Long-Range Electron Transfer in Biology

231 Pages·1991·3.062 MB·English

by  Patrick Bertrand (auth.)

Preface
The possibility that there might be long-range electron transfer between redox-
active centers in enzymes was first suspected by biochemists working on the
mechanism of action of metalloenzymes such as xanthine oxidase which contain
more than one metal-based redox center. In these enzymes electron transfer
frequently proceeds rapidly but early spectroscopic measurements, notably
those by electron paramagnetic resonance, failed to provide any indication that
these centers were close to one another.
However, it took the seminal experiments on the temperature-independent
light-induced oxidation of cytochromes in photosynthetic bacteria by Devault
and Chance in 1966 to persuade physical scientists that long-range electron
transfer in biological systems might be a real phenomenon. The subsequent
theoretical contribution's of Hopfield and Jortner placed a more rigorous focus
on the problem and triggered a substantial effort towards defining the physico-
chemical basis for this phenomenon. This effort proceeds unabated today, as this
issue of Structure dna Bonding testifies.
This field has progressed rapidly in the last decade and consequently it
appeared worthwhile to ask a number of the individuals who have participated
in these advances to provide their perspective, both on the current state of our
knowledge and on a review of the most recent results from their respective
laboratories.
In an interdisciplinary area such as this it is more than natural that the
character of the individual articles differs widely. We start at one extreme with
the contributions of Bertrand and Kuki which contain a great deal of theoretical
and state-of-the-art physics. We then proceed to those of Gray and Hoffman
who cleverly exploit protein chemistry, continue on to Mauk and McLendon
who take advantage of the tools of modern genetic engineering and end with
Sykes who describes the utility of drawing on Nature's engineering in un-
ravelling the details of metalloprotein redox reactivity.
By bringing together such disparate approaches si hoped that this volume
will serve as a convenient starting point for someone, regardless of background,
who is interested in acquiring some appreciation of the origins of this field, of
our current state of knowledge and of the breadth of approaches that have been
successful in bringing this field to its current level of insight.
Houston, February 1991 Graham Palmer
Table of Contents
Application of Electron Transfer Theories to Biological
Systems
P. Bertrand ............................
Electronic Tunneling Paths in Proteins
A. Kuki .............................. 49
Long-Range Electron Transfer Within Metal-Substituted
Protein Complexes
B. M. Hoffman, M. J. Natan, J. M. Nocek, .S A. Wallin.. 85
Long-Range Electron Transfer in Metalloproteins
M. J. Therien, J. Chang, A. L. Raphael, B. E. Bowler,
H. B. Gray ............................ 109
Electron Transfer in Genetically Engineered Proteins.
The Cytochrome c Paradigm
A. G. Mauk ............................ 131
Control of Biological Electron Transport via Molecular
Recognition and Binding: The "Velcro" Model
G. McLendon ........................... 159
Plastocyanin and the Blue Copper Proteins
A. G. Sykes ............................ 175
Author Index Volumes 1-75 ................... 225
Application of Electron Transfer Theories
to Biological Systems
Patrick Bertrand
Laboratoire d'Electronique des Milieux Condenses URA CNRS 784,
Universit6 de Provence, Centre de St Jrrrme, boSte 241, 13397 MARSEILLE Cedex 13, France
In biological systems, the mechanisms of conversion and storage of energy involve sequences of
oxido-reduction reactions in which electrons are transferred along a chain of redox centers embedded
in a protein medium. The theoretical interpretation of the kinetics of these transfers pertains to
Quantum Mechanics, and was developed by chemists and physicists. However, owing to the
fundamental importance of these processes, many biochemists are also concerned with these theories
and their practical application to biological systems. This introductory chapter is an attempt to
clarify the physical basis of current theoretical interpretations of biological electron transfers. It
comprises an account of the standard formalism appropriate for non-adiabatic processes, and a
detailed review of different approaches which have been developed to apply this formalism to the
analysis of kinetic data. Important advances in thi~ field have resulted, on the one hand, from precise
theoretical calculations based on molecular structures, and on the other hand, from implementation
of elaborate experimental metttods based on efficient chemical and biochemical techniques. This
topic is illustrated by many examples taken from the recent literature which concern redox proteins
as well as photosynthetic systems.
Introduction .......................................................... 3
The Physical Basis ..................................................... 5
2.1 Expression of the Electron Transfer Rate for a Non-adiabatic Process ......... 6
2.l.1 Calculation of the Transition Probability ........................... 6
2.1.2 Expression of the Electron Transfer Rate ........................... 9
2.2 Calculation of the Electronic Factor .................................... 12
2.2.1 Semi-empirical Methods ........................................ 13
2.2.2 One-Electron Theoretical Models ................................ 15
2.2.3 Many-Electron Theoretical Models ............................... 17
2.2.4 Experimental Test of Bridge-assisted Electron Transfer Models ......... 19
2.3 Further Developments of the Theory ................................... 20
Application to Biological Systems .......................................... 22
3.1 Nature of the Parameters Involved by the Theory in Biological Systems ........ 23
3.1.1 Contributions to the Nuclear Factor .............................. 23
3.1.2 Contributions to the Electronic Factor ............................ 24
3.2 Study of the Temperature Dependence of the Electron Transfer Rate .......... 25
3.2.1 Classical Treatment of the Nuclear Factor .......................... 25
3.2.2 Quantum Treatment of the Nuclear Factor ......................... 27
3.3 Study of the Driving Force Dependence of the Electron Transfer Rate ......... 29
3.3.1 Introduction ................................................. 29
3.3.2 Classical Treatment of the Nuclear Factor .......................... 30
3.3.3 Quantum Treatment of the Nuclear Factor ......................... 30
erutcurtS and gnidnoB 57
© gadeV-gnirpS nilreB grebledieH 1991
2 Patrick Bertrand
3.4 Variations of the Electron Transfer Rate Due to Modifications of the Medium... 31
3.4.1 Replacement of Specific Residues Through Site-directed Mutagenesis ..... 32
3.4.2 Change of the Linking Site of two Molecules ....................... 32
3.4.3 Conformational Effects ......................................... 33
3.5 Interpretation of the Primary Electron Transfer in Bacterial Photosynthetic
Reaction Centers ................................................... 35
Conclusion ........................................................... 40
Appendix: Relation Between the Driving Force and the Experimental Redox Potentials 42
References ............................................................ 44
noitacilppA fo nortcelE refsnarT seiroehT ot lacigoloiB smetsyS
1 Introduction
The basis of electron transfer theories were established by Marcus at the
beginning of the .s0691 Originally, Marcus' work was concerned with bimolecu-
lar reactions between small inorganic molecules in solution, for which transfer
occurs within a transient complex without disruption of the first coordination
sphere (i.e. an outer sphere process). However, the residual interactions between
the centers which result from the overlap of their orbitals were thought to be
strong enough for the rate of intracomplex electron transfer to be independent of
them (an adiabatic process). When the temperature is sufficiently high so that the
nuclear motions of the system can be described classically, the rate expression
takes an activated form 1, ,2 3:
k = v exp (- (AG O + Bk,~4/2),~ T), )1(
where v is a characteristic frequency for the nuclear motion, AG o is the redox free
energy change, and 2 the reorganization energy. By using some simplifying
assumptions, Marcus derived the well known cross-relations which were success-
fully tested in a number of redox reactions between inorganic complexes in
solution 4.
The field of application of the theory was considerably enlarged after the
contributions first by Levich 5 and Dogonadze et al. 6, and later by Kestner
et al. 7 and Hopfield 8. These authors have shown that when the residual
interactions between the centers are weak enough, owing to their large separ-
ation, the expression of the electron transfer rate depends explicitly on these
interactions through an electronic factor (a non-adiabatic process). This ex-
pression also involves a nuclear factor, whose value is determined by the nuclear
motions of the system. These motions are treated semi-classically by Hopfield,
and quantum-mechanically by Jortner 7 but it is remarkable that the
expressions obtained for the nuclear factor give in both cases an activated-type
behavior similar to Eq. )1( in the high-temperature limit. The theoretical
interpretation of the temperature dependence of the photo-induced cytochrome
oxidation in the bacterium Chromatium ,8-I 9 was the starting point of an
impressive number of theoretical and experimental studies devoted to appli-
cations of electron transfer theories to biological systems.
In these systems, the redox centers are constituted by relatively large
prosthetic groups which are held to the proteins by a complex folding of the
peptide chain. This organization prevents the close approach of the centers, and
results generally in long-range electron transfers. The weak residual interactions
between the orbitals of the centers then place these processes in the non-
adiabatic regime, so that the reaction rate depends on both an electronic and a
nuclear factor. These two factors contribute to many basic properties of
biological electron transfer systems. For example, in electron transport chains
coupled to energy conversion in mitochondria and in photosynthetic systems,
redox proteins are often organized in membrane-bound complexes where
4 kcirtaP dnartreB
specific electron transfer processes are coupled to proton transfers. Since the
electronic factor decreases very rapidly when the intercenter distance increases, a
quasi-linear ordering of the centers within these complexes helps to achieve a
directional electron transfer without short circuits. In photosynthetic systems,
the attainment of high quantum yields requires very high forward rates and
much slower recombination rates, particularly in the first steps of charge
separation. In this case, the role of the nuclear factors si essential. Soluble redox
proteins are also involved in some steps of these electron transport chains, and in
many other electron transfer processes which are coupled to enzymatic reactions.
These proteins participate in bimolecular reactions, whose specific character is
ensured by electrostatic interactions between complementary side chains at the
interface of the two partners. The conformation of the transient complex in
which electron transfer takes place is determined by these interactions, and does
not necessarily optimize either the nuclear factor, or the electronic factor; the
latter is expected to be very sensitive to structural details. In this case, the
selectivity of the reaction may result in a relatively slow transfer rate.
Although electron transfers in biological systems are generally expected to be
non-adiabatic, it is possible for some intramolecular transfers to be close to the
adiabatic limit, particularly in proteins where several redox centers are held in a
very compact arrangement. This situation is found for example in cytochromes
3c of sulfate-reducing bacteria which contain four hemes in a 31 kDa molecule
10, 11, or in Escherichia coli sulfite reductase where the distance between the
siroheme iron and the closest iron of a 4Fe-4S cluster is only 4.4 ,A .21-1 It is
interesting to note that a very fast intramolecular transfer rate of about 10 9 S- 1
was inferred from resonance Raman experiments performed in Desulfovibrio
vulgaris Miyazaki cytochrome c 3 13.
In early applications of the theory, some key parameters of the models, such
as the reorganization energy .~ or the parameter ~ that characterizes the decrease
of the electronic factor with distance, were considered as adjustable or were fixed
rather arbitrarily. Such treatments were not fully convincing, since they some-
times allowed the interpretation of a specific set of experimental data with
different sets of parameters, and the validity of the theory could not even be
considered as really tested. Since the beginning of the ,s0891 important advances
have been made in the theory, concerning, for instance, the influence of the
nature of the intercenter medium on the electronic factor, and the effective
calculation of the reorganization energy. The recent synthesis of model systems,
in which a donor and an acceptor are separated by a bridge of known geometry,
has facilitated a detailed test of theoretical predictions. Moreover, the fruitful
discussions between experimentalists and theoreticians that began in 1979 at the
Philadelphia Conference ,41-1 were the starting point for an impressive number
of experimental studies intended to characterize electron transfer reactions in
biological systems. These studies, which have very remarkably exploited the new
possibilities given by chemical modification and site-directed-mutagenesis tech-
niques, are largely reported in several of the chapters of the present volume. In
addition, the X-ray crystallographic structure of two bacterial photosynthetic
noitacilppA fo nortcelE refsnarT seiroehT ot lacigoloiB smetsyS 5
reaction centers has been recently determined at a resolution of about 3
15-21. This allows precise theoretical calculations to be directly compared to
the numerous kinetic data available on these systems, and many papers on this
topic have already been published.
In biological systems, electron transfer kinetics are determined by many
factors of different physical origin. This is especially true in the case of a
bimolecular reaction, since the rate expression then involves the formation
constant K e of the transient bimolecular complex as well as the rate of the
intracomplex transfer 4. The elucidation of the factors that influence the value
of K e in redox reactions between two proteins, or between a protein and organic
or inorganic complexes, has been the subject of many experimental studies, and
some of them are presented in this volume. The complexation step is essential in
ensuring specific recognition between physiological partners. However, it is not
considered in the present chapter, which deals with the intramolecular or
intracomplex steps which are the direct concern of electron transfer theories.
We will first review in Sect. 2 the physical basis of the theory for a non-
adiabatic process, and show how standard simplifying assumptions result in the
definition of an electronic and a nuclear factor for the reaction rate. The effective
calculation of the electronic fator requires a realistic choice of the wavefunctions
describing the initial and final electronic states of the system, and different
models have been proposed for this. However, all these models lead to very
similar concepts concerning the contribution of the medium orbitals to the value
of the electronic factor. The validity of these concepts is confirmed by experi-
mental data obtained in model systems. Next, we will examine in Sect. 3 how this
general formalism can be used to obtain information from kinetic experiments
performed in biological systems. The theoretical expression of the rate involves
many independent parameters, and their determination requires very careful
investigations. A convenient and often used procedure consists in the study of
rate variations as a function of one parameter, the others being maintained
constant. In practice, this last condition is very difficult to achieve in biological
systems, and a definite interpretation of the results must await further ex-
periments. The different approaches which have been used are illustrated by
many examples concerning redox proteins and photosynthetic systems taken
from the recent literature. Some recent theoretical studies devoted to the primary
step of charge separation in bacterial photosynthetic reaction centers are also
briefly presented.
2 The Physical Basis
Exhaustive reviews dealing with the applications of electron transfer theories to
biological systems have been published recently 4, 22J, and should be consulted
for a general presentation of electron transfer processes as well as detailed
mathematical developments. Shorter reviews are also available ,32-1 24. In this
section, we review the physical basis of the formalism generally used in the case of
6 kcirtaP dnartreB
non-adiabatic processes, following a presentation similar to that given in Refs ,7-I
,52 26, with some modifications. In this formalism, the electron transfer rate is
defined as the Boltzmann average of transition probabilities between two states
represented by Born-Oppenheimer wave functions. This leads to the definition of
a nuclear factor which determines the temperature dependence, and an electronic
factor which plays a central role in the case of long-range electron transfers. The
principal models that have been proposed to calculate these factors are then
reviewed. We distinguish those in which the nuclear motions coupled to the
process are represented by a set of harmonic oscillators, and others where some
motions are treated classically. Turning to the electronic factor, we first examine
the principle of semi-empirical determinations, and then present one-electron
and many-electron theoretical calculations. We show that the validity of the
bridge-assisted electron transfer concept is supported by experimental data
obtained in model systems. At the end of this section, we indicate the various
improvements to the theory which have been proposed over the last years.
All through this chapter, we have avoided the terminology "electron transfer
by tunneling" which is rather confusing though it often appears in the literature
.41- The nature of the different tunneling effects involved in electron transfer
processes is discussed in the previously cited reviews 4, ,22 23.
2.1 Expression of the Electron Transfer Rate
for a Non-adiabatic Process
Electron transfer processes induce variations in the occupancy and/or the nature
of orbitals which are essentially localized at the redox centers. However, these
centers are embedded in a complex dielectric medium whose geometry and
polarization depend on the redox state of the system. In addition, a finite
delocalization of the centers' orbitals through the medium is essential to'promote
long-range electron transfers. The electron transfer process must therefore be
viewed as a transition between two states of the whole system. The expression of
the probability per unit time of this transition may be calculated by the general
formalism of Quantum Mechanics.
1.1.2 noitaluclaC of eht noitisnarT ytilibaborP
Let us consider an electron transfer system, whose Hamiltonian may be written:
,r( Q) = n ,r( Q) + ,NT
where r and Q represent the whole sets of electronic and nuclear coordinates
respectively, H is the electronic Hamiltonian, and ~rT the nuclear kinetic-energy
operator defined by:
NT ~)kM2/2h(,lX ~/2 Q2.
= _
Application of Electron Transfer Theories to Biological Systems 7
The system is assumed to be initially prepared in a vibronic state in which the
donor center is reduced and the acceptor center oxidized, and we intend to find
the transition probability to a vibronic state in which the donor is oxidized and
the acceptor reduced. These two states, which of course are not stationary states
of ~, are written as )~v(Q) a/~ (r, Q) and wbZ (Q) ~b (r, Q) respectively, where ~/a
and qu are normalized with respect to r for any value of Q:
(%l~a) = (~bl~b) = -1
It is convenient to seek a solution q(r, Q, t) of the time-dependent Schr6dinger
equation:
~tg b~ = ifi ~ ~/~t, (2)
in the form:
~(r, Q, t)=x~(Q, t)~a(r, Q)+xb(Q, t)~b(r, Q), (3)
where the functions ,X and Zb are to be determined. By substituting expression
(3) into Eq. (2), one finds that ~X and bX must satisfy the following equation
7, 25, 26:
(TN + T'~'. + H..- ifi ~/~ t) X. = - Vba~b (4)
and Eq. 4' obtained by permutation of a and b in Eq. 4, with:
nij= (~ilHl*j) Sij = (*ilk/j)
Tij ---- (Oij -- Sij Oii)/(1 - 8 2) (5)
T'ij = - ~X (h~/M0(% I~*~/~ Q~ )~/~ O~
it
Tij -- -- k~ (fiZ/2Mk)(*j ~I 2~/i/~ Q2 )
Vii = T,j + T'ij + ~T - Sij ~('T (6)
In the left-hand side of Eq. (4), second-order cross terms have been neglected.
The determination of the functions aX and bX that satisfy Eqs. (4) and (4') and
the boundary conditions:
Z. (Q, 0)= v~Z (Q) zb(Q, 0)=0 (7)
is straightforward when ~a and qb are Born-Oppenheimer type wavefunctions,
which depend only parametrically on the nuclear coordinates. This is the case for
example if q. and ~b are defined as eigenfunctions of two different zero-order
Hamiltonians, H ~ (r, Q) and H b (r, Q). In this case, the Tij given by Eq. (5) reduces
to:
tniH
Tij = ( (~/i Hint ~/j ) - Sij (l/i iJN ))/(1 - S 2) (8)
with
Hin t
= H - H i.
This formulation emphasizes the importance of the residual interactions Hi~t in
the electron transfer process.
8 kcirtaP dnartreB
Another way of defining Born-Oppenheimer wavefunctions is to assume
realistic forms for ~a and b@ and to optimize them variationally, by minimizing
the energies aaH and Hbb for any value of Q. We shall find later several examples
of this procedure. The representation of the initial and final states by Born-
Oppenheimer wavefunctions is of course not the most general, and we shall see in
Sect. 3.2 that its validity has been questioned.
The functions vaZ and wbZ which describe the nuclear motions in the initial
and final states, are defined more precisely as belonging to the basis sets },va~{
and },wbZ{ constituted by the solutions of the eigenvalue equations:
NT( + a'a'T + Haa- ,vaZ),vaE =0
NT( + b~T + Hbb -- Ebw,) ,wbX = 0 )9(
These equations express that (T'a'a+H J (Q) and (T~b+Hbb) (Q) constitute
potential energy surfaces for the nuclear motions represented by ,vaX and gbw,.
The solution to our problem is readily found by substituting in Eqs. )4( and (4')
the following expansions:
ga(Q, t)= Ev, Cav, (t) gay, (Q) exp(-iEav, t/fi)
zb(Q, 0 = ,wE Cbw, (t) ,wbZ (Q) exp(--iEbw, t&)
The coefficient Cbw(t ) is then obtained by a stafidard first-order perturbation
calculation which takes into account the initial conditions defined by Eq. .)7(
This gives the transition probability per unit time from the initial state aqvaX to
the isoenergetic continuum of states b/~wb~ in the form:
W(av, bw) = )h/r72( I oS Z~w(Q)Vab(Q) X~v(Q)dQ 21 )wbE(p (10)
where p(Ebw) represents the density of final states.
It must be emphasized that this "golden rule" formulation is correct to the
extent that baV is weak enough for the perturbation method to be valid, and is
thus appropriate for non-adiabatic processes. We recall that when the inter-
action is strong enough for the reaction to move into the adiabatic regime, the
rate becomes independent of these interactions ,1"1 ,2 3. It is also interesting to
note that expression (10) was obtained by using the sets ),vaX{ and },wbZ{ defined
by Eqs. )9( through the expectation values H,~ and Hbb of the whole electronic
Hamiltonian H. An alternative choice would be to use the zero-order Hamilton-
ians H a and H b that were introduced previously. It is easy to see that such a
procedure would also lead to expression (10) at the first order of the perturbation
calculation. The effective calculation of expression (10) requires several simplify-
ing assumptions, which rest on a weak dependence of ~a (r, Q) and ~b ,r( Q) on
the nuclear coordinates Q. Usually it is postulated that this dependence is weak
enough to ensure that:
)i the different contributions to Vab(Q) vary slowly with Q in the transition
region Q ~ Q* where the nuclear functions vaZ and wbZ overlap significantly.
This assumption enables the factorization of baV (Q*) out of the integral in
expression (10) (Frank-Condon factorization)