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GE 140a 2019 Lecture 12 CHEMICAL KINETIC ISOTOPE EFFECTS

GE 140a 2019 Lecture 12 CHEMICAL KINETIC ISOTOPE EFFECTS. A chemical kinetic isotope effect is a dependence of a rate of a unidirectional chemical process a n the isotopic content or structure of the reactant(s). A. B. A’. B’.

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GE 140a 2019 Lecture 12 CHEMICAL KINETIC ISOTOPE EFFECTS

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  1. GE 140a 2019 Lecture 12 CHEMICAL KINETIC ISOTOPE EFFECTS

  2. A chemical kinetic isotope effect is a dependence of a rate of a unidirectional chemical process an the isotopic content or structure of the reactant(s) A B A’ B’ The ‘kinetic isotope effect’ (KIE) on the reaction can be expressed as an a value relating the abundance ratios of the substituted and unsubstituted reactants ([A’]/[A]) to the ratio of their rates of reaction (d[A’]/d[A]) (d[A’]/d[A]) ([A’]/[A]) ‘KIE’ = a’reacted-reactant = So, our job in understanding this phenomenon is to describe and predict how concentration relates to rate, and how that relationship changes on isotopic substitution

  3. A couple of important examples RuBisCO enzyme • First steps of photosynthetic carbon fixation Ephemeral C6 intermediate CO2 (g) + Rubulose 1,5 Biphosphate 2xPGA + • First steps of pyrolysis of buried organic matter Kerogen Gasses Condensates Oils

  4. You’ve already seen an example of a process that is partly chemical-kinetic, complexly mixed with equilibrium and physical-kinetic processes Vapor Interdiffusion with overlying undersaturated air Condensation of H2O Evaporation of H2O Liquid

  5. Placing the chemical kinetic effect in context Quantum Mechanics Classical Physics Isotope Exchange Rxns • Heterogeneous • Homogenous • VPIE Reversible, Steady State Gravitational A B Diffusion Thermal Escape Unidirectional, Time-Evolving Chemical Kinetic A B

  6. A primer on chemical kinetics The characteristic patterns of time evolution for simple reactions d[product] dt ‘Rate’: Reactant d[i] dt Product [i] or Mi Product [i] Reactant time d[reactant] dt ‘Rate’: ln(rate) time

  7. A primer on chemical kinetics Consider CO2 (g) uptake into aqueous solution: CO2(g) CO2(aq) d[CO2]/dt= -k•[CO2] where k is an empirically observed ‘rate constant’ d[CO2]/[CO2] = -k•dt Integration from some initial time and concentration, t0, [CO2]0, up to some arbitrary time t and concentration [CO2]t yields: = — ln[CO2]t– ln[CO2]0 = -k∆t ln([CO2]t/[CO2]0) = -k∆t [CO2]t/[CO2]0 = exp-k∆t (all [CO2] terms refer to reactant gas) ‘k’ is a fundamental speed of the reaction, normalized for the strength of its driving force (i.e., activity of the reactant or reactants) (∆t is t - t0)

  8. Measuring reaction rate Rates of reactions are generally defined by change over time of amounts or concentrations of reactants or products, multiplied by 1/c, where c is the stoichiometric coefficient Consider carbonate precipitation: CO3=(aq)+ Ca+2(aq) CaCO3 (s) c: -1 -1 +1 Rate = —dMCO3=/dt = —dMCa+2/dt = dMcc/dt An abstract example with funny stoichiometry: 2 x A + 3 x B C c: -2 -3 +1 Rate = —(dMA/dt)/2 = —(dMB/dt)/3 = dMC/dt Thus the rate of reaction will be the same regardless of which reactant or product is observed to accumulate or be consumed

  9. Quantifying integrated reaction progress • • The ‘reaction progress variable’ is a unit for reporting the integrated extent • of reaction; nomenclature is mixed, but it is sometimes given as V, (capital • ‘Chi’), • is generally calculated as total mols of accumulated product or lost reactant (per unit volume, mass or total mols of system), divided by the stoichiometric coefficient for that product or reactant 2 x A + 3 x B C V = [B] – [B]0 -3 Where a critical reactant is being depleted following the Rayleigh distillation law, it is also common todefine ‘reaction progress’ as -ln(F), where F is the fraction of initial reactant remaining; in this case, ‘reaction progress’ will be proportional to how much time has passed as the reaction proceeds. For example above: B B0 -ln(F) = -ln( )

  10. More complex, or ‘higher order’ reactions The preceding introduction focuses on on reactions with the simplest ‘rate laws’ — rate depends linearly on the concentration of each reactant. This is just one of many possible rate laws. A general formula is: Rate = -k P[i]Xi Where k is the experimentally determined rate constant, [i] is the concentration of any of the reactants that participate in the reaction, Xi is an experimentally determined exponent specific to that reactant, i, and the product summation is made over all reactants

  11. More complex, or ‘higher order’ reactions • When all Xi values are 1, rate is a simple function of the activity product of reactants: A + B C Rate = k x [A] x [B] • Xi may be (but need not be) equal to the stoichiometric coefficient: 2 x A + 3 x B C Rate = k x [A]2 x [B]3 • If Xi is zero, [i] has no effect on reaction rate, despite the fact that it is part of the reaction stoichiometry. This sounds weird, but we will see such ‘zero order’ kinetics are an important part of our understanding of enzymatically catalyzed biochemistry Thermal decomposition of acetaldehyde: CH3CHO CH4 + CO Looks simple, but has a peculiar rate law: Rate = k x [CH3CHO]2 Thermal decomposition of N2O5 : 2N2O5 4 NO2 + O2 Complex stoichiometry but a simple rate law: Rate = kx [N2O5]

  12. Elementary reactions and their order The preceding review presents the phenomenological description of the kinetics of net reactions. More useful to us is the treatment of ‘elementary kinetics’. Elementary kinetics describes the time evolution and concentration dependence of individual atomistic steps, several of which might combine to yield a net reaction. The general rule is that rate laws for elementary reactions have Xi values that exactly equal the stoichiometry coefficient for that reactant, and that it is rare for an elementary step to involve more than 1 or 2 molecules. Thus, these are the expected scenarios: A B Rate = k[A] 2A B Rate = k[A]2 A + B C Rate = k[A][B] We are more interested in elementary, ’atomistic’ kinetics because it will be possible for us to call on chemical-physics principles to predict their isotope effects, whereas this generally is not possible for phenomenological net reactions

  13. Photosynthesis provides a good example of the difference between elementary reactions and net phenomenological reactions This is a pair of elementary reactions for which we can describe and predict KIE’s RuBisCO enzyme Ephemeral C6 intermediate CO2 (g) + Rubulose 1,5 Biphosphate 2xPGA + This is a net reaction that has nothing to do with atomistic reactions This is fine; CO2 is a reactant in C fixation Not only is O2 not directly produced at site of fixation, but it actually competes with CO2 for reaction with RuBP CO2 + H2O CH2O + O2 RuBP is omitted because it is in a steady state of consumption and regeneration Eventually C will end up in starches, but several steps intervene (p.s., this Is the formula for formaldehyde ) Water is consumed, but nowhere near site of CO2 fixation

  14. Forward and back reactions If a reaction can run in either direction, it is assigned a different rate constant for the ’forward’ (left to right) and ‘back’ (right to left) directions, each of which can have its own KIE: kf A B kb kf’ A’ B’ kb’ (d[A’]/d[A]) ([A’]/[A]) kf’ kf = = Forward KIE = a’A B (d[B’]/d[B]) ([B’]/[B]) kb’ kb Backward KIE = a’BA = =

  15. Forward and back reactions Note that the abundances of A, A’, B and B’ will evolve over time in a way that reflects both the forward and back reactions: kf A B dA = -kf[A] + kb[B] dB = kf[A] - kb[B] dA’ = -kf’[A’] + kb’[B’] dB’ = kf’[A’] - kb’[B’] kb kf’ A’ B’ kb’ Initially, only rapid forward reaction is observed A steady state is reached when forward and back reactions happen at the same rate As B accumulates, back reaction begins B A [i] or Mi A’ B’ time

  16. The relationship between kinetic and equilibrium fractionations At steady state dA = dA’ = dB = dB’ = 0 kf kf[A] = kb[B] kf[A] = kb[B] A B kf’[A’] = kb’[B’] kf’[A’] = kb’[B’] kb [A] [B] kb kf [A’] [B’] kb’ kf’ kf’ = = A’ B’ kb’ [A] x [B’] [A’] x [B] kb x kf’ kf x kb’ = kf’ kf kb’ kb RB’ RA’ af ab = aB-A at equilibrium = = I.e., the equilibrium fractionation is simply the balance of forward and reverse KIE’s; conversely, we can leverage the knowledge of equilibrium a’s to constrain relative values of af and ab. Also, we might suspect KIE’s are related in some way to Q’/Q values

  17. The Arrhenius relationship and plot Now let’s see how the energetics of k behaves, which will be our route to understanding KIE’s Experimental observations indicate a characteristic dependence of k’s on Temperature: -Ea R x T -Ea kB x T k = A x e k = A x e or A: The pre-exponential factor R: The gas constant kB: The Boltzmann constant Ea: ‘Activation energy’, in molar or atom units (for R or kB, respectively) T: Temperature in K ln(k) 1 T (K-1) • ‘A’ can be interpreted as a frequency of events (e.g., collisions) where reactant(s) take on an orientation that permits reaction • Ea can be thought of as an energy barrier that must be overcome for appropriately oriented reactant(s) to react

  18. KIE’s are generally temperature dependent 12C2H5D vs. 12C2H6 13C12CH6 vs. 12C2H6

  19. This temperature dependence implies that A and Eaare themselves isotope dependent -Ea kB x T k = A x e A k -Ea’ kB x T A’ Slope = Ea/R or Ea/kB k’ = A’ x e ln(k) k’ -∆Ea kB x T A’ A k’ k x e = KIE = Slope = Ea’/R or Ea’/kB Where ∆Ea = Ea’ - Ea 1 T (K-1) This observation, combined with the insight that KIE’s have a connection to equilibria, and thus Q’/Q values, gives us some attractive ‘leads’ for constructing a predictive atomistic model

  20. Transition state theory and the Eyring-Evans-Polanyi equation The reactant and transition state interconvert, establishing a sort of local equilibrium But conversion of the transition state to product is irreversible (for one direction of an elementary rxn) Unimolecular rxn: A [‘activated’ A] Product Bimolecular rxn: A + B [A—B] Product

  21. Transition state theory and the Eyring-Evans-Polanyi equation This is the ArrhenianEa -∆G kB x T k x kB x T h e k = All this is the Arrhenian A k: Rate of reaction k: ‘transmission’ coefficient, which is the probability a TS molecule goes to product ∆G: Gibbs free energy difference between reactant and TS kB: Boltzmann constant h: Plank constant T: Temperature in Kelvin So, question is how to evaluate ∆G and k Note: you will sometimes see ∆G deconvolved into to terms, with ∆S between reactant and TS pulled out and used as part of the Arrhenian ‘A’, with ∆H taking the role of ‘Ea’

  22. Urey-Meyer-Bigeleisen theory provides us a way of evaluating free energy differences between reactant and transition state + + H H B B B B A A A A H H B A ∆Efor transition state A—H—B complex H AH exists in exchange equilibrium with TS TS irreversibly turns into product ∆E for product BH ∆E for reactant AH Reaction coordinate

  23. Urey-Meyer-Bigeleisen theory provides us a way of evaluating free energy differences between reactant and transition state Reactant-TS equilibrium constant -∆G kB x T k x kB x T h e k = -∆∆G kB x T k’ k k’ k x e = KIE = Where ∆∆G = ∆G’ - ∆G -∆∆G kB x T bTS breactant keq for TS-reactant isotope exchange equilibrium = (QTS’/QTS) (QReactant’/QReactant) e = = (under constant P condition, where we can relate Q to ∆G)

  24. Urey-Meyer-Bigeleisen theory provides us a way of evaluating free energy differences between reactant and transition state + + H H B B B B A A A A H H B bTS breactant k’ k k’ k A b for transition state A—H—B complex = KIE = x H AH exists in exchange equilibrium with TS TS irreversibly turns into product b for product BH Two tasks: • How do I evaluate bTS? • What do I do with this k’/k nonsense? b for reactant AH Reaction coordinate

  25. The potential energy landscape of possible inter-atomic geometries Energy Motion 2 Motion 1

  26. Interatomic motions (including, but not limited to bond vibrations) can be understood as a consequence of the shape of the energy landscape d2E dx2 1/2 k µ 1 2π = k — the spring constant for motion along x coordinate n = kpositive; a harmonic ‘real’ frequency Recall n for a motion in the dimension of x is given by the harmonic oscillator eqn.: m1m2 m1+m2 µ = k negative; this ‘frequency’ is an imaginary number; a motion can occur, but does not harmonically repeat – it is an irreversible translation, like a spring-loaded trap that fires

  27. Interatomic motions (including, but not limited to bond vibrations) can be understood as a consequence of the shape of the energy landscape The transition state is the lowest saddle point on the energy surface between the reactant and some other stable (or metastable) product The reaction coordinate: the direction of motion out of the deep, confined ‘well’ of the reactant through the transition state As this irreversible motion occurs, the transition state has stable, bound vibrations in other directions. Thus, one of the reactant molecules degrees of free motions (its fundamental modes) has been turned into a free translation, while the others (or things like them…) continue to harmonically oscillate in the normal way

  28. A concrete example: interconversion of hydrogen cyanide and hydrogen isocyanide Linear tri-atomic 3x3-5= 4 fundamental, modes; all are ’real’ • CN stretch • CH stretch • CH bend (x2 degenerate) Non-linear tri-atomic 3x3-6= 3 modes 2 are real • CN stretch • Another I don’t know 1 is ‘imaginary’ • H migration Linear tri-atomic 3x3-5= 4 fundamental, modes; all are real • CN stretch • NH stretch • NH bend (x2 degenerate)

  29. Reduced partition function ratio of the transition state, considering only its harmonically oscillating ’real’ modes + + H H B B B B A A A A H H bTS breactant k’ k k’ k = KIE = x B A b for transition state A—H—B complex Reduced partition function ratio of the reactant, considering all of itsfundamental modes H AH exists in exchange equilibrium with TS TS irreversibly turns into product b for product BH Relative rates of motion associated with the ‘imaginary’ frequency b for reactant AH Reaction coordinate

  30. The b value of the transition state, ignoring motions along the reaction coordinate Bigeleisen, 1949 ’semiclassical’ theory for KIE’s The b value of the transition state, ignoring motions along the reaction coordinate (3N-6 for linear transition state) The b value of the reactant The ratio of imaginary frequencies for motion across the saddle point; this term should be T-independent and equal to (µ2/µ1)1/2 (3N-5 for linear reactant)

  31. Hopefully you found that coherent, but unfortunately you will find alternate treatments of k’/ k almost everywhere you look Reduced partition function ratio of the transition state, considering only its harmonically oscillating ’real’ modes bTS breactant k’ k k’ k = KIE = x • Ratio of imaginary frequencies • Assumed ~1 • Retrieve from isotope effect on entropies of reactant and TS, (∆S), computed by first-principles models; in this case the b is replace with simply the ∆ZPE Reduced partition function ratio of the reactant, considering all of itsfundamental modes

  32. A primary isotope effectarises when a substitution in a site that gains or lose a bond during the reaction slows or speeds up the reaction; these are usually faster for light species than heavy because the atom is relatively free to move in the TS. This is the origin of the expectation that KIE’s ’normally’ favor light isotopes Disproprtionation H H KIE ~ breactant B B A A A A H H (a bit lower if TS has some sort of harmonic interaction between H and A) Reaction coordinate B A A b for transition state A—H—B complex b for transition state A—H complex H H Atom transfer KIE ~ breactant b for product BH (a bit lower if TS has some sort of harmonic interaction between H and A or B) b for reactant AH b for reactant AH Reaction coordinate

  33. A secondary isotope effectarises when a substitution in a site that doesn’t gain or lose a bond during the reaction slows or speeds up the reaction; generally, this is because of the reactant and TS differ in bonding environment for that atom Here the C-H bond is in a trigonal planar geometry Here the C-H bond is in a tetragonal geometry with a strong C-Br bond Deuteration in this site will likely lead to a secondary isotope effect because the modes of motion involving the C-H bond will have different k’s and µ’s Secondary KIE’s are generally an order of magnitude lower than primary KIEs because differences in bonding environments are subtle

  34. An inverse isotope effectarises when the heavy species reacts faster than the light; this is usually because of a secondary isotope effect where the TS has a tighter bonding environment than the reactant for the atom in question Ea for light > Ea for heavy b for transition state complex Reaction coordinate b for reactant (note in this sort of cartoon, the energy well for the TS is an abstract depiction of all the motions orthogonal to the reaction coordinate) b for product

  35. The ‘sp2-sp3’ transition is the iconic example of a secondary isotope effect that is often ‘inverse’ (kH/kD)

  36. Mapping out primary and secondary isotope effects is a good way to develop an atomistic understanding of a reaction Diels-Alder reaction Isoprene Maleic anhydride Cyclohexene products Assumed 0 ‰ -10 ‰ Fractionation factors 1 ‰ 0 ‰ dD d13C -32 ‰ (evalues; negative is ‘inverse’) 17 ‰ -44 ‰ 22 ‰ -62 ‰ -92 ‰ Thus, the reaction is ‘concerted’ (involves both ends of the isoprene boomarang), and the hydrogens on those carbons undergo conversion from sp2 to sp3 electronic structure Singleton and Thomas, 1995

  37. When you are not sure what to think about the transition state structure, reaction energetics provides a guide TS is ‘ergo neutral’ – reacting atoms relatively free of bonds TS is product-like TS is reactant-like If you ever get serious about a problem like this, use a Density Functional Theory software packages to predict transition state structure (they will even give you KIE’s without any extra effort on your part)

  38. The energy surface and KIE associated with photosynthetic carbon fixation’s first steps K13CO2/K12CO2 ~ 0.97 (unfilled are 13C KIE’s; filled are 18O) The KIE for each version of the RuBisCO enzyme varies with how selective it is for CO2 and against O2 (S). A good guess is that the enzyme is manipulating the transition state structure to select for modes of motion that involve carbon (e.g., bending and asymmetric stretch) Tcherkez et al., 2006

  39. KIE’s are generally strongly position specific in vitro pyruvate decarboxylation Pyruvate 20 Observed 10 (‰) 0 KIE ~25 ‰ -10 d13C (C2) — d13C (C1) -20 Low d13C -30 Equilibrium Low d13C C1 High d13C C2 -40 CO2 10 20 30 40 T (˚C) Acetyl group DeNiro and Epstein, 1977

  40. Position specificity is often ’passed’ from precursor to product, meaning isotopic structures of organics are a record of several steps in their biosynthetic pathways Lipid synthesis inherits the acetyl group ‘fingerprint’ CO2 KIE ~25 ‰ ‘Elongation’ Low d13C Pyruvate Acetyl group High d13C “C=O” C=O C=O C=O C=O C=O C=O C=O “CH3” CH3 CH3 CH3 CH3 CH3 CH3 CH3 d13C DeNiro and Epstein, 1977; Monson and Hayes, 1982

  41. Mass laws of kinetic isotope effects bTS breactant k’ k k’ k = KIE = x This term generally acts like the equilibrium mass law; i.e.: This term generally acts like a diffusive mass law for interdiffusion, where the controlling variable is the reduced mass; i.e.: 1 m1 1 m2 - ln(a2i-j) ln(a3i-j) = l2/3 = 1 m3 1 m1 - At the high-temperature limit, but generally ’flatter’ slopes at lower temperatures li/j= Independent of temperature, but very dependent on reduced mass of reaction coordinate motion

  42. An example of a mass law for a net metabolic reaction: sulfate reduction to H2S

  43. Clumped isotope effects can arise through both distinctive b values of reactants and transition states and differences in reduced mass of motion on reaction coordinate Additionally, mixing/inheritance effects arise, and can be surprisingly strong Pyrolysis of Octadecane to yield methane: CH4 inherits 3 hydrogens from methyl precursor, and one Very low D/H hydrogen from another source; once they are combined they are in symmetrically equivalent sites, yet the probability of two D’s in the same molecule has been suppressed Model prediction Work of Guannan Dong

  44. Some closing comments about how this works in practice • Most KIE’s are constrained by experimental observations of net (not elementary) reactions, or system/organismal level net processes, at a single temperature. • The most common modern approach to estimating KIE’s of elementary reactions is density functional theory modeling, which can tell you about potential energy surfaces, transition state structures and KIE’s with minimal effort. Just don’t get comfortable – it always looks cool but is often wrong. • Linking models of elementary reactions to systems scale processes has to happen through multi-step models of intermediate fluxes and reservoirs; we’ll discuss examples on Wednesday. • The end result is that even well constrained KIE’s have real errors of 10’s of %, relative when you apply them to new experimental or natural contexts. • The material you learned in this lecture is a kind of road map to understanding what these data sources are trying to tell you, and developing your own intuitions about how KIE’s are likely to behave in new contexts. Good luck!

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