## Monday, 11 June 2012

### Fantuzzi: Lecture 1 on Bioenergetics

From Wikipedia
• Marcus Theory is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron transfer reactions
•  the rate at which an electron can move or jump from one chemical species (called the electron donor) to another (called the electron acceptor).

## The One-Electron Redox Reaction

• Chemical reactions may lead to a substitution of a group in a molecule or a ligand in a complex, to the elimination of a group of the molecule or a ligand, or to a rearrangement of a molecule or complex
• Chem reaction may cause exchange of charge between reactants
• Redox reactions without making or breaking bonds are common for ions and complexes
• Change colour
• In redox reaction, one partner is e donor D
• OTher is acceptor A
• For reaction to occur D and A must diffuse together
• Form precursor complex
• Kintic unstable solvated encounter complex
• by which ET if transformed to successor complex
• This separates by diffusion

•  Here k12, k21 and k30 are diffusion constants, k23 and k32 rate constants of activated reactions.
• Total reaction may be diffusion controlled
--- ET step faster than diffusion
--- all encounters lead to reaction
• Reaction may be activation controlled
-- Equilibrium of association is reached
--- ET step is slow
--- separation of sucessor complexis fast.

## Outer Sphere Electron Transfer

• Redox reactions preferably run in polar solvents
• Donor and acceptor have a solvent shell
• Precursor and successor complexes are solvated
• Closest molecules of solvent shell (ligands in complexes) are tightly bound
• constitute inner sphere.
• Reactions where they take part are inner sphere redox reactions
• Free solvent molecules are outer sphere.
• Outer sphere reactions do not change inner sphere
• no bonds made or broken

## The Problem

• In outer sphere redox reactions no bonds are formed or broken
• Only ET occurs
• A quite simple example is the Fe2+/Fe3+ redox reaction, the self exchange reaction which is known to be always occurring in an aqueous solution containing both FeSO4 and Fe2(SO4)3 (of course, with equal and measurable rates in both directions and with Gibbs free reaction energy $\Delta$G0 = 0).
• From the reaction rate's temperature dependence (like e.g. the SN2-substitution reaction of the saponification of a alkyl halide) an activation energy is determined
• activation energy is interpreted as the energy of the transition state in a reaction diagram
• The reaction coordinate describes the minimum energy path from the reactants to the products
• points of this coordinate are combinations of distances and angles between and in the reactants in the course of the formation and/or cleavage of bonds
• the transition state, is characterized by a specific configuration of the atoms
• For outer sphere redox reactions there cannot be such a reaction path
• but activation energy is observed

## The Marcus Model

• ET causes charge rearrangement
• influence solvent env
• For dipolar solvent mols rearrange in direction of field or charges
• Atoms and e in solvent mols are slightly displaced (atomic and e polarisation)
• Solvent polarisation determines free energy of activation and reaction rate
• In outer sphere reactions nuclei displacement in reactants are small
• Solvent has dominant role
• donor- acceptor coupling is weak
• Electron can only jump as a whole (ET)
• If e jumps transfer is faster than movemetn fo large solvent mols
• results: nuclear positions of reactions partners and solvent mols are same before and after e jump
• e jump is governed by quantum mech rules
• only possible is energy of ET system does not change during jump
• Arrangement of solvent mols depends on charge distribution on reactants
• If solvent config must be same before and after jump and energy might not change solvent cannot be in solvation state of precursor or of successor complex
• they are different, must be in between.
• For self-exchange reactions arrange solvent mols in middle of those of precursor and successor complex
• solvent arrangement with 1/2 of e on both D and A is correct env for jumping
• In this state energy of precursor and successor in solvent env would be same
• e- cannot be divided
• must be on D or A
• transition state requires solvent config causing transfer of half an electron ---> impossible
• Solvent can take config corresponding to transition state, even if e is on D or A
• Requires energy
• Thermal energy of solvent proveides energy ---> correct polarisation state
• Then e can jump

Marcus theory
• 2 different electron transfer reactions are represented, one diabatic, and the other adiabatic.
• Both cases: system is represented in 2 states, before ET (R the reactant state) and after ET (P the product state)

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
• 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 kBT/h. exp[-DGact / kBT] = k kBT/h exp[DSact kB] exp[-DHact / kBT]
k in the equation above has a value of 1 if the reaction is adiabatic, less than 1 if non-adiabatic.
1. 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 = XB - XA 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 = kHQ2 / 2
From the diagrams, it can be seen l, Eact and Eo are related, so that substitution using the Hooke's Law equations gives:
DEact = (l - DEo)2 / 4

Note that DE in the diagram and equations above corresponds to -DG.This give:DGact = (l + DGo)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 Eact is zero). Under these conditions, since the activation energy is zero,
l = -DGo
and the reaction proceeds with its maximal rate, with a an intrinsic maximal rate constant (koET) normalized to this condition. Values for (koET) can be found experimentally by measuring the rate constant for a reaction under different conditions, giving different values for DGo. 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 DGo so as to span the range about this value. They have produced an empirical equation (Dutton's Ruler) relating rate to distance:
log10ket = 13 - 0.6 (R - 3.6) - 3.1 (DGo + 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 (koET) 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 bL and bH 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, li and lo. The former (li) 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 (lo) 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; Dop is the square of the refractive index (to give polarizability) of the local medium and Ds 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 kBT), 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 bc1complex. 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:
DGproton transfer = 2.303RT(pKD - pKA)
Nocera found that the rate of electron transfer was strongly dependent on DGproton 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.
outer-sphere electron transfer (IUPAC gold book)
• ET takes place with no or very weak electronic interaction between reactants in transition state
• If donor and acceptor show strong electronic coupling it is inner-sphere electron transfer.
Wikipedia: Outer sphere reactions

• Outer sphere refers to an electron transfer (ET) event that occurs between chemical species that remain separate intact before, during, and after the ET event.
• In contrast, for inner sphere electron transfer the participating redox sites undergoing ET become connected by a chemical bridge.
•  Because the ET in outer sphere electron transfer occurs between two non-connected species, the electron is forced to move through space from one redox center to the other.

## Marcus Theory

• ET rate depends on thermodynamic driving force (diff in redox potentials of e-exchanging sites)
• For most reactions rates increase with increased driving force
• rate of outer-sphere electron-transfer depends inversely on the "reorganizational energy."
• Reorganization energy describes the changes in bond lengths and angles that are required for the oxidant and reductant to switch their oxidation states.
• This energy is assessed by measurements of the self-exchange rates (see below).
• Outer sphere electron transfer is the most common type of electron transfer, especially in biochemistry
• where redox centers are separated by several (up to about 11) angstroms by intervening protein.
• In biochemistry, there are two main types of outer sphere ET: ET between two biological molecules or fixed distance electron transfer, in which the electron transfers within a single biomolecule (e.g., intraprotein)

## Examples

### Self-exchange

Outer sphere electron transfer can occur between chemical species that are identical save for their oxidation state.[3] This process is termed self-exchange. An example is the degenerate reaction between the tetrahedral ions permanganate and manganate:
[MnO4]- + [Mn*O4]2- → [MnO4]2- + [Mn*O4]-
For octahedral metal complexes, the rate constant for self-exchange reactions correlates with changes the population of the eg orbitals, the population of which most strongly affects the length of metal-ligand bonds:
• For the [Co(bipy)3]+/[Co(bipy)3]2+ pair, self exchange proceeds at 109 M-1s-1. In this case, the electron configuration changes from (t2g)6(eg)2 (for Co(I)) to (t2g)5(eg)2 (for Co(II)).
• For the [Co[bipy)3]2+/[Co(bipy)3]3+ pair, self exchange proceeds at 18 M-1s-1. In this case, the electron configuration changes from (t2g)5(eg)2 (for Co(II)) to (t2g)6(eg)0 (for Co(III)).

### Iron-sulfur proteins

• Outer sphere ET is the basis of the biological function of the iron-sulfur proteins
• The Fe centers are typically further coordinated by cysteinyl ligands. T
• he [Fe4S4] electron-transfer proteins ([Fe4S4ferredoxins) may be further subdivided into low-potential (bacterial-type) and high-potential (HiPIP) ferredoxins
• Low- and high-potential ferredoxins are related by the following redox scheme:

• Because of the small structural differences between the individual redox states, ET is rapid between these clusters.
inner-sphere electron transfer (IUPAC gold book)
Historically an electron transfer between two metal centres sharing a ligand or atom in their respective coordination shells. The definition has more recently been extended to any situation in which the interaction between the donor and acceptor centres in the transition state is significant ().

Inner sphere electron transfer
• Inner sphere or bonded electron transfer proceeds via a covalent linkage between the two redox partners, the oxidant and the reductant
• ligand bridges the two metal redox centers during the electron transfer event
•  Inner sphere reactions are inhibited by large ligands, which prevent the formation of the crucial bridged intermediate.
• IS ET is rare in biological systems, where redox sites are often shielded by bulky proteins
• Inner sphere ET is usually used to describe reactions involving transition metal complexes
• Bridging ligand can be anything that can convey e
• Has more than one lone electron pair
• serve as e donor to reductant and oxidant
• Common bridging ligands include the halides and the pseudohalides such as hydroxide and thiocyanate
• Before ET, bridged complex must form
• reversible process
• ET occurs thru bridge once it is established
• Stable bridged structure my be in ground state, a transiently-formed intmt or transition state during reaction.
•  Inner sphere electron transfer is generally enthalpically more favorable than outer sphere electron transfer
•  larger degree of interaction between the metal centers involved,
•  however, inner sphere electron transfer is usually entropically less favorable
•  two sites involved must become more ordered (come together via a bridge) than in outer sphere electron transfer.
Quantum tunnelling
• When a particles tunnels thru a barrier it classically cannot surmount
• Occurs in nuclear fusion

• Particles attempting to travel between potential barriers are a ball trying to roll over a hill
• Classical mech predicts particles that do not have enough energy to surmount a barruer will not reach other side
• Ball without enough energy rolls back down
• Or bounce back or bury itself in the wall (absorption).
• In quantum mech these particles can with a small probability tunnel to other side and cross barrier.
• Ball could borrow energy from surroundings to tunnel thru wall or roll over hill
• pay back --- make reflected electrons more energetic than they would have been
• Why? have properties of waves and particles

### The tunnelling problem

• Solve wave function ---> related to probability density of particle's position
--- describe probability that particle is at any given place
• In limit of large barriers, probability of tunnelling decreases for taller and wider barriers
Franck-Condon Principle
• explains intensity of vibronic transitions
• simultaneous changes in electronic and vibrational energy levels of a mol due to absorption or emission of a photon of appropriate energy
• During electronic transition, a change from one vibrational energy level to another is more likely to happen if 2 vibrational wave functions overlap more significantly.
• Electronic transitions are instantaneous compzred to time scale of nuclear motions
• If mol is to move to new vibr level during electronic transition, new vib level must be instantaneously compatbile with nuclear positions and momenta of vib level of mol in originating electronic state
• In semiclassical picture of vibrations (oscillations) of simple harmonic oscillator, necessary conditions can occur at turning points where momentum is zero
• In quantum mech, vibr levels and wavefunctions are those of quantum harmonic oscillators
• In low temperature approximationmolecule starts out in the v = 0 vibrational level of the ground electronic state
• pon absorbing a photon of the necessary energy, makes a transition to the excited electronic state.
•  may result in a shift of the equilibrium position of the nuclei constituting the molecule.
• The probability that the molecule can end up in any particular vibrational level is proportional to the square of the (vertical) overlap of the vibrational wavefunctions of the original and final state
• In the electronic excited state molecules quickly relax to the lowest vibrational level (Kasha's rule),
• can decay to the lowest electronic state via photon emission. T
• the Franck–Condon principle is applied equally to absorption and to fluorescence.
• Equal spacing between vibrational levels is only the case for the parabolic potential of simple harmonic oscillators,
• in more realistic potentials, such as those shown in Figure 1, energy spacing decreases with increasing vibrational energy.
• Electronic transitions to and from the lowest vibrational states are often referred to as 0-0 (zero zero) transitions and have the same energy in both absorption and fluorescence.