ρ

is thus necessarily non-negative. Also, it clearly appears in Eq (14) that the affinity and hence the EPR diverge if process ρ is irreversible, i.e. J−ρ → 0.

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Reversible MM scheme

In living systems, biochemical reactions are maintained out of equilibrium due to exchanges with the environment (influx of nutrients, excretion, etc), which influences the steady-state of the system or even leads to more complex dynamics such as oscillations [1, 2]. To mimic this feature at the scale of a given pathway or catalytic step, it is usual to assume that the input and output chemicals are chemostatted species, i.e. their concentration is maintained constant in the course of time due to continuous exchanges with the environment and homeostasis. In the MM scheme, we thus consider that S and P are chemostatted, with concentrations [S] and [P], respectively. By doing so, we assume that the dynamics of the global pathway has relaxed to a steady state or oscillates at a frequency much slower than the timescale of the enzymatic reaction of interest.

At steady state (denoted “ss”), Eq (14) is equivalent to

Σ˙ sresv = Jrev ATrev ,

(15)

where Jrev = J1ss = J2ss is given by Eq (5) and Arev = A1 + A2 = RT ln k−k11kk−2[2S[]P] , with indexes 1 and 2 referring to the first and second elementary steps constituting the reversible MM scheme (4).

Put in this form, the affinity and the EPR diverge in the irreversible limit, which corresponds to k−2 [P] → 0. However, the affinity can be rewritten using the definition of chemical potential (Eq (11)). We then have

Arev = µ◦S − µ◦P + RT ln [[PS]] , (16)

where µ◦P − µ◦S = ∆revG◦ is the standard Gibbs free energy of the reaction S → P. It can be expressed as ∆revG◦ = ∆f G◦P − ∆f G◦S, where ∆f G◦P and ∆f G◦S are the standard Gibbs free energies of formation of P and S, respectively. These quantities are thermodynamic data usually available in tables and Arev can thus be calculated for positive and non-zero values of [S] and [P], as is the case in physiological conditions.

Using Eq (16) to circumvent the divergence issues is thus the first milestone towards a thermodynamically-consistent modelling of irreversible enzymatic reactions, as

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discussed in the next Section.

Results

Nonequilibrium thermodynamics of the irreversible MM scheme

We apply the concepts introduced in the previous Section and ref [6] to a catalytic step proceeding according to the MM kinetics as if it were part of a biochemical pathway. To evaluate the steady-state EPR associated with the irreversible MM kinetics, the natural option is to approximate the reversible flux in Eq (15) by its irreversible counterpart:

Σ˙ sirsr = Jirr ATrev ,

(17)

where Jirr is given by (2) and Arev is written as (16) with positive and non-zero concentrations for S and P. The irreversible steady-state EPR can then be computed using the standard Gibbs free energies of formation of S and P and the kinetic parameters involved in Jirr, i.e. KM and V , which are typically available.

In order to check that Eq (17) is consistent with Eq (15), we need to compare the EPR obtained in both cases. However, the comparison is only possible when the kinetic parameters of Jrev (ks, kp, KMs and KMp) are also available, which is usually not the case. A more general and systematic analysis can be performed if the steady-state EPR is written in terms of the kinetic constants {k1, k2, k−1, k−2}, where {k1, k2, k−1} correspond to the available KM and V , while k−2 is set to a positive and arbitrarily small value to mimic irreversibility. k−2 can then be gradually increased to tend to a reversible reaction. Eq (15) becomes

Σ˙ ss

=

[E]0

(k1

k2

[S]

−

k−1

k−2

[P]) R ln

k1 k2 [S]

,

(18)

rev k1 [S] + k−2 [P] + k−1 + k2

k−1 k−2 [P]

and Eq (17)

Σ˙ ss = k1k2 [E]0 [S] R ln k1 k2 [S] .

(19)

irr k1 [S] + k−1 + k2

k−1 k−2 [P]

In Eq (18), the flux and the force terms have always the same sign and the EPR is thus always positive. However, in Eq (19), Jirr is always positive while Arev can take negative values if k1 k2 [S] < k−1 k−2 [P], which leads to a negative EPR, in

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ss 10-6 J

Σirr (

)

min L K

0 5 10 15 20 25 30

1.0

1.0

1.0

0.8

0.8

0.8

[P] (μM) [P] (μM) [P] (μM)

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.0 0.2 0.4 0.6 0.8 1.0

[S] (μM)

(a)

[S] (μM)

(b)

[S] (μM)

(c)

Fig 1. Contour plot of the irreversible steady-state EPR as a function of

substrate and product concentrations. k−2 is 0.01 µM−1 min-1 in (a), 0.10 µM−1 min-1 in (b) and 0.50 µM−1 min-1 in (c), respectively. The other parameter values are

k1 = 1 µM−1 min-1, k−1 = 20 min-1 and k2 = 26 min-1. Such values correspond to KM = 46 µM and V = 26 µM min-1 and are for example of the same order of magnitude

as the kinetic parameters reported for the tyrosine hydroxylase in the synthesis of dopamine from L-tyrosine [19]. The dashed black line corresponds to Σ˙ sirsr = 0.

contradiction with the second law of thermodynamics. We can thus expect that Eq (19) approximates Eq (18) in a thermodynamically-consistent way when k−2 and [P] are small enough, but how can we define a threshold?

To investigate the conditions under which the use of the irreversible MM equation becomes problematic, we plot Σ˙ sirsr as a function of [S] and [P]. The free parameters are the kinetic constants {k1, k2, k−1, k−2}, which are chosen to be of the same order of magnitude as the typical values of KM and V reported in the literature. We also verify that the reaction is spontaneous in standard conditions, i.e. ∆revG◦ < 0. Our results are independent of the enzyme total concentration ([E]0), which is set to 1 µM, and the temperature is set to 298.15 K to match the conditions of the thermodynamic tables used in the following.

As shown in Fig 1, Σ˙ sirsr < 0 for a certain range of concentration of the chemostatted species S and P, but this behaviour tends to disappear as k−2 decreases. More importantly, the boundary defining Σ˙ sirsr = 0 is always a straight line of equation

[P] = [S] exp − ∆revG◦ , (20) RT

which can easily be derived by introducing ∆revG◦ into the last factor of Eq (19) via

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Thermodynamic validity criterion for the irreversible Michaelis-Menten equation

Val´erie Voorsluijs1,2*, Francesco Avanzini1**, Massimiliano Esposito1***

1 Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg. 2 Luxembourg Centre for Systems Biomedicine, University of Luxembourg, L-4365 Esch-sur-Alzette, Luxembourg.

* [email protected] ** [email protected] *** [email protected]

Abstract

Enzyme kinetics is very often characterised by the irreversible Michaelis-Menten (MM) equation. However, in open chemical reaction networks such as metabolic pathways, this approach can lead to significant kinetic and thermodynamic inconsistencies. Based on recent developments in nonequilibrium chemical thermodynamics, we present a validity criterion solely expressed in terms of the equilibrium constant of the enzyme-catalysed reaction. When satisfied, it guarantees the ability of the irreversible MM equation to generate kinetic and thermodynamic data that are quantitatively reliable for reasonable ranges of concentrations. Our validity criterion is thus a precious tool to ensure reliable kinetic and thermodynamic modelling of pathways. We also show that it correctly identifies the so-called irreversible enzymatic reactions in glycolysis and Krebs cycle.

February 23, 2022

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February 23, 2022

Author summary

Living systems need to continuously feed on nutrients present in their surrounding to gather the energy necessary to their survival. That energy is extracted by a complex network of chemical reactions which constitutes metabolism. Enzymes are proteins that speed up and regulate the reactions without being consumed or produced by them. Most enzymes are characterised by only two kinetic parameters which are measured experimentally and reported in tables. By doing so, one implicitly assumes enzymes follow a kinetic mechanism called the irreversible Michaelis-Menten (MM) scheme. While this assumption usually holds in vitro, it is by no means ensured to apply to living systems. Furthermore, the irreversible MM scheme is a priori inconsistent with thermodynamics, the theory describing energy conversion in systems ranging from car engines to molecular motors. In this paper, we propose a way to easily test if the MM equation holds and explain how to make the irreversible MM scheme consistent with thermodynamics. Beside providing precious information about which enzymes need to be characterised beyond the irreversible MM scheme, our work paves the way for more realistic thermodynamic considerations in biochemistry to understand the efficiency with which living systems process energy.

Introduction

Enzymes are ubiquitous proteins catalysing biochemical reactions with very high efficiency. They are involved in a broad array of physiological processes, ranging from metabolism to cell signalling, and usually display a high selectivity with respect to their substrate. Enzyme kinetics can be subject to regulation by activators and/or inhibitors, and the fluxes along the biochemical pathways can hence be modulated in the course of time and, for example, adapt to the metabolic demand. Variations in kinetic rates can also indicate the presence of a disease or be induced by a pharmacological treatment. Appropriate mathematical descriptions of enzyme kinetics are thus crucial to improve the current understanding of physiology in health and disease.

The analysis of complex reactions networks such as biochemical pathways usually relies on reduced mathematical models incorporating a smaller number of variables

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February 23, 2022

and/or parameters [1, 2]. Simplifications can be achieved by nondimensionalisation and scaling of the governing equations [3], and/or by making assumptions regarding the kinetics. More precisely, when the variables of the system evolve on very different timescales, the dynamics is dominated by the behaviour of the slow variables and the fast variables, which are driven towards a pre-equilibrium or a steady state, can be adiabatically eliminated [4]. This idea is at the core of the equilibrium and quasi-steady-state approximations [1, 2]. However, this elimination can lead to thermodynamic inconsistencies in the model [5] and these aspects should be taken into account [6].

In the context of enzyme kinetics, the Michaelis-Menten (MM) equation is extensively used to characterise the rate of enzyme-catalysed reactions [7, 8]. This equation not only relies on the steady-state approximation, but additionally assumes an irreversible mechanism. While some enzymes such as invertases, phosphatases and peptidases can indeed be considered as “one-way” catalysts, where the backward transformation is negligible, most enzymatic reactions are reversible [8, 9]. If the reaction product has not accumulated in sufficiently large quantities to make the reverse reaction significant, as is the case at the beginning of the reaction, the irreversible MM equation can be applied to reversible processes without jeopardising the relevance of kinetic results [9]. For example, enzyme characterisation typically takes place in this regime in order to avoid the effects of the reverse reaction [8]. However, how can we ascertain that the MM equation provides a good description of the product formation rate in reaction pathways [10]? In this case, each reaction can be assimilated to an open system where the substrate and product are continuously injected and removed, which contrasts with the conditions of enzyme characterisation, usually proceeding in closed environments (i.e. without exchange of matter).

In order to provide a more robust framework for enzyme modelling, it thus appears crucial to derive explicit conditions of validity for the irreversible MM equation. Different theoretical approaches have been adopted to this end [11–13], but mainly focussed on kinetic and conformational aspects of catalysis. So far, the thermodynamic validity of the MM equation has been sidestepped, probably because of the theoretical issues raising from the irreversible character of the reaction [14]. In this paper, we address that gap by deriving a nonequilibrium thermodynamic criterion indicating

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whether the irreversible MM equation is a valid model for a given single-substrate enzymatic reaction. The novelty of our approach is twofold. We not only provide modellers with an explicit criterion indicating when the irreversible MM rate is thermodynamically consistent. We also show that nonequilibrium thermodynamics can be applied to irreversible reactions.

This paper is organised as follows. In Methods, we describe and compare the irreversible and reversible MM equations before introducing the entropy production rate (EPR) of reactive systems. This quantity is used to derive our validity criterion, which is one of the key results of this paper. The validity of our approach is examined in two ways. On the one hand, we test our criterion by applying it to well-characterised biochemical reactions. On the other hand, we quantify the error made on the EPR when it is estimated via the irreversible flux. We finally summarise our results and discuss the relevance of our approach to less well characterised pathways.

Methods

The Michaelis-Menten equations

Irreversible case The most common mechanism representing an enzyme-catalysed reaction of the type S → P follows the kinetic scheme:

E + S −−k−1− ES −k→2 E + P,

(1)

k−1

where substrate S binds to enzyme E to form a complex ES that further releases the

intact enzyme and the reaction product P. This mechanism, which neglects the reverse

reaction and regulation processes such as product inhibition, was used to derive the

irreversible MM equation:

Jirr = kcat [E]0 [S] ,

(2)

[S] + KM

where Jirr is the reaction rate, [S] is the concentration of the substrate, [E]0 is the total concentration of the enzyme ([E]0 = [E] + [ES]), kcat is the catalytic constant or turnover number, and KM is the Michaelis constant [8]. As ES is initially absent, [E]0

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also corresponds to the initial concentration of E. For the two-step mechanism shown in (1), kcat = k2 and KM = k−1k+1 k2 , but Eq (2) also applies to more complex kinetic schemes involving multiple steps. The catalytic efficiency of an enzyme, defined as kcat/KM , and the limiting rate, V = kcat [E]0, are also widely used in the literature to characterise enzymes.

The derivation of Eq (2) relies on the assumption that the formation of the complex is fast and the enzyme rapidly saturated, leading to a quasi-steady state for the complex (i.e. dt [ES] = 0) [1, 2, 15]. This condition is fulfilled if the substrate is in large excess with respect to the enzyme or if KM is relatively low compared to the substrate concentration. A deeper timescale analysis of the system provides a refined condition guaranteeing the validity of the quasi-steady-state approximation [1]:

[E]0

1

1,

(3)

[S]0 + KM 1 + (k−1/k2) + (k1 [S]0 /k2)

where [S]0 is the initial substrate concentration.

Reversible case Using the quasi-steady-state approximation for ES, a rate expression can be derived similarly for the reversible scheme

E + S −−k−1− ES −−k−2− E + P,

(4)

k−1

k−2

and reads

Jrev

=

[E]0

(ks

[S]

−

kp

[P]) ,

(5)

1 + K[SM]s + K[PM]p

where [P] is the product concentration, ks = k−k11+k2k2 , kp = kk− −11+k−k22 , KMs = k−1k+1 k2 , KMp = k−k1−+2k2 and the other notations are the same as in Eq (2). Although feasible and based on experimental procedures similar to the handling of irreversible enzymes,

the characterisation of reversible enzymatic processes is less straightforward, since it

requires measurements for backward and forward reactions [9, 16], and tends to be

avoided. Using the irreversible MM equation is thus usually more convenient and we

now examine the thermodynamic conditions under which this approximation can be

made.

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Nonequilibrium thermodynamics of reversible processes

To investigate the validity of the irreversible MM equation, we base our analysis on the EPR, Σ˙ . As a consequence of the second law of thermodynamics, the EPR of a system is always non-negative and is equal to zero at equilibrium. In this Section, we introduce key thermodynamic quantitites involved in the computation of EPR for elementary processes and the reversible MM scheme, before extending this formalism to the irreversible MM scheme, as further detailed in Results. A more systematic description of the nonequilibrium thermodynamics of chemical reaction networks can be found in refs [17] and [6].

Elementary processes

In the absence of mass and heat transport, the EPR associated with a set of N chemical

reactions is given by Σ˙ = N Jρ ATρ ≥ 0, (6)

ρ=1

where T is the absolute temperature while Jρ and Aρ are the net rate and affinity of

reaction ρ, respectively. In nonequilibrium thermodynamics, the latter factor is often

referred to as the force acting on the system while the first factor is interpreted as the

response or the flux of the system. The equilibrium state is characterised by zero forces

and hence zero fluxes.

For elementary processes, the net rate is Jρ = J+ρ − J−ρ and, according to the law

of mass action, the reaction rates associated with the forward (+ρ) and backward (−ρ)

reactions are given by

J±ρ = k±ρ

Z

νσ±ρ σ

,

(7)

σ

where Zσ is the concentration of species σ and νσ±ρ is the stoichiometric coefficient of species σ in reaction ±ρ. The forward and backward reaction currents become equal at

equilibrium (this is the principle of detailed balance), so the net reaction rate becomes

zero. It follows that

k+ρ

eq Sρ

= Zσ σ ,

(8)

k−ρ σ

where Sρσ = νσ−ρ − νσ+ρ is the net stoichiometric coefficient of species σ in reaction ρ and superscript “eq” denotes equilibrium conditions. The right-hand side of Eq (8) is the

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equilibrium constant, Kρ, related to the standard Gibbs free energy of reaction by

∆ρG◦

k+ρ

Kρ = exp −

=,

(9)

RT

k−ρ

where R is the gas constant. On the other hand, the affinity of reaction ρ is defined by [18]

Aρ = − Sρσµσ,

(10)

σ

where µσ is the chemical potential of species σ. Under the hypothesis of local

equilibrium, i.e. state variables such as temperature and pressure relax to equilibrium on

a much faster timescale than reaction rates so the expressions for the thermodynamic

potentials derived at equilibrium still hold locally out-of-equilibrium [18], the chemical

potential µσ is given by

µσ = µ◦σ + RT ln Zσ,

(11)

where µ◦σ denotes the standard chemical potential of species σ. Standard conditions correspond to atmospheric pressure p◦ = 1 bar and molar concentrations Zσ◦ = 1 mol L−1. Finally, standard chemical potentials are directly related to the standard Gibbs free energy of reaction

∆ρG◦ = Sρσµ◦σ.

σ

(12)

Combining equations (7), (9), (10), (11) and (12) yields

k+ρ

Z νσ+ρ

σ

Aρ = RT ln

σ −ρ = RT ln J+ρ

k−ρ Zσνσ

J−ρ

σ

(13)

and the EPR, which can then be rewritten as

Σ˙ = (J+ρ − J−ρ) R ln J+ρ

(14)

ρ J−ρ

is thus necessarily non-negative. Also, it clearly appears in Eq (14) that the affinity and hence the EPR diverge if process ρ is irreversible, i.e. J−ρ → 0.

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February 23, 2022

Reversible MM scheme

In living systems, biochemical reactions are maintained out of equilibrium due to exchanges with the environment (influx of nutrients, excretion, etc), which influences the steady-state of the system or even leads to more complex dynamics such as oscillations [1, 2]. To mimic this feature at the scale of a given pathway or catalytic step, it is usual to assume that the input and output chemicals are chemostatted species, i.e. their concentration is maintained constant in the course of time due to continuous exchanges with the environment and homeostasis. In the MM scheme, we thus consider that S and P are chemostatted, with concentrations [S] and [P], respectively. By doing so, we assume that the dynamics of the global pathway has relaxed to a steady state or oscillates at a frequency much slower than the timescale of the enzymatic reaction of interest.

At steady state (denoted “ss”), Eq (14) is equivalent to

Σ˙ sresv = Jrev ATrev ,

(15)

where Jrev = J1ss = J2ss is given by Eq (5) and Arev = A1 + A2 = RT ln k−k11kk−2[2S[]P] , with indexes 1 and 2 referring to the first and second elementary steps constituting the reversible MM scheme (4).

Put in this form, the affinity and the EPR diverge in the irreversible limit, which corresponds to k−2 [P] → 0. However, the affinity can be rewritten using the definition of chemical potential (Eq (11)). We then have

Arev = µ◦S − µ◦P + RT ln [[PS]] , (16)

where µ◦P − µ◦S = ∆revG◦ is the standard Gibbs free energy of the reaction S → P. It can be expressed as ∆revG◦ = ∆f G◦P − ∆f G◦S, where ∆f G◦P and ∆f G◦S are the standard Gibbs free energies of formation of P and S, respectively. These quantities are thermodynamic data usually available in tables and Arev can thus be calculated for positive and non-zero values of [S] and [P], as is the case in physiological conditions.

Using Eq (16) to circumvent the divergence issues is thus the first milestone towards a thermodynamically-consistent modelling of irreversible enzymatic reactions, as

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February 23, 2022

discussed in the next Section.

Results

Nonequilibrium thermodynamics of the irreversible MM scheme

We apply the concepts introduced in the previous Section and ref [6] to a catalytic step proceeding according to the MM kinetics as if it were part of a biochemical pathway. To evaluate the steady-state EPR associated with the irreversible MM kinetics, the natural option is to approximate the reversible flux in Eq (15) by its irreversible counterpart:

Σ˙ sirsr = Jirr ATrev ,

(17)

where Jirr is given by (2) and Arev is written as (16) with positive and non-zero concentrations for S and P. The irreversible steady-state EPR can then be computed using the standard Gibbs free energies of formation of S and P and the kinetic parameters involved in Jirr, i.e. KM and V , which are typically available.

In order to check that Eq (17) is consistent with Eq (15), we need to compare the EPR obtained in both cases. However, the comparison is only possible when the kinetic parameters of Jrev (ks, kp, KMs and KMp) are also available, which is usually not the case. A more general and systematic analysis can be performed if the steady-state EPR is written in terms of the kinetic constants {k1, k2, k−1, k−2}, where {k1, k2, k−1} correspond to the available KM and V , while k−2 is set to a positive and arbitrarily small value to mimic irreversibility. k−2 can then be gradually increased to tend to a reversible reaction. Eq (15) becomes

Σ˙ ss

=

[E]0

(k1

k2

[S]

−

k−1

k−2

[P]) R ln

k1 k2 [S]

,

(18)

rev k1 [S] + k−2 [P] + k−1 + k2

k−1 k−2 [P]

and Eq (17)

Σ˙ ss = k1k2 [E]0 [S] R ln k1 k2 [S] .

(19)

irr k1 [S] + k−1 + k2

k−1 k−2 [P]

In Eq (18), the flux and the force terms have always the same sign and the EPR is thus always positive. However, in Eq (19), Jirr is always positive while Arev can take negative values if k1 k2 [S] < k−1 k−2 [P], which leads to a negative EPR, in

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February 23, 2022

ss 10-6 J

Σirr (

)

min L K

0 5 10 15 20 25 30

1.0

1.0

1.0

0.8

0.8

0.8

[P] (μM) [P] (μM) [P] (μM)

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.0 0.2 0.4 0.6 0.8 1.0

[S] (μM)

(a)

[S] (μM)

(b)

[S] (μM)

(c)

Fig 1. Contour plot of the irreversible steady-state EPR as a function of

substrate and product concentrations. k−2 is 0.01 µM−1 min-1 in (a), 0.10 µM−1 min-1 in (b) and 0.50 µM−1 min-1 in (c), respectively. The other parameter values are

k1 = 1 µM−1 min-1, k−1 = 20 min-1 and k2 = 26 min-1. Such values correspond to KM = 46 µM and V = 26 µM min-1 and are for example of the same order of magnitude

as the kinetic parameters reported for the tyrosine hydroxylase in the synthesis of dopamine from L-tyrosine [19]. The dashed black line corresponds to Σ˙ sirsr = 0.

contradiction with the second law of thermodynamics. We can thus expect that Eq (19) approximates Eq (18) in a thermodynamically-consistent way when k−2 and [P] are small enough, but how can we define a threshold?

To investigate the conditions under which the use of the irreversible MM equation becomes problematic, we plot Σ˙ sirsr as a function of [S] and [P]. The free parameters are the kinetic constants {k1, k2, k−1, k−2}, which are chosen to be of the same order of magnitude as the typical values of KM and V reported in the literature. We also verify that the reaction is spontaneous in standard conditions, i.e. ∆revG◦ < 0. Our results are independent of the enzyme total concentration ([E]0), which is set to 1 µM, and the temperature is set to 298.15 K to match the conditions of the thermodynamic tables used in the following.

As shown in Fig 1, Σ˙ sirsr < 0 for a certain range of concentration of the chemostatted species S and P, but this behaviour tends to disappear as k−2 decreases. More importantly, the boundary defining Σ˙ sirsr = 0 is always a straight line of equation

[P] = [S] exp − ∆revG◦ , (20) RT

which can easily be derived by introducing ∆revG◦ into the last factor of Eq (19) via

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