Transient kinetic isotope effects (or fractionation) occur when the reaction leading to isotope fractionation does not follow pure first-order kinetics (FOK) and therefore isotopic effects cannot be described with the classical equilibrium fractionation equations or with steady-state kinetic fractionation equations (also known as the Rayleigh equation).
[1] In these instances, the general equations for biochemical isotope kinetics (GEBIK) and the general equations for biochemical isotope fractionation (GEBIF) can be used.
Describing with accuracy isotopic effects, however, depends also on the specific rate law used to describe the chemical or biochemical reaction that produces isotopic effects.
Normally, regardless of whether a reaction is purely chemical or whether it involves some enzyme of biological nature, the equations used to describe isotopic effects base on FOK.
This approach systematically leads to isotopic effects that can be described by means of the Rayleigh equation.
In this case, isotopic effects will always be expressed as a constant, hence will not be able to describe isotopic effects in reactions where fractionation and enrichment are variable or inverse during the course of a reaction.
However, conversely to Michaelis–Menten kinetics, GEBIK and GEBIF equations are solved under the hypothesis of non-steady state.
This characteristic allows GEBIK and GEBIF to capture transient isotopic effects.
The GEBIK and GEBIF equations describe the dynamics of the following state variables 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
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.
defines only one of the two methane forms (either with adjacent or non-adjacent D atoms).
For instance, the isotopomers of the (asymmetric) nitrous oxide molecule
Reactions of asymmetric isotopomers can be written using the partitioning coefficient u as where
More generally, the tracer element does not necessarily occur in only one substrate and one product.
products, each having an isotopic expression of the tracer element, then the generalized reaction notation is For instance, consider the
, bind to an enzyme, E, to form a reversible activated complex, C, which releases one or more products,
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 with the index
, where m depends on the number of possible atomic combinations among all isotopologues and isotopomers.
The reactions can be written as The following isotope mass balances must hold 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.
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
are the molecular weight of each isotopic expression of the substrate and product.
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 Eqs.
(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 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.
(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
according to the simultaneous set of reactions These can be rewritten using the notation introduced before as.