Notation

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 ${\displaystyle {\ce {^{12}C}}}$ and ${\displaystyle {\ce {^{13}C}}}$. The isotopic expression within a molecule is

${\displaystyle _{a}^{b}{\ce {S}}}$

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

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 ${\displaystyle x_{b}}$ and ${\displaystyle y_{d}}$ are the stoichiometric coefficient for ${\displaystyle _{a}^{b}{\ce {S}}}$ substrate and ${\displaystyle _{c}^{d}{\ce {P}}}$ product, a reaction takes the form

${\displaystyle \sum _{b=0}^{a}x_{b}{}_{a}^{b}{\ce {S ->}}\sum _{d=0}^{c}y_{d}{}_{c}^{d}{\ce {P}}.}$

For example, in the reaction ${\displaystyle {\ce {{^{14}NO3^{-}}+^{15}NO3^{-}->{^{14}{N}}{^{15}{NO}}}}}$, the notation is ${\displaystyle {\ce {{^{0}_{1}S}+{^{1}_{1}S}->{^{1}_{2}P}}}}$ with ${\displaystyle x_{0}=x_{1}=1}$ for both isotopologue reactants of the same substrate with substitution number ${\displaystyle b=0}$ and ${\displaystyle b=1}$, and with ${\displaystyle y_{1}=1}$ for ${\displaystyle {\ce {^1_2P}}}$ and ${\displaystyle y_{0}=y_{2}=0}$ because the reaction does not comprise production of ${\displaystyle {\ce {^{0}_{2}P={^{14}N2O}}}}$ and ${\displaystyle {\ce {^{2}_{2}P={^{15}N2O}}}}$.

For isotopomers, the substitution location is taken into account as ${\displaystyle _{a}^{b}{\ce {S}}^{\beta }}$ and ${\displaystyle _{a}^{b}{\ce {S}}^{\gamma }}$, where ${\displaystyle \beta }$ and ${\displaystyle \gamma }$ indicate a different expressions of the same isotopologue ${\displaystyle _{a}^{b}{\ce {S}}}$. Isotopomers only exist when ${\displaystyle 1\leq b<a}$ and ${\displaystyle a\geq 2}$. The substitution location has to be specifically defined depending on the number of tracer atoms a, number of substitutions b, and molecule structure. For multiatomic molecules that are symmetric with respect to tracer position, there is no need to specify the substitution position when ${\displaystyle b=1}$. For example, one substitution of deuterium ${\displaystyle {\ce {D\ =\ {^{2}H}}}}$ in the symmetric methane molecule ${\displaystyle {\ce {CDH3}}}$ does not require the use of the right superscript. In the case that ${\displaystyle b=2}$, the substitution location has to be specified, while for ${\displaystyle {\ce {CHD3}}}$ and ${\displaystyle {\ce {CD4}}}$ it is not required. For example, two D substitutions in ${\displaystyle {\ce {CD2H2}}}$ can occur in adjacent or non-adjacent locations. Using this notation, the reaction ${\displaystyle {\ce {{CD2H2}+ {2O2}-> {H2O}+ {D2O}+ {CO2},}}}$ can be written as

${\displaystyle {\ce {{^{2}_{4}S}^{\beta }->{^{0}_{2}P}+{^{2}_{2}P}}}}$

where ${\displaystyle \beta }$ in ${\displaystyle {\ce {{^{2}_{4}S}^{\beta }}}}$ 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 ${\displaystyle 1\leq b<a}$ and ${\displaystyle a\geq 2}$, definition of the substitution location is always required. For instance, the isotopomers of the (asymmetric) nitrous oxide molecule ${\displaystyle {\ce {N2O}}}$ are ${\displaystyle {\ce {^{1}_{2}S^{\beta }={^{15}N}{^{14}NO}}}}$ and ${\displaystyle {\ce {^{1}_{2}S^{\gamma }={^{14}N}{^{15}NO}}}}$.

Reactions of asymmetric isotopomers can be written using the partitioning coefficient u as

${\displaystyle \sum _{b=0}^{a}\sum _{\beta }x_{b}\ {_{a}^{b}}{\ce {S}}^{\beta }{\ce {->}}\sum _{d=0}^{c}\sum _{\gamma }u_{\gamma }y_{d}\ {_{c}^{d}}{\ce {P}}^{\gamma },}$

where ${\displaystyle u_{\gamma }=1}$. For example, using N isotope tracers, the isotopomer reactions

${\displaystyle {\ce {{^{14}NO3^{-}}+{^{15}NO3^{-}}->{^{14}N}{^{15}NO},}}}$

${\displaystyle {\ce {{^{14}NO3^{-}}+{^{15}NO3^{-}}->{^{15}N}{^{14}NO},}}}$

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

${\displaystyle {\ce {{^{0}_{1}S}+{^{1}_{1}S}->{\mathit {u}}_{\beta }{^{1}_{2}P}^{\beta }+{\mathit {u}}_{\gamma }{^{1}_{2}P}^{\gamma },}}}$

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

${\displaystyle \sum _{j=1}^{n_{S}}\sum _{b_{j}=0}^{a_{j}}\sum _{\beta _{j}}x_{b_{j}}\ {_{a_{j}}^{b_{j}}}{\ce {S}}_{j}^{\beta _{j}}{\ce {->}}\sum _{h=1}^{n_{P}}\sum _{d_{h}=0}^{c_{h}}\sum _{\gamma _{h}}u_{\gamma _{h}}\ y_{d_{h}}\ {_{c_{h}}^{d_{h}}}{\ce {P}}_{h}^{\gamma _{h}}.}$

(1)

For instance, consider the ${\displaystyle {\ce {^{16}O}}}$ and ${\displaystyle {\ce {^{18}O}}}$ tracers in the reaction

${\displaystyle {\ce {{CH2^{18}O}+{^{16}O2}->{H2^{16}O}+C^{18}O^{16}O}}}$

In this case the reaction can be written as

${\displaystyle {\ce {{^{1}_{1}S1}+{^{0}_{2}S2}->{^{0}_{1}P1}+{^{1}_{2}P2}}}}$

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

Biochemical kinetic reactions of type (1) are often catalytic reactions in which one or more substrates, ${\displaystyle {\ce {S}}_{j}}$, bind to an enzyme, E, to form a reversible activated complex, C, which releases one or more products, ${\displaystyle {\ce {P}}_{h}}$, 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

${\displaystyle \sum _{j=1}^{n_{{\ce {S}}}}\sum _{b_{ji}=0}^{a_{ji}}\sum _{\beta _{ji}}x_{b_{ji}}\ {_{a_{j}}^{b_{ji}}}{\ce {S}}_{j}^{\beta _{ji}}+{\ce {E <=>[{k}_{1(i)}][{k}_{2(i)}]}}{\ce {C}}_{i}{\ce {->[{k}_{3(i)}]}}\sum _{h=1}^{n_{{\ce {P}}}}\sum _{d_{hi}=0}^{c_{hi}}\sum _{\gamma _{hi}}u_{\gamma _{hi}}y_{d_{hi}}\ {_{c_{h}}^{d_{hi}}}{\ce {P}}_{h}^{\gamma _{hi}}+{\ce {E}}}$

(2)

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

Isotope mass balance

The following isotope mass balances must hold

${\displaystyle \sum _{j=1}^{n_{S}}\sum _{b_{j}=0}^{a_{j}}\sum _{\beta _{j}}x_{b_{j}}\ a_{j}=\sum _{h=1}^{n_{P}}\sum _{d_{h}=0}^{c_{h}}\sum _{\gamma _{h}}u_{\gamma _{h}}\ y_{d_{h}}\ c_{h},}$

${\displaystyle \sum _{j=1}^{n_{S}}\sum _{b_{j}=0}^{a_{j}}\sum _{\beta _{j}}x_{b_{j}}\ b_{j}=\sum _{h=1}^{n_{P}}\sum _{d_{h}=0}^{c_{h}}\sum _{\gamma _{h}}u_{\gamma _{h}}\ y_{d_{h}}\ d_{h}.}$

General equations for biochemical isotope kinetics (GEBIK)

To solve for the concentration of all components appearing in any general biochemical reaction as in (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

${\displaystyle {\frac {{\ce {d}}[{^{b_{j}}_{a_{j}}}{\ce {S}}_{j}^{\beta _{j}}]}{{\ce {d}}t}}=\sum _{i}x_{b_{ji}}[{k}_{2(i)}C_{i}-{k}_{1(i)}{E{\overline {S}}}_{i}]}$

(3a)

${\displaystyle {\frac {{\ce {d}}C_{i}}{{\ce {d}}t}}={k}_{1(i)}{E{\overline {S}}}_{i}-[{k}_{2(i)}+{k}_{3(i)}]C_{i}}$

(3b)

${\displaystyle {\frac {{\ce {d}}[{^{d_{h}}_{c_{h}}}{\ce {P}}_{h}^{\gamma _{h}}]}{{\ce {d}}t}}=\sum _{i}u_{\gamma _{hi}}y_{d_{hi}}{k}_{3(i)}C_{i}}$

(3c)

${\displaystyle {\frac {{\ce {d}}E}{{\ce {d}}t}}=z{\frac {{\ce {d}}B}{{\ce {d}}t}}-\sum _{i}{\frac {{\ce {d}}C_{i}}{{\ce {d}}t}}}$

(3d)

${\displaystyle {\frac {{\ce {d}}B}{{\ce {d}}t}}=Y\sum _{h}\sum _{d_{h}}\sum _{\gamma _{h}}{\frac {{\ce {d}}[{^{d_{h}}_{c_{h}}}{\ce {P}}_{h}^{\gamma _{h}}]}{{\ce {d}}t}}-\mu B}$

(3e)

with ${\displaystyle i=1,...,m}$, and where ${\displaystyle {\overline {S}}_{i}}$ 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 ${\displaystyle \mu }$ is the biomass mortality rate.^[3]

General equations for biochemical isotope fractionation (GEBIF)

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:

BFEI 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

${\displaystyle {\frac {{\ce {d}}[{^{b_{j}}_{a_{j}}}{\ce {S}}_{j}^{\beta _{j}}]}{{\ce {d}}t}}=\sum _{i}x_{b_{ji}}[{k}_{2(i)}C_{i}-{k}_{1(i)}{E{\overline {S}}}_{i}]}$

(8a)

${\displaystyle {\frac {{\ce {d}}C_{i}}{{\ce {d}}t}}={k}_{1(i)}{E{\overline {S}}}_{i}-[{k}_{2(i)}+{k}_{3(i)}]C_{i}}$

(8b)

${\displaystyle {\frac {{\ce {d}}[{^{d_{h}}_{c_{h}}}{\ce {P}}_{h}^{\gamma _{h}}]}{{\ce {d}}t}}=\sum _{i}u_{\gamma _{hi}}y_{d_{hi}}{k}_{3(i)}{\ce {C}}_{i}}$

(8c)

${\displaystyle {\frac {{\ce {d}}E}{{\ce {d}}t}}=-\sum _{i}{\frac {{\ce {d}}C_{i}}{{\ce {d}}t}}}$

(8d)

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

QSS 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

${\displaystyle {\frac {{\ce {d}}[{_{a_{j}}^{b_{j}}}{\ce {S}}_{j}^{\beta _{j}}]}{{\ce {d}}t}}\simeq -\sum _{i=1}^{m}{\frac {x_{b_{ji}}{k}_{3(i)}E_{0}{\overline {S}}_{i}}{{\overline {S}}_{i}+K_{i}\left(1+\displaystyle \sum _{p\neq i}{\dfrac {{\overline {S}}_{p}}{K_{p}}}\right)}}}$

(9a)

${\displaystyle {\frac {{\ce {d}}[{^{d_{h}}_{c_{h}}}{\ce {P}}_{h}^{\gamma _{h}}]}{{\ce {d}}t}}\simeq \sum _{i=1}^{m}{\frac {u_{\gamma _{hi}}y_{d_{hi}}{k}_{3(i)}E_{0}{\overline {S}}_{i}}{{\overline {S}}_{i}+K_{i}\left(1+\displaystyle \sum _{p\neq i}{\dfrac {{\overline {S}}_{p}}{K_{p}}}\right)}}}$

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