Transformation and Translation

The thermokinetic modeling formalism is a convenient reformulation of the thermodynamic modeling formalism. Both formalisms are basically equivalent and the transformation and translation methods for thermodynamic models (see Section 4.1.1, p. 51) can be applied.

Here we give the transformation and translation rules for thermokinetic models. Since we re-strict ourselves to diagonal resistance matrices, we can only adapt methods where the occurrence of off-diagonal elements in the resistance matrix can be avoided.

Further, this section introduces an additional transformation method, namely the translation of stoichiometric coefficients that is specific for thermokinetic models.

5.4.1. Linear Transformation

Due to the restriction to diagonal resistance matrices, linear transformations of fluxes are not possible. Such transformations lead to non-diagonal resistance matrices. Thus, the following corollary considers only a transformation of concentrations.

Corollary 5.32 (Transformation of a thermokinetic model). Let M be a thermokinetic model (see Definition 5.29, p. 100), and let T_c, T_c,e andT_J,e be square matrices of full rank. Then, the solution of the system Mˆ with

Nˆ_E =T_cN_E, Nˆ_P =T_cN_P, Sˆ_E =T_c,eS_E, Sˆ_P =T_c,eS_P, ˆ

c( ˆξ,ξˆe) =Tcc(T_c^T# ˆξ, T_c,e^T # ˆξe), R( ˆˆ ξ,ξˆe) =R(T_c^T# ˆξ, T_c,e^T # ˆξe), Nˆ_e =T_cN_eT_J,e, ξˆ₀ =T_c^{T ,−1}#ξ₀.

is equivalent to that of M with ˆ

c=T_cc, T_c^T# ˆξ =ξ, T_c,e^T # ˆξ_e=ξ_e, Jˆ=J, Fˆ =F, T_J,eJˆ_e=J_e.

5. Thermokinetic Modeling

Proof. This transformation rule follows from the Corollary 4.2 (p. 52).

§ 5.33 (Negative stoichiometric coefficients). In natural coordinates the entries of the matrices NE, NP, SE and SP are non-negative. After a transformation of concentrations, the entries of the corresponding transformed matrices may be negative.

5.4.2. Translation of the Variables

Section 4.1.2 (p. 54) introduced rules for the translation of the variablesc,µandJ in thermody-namic models. Analog rules can be formulated for thermokinetic models. This section discusses only the translation of chemical potentials because this operation is needed for an appropriate scaling of the state variables of the simulation equations. It is equivalent to a change of the reference state of the Gibbs formation energies.

Corollary 5.34 (Scaling of a TK model). Let M be a thermokinetic model (see Definition 5.29, p. 100), and let δξ ∈ Rⁱ+⁰, δξ_e ∈ Rⁱ+^0,e be vectors with N^T log(δξ) +S^T log(δξ_e) = 0. Then, the solution of the system Mˆ with

Nˆ_E/P =N_E/P, Sˆ_E/P =S_E/P, Nˆ_e=N_e, ˆ

c( ˆξ,ξˆe) =c

ξˆ◦δξ⁽⁻¹⁾,ξˆe◦δξ_e⁽⁻¹⁾

, ξˆ0 =ξ0◦δξ⁽⁻¹⁾ and

R( ˆˆ ξ,ξˆ_e) = N_E^T#δξ◦S_E^T#δξ◦R

ξˆ◦δξ⁽⁻¹⁾,ξˆ_e◦δξ_e⁽⁻¹⁾

=N_P^T#δξ◦S_P^T#δξ◦R

ξˆ◦δξ⁽⁻¹⁾,ξˆ_e◦δξ_e⁽⁻¹⁾ is equivalent to that of M with

ˆ

c=c, ξˆ=ξ◦δξ, ξˆ_e =ξ_e◦δξ_e, Jˆ=J, Fˆ =F, Jˆ_e =J_e.

Proof. The proof follows the lines of the proof of Corollary 4.10 (p. 54). Write down the model equations ofM. Now replaceJ = ˆJ,F = ˆF,ξ = ˆξ◦δξ⁽⁻¹⁾andξ_e= ˆξ_e◦δξe⁽⁻¹⁾. This immediately leads to the functions ˆc( ˆξ,ξˆ_e) and the matrices Nˆ_E/P, Sˆ_E/P and Nˆ_e. To prove the expression for Rˆ we start with the equation Rˆ◦Jˆ= ˆF and prove that it is equivalent to R◦J =F. The equationRˆ◦Jˆ= ˆF can be expanded to

Rˆ

z }| { N_E^T#δξ◦S_E^T#δξ

| {z }

δFE

◦R◦Jˆ= ˆF = ˆN_E^T# ˆξ◦Sˆ_E^T# ˆξ_e−Nˆ_P^T# ˆξ◦Sˆ_P^T# ˆξ_e

=N_E^T#ξ◦S_E^T#ξ_e◦N_E^T#δξ◦S_E^T#δξ_e

| {z }

δFE

−N_P^T#ξ◦S_P^T#ξ_e◦N_P^T#δξ◦S_P^T#δξ_e

| {z }

δFP

. (5.5)

5. Thermokinetic Modeling

Due to the specific choice of δξ and δξ_e, we have N_E^T log(δξ) +S_E^T log(δξ_e) = N_P^T log(δξ) + S_P^T log(δξ_e) and thus δF_E = δF_P. For this reason these terms cancel out and the equation R◦J =F remains.

Example 5.35 (Scaling of capacities and resistances). The above corollary is often neces-sary to numerically scale the simulation equations. Consider the single reaction of glucose-6-phosphate to fructose-6-glucose-6-phosphate (g6p f6p) that is catalyzed by the phosphoglucose iso-merase. Alberty [1] gives transformed Gibbs formation energies∆_fG^0◦_i atpH = 7,T = 298.15 K, I = 0.25 M and c^◦ = 1 M:

∆_fG^0◦_g6p =−1318.92 kJ mol⁻¹, ∆_fG^0◦_f6p =−1315.74 kJ mol⁻¹.

These values may be used as chemical standard potentials µ^◦_i and lead to the capacities (C_i = c^◦mexp(−µ^◦_i/(R^∗T))):

C_g6p = 1.160×10²³¹ M, C_f6p = 3.217×10²³⁰ M.

Clearly, these values are not suited for direct use in numerical simulations. The tables from Alberty [1] are based on the convention that chemical elements in the standard state in their most stable form have ∆fG^0◦_i = 0. Most relevant (bio)chemical species have a rather low, negative ∆_fG_i because the respective compounds are energetically more favorable than an analogous mixture of pure elements.

Using the above corollary, we can scale the capacities to values more suited for numerical computations. The stoichiometric matrix is N^T = (−1,1) and thus the condition for the scaling factors is−log(δξ_g6p) + log(δξ_f6p) = 0 or equivalently δξ_g6p =δξ_f6p.

Assume that g6p has a typical concentration of 1 mM. From a numerical perspective, an optimal value of its capacity is1 mMbecause then the thermokinetic potentialξ_g6pthat is a state variable of the model varies around the typical value1. We chooseδξ_g6p =δξ_f6p = 1.160×10²³⁴ and get the new relation

c_g6p =C_g6pδξ_g6p⁻¹

| {z }

Cˆg6p

ξˆ_g6p c_f6p =C_f6pδξ_f⁻¹_6p

| {z }

Cˆf6p

ξˆ_f6p

with Cˆ_g6p = 1 mM and Cˆ_f6p = 0.2773 mM. These values are moderate and thus can be used for numerical computations.

§ 5.36. The above example shows that tables of Gibbs formation energies can be used to determine the capacities. The standardized and widely accepted definition of the reference state that underlies the tables of Gibbs formation energies allows for the direct exchange of model parameters between different models. These tables should be used if they are available.

This often leads to very large capacities because most Gibbs formation energies are rather low. To avoid numerical problems, the models can be scaled before simulation and analysis by applying the above corollary. This scaling of the model can be performed automatically by the modeling or simulation tool.

5. Thermokinetic Modeling

5.4.3. Translation of the Stoichiometric Coefficients

The thermokinetic modeling formalism explicitly distinguishes the stoichiometric matrices of reactants and products. In the thermodynamic modeling formalism, this distinction is not necessary because the information is coded in the expression for the thermodynamic resistances.

For example, the ideal thermodynamic resistance function for mass-action kinetics depends on the thermodynamic forces exerted by reactants and products∆µ_E/P,j =P

i∈E/P_jν_E/P,ijµ_i,j that depend on the stoichiometric coefficients of reactants and products (see Definition 3.22, p. 47).

The explicit introduction of N_E/P and S_E/P in TKM allows for a further translation operation.

Two reactions with the same overall stoichiometry may differ in their reactant and product stoichiometry. Consider the reaction from A to B catalyzed by X. Its stoichiometry can be described by two equivalent variants: (1) The reaction equation A+X B+X shows that X is a catalyst of the reaction. (2) The formulationA B has the same overall stoichiometry but does not refer to the catalyst X. Assume mass-action kinetics and small concentrations of A and B such that saturation effects of the catalyst X can be neglected. Then, the kinetics of variant (1) can be described by a constant thermokinetic resistance R₁ = const, whereas the resistance of the variant (2) is ξ_X-dependent in order to model the catalyzing activity of X, i. e.R₂ is proportional toξ_X⁻¹. From a formal point of view, both model variants are equivalent.

Depending on the context, the modeler will prefer variant (1) or (2).

In the following corollary we establish the general rules that allow transforming such different representations into each other. These rules are based on a translation of the stoichiometric coefficients.

Corollary 5.37 (Translation of stoichiometric coefficients). Let M be a thermokinetic model (see Definition 5.29, p. 100) and let δN ∈ R^i0×j0 and δS ∈ R^i0,e×j0. Then, the solution of the system Mˆ with

Nˆ_E/P =N_E/P +δN, Sˆ_E/P =S_E/P +δS, Nˆ_e=N_e, ˆ

c( ˆξ,ξˆ_e) = c( ˆξ,ξˆ_e), R( ˆˆ ξ,ξˆ_e) = R( ˆξ,ξˆ_e)◦δN#ξ◦δS#ξ_e, ξˆ₀ =ξ₀ is equivalent to that of M with

ˆ

c=c, ξˆ=ξ, ξˆ_e=ξ_e,

Jˆ=J, Fˆ =F ◦δN#ξ◦δS#ξ_e, Jˆ_e=J_e.

Proof. We compare the model equations of M and Mˆ (see Definition 5.29, p. 100) under appli-cation of the above transformation rules.

Since N =N_P −N_E = ˆN andS =S_P−S_E = ˆS, the mole balance equations are equal c˙ˆ= ˙c. Since cˆ= c, ξˆ= ξ and ξˆe = ξe, the equations ˆc= ˆc( ˆξ,ξˆe) and c = c(ξ, ξe) are also equivalent.

Further, we have thatFˆ = ˆN_E# ˆξ◦Sˆ_E# ˆξ_e−Nˆ_P# ˆξ◦Sˆ_P# ˆξ_e=N_E#ξ◦δN#ξ◦S_E#ξ_e◦δS#ξ_e− N_P#ξ ◦ δN#ξ ◦ S_P#ξ_e ◦ δS#ξ_e = δN#ξ ◦ δS#ξ ◦ (N_E#ξ ◦ S_E#ξ_E −N_P#ξ ◦S_P#ξ_P) = δN#ξ◦δS#ξ◦F. Thus, the forces follow the given transformation rule. With this result it

5. Thermokinetic Modeling

follows that RˆJˆ= ˆF is equivalent to R J = F. Thus, all four parts of the model equations given in Definition 5.29 are equivalent.

Example 5.38(Catalyzed reaction).Let the reactionA+X B+Xhave a constant resistance R. The species X has an overall stoichiometric coefficient of zero (ν_X =ν_X,P −ν_X,E = 0) and is therefore a catalyst of the reaction. With the ordering(A, B, X), we haveN_E^T = (1,0,1)and N_P^T = (0,1,1). By using δN^T = (0,0,−1) we get the reaction A B with Nˆ_E^T = (1,0,0), Nˆ_P^T = (0,1,0)and a resistanceRˆ=R ξ_X⁻¹. Thus, the effect of the catalyst could be moved from the stoichiometric submodel to the kinetic submodel. Observe that in the case of a constant C_X the model Mˆ can be reduced, since the concentration and the thermokinetic potential of the catalyst are constant: ξ_X =C_x⁻¹c_x = const. In the translated model ξ_X, can be treated as a model parameter and not as a state variable.

Im Dokument Thermokinetic modeling and model reduction of reaction networks (Seite 101-105)