The simple form of equation (1) encourages us to use a similar formula for other stoichiometries. For a reaction

A₁ + A₂ + ... ↔ B₁ + B₂ + ...

with concentration vectors a = (a₁, a₂, ...)^T and b = (b₁, b₂, ...)^T, we define the convenience kinetics

v ( a , b ) = E k + c a t ∏ i a ˜ i − k − c a t ∏ j b ˜ j ∏ i ( 1 + a ˜ i ) + ∏ j ( 1 + b ˜ j ) − 1 .       ( 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6B33@

By analogy to the k^M values in Michaelis-Menten kinetics, we have defined substrate constants kaiM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaacbaGae8xyae2aaSbaaWqaaiab=LgaPbqabaaaleaacqWFnbqtaaaaaa@3236@ and product constants kbjM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaacbaGae8Nyai2aaSbaaWqaaiab=PgaQbqabaaaleaacqWFnbqtaaaaaa@323A@ (in mM); just as above, variables with a tilde denote the normalised reactant concentrations ã_i = a_i/kaiM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaacbaGae8xyae2aaSbaaWqaaiab=LgaPbqabaaaleaacqWFnbqtaaaaaa@3236@ and b˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGIbGygaacaaaa@2E08@_j = b_j/kbjM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaacbaGae8Nyai2aaSbaaWqaaiab=PgaQbqabaaaleaacqWFnbqtaaaaaa@323A@. If the denominator is multiplied out, it contains all mathematical products of normalised substrate concentrations and product concentrations, but no mixed terms containing substrates and products together; the term +1 in the denominator is supposed to appear only once, so it is subtracted in the end. If several molecules of the same substance participate in a reaction, that is, for general stoichiometries

α₁ A₁ + α₂ A₂ + ... ↔ β₁ B₁ + β₂ B₂ + ...,

the formula looks slightly different:

v ( a , b ) = E k + c a t ∏ i a ˜ i α i − k − c a t ∏ j b ˜ j β j ∏ i ( 1 + a ˜ i + ... + a ˜ i α i ) + ∏ j ( 1 + b ˜ j + ... + b ˜ j β j ) − 1 = E k + c a t ∏ i a ˜ i α i − k − c a t ∏ j b ˜ j β j ∏ i ( ∑ m = 0 α i ( a ˜ i ) m ) + ∏ j ( ∑ m = 0 β j ( b ˜ j ) m ) − 1 .       ( 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@D4B2@

The stoichiometric coefficients α_i and β_j appear as exponents in the numerator and determine the orders of the polynomials in the denominator.

Reaction velocities do not only depend on reactant concentrations, but can also be controlled by modifiers. For each of them, we multiply eqn. (3) by a prefactor

h A ( d , k A ) = d k A + d o r  alternatively, h A ( d , k A ) = 1 + d k A       ( 4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeGaeaaaaeaaaeaacqWGObaAdaWgaaWcbaacbaGae8xqaeeabeaakiabcIcaOiabdsgaKjabcYcaSiabdUgaRnaaCaaaleqabaGae8xqaeeaaOGaeiykaKcabaGaeyypa0dabaWaaSaaaeaacqWGKbazaeaacqWGRbWAdaahaaWcbeqaaiab=feabbaakiabgUcaRiabdsgaKbaaaeaacqWFVbWBcqWFYbGCcqqGGaaicqqGHbqycqqGSbaBcqqG0baDcqqGLbqzcqqGYbGCcqqGUbGBcqqGHbqycqqG0baDcqqGPbqAcqqG2bGDcqqGLbqzcqqGSbaBcqqG5bqEcqqGSaalaeaacqWGObaAdaWgaaWcbaGae8xqaeeabeaakiabcIcaOiabdsgaKjabcYcaSiabdUgaRnaaCaaaleqabaGae8xqaeeaaOGaeiykaKcabaGaeyypa0dabaGaeGymaeJaey4kaSYaaSaaaeaacqWGKbazaeaacqWGRbWAdaahaaWcbeqaaiab=feabbaaaaaaaOGaaCzcaiaaxMaadaqadaqaaiabisda0aGaayjkaiaawMcaaaaa@671E@

for an activator and

h I ( d , k I ) = k I k I + d       ( 5 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGObaAdaWgaaWcbaacbaGae8xsaKeabeaakiabcIcaOiabdsgaKjabcYcaSiabdUgaRnaaCaaaleqabaGae8xsaKeaaOGaeiykaKIaeyypa0ZaaSaaaeaacqWGRbWAdaahaaWcbeqaaiab=LeajbaaaOqaaiabdUgaRnaaCaaaleqabaGae8xsaKeaaOGaey4kaSIaemizaqgaaiaaxMaacaWLjaWaaeWaaeaacqaI1aqnaiaawIcacaGLPaaaaaa@4253@

for an inhibitor. The activation constants k^A and inhibition constants k^I are measured in mM, and d is the concentration of the modifier.

Like many established rate laws (first of all, irreversible Michaelis-Menten kinetics 20), convenience kinetics can be derived from a molecular enzyme mechanism. We impose three main assumptions: (i) the substrates bind to the enzyme in arbitrary order and are converted into the products, which then dissociate from the enzyme in arbitrary order; (ii) binding of substrates and products is reversible and much faster than the conversion step; (iii) the binding energies of individual reactants do not depend on other reactants already bound to the enzyme.

We shall demonstrate how the convenience rate law is derived for a bimolecular reaction

A + X ↔ B + Y

without enzyme regulation. The reaction mechanism looks as follows:

The letters A, X, B, Y denote the reactants, E₀ is the free enzyme, and E_A, E_X, E_AX, E_B, E_Y, and E_BY denote complexes of the enzyme and different combinations of reactants. We shall denote their concentrations by brackets (e.g., [E_A]), the total enzyme concentration by E, and the concentrations of small metabolites by small letters (e.g., a = [A]).

The reaction proceeds from left to right; the free enzyme E₀ binds to the substrates A and X in arbitrary order, forming the complexes E_A, E_X, and E_AX. The binding of A can be described by an energy, the standard Gibbs free energy ΔGA(0)=GE(0)+GA(0)−GEA(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGhbWrdaqhaaWcbaacbaGae8xqaeeabaGaeiikaGIaeGimaaJaeiykaKcaaOGaeyypa0Jaem4raC0aa0baaSqaaiab=veafbqaaiabcIcaOiabicdaWiabcMcaPaaakiabgUcaRiabdEeahnaaDaaaleaacqWFbbqqaeaacqGGOaakcqaIWaamcqGGPaqkaaGccqGHsislcqWGhbWrdaqhaaWcbaGae8xrau0aaSbaaWqaaiab=feabbqabaaaleaacqGGOaakcqaIWaamcqGGPaqkaaaaaa@4609@ that is necessary to detach A from the complex E_A. The dissociation constant kAM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaacbaGae8xqaeeabaGae8xta0eaaaaa@3067@ = (a [E₀])/[E_A] describes the balance of bound and unbound A in chemical equilibrium and can be computed from the Gibbs free energy (in kJ/mol)

k A M = e − Δ G A ( 0 ) / R T m M       ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaacbaGae8xqaeeabaGae8xta0eaaOGaeyypa0Jae8xzau2aaWbaaSqabeaacqGHsislcqqHuoarcqWGhbWrdaqhaaadbaGae8xqaeeabaGaeiikaGIaeGimaaJaeiykaKcaaSGaei4la8IaemOuaiLaemivaqfaaOGae8xBa0Mae8xta0KaaCzcaiaaxMaadaqadaqaaiabiAda2aGaayjkaiaawMcaaaaa@43D0@

with RT ≈ 2.490 kJ/mol.

We now make a simplifying assumption: the binding energy of A does not depend on whether X is already bound. With analogous assumptions for binding of X and with the abbreviations a˜=a/kAM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGHbqygaacaiabg2da9iabdggaHjabc+caViabdUgaRnaaDaaaleaacqqGbbqqaeaacqqGnbqtaaaaaa@34F3@, x˜=x/kXM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaacaiabg2da9iabdIha4jabc+caViabdUgaRnaaDaaaleaacqqGybawaeaacqqGnbqtaaaaaa@357D@, the equilibrium concentrations of the substrate complexes can be written as [E_A] = ã [E₀], [E_X] = x˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaacaaaa@2E34@[E₀], [E_AX] = ãx˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaacaaaa@2E34@[E₀]. By analogy, we obtain expressions for the product complexes on the right hand side: [E_B] = b˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGIbGygaacaaaa@2E08@[E₀], [E_Y] = y˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG5bqEgaacaaaa@2E36@[E₀], [E_BY] = b˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGIbGygaacaaaa@2E08@y˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG5bqEgaacaaaa@2E36@[E₀]. The total enzyme concentration E is the sum over the concentrations of all enzyme complexes

E = [ E 0 ]   ( 1 + a ˜ + x ˜ + a ˜ x ˜ + b ˜ + y ˜ + b ˜ y ˜ ) .       ( 7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrcqGH9aqpcqGGBbWwcqWGfbqrdaWgaaWcbaGaeGimaadabeaakiabc2faDjabbccaGiabcIcaOiabigdaXiabgUcaRiqbdggaHzaaiaGaey4kaSIafmiEaGNbaGaacqGHRaWkcuWGHbqygaacaiqbdIha4zaaiaGaey4kaSIafmOyaiMbaGaacqGHRaWkcuWG5bqEgaacaiabgUcaRiqbdkgaIzaaiaGafmyEaKNbaGaacqGGPaqkcqGGUaGlcaWLjaGaaCzcamaabmaabaGaeG4naCdacaGLOaGaayzkaaaaaa@4C6E@

We next assume a reversible conversion between the complexes E_AX and E_BY with forward and backward rate constants k+cat MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaey4kaScabaacbaGae83yamMae8xyaeMae8hDaqhaaaaa@3322@ and k−cat MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaeyOeI0cabaacbaGae83yamMae8xyaeMae8hDaqhaaaaa@332D@; this reaction step determines the overall reaction rate. Its velocity reads

v ( a , x , b , y ) = k + cat [ E AX ] − k − cat [ E BY ] = k + cat a ˜ x ˜ [ E 0 ] − k − cat b ˜ y ˜ [ E 0 ] = E k + cat a ˜ x ˜ − k − cat b ˜ y ˜ 1 + a ˜ + x ˜ + a ˜ x ˜ + b ˜ + y ˜ + b ˜ y ˜ = E k + cat a ˜ x ˜ − k − cat b ˜ y ˜ ( 1 + a ˜ ) ( 1 + x ˜ ) + ( 1 + b ˜ ) ( 1 + y ˜ ) − 1 ,       ( 8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@CA29@

which is exactly the convenience rate law (2). The derivation has shown that the turnover rates k±cat MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaeyySaelabaacbaGae83yamMae8xyaeMae8hDaqhaaaaa@342E@ stem from the conversion step, while the reactant constants k^M are actually dissociation constants, related to the binding energies between reactants and enzyme. The terms in the denominator represent the enzyme complexes in the reaction scheme shown above. Equation (8) also shows why the term -1 in formulae (2) and (3) is necessary: the two product terms in the denominator represent all complexes shown in the reaction scheme. However, when summing up the terms from both sides, we counted the free enzyme E₀ twice, so we have to subtract it once.

The same kind of argument can be applied to reactions with other stoichiometries; let us consider a reaction with the left-hand side 2 A + X ↔ ...

The substrate complex E_AAX gives rise to the first term k+cat MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaey4kaScabaacbaGae83yamMae8xyaeMae8hDaqhaaaaa@3322@ã²x˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaacaaaa@2E34@ in the numerator, with the stoichiometric coefficient in the exponent. In the denominator, each term corresponds to one of the enzyme complexes, yielding

1 + a ˜ + x ˜ + a ˜ x ˜ + a ˜ 2 + a ˜ 2 x ˜ + ... = ( 1 + a ˜ + a ˜ 2 ) ( 1 + x ˜ ) + ...       ( 9 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIXaqmcqGHRaWkcuWGHbqygaacaiabgUcaRiqbdIha4zaaiaGaey4kaSIafmyyaeMbaGaacuWG4baEgaacaiabgUcaRiqbdggaHzaaiaWaaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcuWGHbqygaacamaaCaaaleqabaGaeGOmaidaaOGafmiEaGNbaGaacqGHRaWkcqGGUaGlcqGGUaGlcqGGUaGlcqGH9aqpcqGGOaakcqaIXaqmcqGHRaWkcuWGHbqygaacaiabgUcaRiqbdggaHzaaiaWaaWbaaSqabeaacqaIYaGmaaGccqGGPaqkcqGGOaakcqaIXaqmcqGHRaWkcuWG4baEgaacaiabcMcaPiabgUcaRiabc6caUiabc6caUiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaI5aqoaiaawIcacaGLPaaaaaa@5796@

where the dots still denote the terms from the right-hand side. The shape of the two factors, (1 + ã + ã²) and (1 + x˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaacaaaa@2E34@), corresponds to the rows and columns in the above scheme.

The activation and inhibition terms in the prefactor can also be justified mechanistically: in addition to binding sites for reactants, the enzyme contains binding sites for activators and inhibitors. Only those enzyme molecules to which all activators and none of the inhibitors are bound contribute to the reaction mechanism; all other enzyme molecules are inactive. Again, we assume that the Gibbs free energies for binding do not depend on whether other modifiers are bound, and they determine the k^A and k^I values as in eqn. (6).

To define a convenience kinetics for irreversible reactions, we assume that all product constants kbM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaacbaGae8NyaigabaGae8xta0eaaaaa@30A9@ – and thereby the overall equilibrium constant, as will be explained below – go to infinity. In the enzymatic mechanism, binding between products and enzyme becomes energetically very unfavourable. As a consequence, all b˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGIbGygaacaaaa@2E08@_j in eqn. (3) vanish and we obtain the irreversible rate law

v ( a ) = E   k + c a t ∏ i a ˜ i α i ∑ m = 0 α i ( a ˜ i ) m = E   k + c a t ∏ i ( ∑ m = 0 α i ( a ˜ i ) − m ) − 1 .       ( 10 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@789C@

Besides being a dissociation constant, the k^M value in Michaelis-Menten kinetics (1) has a simple mathematical meaning: it denotes the substrate concentration that leads to a half-maximal reaction velocity if the product is absent. A similar rule holds for the substrate and product constants in convenience kinetics. Let us first assume that all stoichiometric coefficients are ±1; if the product concentrations vanish (b_j = 0), then rate law (2) can be factorised into

v ( a , 0 ) = E   k + c a t ∏ i a ˜ i 1 + a ˜ i .       ( 11 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDcqGGOaakieqacqWFHbqycqGGSaalcqaIWaamcqGGPaqkcqGH9aqpcqWGfbqrcqqGGaaiieGacqGFRbWAdaqhaaWcbaGae43kaScabaacbaGae03yamMae0xyaeMae0hDaqhaaOWaaebuaeaadaWcaaqaaiqbdggaHzaaiaWaaSbaaSqaaiabdMgaPbqabaaakeaacqaIXaqmcqGHRaWkcuWGHbqygaacamaaBaaaleaacqWGPbqAaeqaaaaaaeaacqWGPbqAaeqaniabg+GivdGccqGGUaGlcaWLjaGaaCzcamaabmaabaGaeGymaeJaeGymaedacaGLOaGaayzkaaaaaa@4CF5@

If in addition, all substrate concentrations except for a certain a_m are kept fixed, the rate law reads

v ( a , 0 ) = a ˜ m 1 + a ˜ m ⋅ c o n s t .       ( 12 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDcqGGOaakieqacqWFHbqycqGGSaalcqaIWaamcqGGPaqkcqGH9aqpdaWcaaqaaiqbdggaHzaaiaWaaSbaaSqaaiabd2gaTbqabaaakeaacqaIXaqmcqGHRaWkcuWGHbqygaacamaaBaaaleaacqWGTbqBaeqaaaaakiabgwSixJqaaiab+ngaJjab+9gaVjab+5gaUjab+nhaZjab+rha0jabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIYaGmaiaawIcacaGLPaaaaaa@4A88@

For a_m → ∞, the fraction approaches 1, while for a_m = kaM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemyyaegabaacbaGae8xta0eaaaaa@30AB@ it yields 1/2. In particular, if all other substrates are present in high amounts, we obtain the half-maximal velocity, just as in Michaelis-Menten kinetics.

What if the stoichiometric coefficient is larger than one? Applying the same argument for α_m = 2, we obtain the velocity

v ( a , 0 ) = a ˜ m 2 1 + a ˜ m + a ˜ m 2 ⋅ c o n s t .       ( 13 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDcqGGOaakieqacqWFHbqycqGGSaalcqaIWaamcqGGPaqkcqGH9aqpdaWcaaqaaiqbdggaHzaaiaWaa0baaSqaaiabd2gaTbqaaiabikdaYaaaaOqaaiabigdaXiabgUcaRiqbdggaHzaaiaWaaSbaaSqaaiabd2gaTbqabaGccqGHRaWkcuWGHbqygaacamaaDaaaleaacqWGTbqBaeaacqaIYaGmaaaaaOGaeyyXICncbaGae43yamMae43Ba8Mae4NBa4Mae43CamNae4hDaqNaeiOla4IaaCzcaiaaxMaadaqadaqaaiabigdaXiabiodaZaGaayjkaiaawMcaaaaa@5045@

At a_m = kaM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemyyaegabaacbaGae8xta0eaaaaa@30AB@, the ratio is 1/3, so the reaction rate is 1/3 of the maximal rate. Extending this argument to other stoichiometric coefficients α_i, we can conclude: at a_m = kaM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemyyaegabaacbaGae8xta0eaaaaa@30AB@, excess of all other substrates, and vanishing product concentrations, the reaction rate equals the maximal reaction rate divided by 1 + α_i.

To parametrise an entire metabolic network with stoichiometric matrix N and regulation matrix W (for notation, see methods section), it is practical to arrange the kinetic parameters in vectors and matrices. The enzyme concentration of a reaction l reads E_l, and the turnover rates are called k±lcat MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaeyySaeRaemiBaWgabaacbaGae83yamMae8xyaeMae8hDaqhaaaaa@358F@. Each stoichiometric interaction (where n_il ≠ 0) comes with a value kliM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemiBaWMaemyAaKgabaacbaGae8xta0eaaaaa@321C@, while activation (w_li = 1) and inhibition (w_li = -1) are quantified by values kliA MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemiBaWMaemyAaKgabaacbaGae8xqaeeaaaaa@3204@ and kliI MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemiBaWMaemyAaKgabaacbaGae8xsaKeaaaaa@3214@, respectively. The k^M, k^A and k^I values for non-existing interactions (where n_il = 0 or w_li = 0) remain unspecified or can be assigned a value of 1, i.e., a logarithmic value of 0.

With metabolite concentrations arranged in a vector c, the convenience kinetics can now be written as

v l = E l ∏ m h A ( c m , k l m A ) w l m + h I ( c m , k l m I ) w l m − k + l cat ∏ i c ˜ l i n i l − − k − l cat ∏ i c ˜ l i n i l + ∏ i ∑ m = 0 n i l − ( c ˜ l i ) m + ∏ i ∑ m = 0 n i l + ( c ˜ l i ) m − 1 .       ( 14 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@BB93@

with the abbreviation c˜li=ci/kliM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaacamaaBaaaleaacqWGSbaBcqWGPbqAaeqaaOGaeyypa0Jaem4yam2aaSbaaSqaaiabdMgaPbqabaGccqGGVaWlcqWGRbWAdaqhaaWcbaGaemiBaWMaemyAaKgabaacbaGae8xta0eaaaaa@3B38@. For ease of notation here, we defined the matrices N⁺ = (nil+ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGUbGBdaqhaaWcbaGaemyAaKMaemiBaWgabaGaey4kaScaaaaa@31DC@), N^- = (nil− MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGUbGBdaqhaaWcbaGaemyAaKMaemiBaWgabaGaeyOeI0caaaaa@31E7@), which respectively contain the absolute values of all positive and negative elements of N. The matrices W⁺ and W^- are derived from W in the same way.

Let us add some remarks, (i) It is common to describe some of the metabolite concentrations by fixed values rather than by a balance equation. In the present framework, these metabolites are included in the concentration vector c and in the structure matrices N or W. (ii) A reaction is always catalysed by a specific enzyme; we describe isoenzymes by distinct reactions. (iii) If the sign of a regulatory interaction is unknown, we may consider terms for both activation and inhibition. (iv) To describe indirect regulation, e.g. by transcriptional control, the production and degradation of enzymes has to be modelled explicitly by chemical reactions.

The convenience kinetics (14) has a major drawback: its parameters are constrained by the second law of thermodynamics. The equilibrium constant of reaction l is defined as

k l e q = ∏ i ( c i e q ) n i l       ( 15 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemiBaWgabaacbaGae8xzauMae8xCaehaaOGaeyypa0ZaaebuaeaacqGGOaakcqWGJbWydaqhaaWcbaGaemyAaKgabaGae8xzauMae8xCaehaaOGaeiykaKYaaWbaaSqabeaacqWGUbGBdaWgaaadbaGaemyAaKMaemiBaWgabeaaaaaaleaacqWGPbqAaeqaniabg+GivdGccaWLjaGaaCzcamaabmaabaGaeGymaeJaeGynaudacaGLOaGaayzkaaaaaa@4758@

where c^eq is a vector of metabolite concentrations in a chemical equilibrium state. By setting eqn. (3) to zero, we obtain the Haldane relationship 16 for the convenience kinetics,

k e q = ∏ j b j β j ∏ i a i α i = k + c a t k − c a t ∏ j ( k b j M ) β j ∏ i ( k a i M ) α i .       ( 16 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaahaaWcbeqaaGqaaiab=vgaLjab=fhaXbaakiabg2da9maalaaabaWaaebuaeaacqWGIbGydaqhaaWcbaGaemOAaOgabaacciGae4NSdi2aaSbaaWqaaiabdQgaQbqabaaaaaWcbaGaemOAaOgabeqdcqGHpis1aaGcbaWaaebuaeaacqWGHbqydaqhaaWcbaGaemyAaKgabaGae4xSde2aaSbaaWqaaiabdMgaPbqabaaaaaWcbaGaemyAaKgabeqdcqGHpis1aaaakiabg2da9maalaaabaGaem4AaS2aa0baaSqaaiabgUcaRaqaaiab=ngaJjab=fgaHjab=rha0baaaOqaaiabdUgaRnaaDaaaleaacqGHsislaeaacqWFJbWycqWFHbqycqWF0baDaaaaaOWaaSaaaeaadaqeqbqaaiabcIcaOiabdUgaRnaaDaaaleaacqWFIbGydaWgaaadbaGaemOAaOgabeaaaSqaaiab=1eanbaaaeaacqWGQbGAaeqaniabg+GivdGccqGGPaqkdaahaaWcbeqaaiab+j7aInaaBaaameaacqWGQbGAaeqaaaaaaOqaamaarafabaGaeiikaGIaem4AaS2aa0baaSqaaiab=fgaHnaaBaaameaacqWGPbqAaeqaaaWcbaGae8xta0eaaaqaaiabdMgaPbqab0Gaey4dIunakiabcMcaPmaaCaaaleqabaGae4xSde2aaSbaaWqaaiabdMgaPbqabaaaaaaakiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI2aGnaiaawIcacaGLPaaaaaa@74BD@

In the notation of eqn. (14) and by taking the logarithm, the Haldane relationship can be expressed as

ln ⁡ k l e q = ln ⁡ k + l c a t − ln ⁡ k − l c a t + ∑ i n i l ln ⁡ k l i M .       ( 17 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGSbaBcqGGUbGBcqWGRbWAdaqhaaWcbaGaemiBaWgabaacbaGae8xzauMae8xCaehaaOGaeyypa0JagiiBaWMaeiOBa4Maem4AaS2aa0baaSqaaiabgUcaRiabdYgaSbqaaiab=ngaJjab=fgaHjab=rha0baakiabgkHiTiGbcYgaSjabc6gaUjabdUgaRnaaDaaaleaacqGHsislcqWGSbaBaeaacqWFJbWycqWFHbqycqWF0baDaaGccqGHRaWkdaaeqbqaaiabd6gaUnaaBaaaleaacqWGPbqAcqWGSbaBaeqaaaqaaiabdMgaPbqab0GaeyyeIuoakiGbcYgaSjabc6gaUjabdUgaRnaaDaaaleaacqWGSbaBcqWGPbqAaeaacqWFnbqtaaGccqGGUaGlcaWLjaGaaCzcamaabmaabaGaeGymaeJaeG4naCdacaGLOaGaayzkaaaaaa@62E2@

For each reaction, this relationship constitutes a constraint for the kinetic parameters within the reaction. In addition, each equilibrium constant obeys

ln ⁡ k l e q = − ∑ i n i l   G i ( 0 ) / ( R T ) ,       ( 18 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGSbaBcqGGUbGBcqWGRbWAdaqhaaWcbaGaemiBaWgabaacbaGae8xzauMae8xCaehaaOGaeyypa0JaeyOeI0YaaabuaeaacqWGUbGBdaWgaaWcbaGaemyAaKMaemiBaWgabeaakiabbccaGaWcbaGaemyAaKgabeqdcqGHris5aOGaem4raC0aa0baaSqaaiabdMgaPbqaaiabcIcaOiabicdaWiabcMcaPaaakiabc+caViabcIcaOiabdkfasjabdsfaujabcMcaPiabcYcaSiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI4aaoaiaawIcacaGLPaaaaaa@4F9B@

where Gi(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemyAaKgabaGaeiikaGIaeGimaaJaeiykaKcaaaaa@31EB@ is the Gibbs free energy of formation of metabolite i (see methods). Equations (17) and (18) imply that parameters in the entire network are coupled; an arbitrary choice can easily violate the second law of thermodynamics, which is a severe obstacle to parameter optimisation and fitting.

To circumvent this problem, we introduce new, thermodynamically independent system parameters 18. For each substance i, we define the dimensionless energy constant

k i G = e G i ( 0 ) / ( R T )       ( 19 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemyAaKgabaacbaGae83raCeaaOGaeyypa0Jae8xzau2aaWbaaSqabeaacqWGhbWrdaqhaaadbaGaemyAaKgabaGaeiikaGIaeGimaaJaeiykaKcaaSGaei4la8IaeiikaGIaemOuaiLaemivaqLaeiykaKcaaOGaaCzcaiaaxMaadaqadaqaaiabigdaXiabiMda5aGaayjkaiaawMcaaaaa@4243@

with Boltzmann's gas constant R ≈ 8.314 J/(mol K) and given absolute temperature T. For each reaction l, we define the velocity constant

k l V = ( k + l c a t   k − l c a t ) 1 / 2       ( 20 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemiBaWgabaacbaGae8NvayfaaOGaeyypa0JaeiikaGIaem4AaS2aa0baaSqaaiabgUcaRiabdYgaSbqaaiab=ngaJjab=fgaHjab=rha0baakiabbccaGiabdUgaRnaaDaaaleaacqGHsislcqWGSbaBaeaacqWFJbWycqWFHbqycqWF0baDaaGccqGGPaqkdaahaaWcbeqaaiabigdaXiabc+caViabikdaYaaakiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaIWaamaiaawIcacaGLPaaaaaa@4BC3@

as the geometric mean of the forward and backward turn-over rate, measured in s^-1. From now on, we shall use the energy constants and velocity constants as model parameters and treat the equilibrium constants k^eq and the turnover rates k^cat as dependent quantities: the equilibrium constants are computed from eqn. (18), and k^cat values are chosen such that equation (17) is satisfied. Using equations (17) and (18), we can write the turnover rates k±cat MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaeyySaelabaacbaGae83yamMae8xyaeMae8hDaqhaaaaa@342E@ as [See Additional file 1]

k ± l cat = k l V ∏ i ( k i G   k l i M ) ∓ n i l / 2       ( 21 ) = k l V ( ∏ i ( k i G k l i M ) n i l − ∏ i ( k i G k l i M ) n i l + ) ± 1 / 2       ( 22 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8EEB@

Altogether, the convenience kinetics of a metabolic network is characterised by the system parameters listed in table 1. If a reaction network is displayed as a bipartite graph of metabolites and reactions, each of the nodes and each of the arrows in the graph is characterised by one of the parameters, as shown in Figure 1. In addition, each node can carry an enzyme concentration E_l or a metabolite concentration c_i; as these concentrations can fluctuate in time, we shall call them state parameters rather than system parameters.

By taking the logarithm in both sides of eqn. (22), we obtain a linear equation between logarithmic parameters; this handy property also holds for other dependent parameters, as shown in table 2. We can express various kinetic parameters in terms of the system parameters: let θ denote the vector of logarithmic system parameters and x a vector containing various derived parameters in logarithmic form. It can be computed from θ by the linear relation

x ( θ ) = R θ x   θ .       ( 23 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWF4baEcqGGOaakiiGacqGF4oqCcqGGPaqkcqGH9aqpcqWGsbGudaqhaaWcbaGae4hUdehabaGaemiEaGhaaOGaeeiiaaIae4hUdeNaeiOla4IaaCzcaiaaxMaadaqadaqaaiabikdaYiabiodaZaGaayjkaiaawMcaaaaa@3F3A@

The sensitivity matrix Rθx MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaacciGae8hUdehabaGaemiEaGhaaaaa@313C@ is sparse and can be constructed easily from the network structure and the relations listed in table 2 [See Additional file 1].

By inserting the expression (22) for k±lcat MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaeyySaeRaemiBaWgabaacbaGae83yamMae8xyaeMae8hDaqhaaaaa@358F@ into (14), we obtain a rate law in which all parameters can be varied independently, remaining in accordance with thermodynamics. In its thermodynamically independent form, the convenience kinetics reads

v l = E l ∏ m h A ( c m , k l m A ) w l m + h I ( c m , k l m I ) w l m − × k l V ∏ i c ˜ l i n i l − ( k ˜ l i M ) − n i l / 2 − ∏ i c ˜ l i n i l + ( k ˜ l i M ) n i l / 2 ∏ i ∑ m = 0 n i l − ( c ˜ l i ) m + ∏ i ∑ m = 0 n i l + ( c ˜ l i ) m − 1       ( 24 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@CD6E@

with the abbreviations c˜li=ci/kliM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGJbWygaacamaaBaaaleaacqWGSbaBcqWGPbqAaeqaaOGaeyypa0Jaem4yam2aaSbaaSqaaiabdMgaPbqabaGccqGGVaWlcqWGRbWAdaqhaaWcbaGaemiBaWMaemyAaKgabaacbaGae8xta0eaaaaa@3B38@ and k˜liM=kiGkliM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGRbWAgaacamaaDaaaleaacqWGSbaBcqWGPbqAaeaaieaacqWFnbqtaaGccqGH9aqpcqWGRbWAdaqhaaWcbaGaemyAaKgabaGae83raCeaaOGaem4AaS2aa0baaSqaaiabdYgaSjabdMgaPbqaaiab=1eanbaaaaa@3CA6@. Special cases for some simple stoichiometries are listed in table 3.

All system parameters can be expressed in terms of Gibbs free energies: the k^M, k^A, and k^I values represent binding energies, and the energy constants k^G are defined by the Gibbs free energy of formation. Finally, we can also write the velocity constants as

k V = e − Δ G t r ( 0 ) / ( R T ) s − 1 .       ( 25 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaahaaWcbeqaaGqaaiab=zfawbaakiabg2da9iab=vgaLnaaCaaaleqabaGaeyOeI0IaeuiLdqKaem4raC0aa0baaWqaaiab=rha0jab=jhaYbqaaiabcIcaOiabicdaWiabcMcaPaaaliabc+caViabcIcaOiabdkfasjabdsfaujabcMcaPaaakiab=nhaZnaaCaaaleqabaGaeyOeI0IaeGymaedaaOGaeiOla4IaaCzcaiaaxMaadaqadaqaaiabikdaYiabiwda1aGaayjkaiaawMcaaaaa@4931@

To illustrate the meaning of the energy ΔGtr(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGhbWrdaqhaaWcbaacbaGae8hDaqNae8NCaihabaGaeiikaGIaeGimaaJaeiykaKcaaaaa@34D5@, we consider again the bimolecular enzymatic mechanism: in transition state theory 15, the rate constants between the substrate and product complex are formally written as

k + c a t = e − ( G t r ( 0 ) − G E A X ( 0 ) ) / ( R T ) s − 1 k − c a t = e − ( G t r ( 0 ) − G E B Y ( 0 ) ) / ( R T ) s − 1 ,       ( 26 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGadaaabaGaem4AaS2aa0baaSqaaiabgUcaRaqaaGqaaiab=ngaJjab=fgaHjab=rha0baaaOqaaiabg2da9aqaaiabdwgaLnaaCaaaleqabaGaeyOeI0IaeiikaGIaem4raC0aa0baaWqaaiab=rha0jab=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@77DD@

where the quantities G⁽⁰⁾ denote Gibbs free energies of formation for the substrate complex E_AX, the product complex E_BY, and a hypothetical transition state E^tr that has to be crossed on the way from E_AX to E_BY. By inserting eqn. (26) into the definition (20) and defining an energy barrier ΔGtr(0)=Gtr(0)−12(GEAX(0)+GEBY(0)) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGhbWrdaqhaaWcbaacbaGae8hDaqNae8NCaihabaWaaeWaaeaacqaIWaamaiaawIcacaGLPaaaaaGccqGH9aqpcqWGhbWrdaqhaaWcbaGae8hDaqNae8NCaihabaWaaeWaaeaacqaIWaamaiaawIcacaGLPaaaaaGccqGHsisldaWcaaqaaiabigdaXaqaaiabikdaYaaadaqadaqaaiabdEeahnaaDaaaleaacqWGfbqrdaWgaaadbaGae8xqaeKae8hwaGfabeaaaSqaamaabmaabaGaeGimaadacaGLOaGaayzkaaaaaOGaey4kaSIaem4raC0aa0baaSqaaiabdweafnaaBaaameaacqWFcbGqcqWFzbqwaeqaaaWcbaWaaeWaaeaacqaIWaamaiaawIcacaGLPaaaaaaakiaawIcacaGLPaaaaaa@503D@, we obtain eqn. (25).

We introduced the energy constants kiG MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemyAaKgabaacbaGae83raCeaaaaa@30AF@ as model parameters for two reasons: first, they provide a consistent way to describe the equilibrium constants; secondly, if Gibbs free energies of formation are known from experiments, they can be used for fitting the energy constants and will thus contribute to a good choice of equilibrium constants. However, if no such data are available, the second reason becomes redundant, and a different choice of the system parameters may be appropriate: instead of the energy constants, we employ a set of independent equilibrium constants. If the stoichiometric matrix N has full column rank, then the equilibrium constants are independent anyway because for given k^eq, eqn. (18) can always be satisfied by some choice of the Gi(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemyAaKgabaGaeiikaGIaeGimaaJaeiykaKcaaaaa@31EB@; in this case, the equilibrium constants can be directly used as model parameters. Otherwise, we can choose a set of reactions with the following property: their equilibrium constants (collected in a vector k^ind) are thermodynamically independent, and they determine all other equilibrium constants in the model via a linear equation

l n   k eq = R i n d e q  ln  k ind .       ( 27 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieaacqWFSbaBcqWFUbGBcqqGGaaiieqacqGFRbWAdaahaaWcbeqaaiabbwgaLjabbghaXbaakiabg2da9iabdkfasnaaDaaaleaacqWFPbqAcqWFUbGBcqWFKbazaeaacqWFLbqzcqWFXbqCaaGccqqGGaaicqqGSbaBcqqGUbGBcqqGGaaicqGFRbWAdaahaaWcbeqaaiabbMgaPjabb6gaUjabbsgaKbaakiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaI3aWnaiaawIcacaGLPaaaaaa@4D3F@

The choice of independent reactions and the computation of Rindeq MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaacbaGae8xAaKMae8NBa4Mae8hzaqgabaGae8xzauMae8xCaehaaaaa@34CA@ are explained in the methods section. Given the equilibrium and velocity constants, the turnover rates can be expressed as

k ± l c a t = k l V ( k l e q ) ± 1 / 2 ,       ( 28 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaeyySaeRaemiBaWgabaacbaGae83yamMae8xyaeMae8hDaqhaaOGaeyypa0Jaem4AaS2aa0baaSqaaiabdYgaSbqaaiab=zfawbaakmaabmaabaGaem4AaS2aa0baaSqaaiabdYgaSbqaaiab=vgaLjab=fhaXbaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyySaeRaeGymaeJaei4la8IaeGOmaidaaOGaeiilaWIaaCzcaiaaxMaadaqadaqaaiabikdaYiabiIda4aGaayjkaiaawMcaaaaa@4C87@

or equivalently as

ln ⁡   k ± l cat = ln ⁡   k l V ± 1 2 ln ⁡   k l eq       ( 29 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGSbaBcqGGUbGBcqqGGaaiieGacqWFRbWAdaqhaaWcbaGaeyySaeRae8hBaWgabaGaee4yamMaeeyyaeMaeeiDaqhaaOGaeyypa0JagiiBaWMaeiOBa4MaeeiiaaIae83AaS2aa0baaSqaaiab=XgaSbqaaiabbAfawbaakiabgglaXoaalaaabaGaeGymaedabaGaeGOmaidaaiGbcYgaSjabc6gaUjabbccaGiab=TgaRnaaDaaaleaacqWFSbaBaeaacqqGLbqzcqqGXbqCaaGccaWLjaGaaCzcamaabmaabaGaeGOmaiJaeGyoaKdacaGLOaGaayzkaaaaaa@53B0@

and be inserted into eqn. (14).

To check whether the convenience kinetics yields any unusual results, we compared it to two established rate laws, namely the ordered and ping-pong mechanisms for bimolecular reactions. In both mechanisms, binding and dissociation occur in a fixed order:

Besides the turnover rates and k^M values, their kinetic laws also contain product inhibition constants. For the comparison, we made the simplifying (yet biologically realistic) assumption that these inhibition constants equal the respective k^M values, which yields the following rate laws 8

Ordered mechanism : v = E k + cat a ˜ x ˜ − k − cat b ˜ y ˜ 1 + a ˜ + x ˜ + a ˜ x ˜ + b ˜ + y ˜ + b ˜ y ˜ + a ˜ b ˜ + x ˜ y ˜ + a ˜ x ˜ b ˜ + x ˜ b ˜ y ˜       ( 30 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@83B3@

Ping-pong mechanism : v = E k + cat a ˜ x ˜ − k − cat b ˜ y ˜ a ˜ + x ˜ + a ˜ x ˜ + b ˜ + y ˜ + b ˜ y ˜ + a ˜ b ˜ + x ˜ y ˜ .       ( 31 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@79F2@

In contrast to the convenience rate law (8), the denominators contain mixed terms between substrates and products, and in the ping-pong kinetics, the term +1 is missing. The ordered mechanism yields smaller reaction rates than the ping-pong and the convenience kinetics because its denominator is always larger. To compare the three rate laws, we sampled metabolite concentrations and k^M values from a random distribution and computed the resulting reaction velocities. Parameters and concentrations were independently sampled from a uniform distribution in the interval [0.001, 1000] and from a log-uniform distribution on the same interval. Figure 2 shows scatter plots between reaction velocities computed from the different rate laws. For the uniform distribution, the results from convenience kinetics resemble those from ordered and ping-pong kinetics; they are about as similar as the ordered and ping-pong kinetics. With the log-uniform distribution, the correlations between all three kinetics become smaller, and ping-pong kinetics is more similar to convenience than to ordered kinetics. We conclude that erroneously choosing convenience kinetics instead of the other kinetic laws is just as risky as a wrong choice between the two other mechanisms.

The parameters in convenience kinetics – the independent and the resulting dependent ones – can be measured in experiments. The linear relationship (23) makes it particularly easy to use such experimental values for parameter fitting: given a metabolic network, we mine the literature for thermodynamic and kinetic data, in particular Gibbs free energies of formation, reaction Gibbs free energies, equilibrium constants, k^M values, k^I values, k^A values, and turnover rates, and merge their logarithms in a large vector x*. The vector can contain multiple values for a parameter, it can contain thermodynamically dependent parameters, and of course, many parameters from the model will be missing. We try to determine a vector θ of logarithmic system parameters that yields a good match between the resulting parameter predictions x (θ) and the data x*. Solving x* ≈ Rθx MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaacciGae8hUdehabaGaemiEaGhaaaaa@313C@θ for θ by the method of least squares yields an estimate of the system parameters. Using eqn. (23) again, consistent values of all kinetic parameters can be computed from the estimated system parameters. Contradictions in the original data are resolved; in addition, we can employ a prior distribution representing typical parameter ranges to compensate for missing data. A more general estimation procedure, which can also integrate measured metabolic concentrations and fluxes, is described in the companion article 19.

The structure of a metabolic network is defined by the lists of metabolites and reactions and by two structural matrices, N and W. The coefficients n_il contained in the stoichiometric matrix N describe how many molecules of type i are produced in reaction l; negative elements describe consumption of molecules. The elements of the regulation matrix W describe enzyme regulation between metabolites i and enzymes l: w_li = 1 indicates activation, w_li = -1 represents inhibition, and w_li = 0 no interaction.

In the setting of deterministic differential equations, the substance concentrations in a biochemical system follow the balance equations

d d t c = N   v ( c , k ) .       ( 32 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaGqaaiab=rgaKbqaaiab=rgaKjabdsha0baaieqacqGFJbWycqGH9aqpcqWGobGtcqqGGaaicqGF2bGDdaqadaqaaiab+ngaJjabbYcaSiab+TgaRbGaayjkaiaawMcaaiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIZaWmcqaIYaGmaiaawIcacaGLPaaaaaa@4129@

The vectors c, v, and k contain the metabolite concentrations (in mM), reaction velocities (in mM/s), and system parameters, respectively. External or buffered metabolites with fixed concentrations are contained in the parameter vector k.

To relate activation and inhibition (as stated in W) to the reaction kinetics, we first assume a hypothetical kinetic law without regulation; in this law, the reaction velocity depends only on the substrate and product concentrations. In the real rate law, a metabolite is an activator if (i) it increases the rate although it is not a reactant, or (ii) it increases the rate more strongly than it would by just being a reactant. Inhibition is defined analogously.

The kinetic laws v_l(c, k) are constrained by fundamental thermodynamic laws that relate the metabolite concentrations in steady state to molecular energies 15. A single reaction event of reaction l changes the Gibbs free energy of the system by

Δ G l = ∑ i n i l   μ i       ( 33 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGhbWrdaWgaaWcbaGaemiBaWgabeaakiabg2da9maaqafabaGaemOBa42aaSbaaSqaaiabdMgaPjabdYgaSbqabaaabaGaemyAaKgabeqdcqGHris5aOGaeeiiaaccciGae8hVd02aaSbaaSqaaiabdMgaPbqabaGccaWLjaGaaCzcamaabmaabaGaeG4mamJaeG4mamdacaGLOaGaayzkaaaaaa@425A@

where the sum runs over all metabolites and μ_i denotes the chemical potential of metabolite i (in kJ/mol). In an ideal mixed phase at pressure P and absolute temperature T, the chemical potential of substance i with concentration c_i reads

μ i ( P , T , c i ) = μ i ( 0 ) ( P , T ) + R T  ln  c i       ( 34 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF8oqBdaWgaaWcbaGaemyAaKgabeaakmaabmaabaGaemiuaaLaeiilaWIaemivaqLaeiilaWIaem4yam2aaSbaaSqaaiabdMgaPbqabaaakiaawIcacaGLPaaacqGH9aqpcqWF8oqBdaqhaaWcbaGaemyAaKgabaWaaeWaaeaacqaIWaamaiaawIcacaGLPaaaaaGcdaqadaqaaiabdcfaqjabcYcaSiabdsfaubGaayjkaiaawMcaaiabgUcaRiabdkfasjabdsfaujabbccaGiabbYgaSjabb6gaUjabbccaGGqaciab+ngaJnaaBaaaleaacqGFPbqAaeqaaOGaaCzcaiaaxMaadaqadaqaaiabiodaZiabisda0aGaayjkaiaawMcaaaaa@5328@

where μi(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF8oqBdaqhaaWcbaGaemyAaKgabaWaaeWaaeaacqaIWaamaiaawIcacaGLPaaaaaaaaa@3268@ denotes the chemical potential of the pure substance at infinite dilution, and R ≈ 8.314 J/(mol K) is Boltzmann's gas constant. In (34), the c_i are dimensionless numbers denoting concentrations in mM. In real mixed phases, there would be an additional term +RT ln fi0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzdaqhaaWcbaGaemyAaKgabaGaeGimaadaaaaa@3077@ with the activity coefficient f_i. We neglect this term, assuming an ideal mixed phase without mixture effects on volume or energy; we also neglect effects of changing pressure or electric charges.

The equilibrium constant of reaction l is defined as

k l e q = ∏ i ( c i e q ) n i l       ( 35 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaqhaaWcbaGaemiBaWgabaacbaGae8xzauMae8xCaehaaOGaeyypa0ZaaebuaeaadaqadaqaaiabdogaJnaaDaaaleaacqWGPbqAaeaacqWFLbqzcqWFXbqCaaaakiaawIcacaGLPaaaaSqaaiabdMgaPbqab0Gaey4dIunakmaaCaaaleqabaGaemOBa42aaSbaaWqaaiabdMgaPjabdYgaSbqabaaaaOGaaCzcaiaaxMaadaqadaqaaiabiodaZiabiwda1aGaayjkaiaawMcaaaaa@473D@

where c^eq is the vector of metabolite concentrations in a chemical equilibrium state. According to the second law of thermodynamics, the equilibrium state of a chemical system is characterised by a minimum of the Gibbs free energy. This implies that each chemical reaction in equilibrium satisfies ΔG_l = 0. From eqs. (33), (34), and (35) follows

ln ⁡ k l e q = ∑ i n i l ln ⁡ c i e q = − Δ G l ( 0 ) R   T ,       ( 36 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGSbaBcqGGUbGBcqWGRbWAdaqhaaWcbaGaemiBaWgabaacbaGae8xzauMae8xCaehaaOGaeyypa0ZaaabuaeaacqWGUbGBdaWgaaWcbaGaemyAaKMaemiBaWgabeaakiGbcYgaSjabc6gaUjabdogaJnaaDaaaleaacqWGPbqAaeaacqWFLbqzcqWFXbqCaaGccqGH9aqpcqGHsisldaWcaaqaaiabfs5aejabdEeahnaaDaaaleaacqWGSbaBaeaadaqadaqaaiabicdaWaGaayjkaiaawMcaaaaaaOqaaiabdkfasjabbccaGiabdsfaubaacqGGSaalcaWLjaGaaCzcamaabmaabaGaeG4mamJaeGOnaydacaGLOaGaayzkaaaaleaacqWGPbqAaeqaniabggHiLdaaaa@57AF@

where ΔGl(0)=∑inil μi(0)(P,T) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGhbWrdaqhaaWcbaGaemiBaWgabaWaaeWaaeaacqaIWaamaiaawIcacaGLPaaaaaGccqGH9aqpdaaeqbqaaiabd6gaUnaaBaaaleaacqWGPbqAcqWGSbaBcqqGGaaiaeqaaGGacOGae8hVd02aa0baaSqaaiabdMgaPbqaamaabmaabaGaeGimaadacaGLOaGaayzkaaaaaaqaaiabdMgaPbqab0GaeyyeIuoakmaabmaabaGaemiuaaLaeiilaWIaemivaqfacaGLOaGaayzkaaaaaa@4758@ is called the standard reaction Gibbs free energy and the concentrations are measured in mM. It can also be expressed as

Δ G l ( 0 ) = ∑ i n i l   G i ( 0 )       ( 37 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGhbWrdaqhaaWcbaGaemiBaWgabaWaaeWaaeaacqaIWaamaiaawIcacaGLPaaaaaGccqGH9aqpdaaeqbqaaiabd6gaUnaaBaaaleaacqWGPbqAcqWGSbaBcqqGGaaiaeqaaaqaaiabdMgaPbqab0GaeyyeIuoakiabdEeahnaaDaaaleaacqWGPbqAaeaadaqadaqaaiabicdaWaGaayjkaiaawMcaaaaakiaaxMaacaWLjaWaaeWaaeaacqaIZaWmcqaI3aWnaiaawIcacaGLPaaaaaa@46AC@

in terms of the Gibbs free energies of formation Gi(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemyAaKgabaGaeiikaGIaeGimaaJaeiykaKcaaaaa@31EB@ for a standard state, typically P = 1.015 bar and T = 298.15 K. Equations (36) and (37) constitute the relation (18) between equilibrium constants and the Gibbs free energies of formation.

Dependencies between equilibrium constants can be treated in a similar manner to the linear dependencies that constitute the conservation relations between metabolites 25. To choose a set of reactions with independent equilibrium constants – for brevity, we shall call them independent reactions – we collect a maximal number of linearly independent columns of N and join them in a matrix N˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGobGtgaacaaaa@2DE0@. The chosen columns correspond to the independent reactions, and their choice need not be unique. By construction, N˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGobGtgaacaaaa@2DE0@ has full column rank, and we can split N into a matrix product N = N˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGobGtgaacaaaa@2DE0@L˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGmbatgaacaaaa@2DDC@, by analogy to the splitting N = L N_R that is used in metabolic control analysis to remove dependent metabolites.

To be thermodynamically feasible, the equilibrium constants have to satisfy eqn. (18) or, in vector form,

ln k^eq = -N^T ln k^G     (38)

for at least one choice of the vector k^G. Let us first assume that k^G is given; then the equilibrium constants of the independent reactions read

ln k^ind = -N˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGobGtgaacaaaa@2DE0@^T ln k^G,     (39)

and with the definition

R i n d e q = L ˜ T       ( 40 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaacbaGae8xAaKMae8NBa4Mae8hzaqgabaGae8xzauMae8xCaehaaOGaeyypa0JafmitaWKbaGaadaahaaWcbeqaaiab=rfaubaakiaaxMaacaWLjaWaaeWaaeaacqaI0aancqaIWaamaiaawIcacaGLPaaaaaa@3D1F@

we can write

ln ⁡ k e q = − L ˜ T N ˜ T ln ⁡ k G = R i n d e q ln ⁡ k i n d .       ( 41 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGSbaBcqGGUbGBieqacqWFRbWAdaahaaWcbeqaaGqaaiab+vgaLjab+fhaXbaakiabg2da9iabgkHiTiqbdYeamzaaiaWaaWbaaSqabeaacqGFubavaaGccuWGobGtgaacamaaCaaaleqabaGae4hvaqfaaOGagiiBaWMaeiOBa4Mae83AaS2aaWbaaSqabeaacqGFhbWraaGccqGH9aqpcqWGsbGudaqhaaWcbaGae4xAaKMae4NBa4Mae4hzaqgabaGae4xzauMae4xCaehaaOGagiiBaWMaeiOBa4Mae83AaS2aaWbaaSqabeaacqGFPbqAcqGFUbGBcqGFKbazaaGccqGGUaGlcaWLjaGaaCzcamaabmaabaGaeGinaqJaeGymaedacaGLOaGaayzkaaaaaa@5763@

Hence, eqn. (27) is satisfied and the matrix Rindeq MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaacbaGae8xAaKMae8NBa4Mae8hzaqgabaGae8xzauMae8xCaehaaaaa@34CA@ is known. It remains to be shown that the equilibrium constants contained in k^ind are indeed thermodynamically independent; or in other words, that for any vector ln k^ind, there exists a vector k^G such that eqn. (39) holds; and this is indeed the case because N˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGobGtgaacaaaa@2DE0@^T has full row rank.