Lecture 19

Marcus theory

(Historical aside- Robert Hooke was a contempary or Newton's, and afraid that he would be scooped on his Law. He hid a preliminary formulation (CEIIOSSOTTUU, an anagram) up the chimney in his house,- a coded version of the Latin "ut tensio sic vis" - "as the extension, so is the force". Hooke's worries about Newton were probably well founded. Robert Hooke is better known for his discovery of cells.)

In Hooke's Law, the relation between energy and bond length gives a parabolic curve, and provides the framework for discussion of the dependence of energy on vibrational state, and hence on temperature. As the temperature increased, the increased vibrational energy allows the molecule to "swing" along the parabola, so that it visits the higher energy levels more frequently.

This Hooke's Law description is useful in discussion of the energy levels in more complicated molecules. The distance is replaced by a nuclear coordinate, which lumps together all the distances in all the bonds, and a single representative parabola is used to represent the parabolas of all the bonds. This is obviously a gross simplification,- the real picture would require a multidimensional representation,- but it provides a handy frame of reference.

In the diagram below, two different electron transfer reactions are represented, one diabatic, and the other adiabatic. In both cases, the system is represented in two states, that before electron transfer (R the reactant state), and that after electron transfer (P the product state). It is important to realize that these represent two different states of the same system.

1. Parabolas, because nuclear vibrations are harmonic oscillators, and obey Hooke's Law.
   
2. Electron jumping from R to P has to occur at cross-over point (C) because of:
   • a) Frank-Condon principle. Electron transfer occurs so rapidly (in a vibrational frequency) that no change in nuclear configuration can occur during the transfer. This requires that the transfer is a vertical transition in the diagram.
   • b) Conservation of energy requires that the transition is a horizontal line on the diagram.

     The only place where both conditions are fulfilled is where the nuclear energy profiles cross (C). The crossing point represents the energy level to which the reactant state must be raises before progressing to the product state. Effectively, this is equivalent to the top of the activation barrier in the Arhenius, Eyring, Randall-Wilkins treatment
     

3. Diabatic and adiabatic processes:
   • Diabatic (the term more often used is non-adiabatic) - electron transfer is a quantum jump from one curve to the other (curves cross).
   • Adiabatic - (impassable to heat; involving neither loss nor acquisition of heat; - OED). In thermodynamics, an adiabatic process is one in which no exchange of heat with the environment occurs. In the Carnot cycle of an ideal gas engine, the steps in which expansion or contraction occur without exchange of heat are adiabatic. In the electron transfer context, an adiabatic process is one in which no quantum jump occurs, - the electron lingers at the barrier, and the curves representing the two states smooth to form a continuum, with a quasi-state at the top of the activation barrier. DeVault explains the use of this term as follows: "Briefly, since nuclear motion is generally much slower than electronic motion, one can approximate the electronic part of the wave-function of a molecular system by solving for it with nuclei fixed in position. The electronic energy eigenvalues obtained this way , when plotted as a function of the nuclear positions, form adiabatic surfaces which become potential-energy surfaces for nuclear motion. ..... However, when the nuclei are allowed to move, the wave-functions arrived at by this approximation are no longer exactly eigenfunctions and they can change spontaneously from one to another. The matrix elements causing the changes are made from the terms neglected in the approximation and are called the 'non-adiabaticity operator'. This operator involves derivatives of both the electronic and the nuclear wave-functions with respect to nuclear coordinates" (see ref. 1a, p. 101). Devault has a more extensive discussion for the quantum-mechanically thirsty.

     The type of process sets a limit to the value of the rate constant:

     k₁ = k k_BT/h. exp[-DG^act / k_BT] = k k_BT/h exp[DS_act / k_B] exp[-DH^act / k_BT]

     k in the equation above has a value of 1 if the reaction is adiabatic, less than 1 if non-adiabatic.
     

   
4. Coupling the process to the environment.
   (See diagram above for terms)
   l is the coupling, or reorganizational, energy. It is the energy required to displace the system an amount Q = X_B - X_A without electron transfer. This is the energy required to transfer the electron from the bottom of the energy profile of the acceptor (product) state up to the energy profile of the acceptor state in the same nuclear configuration as the energy minimum of the donor state.
   

   Value for l comes from Hooke's Law
   

   l = k_HQ² / 2
   

   From the diagrams, it can be seen l, E_act and E^o are related, so that substitution using the Hooke's Law equations gives:
   

   DE_act = (l - DE^o)² / 4l
   

   
   Note that DE in the diagram and equations above corresponds to -DG.This give:

   DG_act = (l + DG^o)² / 4l

   From this Marcus term, the reorganizational energy depends on the relative positions of the parabolas in both reaction coordinate and energy dimensions.

   An important point in this space is the condition under which the Products parabola intersects the Reactant parabola at the minimum (when E_act is zero). Under these conditions, since the activation energy is zero,
   

   l = -DG^o       

   and the reaction proceeds with its maximal rate, with a an intrinsic maximal rate constant (k^o_ET) normalized to this condition. Values for (k^o_ET) can be found experimentally by measuring the rate constant for a reaction under different conditions, giving different values for DG^o. The theoretical curve is shown below:

   

   An important aspect of this curve is that it goes through a maximum at the value where the above equation holds, and this imples that a value for l could be determined by experiment. A second important characteristic is the bell-shape, which implies that the rate constant decreases as the driving force (-DG) increases beyond the value at which it is equal to l. The conditions under which this dropping-off of rate with increased driving force occurs is known as the Marcus inverted region, and was an important prediction of the theory subject to experimental test. Several groups have explored this relationship in biological systems. Dutton and colleagues have measured reactions in photochemical reaction centers, and adjusted values for DG^o so as to span the range about this value. They have produced an empirical equation (Dutton's Ruler) relating rate to distance:

   log₁₀k_et = 13 - 0.6 (R - 3.6) - 3.1 (DG^o + l)² / l

   where R is the edge-to-edge distance in Angstroms, and DG and l are expressed in eV (see here for recent revisions).

   Note that Page et al. suggest a different equation for application to endergonic reactions. This is misleading and unecessary.

   Similar work in Harry Gray's lab has led to some refinement of this picture. They have measured electron transfer rates from ruthenium complexes attached covalently at histines, either native, or positionned by site directed mutagenesis, at different positions to redox proteins with different secondary structures (see example for plastocyanin below).

   Photoactivation of the ruthenium complex leads to electron transfer to the redox metal center of the protein. By measuring the rate and examining the structures, they have been able to determine how the structure modulates the rate. Typical results are shown in the following Fig.

   By measuring values for (k^o_ET) for different positions in different proteins, the contribution of the secondary structre to the reaction rate could be determined, givinng diifferent slopes for a-helices and b-strands.

   

   Nature of the reorganization energy, l

   The physical effects underlying the reorganization energy, l, are complex, but a brief analysis provides some useful insights into the electron transfer process. When an electron is transferred in an intramolecular process (such as occurs in photochemical reaction centers, or between heme b_L and b_H in the bc₁ complex), a charge is transfered through the protein matrix, and the distribution of charge is different before and after the transfer. This change in charge distribution has to be accomodated by the local dielectric properties. These are contributed by polarizability in local bonds, reorientation of polar side chains, dissociation (or association) of protolytic groups, movement of ions in the solvent, reorientation of solvent dipoles, etc. In addition to these "solvent" effects, the structure local to the redox center might undergo changes in configuration. To deal with this set of responses, it should be recognized that:
   1. l is a composite term. Marcus originally distinguished between two components, l_i and l_o. The former (l_i) referred to the reorganizational energy of the inner shell of atoms, close to the redox center. It is directly related to the parabolas of the Marcus diagram, and was calculated from parameters of the inner-shell vibrational modes: as given by the Hook's law treatment above.
      The latter (l_o) referred to atoms further out, called "solvent", and is estimated from the polarizability of the local milieu:
      where De is the charge transferred from donor to acceptor; r₁ and r₂ are the radii of the two reactants when in contact; r₁₂ is r₁ + r₂; D_op is the square of the refractive index (to give polarizability) of the local medium and D_s its static dielectric constant; and e is the permittivity of space to give SI units (adapted from ref. 1). Cukier and Nocera (4) note that "The Marcus form of the activation energy ... is obtained by a classical treatment of solvent (its characteristic fequencies should be small compared with k_BT), and the assumption that the solvation surfaces are quadratic. The latter assumption is a consequence of the assumed linear response of the solvent to the prescence of the charge distribution of the solute.".
      In the case of intramolecular electron transfer, the protein matrix provides the "solvent", but dielctric responses might be felt out to considerable distance because of the low dielectric constant inside the protein, and might reach the surrounding phase. For membrane proteins, the effect will be compicated by the multiple phases of different dielectric constant resulting from the membrane environment. The dielectric response will be very different depending on the location of the redox center. If this is in the membrane-spanning region, the lipid phase will have an even lower dielectric constant than the protein; near the aqueous interface, the phospholipid head groups will contribute a substantial polarity, raising the dielectric constant. If the redox center is close to the aqueous phase, the side chains may be predominatly polar or charged, and the high dieltric constant of water may also play a significant or even dominant role.
   2. Because of the different time constants for polarization effects (bond polarizability, protolytic reactions, ion movements, and sidechain movements), the dielectric reorganization, and the component of l that it contributes to, will show a time dependence. The consequence of this temporal evolution have not been explored in any detail, except in photochemical reaction centers, where the stability of the product of the 3 ps phase has been shown to evolve over a ns time scale, and the stability of later photoproducts increases on the µs to ms time scale.
   3. An additional manifestation of these dielectric responses is in the design of intramolecular redox chains. Because the interior of the protein is hydrophobic, the work required to separate charge reflects the low dielectric constant, as determined by Coulomb's law. As we have seen in the context of ionophores and protonophoric uncouplers such as dinitrophenol, the work required to transfer a charged species into the low dielectric phase can be greatly decreased by speading the charge over a larger volume. This was put in quantitative terms by Born, and this Born effect applies with similar treatment in electron transfer processes. It is noteworthy that evolution has selected large p-bonded ring structures, - the cytochromes and chlorophylls (Chl) are obvious examples, - as the main elements involved in the long distance electron transfer reactions within proteins. Because the electrons involved in transfer come from the low-energy p-orbitals, their charge is shared over the volume of the p-conjugated system, allowing a greater volume of the low dielectric phase to participate in the dielectric response. This might be a reason for evolution of the dimeric structure observed in all photochemical reaction centers. The dimer brings two Chl (or BChl) molecules together to form the "special pair" of the primary donor. This allows the charge on P⁺ to be spread over twice the volume of a monomeric species, facilitating charge separation through a stronger Born effect.
   4. The use of chlorophylls and cytochromes in intramolecular electron transfer, especially in the transmembrane reactions of bioenergetic systems, has an additional important consequence for electron transfer. Because the electron can transfer across the conjugated system virtually instantaneously, electron transfer across the distance spanned by the ring systems does not contribute significantly to the electron transfer time. This is implicit in the treatment summarized in Dutton's ruler, where the b parameter for the distance dependence of the rate constant has a value of 0 for transfer through conjugate bonds, and all electron transfer distances are measured edge-to-edge between the nearest atoms in the conjugate systems of donor and acceptor. For reaction centers and the bc₁ complex, a major fraction of the electron transfer distance across the insulating phase is contributed by the conjugate systems.
   5. The benefits of the Born effect and of participation of large conjugate ring structures in electron transfer are dependent on the quantum-mechanical properties of the electron. No such similar benefits ameliorate the constraints of Coulomb's law for proton transfer, because the proton charge is nuclear, and as a consequence, the proton is very different quantum-mechanical beast, and heavily constrained by its mass. This has important consequences for application of Marcus theory to proton-coupled electron transfer reactions, which require more complex treatment (see ref. 4 below). Much recent interest has centered on this type of reaction, which is of great importance in bioenergetics because of the role of protolytic processes in the coupling of electron transfer to generation of the proton gradient. Of particular interest are the reactions of O₂ reduction to H₂O in cytochrome oxidase, the oxidation of H₂O to O₂ in photosytem II, and the reduction and oxidation of quinones in bacterial reaction centers, photosystem II, and the bc₁ complex. Some recent progress in understanding proton-coupled electron transfer reactions has come from the development of model systems by Nocera. He studied such reactions in complexes in which a photoactivatable donor group was joined to an acceptor group through H-bonding between a diamine and a carboxylate group. The driving force for the proton transfer component could be varied in otherwise similar complexes by changing the orientation of the H-bonding pair. The driving force is given by the Bronstedt relationship: DG_{proton transfer} = 2.303RT(pK_D - pK_A) Nocera found that the rate of electron transfer was strongly dependent on DG_{proton transfer}, reflecting a slowing effect if the proton transfer was unfavorable, with rate constant decreased by several orders of magnitude, because of a much higher reorganization energy (see ref. 4 for further discussion and theoretical treatment).
   6. A final note on a related topic is the question of dissipation of heat in electron transfer processes. For an exoergic reaction, the change in state involves a loss of work capacity, and the entropy increase leads to transfer of heat to the environment. This is particularly notable in the photochemical processes of photosynthetic reaction centres, where the operation is essentially that of a heat engine. The energy input as light generates an excited state in thermal equilibrium with the light source (>1000 K). The more stable states are in thermal equilibrium with the environment (~300 K). Stabilization must be accompanied by an entropy increase, and by the transfer of thermal energy to the environment through vibronic interactions with neighboring molecules. For the initial steps in photosynthetic charge separation, the timescale for this thermal equilibration is in the same range as the electron transfer process, so that the rate of the reaction is likely to some extent modulated by this heat transfer. ARC is not aware of any formal treatment of this topic.

   References

   1. DeVault, D. (1980) Quantum-mechanical tunnelling in biological systems. Q. Rev. Biophys. 13, 387-564.

      (1a. also in more extended form as a book with same title, pp. 207. Cambridge University Press, 1984.)

   2. Moser, C.C., Page, C.C., Farid, R. and Dutton, P.L. (1995) Biological electron transfer. J. Bioenergetics and Biomembranes 27, 263-274.
   3. Gray, H.B. and Winkler, J.R. (1996) Electron transfer in proteins. Annu. Rev. Biochem. 65, 537-561
   4. Cukier, R.I. and Nocera, D.G. (1998) Proton-coupled electron transfer. Ann. Rev. Physical Chemistry 49, 337-369.

   

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   ©Copyright 1996, Antony Crofts, University of Illinois at Urbana-Champaign, a-crofts@uiuc.edu