Transient kinetic isotope fractionation

Short description: Isotope separation by unusual processes

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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.

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.[2] 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.

## Contents

* 1 Mathematical description of transient kinetic isotope effects
* 1.1 Notation
* 1.1.1 Example
* 1.2 Isotope mass balance
* 1.3 General equations for biochemical isotope kinetics (GEBIK)
* 1.4 General equations for biochemical isotope fractionation (GEBIF)
* 1.4.1 Isotopic ratio – definition 1
* 1.4.2 Isotopic ratio – definition 2
* 1.4.3 Isotopic ratio – definition 3
* 1.4.4 Isotopic composition
* 1.4.5 Fractionation factor
* 1.4.6 Isotopic enrichment
* 2 Simplified forms of GEBIK and GEBIF
* 2.1 BFEI hypothesis
* 2.2 QSS hypothesis
* 3 Example of application of GEBIK and GEBIF
* 4 Example of application of GEBIK and GEBIF under BFEI and QSS hypotheses
* 5 See also
* 6 Notes
* 7 References

## 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

substrate concentration

product concentration

enzyme concentration

complex concentration

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 A12A2212C and A13A2213C. The isotopic expression within a molecule is

abS

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≤b≤a must be satisfied. For example, the NA2 product in which 1 isotopic substitution occurs (e.g., A15A2215NA14A2214N) will be described by A21A2221P.

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 xb and yd are the stoichiometric coefficient for  abS substrate and  cdP product, a reaction takes the form

∑b=0axbabS⟶∑d=0cydcdP.

For example, in the reaction 14NO3-+A15A2215NOA3A−⟶14N15NO, the notation is 01S+11S⟶12P with x0=x1=1 for both isotopologue reactants of the same substrate with substitution number b=0 and b=1, and with y1=1 for A21A2221P and y0=y2=0 because the reaction does not comprise production of A20A2220P=14N2O and A22A2222P=15N2O.

For isotopomers, the substitution location is taken into account as  abSβ and  abSγ, where β and γ indicate a different expressions of the same isotopologue  abS. Isotopomers only exist when 1≤b<a and a≥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 b=1. For example, one substitution of deuterium D = 2H[lower-alpha 1] in the symmetric methane molecule CDHA3 does not require the use of the right superscript. In the case that b=2, the substitution location has to be specified, while for CHDA3 and CDA4 it is not required. For example, two 2H substitutions in CDA2HA2 can occur in adjacent or non-adjacent locations. Using this notation, the reaction CD2H2+2O2⟶H2O+D2O+CO2, can be written as

24SAβ⟶02P+22P

where β in 24SAβ 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 1≤b<a and a≥2, definition of the substitution location is always required. For instance, the isotopomers of the (asymmetric) nitrous oxide molecule NA2O are A21A2221SAβ=15N14NO and A21A2221SAγ=14N15NO.

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

∑b=0a∑βxb abSβ⟶∑d=0c∑γuγyd cdPγ,

where uγ=1. For example, using N isotope tracers, the isotopomer reactions

14NO3-+15NO3-⟶14N15NO,
14NO3-+15NO3-⟶15N14NO,

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

01S+11S⟶uAβ12PAβ+uAγ12PAγ,

with uγ=1−uβ. More generally, the tracer element does not necessarily occur in only one substrate and one product. If nS substrates react releasing nP products, each having an isotopic expression of the tracer element, then the generalized reaction notation is

∑j=1nS∑bj=0aj∑βjxbj ajbjSjβj⟶∑h=1nP∑dh=0ch∑γhuγh ydh chdhPhγh.

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(1)

For instance, consider the A16A2216O and A18A2218O tracers in the reaction

CH218OA+16O2⟶H216O+CA18A2218OA16A2216O

In this case the reaction can be written as

11S1+02S2⟶01P1+12P2

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, Sj, bind to an enzyme, E, to form a reversible activated complex, C, which releases one or more products, Ph, 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

∑j=1nS∑bji=0aji∑βjixbji ajbjiSjβji+E↽−−⇀kA1(i)kA2(i)Ci→kA3(i)∑h=1nP∑dhi=0chi∑γhiuγhiydhi chdhiPhγhi+E

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(2)

with the index i=1,...,m, where m depends on the number of possible atomic combinations among all isotopologues and isotopomers. Here, k1(i), k2(i), and k3(i) are the rate constants indexed for each of the m reactions.

#### Example

The reactions

2A14A2214NOA3A−⟶14N2O
14NO3-+15NO3-⟶14N15NO
14NO3-+15NO3-⟶15N14NO
2A15A2215NOA3A−⟶15N2O

can be written as

210S+E↽−−⇀kA1(1)kA2(1)C1→kA3(1)02P+E
01S+11S+E↽−−⇀kA1(2)kA2(2)C2→kA3(2)uAβ12PAβ+uAγ12PAγ+E
211S+E↽−−⇀kA1(3)kA2(3)C3→kA3(3)22P+E

### Isotope mass balance

The following isotope mass balances must hold

∑j=1nS∑bj=0aj∑βjxbj aj=∑h=1nP∑dh=0ch∑γhuγh ydh ch,
∑j=1nS∑bj=0aj∑βjxbj bj=∑h=1nP∑dh=0ch∑γhuγh ydh dh.

### 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

d[bjajSjβj]dt=∑ixbji[k2(i)Ci−k1(i)ES‾i]

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(3a)

dCidt=k1(i)ES‾i−[k2(i)+k3(i)]Ci

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(3b)

d[dhchPhγh]dt=∑iuγhiydhik3(i)Ci

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(3c)

dEdt=zdBdt−∑idCidt

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(3d)

dBdt=Y∑h∑dh∑γhd[dhchPhγh]dt−μB

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(3e)

with i=1,...,m, and where 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 μ is the biomass mortality rate.[4]

### 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:

#### Isotopic ratio – definition 1

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

RSbj,βj**(t)=bjajSjβj(t)0ajSj(t)

RPdh,γh**(t)=dhchPhγh(t)0chPh(t)

#### Isotopic ratio – definition 2

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

RSj*(t)=∑bj≠0∑βjbjqbjMSj bjajSjβj(t)∑bj≠aj∑βj(aj−bj)pbjMSj bjajSjβj(t)

RPh*(t)=∑dh≠0∑γhdhqdhMPh dhchPhγh(t)∑dh≠ch∑γh(ch−dh)pdhMPh dhchPhγh(t)

where,  bjMSj and  dhMPh are the molecular weight of each isotopic expression of the substrate and product.

#### Isotopic ratio – definition 3

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

RS(t)=∑j∑bj≠0∑βjbjqbjMSj bjajSjβj(t)∑j∑bj≠aj∑βj(aj−bj)pbjMSj bjajSjβj(t),

RP(t)=∑h∑dh≠0∑γhdhqdhMPh dhchPhγh(t)∑h∑dh≠ch∑γh(ch−dh)pdhMPh dhchPhγh(t).

#### Isotopic composition

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

δS(t)=(RS(t)Rstd−1)1000,

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(4a)

δP(t)=(RP(t)Rstd−1)1000.

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(4a)

where Rstd 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.

#### Fractionation factor

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

IRP(t)=∑h∑dh≠0∑γhdhqdhMPh d[dhchPhγh(t)]dt∑h∑dh≠ch∑γh(ch−dh)pdhMPh d[dhchPhγh(t)]dt

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(5)

and the time-dependent fractionation factor

α(t)=IRP(t)RS(t)

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(6)

#### Isotopic enrichment

The time-dependent isotopic enrichment is simply defined as

ϵ(t)=1−α(t)

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(7)

## Simplified forms of GEBIK and GEBIF

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.

### 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

d[bjajSjβj]dt=∑ixbji[k2(i)Ci−k1(i)ES‾i]

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(8a)

dCidt=k1(i)ES‾i−[k2(i)+k3(i)]Ci

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(8b)

d[dhchPhγh]dt=∑iuγhiydhik3(i)Ci

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(8c)

dEdt=−∑idCidt

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(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

d[ajbjSjβj]dt≃−∑i=1mxbjik3(i)E0S‾iS‾i+Ki(1+∑p≠iS‾pKp)

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(9a)

d[dhchPhγh]dt≃∑i=1muγhiydhik3(i)E0S‾iS‾i+Ki(1+∑p≠iS‾pKp)

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(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.

## Example of application of GEBIK and GEBIF

An example is shown where GEBIK and GEBIF equations are used to describe the isotopic reactions of NA2O consumption into NA2 according to the simultaneous set of reactions

A14A2214NA2O⟶14N2,
14N15NO⟶14N15N,
15N14NO⟶14N15N,

These can be rewritten using the notation introduced before as.

02S+E↽−−⇀kA2(1)kA1(1)CA1→kA3(1)02P+E,
12SAβ+E↽−−⇀kA2(2)kA1(2)CA2→kA3(2)12P+E,
12SAγ+E↽−−⇀kA2(3)kA1(3)CA3→kA3(3)12P+E,

The substrate A22A2222S = 15N2O has not been included due to its scarcity. In addition, we have not specified the isotopic substitution in the NA2 product of the second and third reactions because NA2 is symmetric. Assuming that the second and third reactions have identical reaction rates (k1(3)≡k1(2), k2(3)≡k2(2), and k3(3)≡k3(2)), the full GEBIK and GEBIF equations are

d[A20A2220S]dt=k2(1)CA1−k1(1)A20A2220SE

d[A21A2221SAβ]dt=k2(2)CA2−k1(2)A21A2221SAβE

d[A21A2221SAγ]dt=k2(2)CA3−k1(2)A21A2221SAγE

dCA1dt=k1(1)A20A2220SE−(k2(1)+k3(1))CA1

dCA2dt=k1(2)A21A2221SAβE−(k2(2)+k3(2))CA2

dCA3dt=k1(2)A21A2221SAγE−(k2(2)+k3(2))CA3

d[A20A2220P]dt=k3(1)CA1

d[A21A2221P]dt=k3(2)(CA2+CA3)

dEdt=zdBdt−dCA1dt−dCA2dt−dCA3dt

dBdt=Y(d[A20A2220P]dt+d[A21A2221P]dt)−μB

RP(t)=15 A21A2221P(t)14 A21A2221P(t)+29 A20A2220P(t)

IRAP(t)=15 (CA2+CA3)k3(2)29 CA1k3(1)+14 (CA2+CA3)k3(2),

RS(t)=165 A21A2221S154 A21A2221S+315 A20A2220S

α(t)=7 (CA2+CA3)k3(2)[45 A20A2220S+22 A21A2221S]11 [29 CA1k3(1)+14 (CA2+CA3)k3(2)] 21S

## Example of application of GEBIK and GEBIF under BFEI and QSS hypotheses

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

d[02S]dt≃−k3(1)E002S02S+KA1(1+12SβKA2+12SγKA2)

d[12Sβ]dt≃−k3(2)E012Sβ12Sβ+KA2(1+02SKA1+12SγKA2)

d[12Sγ]dt≃−k3(2)E012Sγ12Sγ+KA2(1+02SKA1+12SβKA2)

d20Pdt=−d[02S]dt

d21Pdt=−d[12Sβ]dt−d[12Sγ]dt

RP(t)=1521P1421P+29 20P

IRAP(t)=15KA1k3(2)21S29KA2k3(1)20S+14KA1k3(2)21S

RS(t)=465 21S14[63 20S+31 21S]

α(t)=14KA1k3(2)[63 20S+31 21S]31[29KA2k3(1) 20S+14KA1k3(2) 20S]

where KA3 has been substituted with KA2 because the rate constants in the third reaction have been assumed to equal those of the second reaction.

## See also

* Equilibrium fractionation
* Isotope electrochemistry

## Notes

1. ↑ IUPAC recommends that the symbol for deuterium should be 2H, rather than D.[3]

## References

1. ↑ Mariotti A., J.C. Germon, P. Hubert, P. Kaiser, R. Letolle, A. Tardieux, P. Tardieux, (1981), Experimental determination of nitrogen kinetic isotope fractionation – Some principles – Illustration for the denitrification and nitrification processes, Plant and Soil 62(3), 413–430.
2. ↑ Maggi F. and W. J. Riley, (2010), Mathematical treatment of isotopologue and isotopomer speciation and fractionation in biochemical kinetics, Geochim. Cosmochim. Acta, doi:10.1016/j.gca.2009.12.021
3. ↑ "Provisional Recommendations". IUPAC. http://www.iupac.org/reports/provisional/abstract04/connelly_310804.html.
4. ↑ Monod J. (1949) The growth of bacterial cultures. Annu. Rev. Microbial. 3, 371–394.

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