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). In these instances, the general equations for biochemical isotope kinetics (GEBIK) and the general equations for biochemical isotope fractionation (GEBIF) can be used. The GEBIK and GEBIF equations are the most generalized approach to describe isotopic effects in any chemical, catalytic reaction and biochemical reactions because they can describe isotopic effects in equilibrium reactions, kinetic chemical reactions and kinetic biochemical reactions. In the latter two cases, they can describe both stationary and non-stationary fractionation (i.e., variable and inverse fractionation). In general, isotopic effects depend on the number of reactants and on the number of combinations resulting from the number of substitutions in all reactants and products. 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. Most chemical reactions do not follow FOK; neither biochemical reactions can normally be described with FOK. To properly describe isotopic effects in chemical or biochemical reactions, different approaches must be employed such as the use of Michaelis–Menten reaction order (for chemical reactions) or coupled Michaelis–Menten and Monod reaction orders (for biochemical reactions). 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.
Mathematical description of transient kinetic isotope effects The GEBIK and GEBIF equations are introduced here below.
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 C 12 {\displaystyle {\ce {^{12}C}}} and C 13 {\displaystyle {\ce {^{13}C}}} . The isotopic expression within a molecule is
a b S {\displaystyle _{a}^{b}{\ce {S}}}
where a {\displaystyle _{a}} is the number of tracer atoms within S, while b {\displaystyle ^{b}} is the number of isotopic substitutions in the same molecule. The condition 0 ≤ b ≤ a {\displaystyle 0\leq b\leq a} must be satisfied. For example, the N 2 {\displaystyle {\ce {N2}}} product in which 1 isotopic substitution occurs (e.g., N 14 15 N {\displaystyle {\ce {^{15}N^{14}N}}} ) will be described by P 2 1 {\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 x b {\displaystyle x_{b}} and y d {\displaystyle y_{d}} are the stoichiometric coefficient for a b S {\displaystyle _{a}^{b}{\ce {S}}} substrate and c d P {\displaystyle _{c}^{d}{\ce {P}}} product, a reaction takes the form
∑ b = 0 a x b
a b S ⟶ ∑ d = 0 c y d
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