Transient kinetic isotope fractionation

The GEBIK and GEBIF equations are introduced here below.

The GEBIK and GEBIF equations describe the dynamics of the following state variables

:; S : substrate concentration

:; P : product concentration

:; E : enzyme concentration

:; C : complex concentration

:; B : biomass concentration

Both S and P contain at least one isotopic expression of a tracer atom. For instance, if the carbon element is used as a tracer, both S and P contain at least one C atom, which may appear as ^{12}C and ^{13}C. The isotopic expression within a molecule is

:$\_a^b\backslash ce\{S\}$

where $\_a$ is the number of tracer atoms within S, while $^b$ is the number of isotopic substitutions in the same molecule. The condition $0\; \backslash leq\; b\; \backslash leq\; a$ must be satisfied. For example, the N2 product in which 1 isotopic substitution occurs (e.g., ^{15}N^{14}N) will be described by ^1_2P.

Substrates and products appear in a chemical reaction with specific stoichiometric coefficients. When chemical reactions comprise combinations of reactants and products with various isotopic expressions, the stoichiometric coefficients are functions of the isotope substitution number. If $x\_b$ and $y\_d$ are the stoichiometric coefficient for $\_a^b\backslash ce\{S\}$ substrate and $\_c^d\backslash ce\{P\}$ product, a reaction takes the form

:$\backslash sum\_\{b=0\}^\{a\}\; x\_b\; \{\}\_\{a\}^\{b\}\backslash ce\{S\; ->\}\; \backslash sum\_\{d=0\}^c\; y\_d\; \{\}\_c^d\backslash ce\{P\}.$

For example, in the reaction {^{14}NO3^-} + ^{15}NO3^- -> {^{14}{N}}{^{15}{NO}}, the notation is {^0_1S} + {^1_1S} -> {^1_2P} with $x\_0=x\_1=1$ for both isotopologue reactants of the same substrate with substitution number $b=0$ and $b=1$, and with $y\_1=1$ for ^1_2P and $y\_0=y\_2=0$ because the reaction does not comprise production of ^0_2P = {^{14}N2O} and ^2_2P = {^{15}N2O}.

For isotopomers, the substitution location is taken into account as $\_a^b\backslash ce\{S\}^\backslash beta$ and $\_a^b\backslash ce\{S\}^\backslash gamma$, where $\backslash beta$ and $\backslash gamma$ indicate a different expressions of the same isotopologue $\_a^b\backslash ce\{S\}$. Isotopomers only exist when

:{^2_4S}^{\beta} -> {^0_2P} + {^2_2P}

where $\backslash beta$ in {^2_4S}^{\beta} defines only one of the two methane forms (either with adjacent or non-adjacent D atoms). The location of D in the two isotopologue water molecules produced on the right-hand side of the reaction has not been indicated because D is present in only one water molecule at saturation, and because the water molecule is symmetric. For asymmetric and multiatomic molecules with

Reactions of asymmetric isotopomers can be written using the partitioning coefficient {{mvar|u}} as

:$\backslash sum\_\{b=0\}^\{a\}\; \backslash sum\_\{\backslash beta\}\; x\_b\; \backslash \; \{\_\{a\}^\{b\}\}\backslash ce\{S\}^\{\backslash beta\}\; \backslash ce\{->\}\; \backslash sum\_\{d=0\}^c\; \backslash sum\_\{\backslash gamma\}\; u\_\{\backslash gamma\}\; y\_d\; \backslash \; \{\_c^d\}\backslash ce\{P\}^\{\backslash gamma\},$

where $u\_\{\backslash gamma\}=1$. For example, using N isotope tracers, the isotopomer reactions

: {^{14}NO3^-} + {^{15}NO3^-} -> {^{14}N}{^{15}NO},

: {^{14}NO3^-} + {^{15}NO3^-} -> {^{15}N}{^{14}NO},

can be written as one reaction in which each isotopomer product is multiplied by its partition coefficient as

:{^0_1S} + {^1_1S} -> \mathit{u}_\beta {^1_2P}^\beta + \mathit{u}_\gamma {^1_2P}^\gamma,

with $u\_\backslash gamma=1-u\_\backslash beta$. More generally, the tracer element does not necessarily occur in only one substrate and one product. If $n\_\backslash ce\{S\}$ substrates react releasing $n\_\backslash ce\{P\}$ products, each having an isotopic expression of the tracer element, then the generalized reaction notation is

{{NumBlk|:|$\backslash sum\_\{j=1\}^\{n\_S\}\; \backslash sum\_\{b\_j=0\}^\{a\_j\}\; \backslash sum\_\{\; \backslash beta\_j\; \}\; x\_\{b\_j\}\; \backslash \; \{\_\{a\_j\}^\{b\_j\}\}\backslash ce\{S\}\_j^\{\backslash beta\_j\}\; \backslash ce\{->\}\; \backslash sum\_\{h=1\}^\{n\_P\}\; \backslash sum\_\{\; d\_h=0\; \}^\{c\_h\}\; \backslash sum\_\{\; \backslash gamma\_h\; \}\; u\_\{\backslash gamma\_h\}\; \backslash \; y\_\{d\_h\}\; \backslash \; \{\_\{c\_h\}^\{d\_h\}\}\backslash ce\{P\}\_h^\{\backslash gamma\_h\}.$|{{EquationRef|1}}}}

For instance, consider the ^{16}O and ^{18}O tracers in the reaction

:{CH2^{18}O}+{^{16}O2} -> {H2^{16}O} + C^{18}O^{16}O

In this case the reaction can be written as

:{^1_1S1} + {^0_2S2} -> {^0_1P1} + {^1_2P2}

with two substrates and two products without indication of the substitution location because all molecules are symmetric.

Biochemical kinetic reactions of type ({{EquationNote|1}}) are often catalytic reactions in which one or more substrates, $\backslash ce\{S\}\_j$, bind to an enzyme, E, to form a reversible activated complex, C, which releases one or more

products, $\backslash ce\{P\}\_h$, and free, unchanged enzyme. These reactions belong to the type of reactions that can be described by Michaelis–Menten kinetics. Using this approach for substrate and product isotopologue and isotopomer expressions, and under the prescribed stoichiometric relationships among them, leads to the general reactions of the Michaelis–Menten type

{{NumBlk|:||{{EquationRef|2}}}}

with the index $i=1,...,m$, where {{mvar|m}} depends on the number of possible atomic combinations among all isotopologues and isotopomers. Here, $k\_\{1(i)\}$, $k\_\{2(i)\}$, and $k\_\{3(i)\}$ are the rate constants indexed for each of the m reactions.

The reactions

: 2 ^{14}NO3^- -> {^{14}N2O}

:{^{14}NO3^-} + {^{15}NO3^-} -> {^{14}N}{^{15}NO}

:{^{14}NO3^-} + {^{15}NO3^-} -> {^{15}N}{^{14}NO}

: 2 ^{15}NO3^- -> {^{15}N2O}

can be written as

:{2 ^0_1S} + {E} <=>[{k}_{1(1)}][{k}_{2(1)}] {C1} ->[{k}_{3(1)}] {^0_2P} + {E}

:{^0_1S} + {^1_1S} + {E} <=>[{k}_{1(2)}][{k}_{2(2)}] {C2} ->[{k}_{3(2)}] \mathit{u}_\beta {^1_2P}^\beta + \mathit{u}_\gamma{^1_2P}^\gamma + {E}

:{2 ^1_1S} + {E} <=>[{k}_{1(3)}][{k}_{2(3)}] {C3} ->[{k}_{3(3)}] {^2_2P} + {E}

The following isotope mass balances must hold

:$\backslash sum\_\{j=1\}^\{n\_S\}\; \backslash sum\_\{b\_j=0\}^\{a\_j\}\; \backslash sum\_\{\; \backslash beta\_j\; \}\; x\_\{b\_j\}\; \backslash \; a\_j\; =\; \backslash sum\_\{h=1\}^\{n\_P\}\; \backslash sum\_\{\; d\_h=0\; \}^\{c\_h\}\; \backslash sum\_\{\; \backslash gamma\_h\; \}\; u\_\{\backslash gamma\_h\}\; \backslash \; y\_\{d\_h\}\; \backslash \; c\_h,$

:$\backslash sum\_\{j=1\}^\{n\_S\}\; \backslash sum\_\{b\_j=0\}^\{a\_j\}\; \backslash sum\_\{\; \backslash beta\_j\; \}\; x\_\{b\_j\}\; \backslash \; b\_j\; =\; \backslash sum\_\{h=1\}^\{n\_P\}\; \backslash sum\_\{\; d\_h=0\; \}^\{c\_h\}\; \backslash sum\_\{\; \backslash gamma\_h\; \}\; u\_\{\backslash gamma\_h\}\; \backslash \; y\_\{d\_h\}\; \backslash \; d\_\{h\}.$

To solve for the concentration of all components appearing in any general biochemical reaction as in ({{EquationNote|2}}), the Michaelis–Menten kinetics for an enzymatic reaction are coupled with the Monod kinetics for biomass dynamics. The most general case is to assume that the enzyme concentration is proportional to the biomass concentration and that the reaction is not in quasi-steady state. These hypotheses lead to the following system of equations

{{NumBlk|:|$\backslash frac\{\backslash ce\{d\}[\{^\{b\_j\}\_\{a\_j\}\}\backslash ce\; S^\{\backslash beta\_j\}\_j]\}\{\backslash ce\{d\}t\}\; =\; \backslash sum\_i$

x_{b_{ji}} [{k}_{2(i)} C_i -{k}_{1(i)} {E \overline{S}}_i]|{{EquationRef|3a}}}}

{{NumBlk|:|$\backslash frac\{\backslash ce\{d\}C\_i\}\{\backslash ce\{d\}t\}\; =$

{k}_{1(i)} {E\overline{S}}_i - [{k}_{2(i)} + {k}_{3(i)}] C_i|{{EquationRef|3b}}}}

{{NumBlk|:|$\backslash frac\{\backslash ce\{d\}[\{^\{d\_h\}\_\{c\_h\}\}\backslash ce\; P^\{\backslash gamma\_h\}\_h]\}\{\backslash ce\{d\}t\}\; =$

\sum_i u_{\gamma_{hi}} y_{d_{hi}} {k}_{3(i)} C_i|{{EquationRef|3c}}}}

{{NumBlk|:|$\backslash frac\{\; \backslash ce\{d\}\; E\; \}\{\; \backslash ce\{d\}t\; \}\; =\; z\; \backslash frac\{\backslash ce\{d\}\; B\}\{\backslash ce\{d\}t\}$

-\sum_i \frac{\ce{d} C_i}{\ce{d}t}|{{EquationRef|3d}}}}

{{NumBlk|:|$\backslash frac\{\; \backslash ce\{d\}\; B\; \}\{\; \backslash ce\{d\}t\; \}\; =\; Y\; \backslash sum\_h\; \backslash sum\_\{d\_h\}$

\sum_{\gamma_h} \frac{\ce{d}[{^{d_h}_{c_h}}\ce P^{\gamma_h}_h]}{\ce{d}t} - \mu B|{{EquationRef|3e}}}}

with $i=1,...,m$, and where $\backslash overline\{S\}\_i$ is the concentration of the most limiting substrate in each reaction i, z is the enzyme yield coefficient, Y is the yield coefficient expressing the biomass gain per unit of released product and $\backslash mu$ is the biomass mortality rate.Monod J. (1949) The growth of bacterial cultures. Annu. Rev.

Microbial. 3, 371–394.

The isotopic composition of the components in a biochemical system can be defined in different ways depending on the definition of isotopic ratio. Three definitions are described here:

Isotopic ratio relative to each component in the system, each with its isotopic expression, with respect to the concentration of its most abundant isotopologue

:$R^\{**\}\_\{S\_\{b\_j,\backslash beta\_j\}\}(t)=\; \backslash frac\{\{^\{b\_j\}\_\{a\_j\}\}S^\{\backslash beta\_j\}\_j\; (t)\}\{^0\_\{a\_j\}S\_j(t)\}$

:$R^\{**\}\_\{P\_\{d\_h,\backslash gamma\_h\}\}(t)=\; \backslash frac\{\{^\{d\_h\}\_\{c\_h\}\}P^\{\backslash gamma\_h\}\_h(t)\}\{^0\_\{c\_h\}P\_h(t)\}$

Isotopic ratio relative to the mass of the tracer element in each component;

:$R^*\_\{S\_j\}(t)=\; \backslash frac\{\; \backslash displaystyle\; \backslash sum\_\{b\_j\backslash neq0\}\; \backslash sum\_\{\backslash beta\_j\}\; \backslash frac\{b\_j\; q\}\{\; ^\{b\_j\}M\_\{S\_j\}\}\; \backslash \; \{^\{b\_j\}\_\{a\_j\}\}S^\{\backslash beta\_j\}\_j\; (t)\}\; \{\; \backslash displaystyle\; \backslash sum\_\{b\_j\backslash neq$

a_j} \sum_{\beta_j} \frac{(a_j-b_j) p}{^{b_j}M_{S_j}} \ {^{b_j}_{a_j}}S^{\beta_j}_j (t)

}

:$R^*\_\{P\_h\}(t)=\; \backslash frac\{\; \backslash displaystyle\; \backslash sum\_\{d\_h\backslash neq\; 0\}\; \backslash sum\_\{\backslash gamma\_h\}\; \backslash frac\{d\_h\; q\}\{^\{d\_h\}M\_\{P\_h\}\}\; \backslash \; \{^\{d\_h\}\_\{c\_h\}\}P^\{\backslash gamma\_h\}\_h\; (t)\; \}\{\; \backslash displaystyle\; \backslash sum\_\{d\_h\backslash neq$

c_h} \sum_{\gamma_h} \frac{(c_h-d_h) p }{^{d_h}M_{P_h}} \ {^{d_h}_{c_h}}P^{\gamma_h}_h (t) }

where, $^\{b\_j\}M\_\{S\_j\}$ and $^\{d\_h\}M\_\{P\_h\}$ are the molecular weight of each isotopic expression of the substrate and product.

Isotopic ratio relative to the mass of the tracer element in the accumulated substrates and products

:$R\_\{S\}(t)=\; \backslash frac\{\backslash displaystyle\; \backslash sum\_j\; \backslash sum\_\{b\_j\backslash neq0\}\; \backslash sum\_\{\backslash beta\_j\}\; \backslash frac\{b\_j\; q\}\{\; ^\{b\_j\}M\_\{S\_j\}\}\; \backslash \; \{^\{b\_j\}\_\{a\_j\}\}S^\{\backslash beta\_j\}\_j\; (t)\}\; \{\; \backslash displaystyle\backslash sum\_j$

\sum_{b_j\neq a_j} \sum_{\beta_j} \frac{(a_j-b_j) p}{^{b_j}M_{S_j}} \ {^{b_j}_{a_j}}S^{\beta_j}_j (t) },

:$R\_\{P\}(t)=\; \backslash frac\{\; \backslash displaystyle\backslash sum\_h\; \backslash sum\_\{d\_h\backslash neq\; 0\}\; \backslash sum\_\{\backslash gamma\_h\}\; \backslash frac\{d\_h\; q\}\{^\{d\_h\}M\_\{P\_h\}\}\; \backslash \; \{^\{d\_h\}\_\{c\_h\}\}P^\{\backslash gamma\_h\}\_h\; (t)\; \}\{\; \backslash displaystyle\; \backslash sum\_h$

\sum_{d_h\neq c_h} \sum_{\gamma_h} \frac{(c_h-d_h) p }{^{d_h}M_{P_h}} \ {^{d_h}_{c_h}}P^{\gamma_h}_h (t) }.

Regardless of the definition of the isotopic ratio, the isotopic composition of substrate and product are expressed as

{{NumBlk|:|$\backslash delta\_\{S\}(t)=\backslash left(\; \backslash frac\{R\_\{S\}(t)\}\{R\_\{std\}\}-1\; \backslash right)\; 1000$,|{{EquationRef|4a}}}}

{{NumBlk|:|$\backslash delta\_\{P\}(t)=\; \backslash left(\; \backslash frac\{R\_\{P\}(t)\}\{R\_\{std\}\}\; -1\; \backslash right)\; 1000$.|{{EquationRef|4a}}}}

where $R\_\{std\}$ is a standard isotopic ration. Here, definition 3 of isotopic ratio has been used, however, any of the three definitions of isotopic ratio can equally be used.

The isotopic ratio of the product can be used to define the instantaneous isotopic ratio

{{NumBlk|:|$\backslash ce\{IR\}\_\backslash ce\{P\}(t)=\; \backslash cfrac\{\; \backslash displaystyle\backslash sum\_h\; \backslash sum\_\{d\_h\backslash neq\; 0\}\; \backslash sum\_\{\backslash gamma\_h\}\; \backslash cfrac\{d\_h\; q\}\{^\{d\_h\}M\_\{\backslash ce\; P\_h\}\}\; \backslash \; \backslash cfrac\{\backslash ce\{d\}\; [\{^\{d\_h\}\_\{c\_h\}\}\backslash ce\; P^\{\backslash gamma\_h\}\_h\; (t)]\; \}\{\backslash ce\{d\}\; t\}\; \}\{\backslash displaystyle\; \backslash sum\_h\; \backslash sum\_\{d\_h\backslash neq\; c\_h\}\; \backslash sum\_\{\backslash gamma\_h\}\; \backslash cfrac\{(c\_h-d\_h)\; p\; \}\{^\{d\_h\}M\_\{\backslash ce\; P\_h\}\}\; \backslash \; \backslash cfrac\{\backslash ce\{d\}\; [\{^\{d\_h\}\_\{c\_h\}\}\backslash ce\; P^\{\backslash gamma\_h\}\_h\; (t)]\; \}\{\backslash ce\{d\}\; t\}\; \}$|{{EquationRef|5}}}}

and the time-dependent fractionation factor

{{NumBlk|:|$\backslash alpha(t)=\backslash frac\{\backslash ce\{IR\}\_\backslash ce\{P\}(t)\}\{R\_\backslash ce\{S\}(t)\}$|{{EquationRef|6}}}}

The time-dependent isotopic enrichment is simply defined as

{{NumBlk|:|$\backslash epsilon(t)=1-\; \backslash alpha(t)$|{{EquationRef|7}}}}

Under specific assumptions, the GEBIK and GEBIF equations become equivalent to the equation for steady-state kinetic isotope fractionation in both chemical and biochemical reactions. Here two mathematical treatments are proposed: (i) under biomass-free and enzyme-invariant (BFEI) hypothesis and (ii) under quasi-steady-state (QSS) hypothesis.

In instances where the biomass and enzyme concentrations are not appreciably changing in time, we can assume that biomass dynamics is negligible and that the total enzyme concentration is constant, and the GEBIK equations become

{{NumBlk|:|$\backslash frac\{\backslash ce\{d\}[\{^\{b\_j\}\_\{a\_j\}\}\backslash ce\; S^\{\backslash beta\_j\}\_j]\}\{\backslash ce\{d\}t\}\; =\; \backslash sum\_i\; x\_\{b\_\{ji\}\}\; [\{k\}\_\{2(i)\}\; C\_i\; -\{k\}\_\{1(i)\}\; \{E\; \backslash overline\{S\}\}\_i]$|{{EquationRef|8a}}}}

{{NumBlk|:|$\backslash frac\{\backslash ce\{d\}C\_i\}\{\backslash ce\{d\}t\}\; =\; \{k\}\_\{1(i)\}\; \{E\; \backslash overline\{S\}\}\_i\; -\; [\{k\}\_\{2(i)\}\; +\; \{k\}\_\{3(i)\}]\; C\_i$|{{EquationRef|8b}}}}

{{NumBlk|:|$\backslash frac\{\backslash ce\{d\}[\{^\{d\_h\}\_\{c\_h\}\}\backslash ce\; P^\{\backslash gamma\_h\}\_h]\}\{\backslash ce\{d\}t\}\; =\; \backslash sum\_i\; u\_\{\backslash gamma\_\{hi\}\}\; y\_\{d\_\{hi\}\}\; \{k\}\_\{3(i)\}\; \backslash ce\; C\_i$|{{EquationRef|8c}}}}

{{NumBlk|:|$\backslash frac\{\; \backslash ce\{d\}\; E\; \}\{\; \backslash ce\{d\}t\; \}\; =\; -\; \backslash sum\_i\; \backslash frac\{\backslash ce\{d\}\; C\_i\}\{\backslash ce\{d\}t\}$|{{EquationRef|8d}}}}

Eqs. ({{EquationNote|4}}) for isotopic compositions, Eq. ({{EquationNote|6}}) for the fractionation factor and Eq. ({{EquationNote|7}}) for the enrichment factor equally applies to the GEBIK equations under the BFEI hypothesis.

If the quasi-steady-state hypothesis is assumed in addition to BFEI hypothesis, then the complex concentration can be assumed to be in a stationary (steady) state according to the Briggs–Haldane hypothesis, and the GEBIK equations become

{{NumBlk|:|$\backslash frac\{\; \backslash ce\{d\}\; [\{\_\{a\_j\}^\{b\_\{j\}\}\}\backslash ce\; S\_j^\{\backslash beta\_\{j\}\}\; ]\; \}\{\; \backslash ce\{d\}t\; \}\; \backslash simeq\; -\; \backslash sum\_\{i=1\}^m\; \backslash frac\{\; x\_\{b\_\{ji\}\}\; \{k\}\_\{3(i)\}\; E\_0\; \backslash overline\{S\}\_i\; \}\{\; \backslash overline\{S\}\_i\; +\; K\_i\; \backslash left(\; 1+\; \backslash displaystyle\backslash sum\_\{p\backslash neq\; i\}\; \backslash dfrac\{\backslash overline\{S\}\_p\}\{K\_p\}\; \backslash right)\; \}$|{{EquationRef|9a}}}}

{{NumBlk|:|$\backslash frac\{\backslash ce\{d\}\; [\{^\{d\_h\}\_\{c\_h\}\}\backslash ce\; P^\{\backslash gamma\_h\}\_h\; ]\}\{\backslash ce\{d\}t\}\; \backslash simeq\; \backslash sum\_\{i=1\}^m\; \backslash frac\{\; u\_\{\backslash gamma\_\{hi\}\}\; y\_\{d\_\{hi\}\}\; \{k\}\_\{3(i)\}\; E\_0\; \backslash overline\{S\}\_i\; \}\{\; \backslash overline\{S\}\_i\; +\; K\_i\; \backslash left($

1+ \displaystyle\sum_{p\neq i} \dfrac{\overline{S}_p}{K_p} \right) }|{{EquationRef|9a}}}}

which are written in a form similar to the classical Micaelis-Menten equations for any substrate and product. Here, the equations also show that the various isotopologue and isotopomer substrates appear as competing species. Eqs. ({{EquationNote|4}}) for isotopic compositions, Eq. ({{EquationNote|6}}) for the fractionation factor and Eq. ({{EquationNote|7}}) for the enrichment factor equally applies to the GEBIK equations under the BFEI and QSS hypothesis.

An example is shown where GEBIK and GEBIF equations are used to describe the isotopic reactions of N2O consumption into N2 according to the simultaneous set of reactions

: ^{14}N2O -> {^{14}N2},

: {^{14}N}{^{15}NO} -> {^{14}N}{^{15}N},

: {^{15}N}{^{14}NO} -> {^{14}N}{^{15}N},

These can be rewritten using the notation introduced before as.

: {^0_2S} + {E} <=>[{k}_{2(1)}][{k}_{1(1)}] C1 ->[{k}_{3(1)}] {^0_2P} + {E},

: {^1_2S}^\beta + {E} <=>[{k}_{2(2)}][{k}_{1(2)}] C2 ->[{k}_{3(2)}] {^1_2P} + {E},

: {^1_2S}^\gamma + {E} <=>[{k}_{2(3)}][{k}_{1(3)}] C3 ->[{k}_{3(3)}] {^1_2P} + {E},

The substrate ^2_2S\ =\ {^{15}N2O} has not been included due to its scarcity. In addition, we have not specified the isotopic substitution in the N2 product of the second and third reactions because N2 is symmetric. Assuming that the second and third reactions have identical reaction rates $(k\_\{1(3)\}\backslash equiv\; k\_\{1(2)\}$, $k\_\{2(3)\}\backslash equiv\; k\_\{2(2)\}$, and $k\_\{3(3)\}\backslash equiv\; k\_\{3(2)\})$, the full GEBIK and GEBIF equations are

:$\backslash frac\{\backslash ce\{d\}\; [\backslash ce\{^0\_2S\}]\}\{\backslash ce\{d\}t\}\; =\; \{k\}\_\{2(1)\}\; \backslash ce\{C1\}\; -\{k\}\_\{1(1)\}\; \backslash ce\{^0\_2S\; E\}$

:$\backslash frac\{\backslash ce\{d\}\; [\backslash ce\{^1\_2S^\backslash beta\}]\}\{\backslash ce\{d\}t\}\; =\; \{k\}\_\{2(2)\}\; \backslash ce\{C2\}\; -\{k\}\_\{1(2)\}\; \backslash ce\{^1\_2S^\backslash beta\; E\}$

:$\backslash frac\{\backslash ce\{d\}\; [\backslash ce\{^1\_2S^\backslash gamma\}]\}\{\backslash ce\{d\}t\}=\; \{k\}\_\{2(2)\}\; \backslash ce\{C3\}\; -\{k\}\_\{1(2)\}\; \backslash ce\{^1\_2S^\backslash gamma\; E\}$

:$\backslash frac\{\backslash ce\{d\}\backslash ce\{C1\}\}\{\backslash ce\{d\}t\}\; =\; \{k\}\_\{1(1)\}\; \backslash ce\{^0\_2S\; E\}\; -\; (\{k\}\_\{2(1)\}\; +\; \{k\}\_\{3(1)\})\; \backslash ce\{C1\}$

:$\backslash frac\{\backslash ce\{d\}\backslash ce\{C2\}\}\{\backslash ce\{d\}t\}=\; \{k\}\_\{1(2)\}\; \backslash ce\{^1\_2S^\backslash beta\; E\}\; -\; (\{k\}\_\{2(2)\}+\; \{k\}\_\{3(2)\})\; \backslash ce\{C2\}$

:$\backslash frac\{\backslash ce\{d\}\backslash ce\{C3\}\}\{\backslash ce\{d\}t\}=\; \{k\}\_\{1(2)\}\; \backslash ce\{^1\_2S^\backslash gamma\; E\}\; -\; (\{k\}\_\{2(2)\}+\; \{k\}\_\{3(2)\})\; \backslash ce\{C3\}$

:$\backslash frac\{\backslash ce\{d\}\; [\backslash ce\{^0\_2P\}]\}\{\backslash ce\{d\}t\}\; =\; \{k\}\_\{3(1)\}\; \backslash ce\{C1\}$

:$\backslash frac\{\backslash ce\{d\}\; [\backslash ce\{^1\_2P\}]\}\{\backslash ce\{d\}t\}\; =\; \{k\}\_\{3(2)\}\; (\backslash ce\{C2\}\; +\; \backslash ce\{C3\})$

:$\backslash frac\{\; \backslash ce\{d\}\backslash ce\; E\; \}\{\; \backslash ce\{d\}t\; \}\; =\; z\; \backslash frac\{\backslash ce\{d\}B\}\{\backslash ce\{d\}t\}\; -\; \backslash frac\{\backslash ce\{d\}\backslash ce\{C1\}\}\{\backslash ce\{d\}t\}\; -\; \backslash frac\{\backslash ce\{d\}\backslash ce\{C2\}\}\{\backslash ce\{d\}t\}\; -\; \backslash frac\{\backslash ce\{d\}\backslash ce\{C3\}\}\{\backslash ce\{d\}t\}$

:$\backslash frac\{\; \backslash ce\{d\}B\; \}\{\; \backslash ce\{d\}t\; \}\; =\; Y\; \backslash left(\; \backslash frac\{\backslash ce\{d\}\; [\backslash ce\{^0\_2P\}]\}\{\backslash ce\{d\}t\}\; +\; \backslash frac\{\backslash ce\{d\}\; [\backslash ce\{^1\_2P\}]\}\{\backslash ce\{d\}t\}\; \backslash right)\; -\; \backslash mu\; B$

:$R\_P(t)\; =\; \backslash frac\{\; 15\; \backslash \; \backslash ce\{^1\_2P\}(t)\; \}\{\; 14\; \backslash \; \backslash ce\{^1\_2P\}(t)\; +\; 29\; \backslash \; \backslash ce\{^0\_2P\}(t)\; \}$

:$\backslash ce\{IR\_P\}(t)\; =\; \backslash dfrac\{\; 15\; \backslash \; (\backslash ce\{C2\}\; +\; \backslash ce\{C3\})\; \{k\}\_\{3(2)\}\; \}\; \{\; 29\; \backslash \; \backslash ce\{C1\}\; \{k\}\_\{3(1)\}\; +\; 14\; \backslash \; (\backslash ce\{C2\}\; +\; \backslash ce\{C3\})\; \{k\}\_\{3(2)\}\; \},$

:$R\_\{S\}(t)\; =\; \backslash frac\{\; 165\; \backslash \; \backslash ce\{^1\_2S\}\; \}\{\; 154\; \backslash \; \backslash ce\{^1\_2S\}\; +\; 315\; \backslash \; \backslash ce\{^0\_2S\}\; \}$

:$\backslash alpha(t)\; =\; \backslash frac\{\; 7\; \backslash \; (\backslash ce\{C2\}\; +\; \backslash ce\{C3\})\; \{k\}\_\{3(2)\}\; [45\; \backslash \; \backslash ce\{^0\_2S\}\; +\; 22\; \backslash \; \backslash ce\{^1\_2S\}]\; \}\; \{\; 11\; \backslash \; [29\; \backslash \; \backslash ce\{C1\}\; \{k\}\_\{3(1)\}\; +\; 14\; \backslash \; (\backslash ce\{C2\}\; +\; \backslash ce\{C3\})\; \{k\}\_\{3(2)\}\; ]\; \backslash \; \{\_2^1\}\backslash ce\; S\; \}$

The same reaction can be described with the GEBIK and GEBIF equations under the BFEI and QSS approximations as

:$\backslash frac\{\backslash ce\{d\}[\{^0\_2\}\backslash ce\; S]\}\{\backslash ce\{d\}t\}\; \backslash simeq\; -\; \backslash frac\{\{k\}\_\{3(1)\}\; E\_0\; \{^0\_2\}\backslash ce\; S\; \}\{\; ^0\_2\backslash ce\; S\; +\; \backslash ce\{K1\}\; \backslash left(\; 1+\; \backslash dfrac\{\{^1\_2\}\backslash ce\; S^\backslash beta\; \}\{\; \backslash ce\{K2\}\}\; +\; \backslash dfrac\{\; \{^1\_2\}\backslash ce\; S^\backslash gamma\; \}\{\; \backslash ce\{K2\}\}\; \backslash right)\}$

:$\backslash frac\{\backslash ce\{d\}[\{^1\_2\}\backslash ce\; S^\backslash beta]\}\{\backslash ce\{d\}t\}\; \backslash simeq\; -\; \backslash frac\{\; \{k\}\_\{3(2)\}\; E\_0\; \{^1\_2\}\backslash ce\; S^\backslash beta\; \}\{\; ^1\_2\backslash ce\; S^\backslash beta\; +\; \backslash ce\{K2\}\; \backslash left(\; 1+\; \backslash dfrac\{\; \{^0\_2\}\backslash ce\; S\; \}\{\; \backslash ce\{K1\}\}\; +\; \backslash dfrac\{\; \{^1\_2\}\backslash ce\; S^\backslash gamma\; \}\{\; \backslash ce\{K2\}\}\; \backslash right)\; \}$

:$\backslash frac\{\backslash ce\{d\}[\{^1\_2\}\backslash ce\; S^\backslash gamma]\}\{\backslash ce\{d\}t\}\; \backslash simeq\; -\; \backslash frac\{\; \{k\}\_\{3(2)\}\; E\_0\; \{^1\_2\}\backslash ce\; S^\backslash gamma\; \}\{\; ^1\_2\backslash ce\; S^\backslash gamma\; +\; \backslash ce\{K2\}\; \backslash left(\; 1+\; \backslash dfrac\{\; \{^0\_2\}\backslash ce\; S\; \}\{\backslash ce\{K1\}\}\; +\; \backslash dfrac\{\; \{^1\_2\}\backslash ce\; S^\backslash beta\; \}\{\; \backslash ce\{K2\}\}\; \backslash right)\; \}$

:$\backslash frac\{\backslash ce\{d\; \}\; \{\_2^0\}\backslash ce\; P\}\{\backslash ce\{d\}t\}\; =\; -\; \backslash frac\{\backslash ce\{d\}[\{^0\_2\}\backslash ce\; S]\}\{\backslash ce\{d\}t\}$

:$\backslash frac\{\backslash ce\{d\; \}\; \{\_2^1\}\backslash ce\; P\}\{\backslash ce\{d\}t\}\; =\; -\; \backslash frac\{\backslash ce\{d\}[\{^1\_2\}\backslash ce\; S^\backslash beta]\}\{\backslash ce\{d\}t\}\; -\; \backslash frac\{\backslash ce\{d\}[\{^1\_2\}\backslash ce\; S^\backslash gamma]\}\{\backslash ce\{d\}t\}$

:$R\_P(t)\; =\; \backslash frac\{\; 15\; \{\_2^1\}\backslash ce\; P\; \}\{\; 14\; \{\_2^1\}\backslash ce\; P\; +\; 29\; \backslash \; \{\_2^0\}\backslash ce\; P\; \}$

:$\backslash ce\{IR\_P\}(t)\; =\; \backslash dfrac\{\; 15\; \backslash ce\{K1\}\; \{k\}\_\{3(2)\}\; \{\_2^1\}\backslash ce\; S\; \}\{\; 29\; \backslash ce\{K2\}\; \{k\}\_\{3(1)\}\; \{\_2^0\}\backslash ce\; S\; +\; 14\; \backslash ce\{K1\}\; \{k\}\_\{3(2)\}\; \{\_2^1\}\backslash ce\; S\; \}$

:$R\_\{S\}(t)\; =\; \backslash frac\{\; 465\; \backslash \; \{\_2^1\}\backslash ce\; S\; \}\{\; 14\; [63\; \backslash \; \{\_2^0\}\backslash ce\; S\; +\; 31\; \backslash \; \{\_2^1\}\backslash ce\; S]\; \}$

:$\backslash alpha(t)\; =\; \backslash frac\{\; 14\; \backslash ce\{K1\}\; \{k\}\_\{3(2)\}\; [63\; \backslash \; \{\_2^0\}\backslash ce\; S\; +\; 31\; \backslash \; \{\_2^1\}\backslash ce\; S]\; \}\{\; 31\; [29\; \backslash ce\{K2\}\; \{k\}\_\{3(1)\}\; \backslash \; \{\_2^0\}\backslash ce\; S\; +\; 14\; \backslash ce\{K1\}\; \{k\}\_\{3(2)\}\; \backslash \; \{\_2^0\}\backslash ce\; S\; ]\; \}$

where K3 has been substituted with K2 because the rate constants in the third reaction have been assumed to equal those of the second reaction.

• Equilibrium fractionation
• Isotope electrochemistry

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Category:Chemical kinetics

Category:Isotope separation

Category:Fractionation