Theoretical Studies on Coupled Electron and
Proton Transfer in Cytochrome c Oxidase
Ville R. I. Kaila
Institute of Biotechnology
and
Division of Biochemistry
Department of Biological and Environmental Sciences
Faculty of Biosciences
University of Helsinki
and
Helsinki Graduate School in Biotechnology and Molecular Biology
ACADEMIC DISSERTATION
To be presented for public examination with permission of the Faculty of
Biosciences of the University of Helsinki in Auditorium 1041 of Biocenter 2,
Viikinkaari 5, Helsinki, on February 13th 2009, at 12 noon.
Helsinki, 2009
Supervisor:
Prof. Mårten Wikström
Institute of Biotechnology
University of Helsinki
Finland
Reviewers:
Prof. Ulf Ryde
Department of Theoretical Chemistry
Lund University
Lund, Sweden
Prof. Peter R. Rich
Glynn Laboratory of Bioenergetics
Department of Biology
University College London
London, UK
Opponent:
Prof. Régis Pomès
Department of Biochemistry
University of Toronto
Canada
Custos:
Prof. Carl G. Gahmberg
Division of Biochemistry
Department of Biological and Environmental Sciences
University of Helsinki
Finland
Cover Art: The figure illustrates how the dissociation of the Arg-438/heme a3
D-propionate ion pair triggers a concerted proton transfer from Glu-242 to the
same site, which further leads to reduction of heme a3 .
ISBN 978-952-10-5263-7 (paperback)
ISBN 978-952-10-5264-4 (PDF, http://ethesis.helsinki.fi)
ISSN 1795-7079
Yliopistopaino, Helsinki University Printing House
Helsinki, 2009
”It doesn’t matter how beautiful your theory is,
it doesn’t matter how smart you are.
If it doesn’t agree with experiment,
it’s wrong”.
— Richard P. Feynman
”This result is too beautiful to be false;
it is more important to have beauty in one’s equations
than to have them fit experiment”.
— P. A. M. Dirac
Contents
Contents
i
List of Publications
iii
Abstract
v
Acknowledgments
vi
Abbreviations and symbols
viii
1 Introduction
1.1 Energy transduction in biology . . . . . . . . . . . . . . . . . . . . .
1.2 Respiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Theoretical biochemistry . . . . . . . . . . . . . . . . . . . . . . . . .
2 Structure and function of cytochrome c oxidase
2.1 Function of cytochrome c oxidase . . . . . . . .
2.2 Structure and physical properties of CcO . . .
2.2.1 Subunit I . . . . . . . . . . . . . . . . . .
2.2.2 Subunit II . . . . . . . . . . . . . . . . .
2.2.3 Subunit III . . . . . . . . . . . . . . . . .
2.2.4 Other subunits . . . . . . . . . . . . . .
2.2.5 Post-translational modifications . . . .
2.3 The heme-copper oxidase family . . . . . . . .
2.4 Pathways and channels . . . . . . . . . . . . . .
2.4.1 Foundations of eT theory . . . . . . . .
2.4.2 Electron transfer pathways in CcO . . .
2.4.3 Proton transfer channels and pathways
2.4.4 Oxygen channels . . . . . . . . . . . . .
2.4.5 Water exit channel . . . . . . . . . . . .
2.5 The catalytic cycle . . . . . . . . . . . . . . . . .
2.6 Proton translocation . . . . . . . . . . . . . . .
2.7 Mechanism of proton translocation . . . . . . .
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1
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3 Foundations of theoretical biochemistry
31
3.1 The Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . 31
3.3 The Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . . . 32
i
3.4
3.5
3.6
3.7
3.8
Beyond HF-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Early models . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 The Hohenberg-Kohn formulation of DFT . . . . . . . . . .
3.5.3 The Kohn-Sham theorem . . . . . . . . . . . . . . . . . . . .
3.5.4 Exchange-correlation functionals . . . . . . . . . . . . . . . .
3.5.5 Performance of DFT in modeling biomolecular systems . . .
The basis set approximation . . . . . . . . . . . . . . . . . . . . . . .
Classical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Classical equations of motion . . . . . . . . . . . . . . . . . .
3.7.2 Classical force-fields . . . . . . . . . . . . . . . . . . . . . . .
3.7.3 Velocity Verlet . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.4 The Langevin equations of motion in computer simulations
3.7.5 Simulation ensembles . . . . . . . . . . . . . . . . . . . . . .
3.7.6 Dynamical properties . . . . . . . . . . . . . . . . . . . . . .
3.7.7 Free energy calculations . . . . . . . . . . . . . . . . . . . . .
3.7.8 Electrostatic calculations . . . . . . . . . . . . . . . . . . . . .
Combined QM/MM calculations . . . . . . . . . . . . . . . . . . . .
4 Aims of the Study
5 Models and Methods
5.1 Quantum chemical models . .
5.2 Classical molecular dynamics
5.3 Free energy calculations . . .
5.4 Other theoretical methods . .
34
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6 Results and Discussion
6.1 Charge parametrization of the metal centers in CcO . .
6.2 Dynamics of the Glu-242 side chain in CcO . . . . . . .
6.3 Glu-242 is a valve in the proton pump of CcO . . . . . .
6.4 The chemistry of the CuB site in CcO . . . . . . . . . . .
6.5 Mechanism and energetics for preventing leaks in CcO
6.6 Formation of redox-linked proton pathways in CcO . .
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7 Conclusions
85
Bibliography
86
List of Publications
List of publications included in the thesis
I. Johansson M.P., Kaila V.R.I., Laakkonen L. (2008)
Charge parameterization of the metal centers in cytochrome c oxidase.
J. Comput. Chem. 29:753-767.
II. Tuukkanen A.∗ , Kaila V.R.I.∗ , Laakkonen L., Hummer G., Wikström M. (2007)
Dynamics of the glutamic acid 242 side chain in cytochrome c oxidase.
Biochim. Biophys. Acta - Bioenergetics 1767:1102-1106.
III. Kaila V.R.I., Verkhovsky M.I., Hummer G., Wikström M. (2008)
Glutamic acid 242 is a valve in the proton pump of cytochrome c oxidase.
Proc. Natl. Acad. Sci. USA 105:6255-6259.
IV. Kaila V.R.I., Johansson M.P., Sundholm D., Laakkonen L., Wikström M. (2008)
The chemistry of the CuB site in cytochrome c oxidase and the importance of its
unique His-Tyr bond.
Biochim. Biophys. Acta - Bioenergetics (in press)
V. Kaila V.R.I., Verkhovsky M.I., Hummer G., Wikström M. (2008)
Mechanism and energetics by which glutamic acid 242 prevents leaks in cytochrome
c oxidase.
Biochim. Biophys. Acta - Bioenergetics (submitted)
VI. Kaila V.R.I., Hummer G., Wikström M. (2008)
Water-gated proton transfer in cell respiration.
(manuscript)
∗ authors
have equal contribution
iii
iv
List of Other Publications
VII. Kaila V.R.I., Verkhovsky M.I., Hummer G., Wikström M. (2008)
How leaks are prevented in the proton-pumping by cytochrome c oxidase.
Biochim. Biophys. Acta - Bioenergetics, EBEC Special Issue 1777:890-892.
VIII. Kaila V.R.I., Annila A. (2008)
Natural Selection - Least Action.
Proc. Roy. Soc. A 464:3055-3070.
IX. Sharma V., Kaila V.R.I., Annila A. (2008)
Protein folding as an evolutionary process.
Physica A 388:851-862.
X. Kaila V.R.I., Taylor P.R.
DFT and ab inito quantum chemical studies on the His-Tyr dimer in heme-copper
oxidases.
(in preparation)
Abstract
The complexity of life is based on an effective energy transduction machinery,
which has evolved during the last 3.5 billion years. In aerobic life, the utilization of the high oxidizing potential of molecular oxygen powers this machinery.
Oxygen is safely reduced by a membrane bound enzyme, cytochrome c oxidase
(CcO), to produce an electrochemical proton gradient over the mitochondrial or
bacterial membrane. This gradient is used for energy-requiring reactions such as
synthesis of ATP by F0 F1 -ATPase and active transport.
In this thesis, the molecular mechanism by which CcO couples the oxygen reduction chemistry to proton-pumping has been studied by theoretical computer
simulations. By building both classical and quantum mechanical model systems
based on the X-ray structure of CcO from Bos taurus, the dynamics and energetics
of the system were studied in different intermediate states of the enzyme.
As a result of this work, a mechanism was suggested by which CcO can prevent protons from leaking backwards in proton-pumping. The use and activation of two proton conducting channels were also enlightened together with a
mechanism by which CcO sorts the chemical protons from pumped protons. The
latter problem is referred to as the gating mechanism of CcO, and has remained a
challenge in the bioenergetics field for more than three decades. Furthermore, a
new method for deriving charge parameters for classical simulations of complex
metalloenzymes was developed.
v
Acknowledgments
The research of this work was performed in Helsinki Bioenergetics Group, at
the Institute of Biotechnology, and the Division of Biochemistry at the Department of Biological and Environmental Sciences, University of Helsinki during
2006-2008. This work was financially supported by the Finnish Cultural Foundation, the Graduate School of Biotechnology and Molecular Biology, the Academy
of Finland, Sigrid Juselius Foundation, Biocentrum Helsinki, the University of
Helsinki, and Magnus Ehrnrooth Foundation. CSC - the Finnish IT Center for
Science is acknowledged for providing computational resources.
First of all, I would like to express my deep gratitude to my mentor and supervisor, prof. Mårten Wikström. I’m grateful that you convinced me in the first place
to apply for the biochemistry program at the University of Helsinki instead of
going to medical school. I’m also grateful that you first suggested me to become
a theoretician, which combined my interests in chemistry, physics, and biology
in an optimal way. Thank you also for letting me solve things "my way" - retrospectively, this has been the only right way for me.
I would also like to thank all present and past members of Helsinki Bioenergetics Group for enlightening discussions during many occasions. Liisa Laakkonen
is especially acknowledged for the tight collaboration in my early projects, and
for many helpful discussions. I also acknowledge you for valuable comments on
language related aspects of this thesis. Thanks to my present and former office
mates - Ilya Belevich, Vivek Sharma, and Lilya Euro for sharing the office space
with me. Vivek - you deserve special thanks for being a good friend to me during
your stay in Finland. Many thanks also to our former secretary Satu Sankkila for
many stimulating discussions.
I wish to acknowledge the co-authors of my publications: Mikael P. Johansson,
Anne Tuukkanen, Michael Verkhovsky, Gerhard Hummer, and Dage Sundholm.
Dage and Mikael, together with the personnel of Svenska Kemen are especially
acknowledged for collaboration and helpful discussions on quantum chemistry
and a diverse variety of other topics on many different occasions. Gerhard Hummer is acknowledged for introducing me to state-of-the-art techniques in free energy calculations. The people from the NMR-lab, especially Arto Annila, Peter
Würtz, and Kai Fredriksson are acknowledged for many discussions during the
years. A special thanks also to Peter R. Taylor for my most enjoyable visit in your
lab at University of Warwick during the spring 2008 - thank you for introducing
me to the secrets of hard-core quantum chemistry.
vi
vii
My pre-examiners, professors Peter R. Rich and Ulf Ryde are acknowledged for
many insightful comments on my thesis.
I also wish to thank all my former students who have taken part in my course
"Introduction to Chemistry" during the last three years. You have provided me
an opportunity to understand the "basics" on a much deeper level than otherwise. It is first when you teach something to someone else, that you understand
it yourself.
A big thanks to all my friends. Especially Klaus Ittonen, you know your importance, Mahesh Karnani, for many interesting discussion in neurobiophysics
and other topics, and Tapio Salminen, for friendship and many both scientific
non-scientific discussions. My violin teacher professor Mari Tampere-Bezrodny
is also sincerely acknowledged for continuing to work with me, despite my decision of becoming a scientist.
My family deserves a very special thanks for a great support; my mom Nalle,
Lasse, and my father Upi! Last but not least, my beloved darling fiancée AnnaKaisa; I partially also blame you for ending up as a biochemist, instead of many
other things. I always enjoy our time together and the interesting discussions
we have about practically everything - even the proton-pumping mechanism of
CcO. I highly appreciate the support you give me for anything I decide to do.
Without you this all would be meaningless!
Helsinki, November, 2008.
V.R.I.K.
Abbreviations and symbols
ATP
B3LYP
BNC
CcO
DFT
EPR
eT
FADH2 /FAD+
FMN
FTIR
MD, MM
MV
N-side
NADH/NAD+
NMR
P-side
pmf
PMF
pT
Q/QH2
QM
QM/MM
ROS
su
TM
adenosine triphosphate
Becke-3 Lee-Yang-Parr, a DFT hybrid functional
binuclear site
cytochrome c oxidase
density functional theory
electron paramagnetic resonance (spectroscopy)
electron transfer
flavine adenine dinucleotide (red/ox)
flavine mononucleotide
fourier transformation infra red (spectroscopy)
molecular dynamics, molecular mechanics
mixed valence state
negatively charged side of the membrane
nicotine adenine dinucleotide (red/ox)
nuclear magnetic resonance (spectroscopy)
positively charged side of the membrane
proton motive force
potential of mean force
proton transfer
quinone/quinol
quantum mechanics
hybrid/mixed QM/MM methods
reactive oxygen species
subunit
transmembrane
Intermediates of the binuclear site
A
E/EH
F
O/OH
PM
PR
R
oxygen bound ferrous/cuprous state
one-electron reduced state, non-pumping/pumping
ferryl state
fully oxidized state, non-pumping/pumping
"peroxy" intermediate prepared from mixed valence state
"peroxy" intermediate prepared from fully reduced state
fully reduced state
Amino acid numbering refers to the amino acid sequence of the Bos taurus enzyme, except when otherwise stated.
viii
Chapter 1
Introduction
1.1 Energy transduction in biology
Bioenergetics is the study of energy transduction mechanisms by which a living
organism exchanges both matter and energy with its surroundings [1]. This
drives the self-organization of the open system and prevents it from reaching
thermodynamic equilibrium.
The first signs of life can be traced back to 3.5 billion years ago. Energy
transduction of these organisms was probably based on the oxidation of simple
inorganic compounds [2], such as NO, H2 , H2 S, NH3 , and CH4 , which were
present in the Earth’s early atmosphere [3]. The energy metabolism of certain
"modern" bacteria living in hot sulfuric springs or lakes with high salt concentration, has been thought to resemble the metabolism of these primordial organisms.
The cyanobacteria evolved approximately 2.7 billion years ago with an ability
to utilize the light energy of the sun. In this process, known as photosynthesis,
electrons are taken from water and incorporated with carbon dioxide to produce
carbohydrates,
6 CO2 + 6 H2 O → C6 H12 O6 + 6 O2 .
(1.1)
As a result of the oxidation of water, molecular oxygen was produced as a
"waste product", which led to a steady increase of the oxygen levels in the
atmosphere. Oxygen is, however, an extremely toxic compound especially for
the anaerobic organism. This is due to reactive oxygen species, ROS (e.g. O•−
2 ,
H2 O2 , OH• ) which can form when oxygen reacts with other compounds in
an uncontrolled manner. Approximately 2-2.5 billion years ago, an organism
which could utilize the high oxidizing power of O2 to oxidize complex organic
compounds, evolved. This process, which is in fact the reverse of photosynthesis,
is know as respiration,
C6 H12 O6 + 6 O2 → 6 CO2 + 6 H2 O.
1
(1.2)
CHAPTER 1. INTRODUCTION
2
Evolution of the respiratory machinery increased the efficiency of energy transduction almost 20-fold [4]. This was the probable cause of the exponential appearance of increasingly complex organisms during the Cambrian era, 500 million years ago. Photosynthesis and respiration have formed ever since the main
energy transduction systems on Earth.
1.2 Respiration
The primary energy sources of eukaryotic cells are carbohydrates, fatty acids,
and proteins. In glycolysis, which takes place in the cytoplasmic compartment of
the cell, carbohydrates (C6 compounds) are split into pyruvate (C3 compounds).
This is coupled to the synthesis of two molecules of ATP and NADH. If oxygen
is present, pyruvate is catabolized further in the citric acid cycle taking place
in mitochondria. These organelles constitute two membranes; a highly invaginated inner membrane, which encloses the mitochondrial matrix, and an outer
membrane enclosing the former. The inner membrane is impermeable to solutes,
whereas the outer membrane is permeable to small molecular compounds (< 1.5
kDa) due to their membrane pore complexes, porins. One turnover of the citric
acid cycle produces one molecule of ATP (GTP), three molecules of NADH, one
molecule of FADH2 , and two molecules of carbon dioxide.
The inner membrane comprises the respiratory complexes (Fig. 1.1). Complex I
(NADH-ubiquinone-oxidoreductase), III (cytochrome bc1 ), and IV (cytochrome
c oxidase) generate a proton motive force by pumping protons from the matrix
side (N-side) to the intermembrane space (P-side) [5, 6]. This generates a
potential difference between the two aqueous compartments, in analogy to a
charged battery. Proton-pumping is driven by electron transfer through metal
centers in the respiratory complexes. Electrons are stepwise transferred from
a lower to a higher redox potential, every step releasing the energy needed for
proton-pumping. The terminal electron acceptor is cytochrome c oxidase, which
catalyzes the reduction of molecular oxygen to water. The stepwise electron
transfer is important, because if electrons were accepted by oxygen at once,
the dissipation of energy would be very large, and not much could be used
for producing electrochemical work. In addition, an uncontrolled reduction of
oxygen in one step would produce a high amount of ROS [7].
Complex I or NADH-ubiquinone-oxidoreductase transfers electrons from
NADH, via its bound flavine mononucleotide (FMN), and eight iron-sulfur
centers, to ubiquinone (Q), reducing it to an ubiquinol (QH2 ) [8]. This process
is linked to the translocation of two protons over the membrane per electron
[9]. Complex I is a huge membrane protein; the mammalian complex comprises
46 subunits and has a molecular weight of 1000 kDa. The hydrophilic domain,
which houses the Fe-S centers, has recently been solved to atomic resolution
[10]. However, it is the membrane domain, for which only low resolution
cryo-electron microscopy structures are available [11], that is responsible for
the proton translocation [12, 13]. Therefore detailed understanding of the
CHAPTER 1. INTRODUCTION
3
proton-pumping mechanism in Complex I is so far unknown.
Complex II or succinate dehydrogenase is another protein which contributes
to generation of the quinol pool by oxidizing succinate to fumarate. Succinate
dehydrogenase also comprises redox active groups; a b-heme with a regulatory
function, a flavine-adenine dinucleotide (FAD), and three iron-sulfur centers.
Although this complex does not contribute to the generation of the proton
motive force, it provides electrons from the citric acid cycle and re-oxidizes
FADH2 to FAD [14].
Electrons and protons of the quinols are transferred to complex III, or the
bc1 -complex, by a mechanism called the Q-cycle. This complex comprises b-, and
c-type hemes, and an iron-sulfur Rieske protein. The Rieske protein is used for
passing the electrons from the quinols to cytochrome c, in a process linked to
translocation of one proton across the membrane per transferred electron [15].
The terminal complex in the respiratory chain is complex IV or cytochrome
c oxidase. It oxidizes cytochrome c, and passes the electrons via a bimetallic
copper-center and an a-type heme to molecular oxygen, located at the binuclear
heme a3 /CuB site. Cytochrome oxidase thus drives the complete electron
transport chain by serving as the terminal acceptor of the electrons. Each
electron transfer step couples to the uptake of two protons from the N-side of the
membrane, one of which is pumped across the membrane (to the P-side), and
the other is transferred to molecular oxygen together with the electron [5, 16, 17].
The electron transfer chain provides an electrochemical proton gradient,
∆µ+
H which is the sum of the pH gradient and the voltage across the membrane
caused by the unequal charge distribution,
∆µH + = 2.303
RT
∆pH + ∆Ψ.
zF
(1.3)
The ∆µH + is used for synthesis of ATP catalyzed by F0 F1 -ATPase, and for
active transport [18]. Proton flux from the P-side to the N-side through the
F0 F1 -ATPase complex couples to synthesis of ATP from ADP and inorganic
phosphate [19]. Interestingly, it has been shown that the maximal efficiency
of the electron transfer-chain is reached by partial uncoupling of electron and
proton transfer (eT/pT) [20]. Total uncoupling can be achieved by synthesis of
an uncoupling protein, thermogenin. This protein works as a proton conducting
channel in the membrane, and thus the energy released by eT is used to produce
heat instead of ATP. This mechanism is important for hibernating animals and in
brown adiopose tissue in human infants [21, 22].
CHAPTER 1. INTRODUCTION
4
Figure 1.1. The respiratory chain. The figure shows the mitochondrial inner membrane
and the five respiratory complexes embedded in the membrane. Blue arrows indicate
the pathway of electron transfer, black arrows indicate the direction of proton-pumping,
and green arrows indicate chemical reactions. Protein structures are based on PDB entries 2FUG (soluble part of complex I) [10], 1YQ3 (complex II) [23], 1BGY (complex III)
[24], 1V54 (complex IV) [25], and 1E79/1YCE (complex V) [26, 27]. The membrane domain of complex I is based on a picture from a cryo-EM structure of complex I [28], and
cytochrome c is taken from PBD entry 2V23 [29].
CHAPTER 1. INTRODUCTION
5
1.3 Theoretical biochemistry
Theoretical biochemistry employs the methods of computational chemistry
to gain insight in the molecular principles governing biochemical processes.
One of the main interests of theoretical biochemistry is to understand the high
catalytic power of enzymes. Enzymes can accelerate reaction rates even by
eight orders of magnitude compared to the uncatalyzed reaction in solution and
in contrast to the non-biological catalyst, enzymatic reactions are often highly
selective for their substrates. These effects can derive from a complex chemical
environment where the enzyme binds to the substrate with many weak bonds,
a hydrophobic or non-polar environment, or a metal center which provides
electrostatic stabilization for the reaction. Linus Pauling realized already in the
1940s that the transition state stabilization is of major importance in enzyme
catalysis [30]. However, this insight does not necessary explain the enzymatic
reaction mechanism on the molecular level [31]. Thus, one of the main aims
of theoretical biochemistry is to establish a structure-function relation, because
high resolution X-ray structures, spectroscopic data and biochemical mutational
analysis seldom provide complete understanding of the reaction. The pumping
mechanism of cytochrome c oxidase illustrates this clearly; high resolution
X-ray structures and a wealth of biochemical and spectroscopic data have been
available for more than a decade, but the pumping mechanism is still not fully
understood [5, 16].
The philosophy of theoretical studies is to build an approximate model
system which can be understood and controlled to a high degree. When
predictions of the model agree with experiments this leads to a more detailed
understanding of the system. Although the experimental information obtained
from a measurement is exact (to the extent of the experimental error marginal),
a model must similarly be used to interpret the results. A theoretician and an
experimentalist should therefore work in close collaboration; the outcome of the
calculations should motivate designing new experiments, which can further be
re-iterated by theoretical predictions to build better models of Nature.
Theoretical structural biology originates from the pioneering work of Shneior
Lifson, who was the first to write down the classical force field equation
(reviewed in Chapter 3) used in biomolecular simulations [32]. This led to
development of the consistent force field (CFF) in 1967, which was applied
to organic molecules and energy minimization of the first protein structures
(lysozyme and myoglobin) by his PhD student Arieh Warshel, and by Michael
Levitt, who visited Lifson’s group in the Weizmann Institute at that time. The
original CFF-program was taken by Arieh Warshel to Harvard, where he did his
post-doctoral studies with Martin Karplus. The method was further developed
and combined with the methods of computational chemistry, and it was soon
applied to studies of many enzymatic reaction and biochemical systems, e.g.
hinge-movements and the hexa-N-acetylglucoseamine cleavage reaction of
lysozyme [33, 34], the photoisomerization reaction in retinal [35], the dynamics
of bovine pancreatic trypsin inhibitor [36], and protein folding [37]. It was
not until the early 1980s when the statistical free-energy calculations methods
CHAPTER 1. INTRODUCTION
6
developed by Kirkwood [38], Zwanzig [39], Valleu and Torrie [40] already since
the 1930s, were also applied to study biological systems. The development of
new density functionals in the late 1980s had a major contribution in theoretical
biochemistry from the quantum chemistry side [41, 42, 43, 44]. DFT overcomes
the high scaling of correlated ab initio methods without losing chemical accuracy. Many enzymatic reactions have been described by DFT-methods, either
by studying model systems of the active site [45], or by combining the DFT
method with a classical description of the rest of the system [46]. Quantum
chemical methods are also important in parametrizing the force fields. Before
the development of DFT, the semi-empirical methods were of major importance
in describing bond breaking and bond making processes. Some semi-empirical
methods are still widely used in the study of enzymatic reactions.
Biomolecular modeling differs in some respects from modeling for example reactions in gas or aqueous phase:
• The protein medium is highly anisotropic. This makes it difficult to cut out
only a small part of the system without considering the rest of the protein.
Therefore, biochemical model systems are often very large. In MD-studies
this does not usually cause a problem, whereas building a sufficiently sized
QM-system can be challenging.
• The biochemical system might be in a interface of many phases. For example, a membrane protein is embedded in a lipid membrane, surrounded
by water compartments on both sides. Ions might also be present in the
solution.
• Protein dynamics and conformational changes might play an important
role in the reaction mechanism. These factors are not always easily captured by free energy calculations or by static QM-models as the real protein
dynamics can reach up to the µs-ms time domain.
• The protonation state of ionizable residues can be difficult to determine,
and it can have a large impact on the dynamics of the system. In principle, classical electrostatic calculations combined with Monte Carlo sampling provide ways to determine the pKa -values of residues in a protein.
• Amino acid residues and other chemical groups are often structurally constrained if they form a part of a helix or beta sheet structure. Especially in
membrane proteins, the transmembrane helices are relatively rigid structures. Freezing terminal groups in structure optimization (e.g. alpha or
beta carbons) are therefore usually important when building QM-models.
• The active site, if located inside the protein, is usually in an low dielectric
medium. Proteins might also contain water filled cavities (which are not always resolved in the X-ray structure) with somewhat higher dielectric constants. Implicit solvation models such as PCM [47] or COSMO [48], which
introduce a dielectric constant, can be used in quantum chemical models
CHAPTER 1. INTRODUCTION
7
and electrostatic calculations. It is also possible to predict the hydration
state of empty protein cavities on different levels of accuracy [49].
• Specific interactions; a group of interest, for example a heme group or a substrate molecule, form specific interactions with surrounding amino acids,
usually by hydrogen bonding. These must be taken into account when
building QM-models by considering biochemical mutational data and conserved residues by bioinformatic analysis.
• Even the dipole field of helices might be of importance for enzyme catalysis
[50, 51]. This effect can be included by using model system which are large
enough or consider a QM/MM approach.
• It can be very challenging to estimate the free energy of a process in a
macromolecular system with an enormous amount of degrees of freedom.
The free energy calculation requires accurate sampling of the phase-space,
which often requires addition of biasing potentials to drive the system towards interesting and possibly rare configurations. However, it can be challenging to find optimal biasing potentials.
Chapter 2
Structure and function of
cytochrome c oxidase
2.1 Function of cytochrome c oxidase
Cytochrome c oxidase (CcO) is the terminal enzyme in the respiratory chain. It
reduces oxygen to water, and generates an electrochemical proton gradient across
the mitochondrial or bacterial membrane. The overall reaction catalyzed by CcO
is,
+
ox
4 cyt cred + O2 + 8 H+
N−side → 4 cyt c + 4 HP−side + 2 H2 O.
(2.1)
Electrons are conducted, one at the time, from the soluble cytochrome c (cyt c)
through the metal centers of CcO, to molecular oxygen. Each electron transfer
step is coupled to translocation of one proton from the negatively charged side
(N-side) to the positively charged side (P-side) of the membrane, and to transfer of one proton from the N-side to oxygen. By working as a terminal electron acceptor, CcO drives the electron transport chain and generates a ∆µH +
through proton-pumping [5, 16, 17]. Wikström showed for the first time in 1977
that CcO is a true proton pump which violates a classical Mitchell-Lundegårdh
redox-loop mechanism [52]. Transfer of electrons from one side of the membrane
and protons from the other, would alone generate an electrical gradient across
the membrane. However, proton-pumping immediately poses mechanistic questions about the pumping mechanism, which still, three decades after the initial
discovery of the pumping activity, are partially unsolved. Proton-pumping by
CcO is thus one of the most crucial questions in molecular bioenergetics.
CcO has also an important role in providing a machinery for the safe reduction
of oxygen. Without this, reactive oxygen species (ROS) would accumulate in
mitochondria [7], which would lead to pathological conditions, such as muscle
myopathies and Parkinson’s disease [53]. In addition, there seems to be a correlation between the amount of ROS produced in mitochondria and the rate of aging
[53, 54].
9
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 10
2.2 Structure and physical properties of CcO
Structural information of cytochrome oxidase is based on X-ray structures from
four different organisms: Bos taurus [25], Paracoccus denitrificans [55], Rhodobacter
sphaeroides [56], and Thermus thermophilus (ba3 -type) [57]. In addition, the structure of quinol oxidase from Escherichia coli (bo3 -type) [58] has been resolved. CcO
from Bos taurus consists of three mitochondrially expressed subunits (I-III), and
ten nuclear encoded subunits [59]. The core subunits have a high degree of sequence homology to the bacterial enzyme [60], whereas the ten nuclear encoded
subunits are unique for the eukaryotic enzyme. In addition to the crystallographic data, structural changes occurring in the reaction cycle of CcO have been
highlighted by many spectroscopic techniques.
Figure 2.1. Structure and location of redox active groups CuA , heme a, and heme a3 /CuB
in CcO. The possible location of the proton loading site (PLS), Glu-242, and the two proton conducting channels D- and K- are also indicated in the figure. Subunit I is shown in
blue and subunit II in red. The figure is reproduced from paper [V].
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 11
2.2.1 Subunit I
Subunit I consists of 12 transmembrane (TM) helices organized in three halfcircles, four TM-helices in each. Subunit I comprises three out of four redox active
metal centers of CcO; a low spin heme a, a high spin heme a3 , and a CuB -center.
The metals of the redox active groups are located at a depth of approximately
1/3 of the membrane dielectric from the P-side of the membrane. The distance
between heme irons is approximately 14 Å, whereas the edge-to-edge distance is
7 Å.
Heme a is a six-coordinated low spin heme, which is ligated to His-61 and His-378
(Fig. 2.2). The D-propionate of heme a forms an ion pair with Arg-439, and hydrogen bond to Trp-126, whereas the A-propionate of heme a is hydrogen bonded
to Tyr-54. Arg-38 binds to the vinyl group heme a, which up-shifts its redox potential from 50 mV to 170 mV [61, 62]. This has also important implications for
the electron transfer; as Arg-38 is mutated, the reduction of heme a is severely
delayed [63]. Heme a cycles between a doublet (S = 12 ) oxidized state, with a
ferric (Fe3+ ) iron, and a singlet (S = 0) reduced state with a ferrous (Fe2+ ) iron
[64, 65]. The "real" charge on the iron is however quite far from +2e or +3e, in the
respective states, because both charge and spin are delocalized in the heme ring
[66, 67], [Paper II].
Figure 2.2. The structure of heme a and its immediate surroundings. The iron (sphere
in silver) is coordinated to His-61 and His-378, the vinyl group is hydrogen bonded to
Arg-38, and the propionates interact with Tyr-54, Trp-126, Arg-438, and Arg-439.
Heme a3 is a five-coordinated high spin heme, which ligates to His-376 (Fig. 2.3).
In the reaction cycle it becomes six-coordinated by binding the oxygenous intermediate. Heme a3 interacts with several groups; its D-propionate forms an ion
pair with Arg-438, and a hydrogen bond to Trp-126, the A-propionate is hydrogen bonded to His-368 and Asp-364. The hydroxyethyl-farnesyl group, which
also anchors the heme to the protein, hydrogen bonds to Tyr-244 of CuB . In its
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 12
oxidized form, heme a3 is in a sextet (S = 25 ) state with a ferric iron (Fe3+ ) [65].
The reduced form is a quintet (S = 2) state with a ferrous iron (Fe2+ ) [68]. In the
reaction cycle, heme a3 visits also a triplet (S = 1) ferryl state (FeIV =O) [69, 70].
Heme a3 can bind several different ligands; as reduced to O2 (its natural ligand)
and CO, and as oxidized to HCN, NH3 and H2 S [71]. It has been shown that
binding of cyanide alters the spin state of heme a3 from high spin to low spin
[65]. Heme a and heme a3 interact anti-cooperatively, i.e., the reduction of heme
a is more difficult when heme a3 is reduced, and vice versa [72].
Heme groups show normally an absorption spectrum due to π → π ∗ transitions with absorption bands around 400-450 nm (the Soret or γ- region), and an
α-band around 600 nm [73]. The 0-1 vibronic coupling gives rise to a less intense band around 560 nm (the β-band). The changes in absorption spectrum of
heme a3 which take place upon reduction have characteristic peaks in the Soret
region with maximum absorption at 445 nm and minimum absorption at 414
nm, whereas the α-band can be found between 588 and 609 nm. For heme a the
reduced-minus-oxidized spectrum have maximum absorption in the Soret region
at 445 nm and minimum absorption at 430 nm, whereas the α-band absorbs at
605 nm. The extinction coefficients of both hemes are nearly equal in the Soret
regions, whereas the α region is particulary weak in heme a3 . A charge transfer
band at 660 nm is also characteristic for heme a3 , which has been suggested to
arise from charge transfer from heme a3 to CuB , when both centers are oxidized
[74, 75]. The absorption properties of the hemes are useful when studying them
by spectroscopic techniques.
The CuB site of CcO is a unique center for heme-copper oxidases, and has so far
not been found in any other protein, although the related NO-reductases have
an FeB center, probably with a similar structure to the CuB -center [76, 77]. The
copper atom of CuB is located 4-5 Å from the iron atom of heme a3 (Fig 2.3).
The copper atom binds to His-240, His-290, and His-291. In addition, Tyr-244 is
covalently bound by its side chain to His-240, which has been verified by protein chemical [78] and X-ray studies [79]. The His-Tyr cross-link has obtained a
lot of attention. It connects the aromatic systems of the histidine and tyrosine
rings together, and it has been suggested to decrease the proton affinity of the
Tyr [79, 80, 81]. The CuB center is spectroscopically invisible in most states due
to spin coupling to heme a3 . However, under certain conditions, it shows an EPR
signal with four peaks around g=2.24 [82]. It is unclear if the spin coupling between CuB and heme a3 is ferromagnetic or antiferromagetic. The copper of CuB
can exist in a doublet cupric (Cu2+ ) and singlet cuprous (Cu1+ ) states. In addition, Tyr-244 can be protonated, anionic [83] or in a neutral radical state [69, 80],
which makes singlet, doublet, and triplet states accessible for the CuB -center.
A non-redox active Mg2+ or Mn2+ metal center is located above the propionates
of heme a3 at a distance of 6 Å. The metal binds to His-368, Asp-369 and Glu-198
from subunit II. Although the function of the metal center is not completely clear,
it has been suggested to be active in the ejection of protons to the P-side of the
membrane [85]. Knockout of this site does not seem to lead to alteration of the
enzyme activity or proton-pumping at zero proton motive force (pmf) [86, 87],
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 13
Figure 2.3. The binuclear site heme a3 /CuB site for oxygen reduction and its immediate
surroundings. The iron of heme a3 (sphere in silver) is coordinated to His-376 whereas
the porphyrin ring (shown in orange) is hydrogen bonded to Trp-126, Arg-438, His-368,
Asp-364. The copper atom of CuB (sphere in orange) is ligated to His-291, His-290, and
His-240, which is covalently bound to Tyr-244. Tyr-244 hydrogen bonds to the farnesyl
group of heme a3 . Glu-242 is also shown together with four predicted water molecules
[84] in the non-polar cavity above Glu.
although, the center might become important under high pmf 1 .
Another non-redox active metal center is located in the cross region of subunits
I and II. Mitochondrial oxidases probably bind two sodium ions [88], whereas
bacterial oxidases bind a Ca2+ [55, 56]. This ion binding site has been suggested
to stabilize the structure, but also to down-shift the Em value of heme a by ca. 16
mV [88].
2.2.2 Subunit II
Subunit II (su II) consist of two alpha helices and a hydrophilic anti-parallel betasheet. The hydrophilic domain houses the CuA site, which is the primary electron
acceptor of CcO. This site constitutes two copper atoms, ligating to Cys-196, Cys200, His-161 and His-204 (Fig. 2.4). In addition Met-207 and Glu-198 bind the
copper. Structurally, the center can be classified as variant of a type I copper site,
found in e.g. azurin. The copper center cycles between an open shell doublet state
CuII CuI , and a closed shell singlet CuI CuI . The copper center has a high intensity
absorption band around 500 nm and a low intensity band at 800 nm. The former
band overlaps however with the absorption band of the hemes, which makes
it difficult to use for spectroscopic studies. The CuA site has also been studied
1
Measuring proton-pumping efficiency in presence of a membrane potential is challenging as
already one charge transfer across the membrane generates a high membrane potential. Protonpumping is therefore measured in presence of e.g. valinomycin which transports K+ across the
membrane and abolishes the pmf.
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 14
by EPR spectroscopy, providing valuable information about its redox properties.
Bacterial quinol oxidases lack the CuA site. Nevertheless, mutational insertion of
the su II renders the quinol oxidase to a cytochrome c oxidase [89].
Figure 2.4. The structure of the CuA and its immediate surroundings. The two copper
atoms are shown as orange spheres, coordinated to His-161, His-204, Cys-196, Cys-200,
Met-207, and the backbone of Glu-198.
Mutational experiments [90, 91] and computational docking [92] studies have
indicated that conserved lysines in cytochrome c interact by electrostatic forces
with carboxylate groups in su II. Also ionic strength affects the binging rate of
cytochrome c, which further supports the electrostatic nature of the docking [93,
94, 95, 96].
2.2.3 Subunit III
Subunit III (su III) consist of seven TM-helices arranged into two bundles. The
bovine enzyme binds strongly two phosphatidyle ethanolamine (PE) and one
phosphatidyle glycerol (PG) lipid [97]. Subunit III lacks redox active groups but
it has been suggested to have stabilizing effects and to contribute in the correct
folding of the enzyme [98, 99, 100, 101]. Knockout of su III increases the amount
of inactive CcO during turnover; the turnover rate decreases from over 10000 s−1
to 350 s−1 without su III in pH=8 [102]. The oxygen channel is located in the
interface of su III, and might thus have importance in oxygen diffusion to the
BNC [103].
2.2.4 Other subunits
In addition to the three core subunits, the bacterial CcOs have a forth subunit
[100, 104]. This is a single TM-helix subunit, with no found effects on enzyme
activity. The eukaryotic enzymes have ten additional nuclear encoded subunits,
of which seven are single TM-helices. The function of the additional subunits are
possibly regulatory, and might increase the enzyme stability. Tissue-specific subunit VIa, VIIa, and VIII have been found in the bovine enzyme; a specific liver
(L-form) and heart (H-form) can be segregated [105].
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 15
2.2.5 Post-translational modifications
In general, post-translational modifications modify enzyme activity, the rate of
re-/de-assembly, and the transport of the protein to a specific cellular compartments. Bovine CcO contains many sites for post-translational modifications, some
of which are crucial for the function whereas others are poorly understood. The
most prominent post-translational modification of CcO is the covalent bond between side chains of His-240 and Tyr-244, which might form autocatalytically in
the first turnover of the enzyme. Mutation of Tyr-244 to Phe inhibits the enzyme
completely [106]. In addition, sequence patterns for several phosphorylation, Nglycosylation, amidation and myristolations sites can be found2 . In the structure
of the bovine-enzyme, two phosphorylated threonines, two acetylated serines,
and four formylated methionines are found. It has been shown that phosphorylation of su I, IV, and V inhibits CcO [107, 108], which leads to an increased
production of ROS by complex I and III. This cAMP/protein kinase A-mediated
phosphorylation of CcO is induced by hypoxia, and might be a molecular cause
to the tissue damage during ischemia and oxidative stress [107]. Although not a
post-translational modification, cardiolipin, which might have a possible role in
proton uptake, binds strongly to CcO [97].
2.3 The heme-copper oxidase family
CcO is a part of a larger protein family called the heme-copper oxidases (EC
1.9.3.1). All members of this family reduce oxygen to water, and couple this to
proton-pumping across a membrane [109] 3 . The proton-pumping stochiometry
however differs between some members of heme-copper oxidases. All hemecopper oxidases have at least six conserved histidines in su I (ligands of the
hemes and CuB ), a tyrosine (a part of CuB ) and a tryptophan (forming π-stacking
interaction to one of the histidines of CuB ) [109]. In contrast to the eukaryotic
oxidases which express only aa3 -type oxidase, the bacterial enzymes can switch
between different type of oxidase expression depending on external conditions
[111].
Cytochrome oxidases have different type of hemes; A-, B-, C-, and O-type which
can be used to classify the oxidases in different subclasses; aa3 -, bo3 -, ba3 -, and
cbb3 -type [112]. A classification based on sequence similarity is more rigorous,
although correlation between sequence and heme-type exist [113, 114, 115]. Sequence comparison divides the oxidase family into three large groups; the first
group consist of aa3 -, bo3 -, and caa3 -type oxidases which have two proton conducting channels. Some members of this family lack Glu-242, but a conserved tyrosine, which also can function as a proton donor, probably replaces this residue
[109]. The bo3 -type oxidases or quinol oxidases lack the CuA center which is
present in all aa3 -type oxidases [109]. The caa3 -type oxidases have an additional
bound cytochrome c domain which receives electrons either from an other soluble cytochrome c, or is associated with the bc1 -complex to form a supra-molecular
2
Results obtained from PredictProtein, http://www.predictprotein.org. The amino acid sequence of Bos taurus CcO was used in the search.
3
With the exception of NO-reductases, which do not pump protons over the membrane [110].
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 16
complex.
The second big group of the heme-copper oxidase family is formed by the ba3 type oxidases, in which the proton conducting channels are not as clear as in the
aa3 -type oxidases. These enzymes most probably pump protons with a stochiometry of 2 H+ /4 e− , i.e. only half of the proton-pumping activity of the aa3 -type
oxidases [109, 116]. Recent mutational data have indicated that the ba3 -type oxidases might have only one proton conducting channel in contrast to the aa3 -type
oxidases with two proton conducting channels [117]. Inspection of the structure
shows that in contrast to the aa3 -type oxidases, the K-channel is rather well defined in ba3 -type oxidases with many tyrosines and threonines [57]. Interestingly,
mutation of many residues in the channel analogous to the D-channel do not inhibit the enzyme in the oxidative phase, as in aa3 -type (see section 2.5), whereas
mutation of many amino acids in the "K-channel" renders the enzyme inactive in
the oxidative phase [117]. The two helices of aa3 - and bo3 -type oxidase su II, are
in ba3 -type oxidases divided into two separate subunits [57].
The cbb3 -oxidases, also known as FixY (Y=N, O, P, Q), form the third group of
heme-copper oxidases. These oxidases are expressed in proteobacteria under microaerobic conditions [118]. Due to their ability to reduce both O2 and NO, it
is thought that they where among the first oxidases to evolve [2], although this
view has recently been challenged [115]. The cbb3 -oxidases consist of four subunits. The catalytically active subunit, ccoN is homologous to CcO subunit I, and
comprises the redox active groups; heme b3 and CuB . In addition, subunit ccoO
and ccoP are membrane bound subunits, comprising a mono-heme, and di-heme
groups, respectively. Subunits N and O are homologous to the NO-reductase
subunits NorB and NorC, whilst subunits P and Q are unique for the cbb3 -family.
Interestingly, the tyrosine of the CuB center, which has analogously to the aa3 type oxidases been shown to form a crosslink to the histidine ligand of the copper [119], is located in a different helix than the histidine [113, 120]. In aa3 -type
oxidases these residues are both located in helix 7. It has been suggested that
the tyrosine is in a different orientation compared to the binuclear site in the cbb3
oxidases compared to the aa3 -type oxidases, which might explain the differences
in oxygen biding affinity between the two oxidases [114].
2.4 Pathways and channels
Specific channels and pathways have evolved in CcO to secure a rapid transfer
of substrates (H+ , e− , O2 ) into the protein, and products (H+
pumped , H2 O) out of
the protein.
2.4.1 Foundations of eT theory
According to the Fermi Golden rule [121, 122], the rate of electron transfer can be
written as,
keT =
2π
|VDA |2 (F C),
h̄
(2.2)
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 17
where |VDA |2 describes the electronic coupling (mixing) between the donor and
acceptor states. According to the Marcus theory of electron transfer, the FranckCondon factor F C can be written as [123],
FC = √
h̄
exp(−∆G† /kB T ).
4πλkB T
(2.3)
(∆G0 + λ)2
.
4λ
(2.4)
where,
∆G† =
The rate of eT thus depends on the driving force ∆G0 , temperature T , and the
reorganization energy λ, which is defined as the energy to produce the oxidized
state in the reduced structure and vice versa. The Marcus expression is derived
from considering a two state system (reduced and oxidized) coupled to a classical heat bath. Marcus parabolas of an electron-transfer process are presented
in Fig. 2.5. When nuclear vibrations become much larger than kB T , the Marcus
treatment fails. The F C can then be written as [124, 125],
FC = (
where,
p
v + 1 P/2
) exp[−S(2v + 1)]Iα [2S v(v + 1)],
v
S=
P =−
v=
(2.5)
λ
,
h̄ω
(2.6)
∆G0
,
h̄ω
(2.7)
1
exp( kh̄ω
)−
BT
1
(2.8)
,
where S and P are the reorganization energy and driving force, normalized by
the characteristic nuclear vibrational frequency h̄ω, v is the average vibrational
energy in temperature T , and Iα is a modified Bessel-function4 of order α. The
difference between the quantized and classical Marcus treatment is illustrated in
Fig. 2.5. As the rate of electron transfer becomes relatively temperature insensitive in regions where ∆G0 ≈ −λ in the Marcus picture, the quantized picture
predicts that rate of electron transfer is relatively temperature insensitive on a
broad range of ∆G0 s when the nuclear vibrational energy h̄ω >> kB T . The two
theories are equal when h̄ω ≃ kB T .
4
2
dy
d y
2
2
These functions are solutions to differential equations of type x2 dx
2 + x dx − (x + n )y = 0,
1 z 2 )k
P
(
4
which for real numbers α are Iα (z) = ( 21 z)α ∞
k=0 k!(α+k+1) [126].
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 18
Figure 2.5. Illustration of the reorganization energy λ (above), and the difference between semi-classical and quantized eT-theory (below). The above figure show Marcus
parabolas for oxidized (blue) and reduced state (red) in different driving forces ∆G0 ,
and activation energies ∆G† . The reorganization energy λ of these states are also shown.
Activationless electron transfer takes place when ∆G0 = -λ. The lower figure shows
that quantized-Marcus theory predicts larger temperature independence compared to
the classical-Marcus theory.
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 19
Biological electron transfer is described by two theories. One of the theories argues that electron transfer occurs via specific pathways [127]. Tunneling can take
place either through-space or through chemical bonds. The coupling term VDA is
in the pathway theory written as,
VDA = A
Y
i
ǫC (i)
Y
ǫH (j)
j
Y
ǫS (k)
(2.9)
k
where A is a pre-factor, and ǫ describes the tunneling through covalent bonds
(C), hydrogen bonds (H), and empty space (S). The rates are experimentally
found to be,
ǫC (i) = 0.6,
(2.10)
ǫH (j) = 0.62 exp[−1.7(R − 2.8)],
(2.11)
ǫS (k) =
1
exp[−1.7(R − 1.4)],
2
(2.12)
which suggest that tunneling through covalent bonds is much more probable
than tunneling through space [127]. Electron transfer paths are studied by pathway analysis with graph-theoretical methods to define most probable paths of eT.
The theory by Moser and Dutton suggests that eT takes place via non-specific
pathways and can be approximated by the edge-to-edge distance between the
donor and the acceptor [128, 129]. The rate of (exergonic) eT is according to the
"Moser-Dutton ruler",
log(k)exer = 13 − (1.2 − 0.8ρ)(R − 3.6) − 3.1
(∆G0 + λ)2
,
λ
(2.13)
where ρ is the packing density (0.76 is a popular value), and R is the distance
between the donor and acceptor. In case of an endergonic eT, the rate of eT takes
the following form,
log(k)ender = 13 − (1.2 − 0.8ρ)(R − 3.6) − 3.1
(∆G0 + λ)2 ∆G0
−
.
λ
0.06
(2.14)
Predictions made by the two theories are in most cases equivalent, although there
are some interesting cases where discrepancies have been found. Recently, it has
been argued that the heme-heme eT in CcO is an example of such a case [130].
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 20
2.4.2 Electron transfer pathways in CcO
The reaction cycle of CcO involves four electron transfer steps; from cytochrome
c through CuA and heme a to the binuclear heme a3 /CuB site for oxygen reduction. As discussed earlier, electron transfer from cytochrome c to CuA , involves
docking of cyt c to CcO, probably by electrostatic forces. The experimentally
found rate of eT is ∼ 105 s−1 [131, 132, 133], whereas the rate of eT from a docked
cyt c-CcO structure can be calculated to 7.6x105 s−1 , which agrees relatively well
with the experimental value5 . From CuA , electron transfer can theoretically take
place both to heme a and heme a3 , which are on an almost equidistant edge-toedge distance from CuA , 16.1 Å and 18.9 Å, respectively. The observed eT-rates
are 2x104 s−1 for the CuA → heme a eT, and 4 s−1 for the CuA → heme a3 eT [134],
respectively. The low eT-rate of the latter can be reproduced if a λ=1.3 eV is used
[135]. However, it is also known that reduction of heme a3 cannot take place prior
to a coupled proton transfer event (see section 2.7). Prior to proton uptake from
the D-channel, heme a has a mid-point potential 270 mV, whereas the mid-point
potential of heme a3 is ≤ 150 mV [136]. This would predict rates of 3x103 s−1 and
0.2 s−1 , for eT to heme a and a3 , respectively, using default λ=0.7 and ρ=0.76.
Electron transfer between heme a and heme a3 takes place after a coupled proton
transfer reaction [137] (see section 2.7). Several different values, ranging from µs
to s time scale, have been suggested for the eT rate [138, 139, 140]. However, recent CO-photolysis experiments have indicated that an ultra-fast eT event takes
place [141] on a time scale of 1.2 ns [134]. Moser-Dutton theory predicts a rate of
7.6x108 which agrees closely to the experimental value using the standard values
for the reorganization energy and packing density. An experimental estimate of
the reorganization energy based on fitting eT-rates measured in different temperatures to eqn. 2.5, however suggested that λ<0.2 eV [142]. This suggests that the
packing density should be much lower than 0.76, on the order of 0.2 [143] or that
eT-takes place through specific pathways [130, 144]. The data obtained by fitting
the eT-rates were however performed on a very narrow temperature range and
therefore might give an unphysically low λ. Combined MD- and QM-simulations
support a λ on the order of 0.8 eV [145].
2.4.3 Proton transfer channels and pathways
CcO both consumes protons as a substrate and transfers them across the membrane. The redox active groups, which are buried inside the protein, therefore
need to be in contact with the N- and P-sides of the membrane by proton conducting channels to ensure a high rate of proton conductivity. Proton transfer (pT) is described by a Grotthuss mechanism in which water molecules (or
other electronegative X-H groups) with a solvated proton alter their hydrogen
bonding pattern and thus do not transfer the actual proton, but rather its charge
[146, 147, 148, 149]. In contrast to electron transfer, proton tunneling is not of
very high importance [150], as the probability of tunneling is proportional to
√
1/ mH + . After altering the bonding pattern the water molecules reorientate.
5
The following values were used; R=12.2 Å, ∆G0 = -0.02 eV, λ=0.7 eV, and ρ=0.76. The edge-toedge distance were taken from the docked cyt c structure with CcO by Roberts et al. [92]
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 21
The Grotthuss-proton transfer is illustrate in Fig. 2.6 by a short ab initio molecular dynamics simulation.
Figure 2.6. Grotthuss proton transfer illustrated by a short ab initio MD simulation. The
figure illustrates that it is the charge rather than the proton itself which is conducted.
After the structural diffusion of the proton, the water chain is reoriented.
Two proton conducting channels have been found in CcO subunit I; the D- and
the K-channel, named after conserved Asp-91 and Lys-319, respectively [56, 58,
86, 104, 106, 151, 152, 153] (Fig. 2.7). In the early pumping-models it was suggested that one channel is used only for translocating protons to the P-side of
the membrane, while the other is used for oxygen reduction chemistry [52, 154].
This was later shown to be wrong, as it was demonstrated that the D-channel is
responsible for the transfer of all pumped protons and two out of four chemical
protons, while the K-channel provides the two last ones in the reductive part of
the cycle [155, 156, 157].
The D-channel is rather well defined, comprising many water molecules [25, 55,
56, 58]. The channel starts on the N-side of the membrane with Asp-124, Thr-167,
and Asn-163, and leads via a center region with many polar residues, to the conserved Glu-242 at the end of the D-channel. It has been recently suggested that
the D-channel may hold an additional solvated proton between Tyr-19 and Glu242 to account for the fast proton transfer of the channel [158]. Interestingly, there
are some mutants which uncouple oxygen reduction from proton translocation
e.g. N139D (Rh. sphaeroides numbering, N98D in B. taurus) [159, 160]. The mutations have been suggested to alter the pKa of Glu-242 although they are located
ca. 20 Å apart from this residue. Recent continuum-electrostatics calculations
suggested that the pKa of Glu-242 is upshifted by two pK-units in these mutants
[161]. The hypothesis of a thermodynamic uncoupling mechanism is challenged
in papers III and V. It has been shown by mutational studies that Glu-242 donates
protons both to the pump-site and the BNC. However, there is no apparent connectivity to these sites in the X-ray structures. Potential of mean force calculations
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 22
Figure 2.7. The proton conducting D- and K-channels, and some functionally important
residues within these (see text).
indicate that a non-polar cavity above Glu-242 and beside heme a3 /CuB , comprises 3-4 water molecules [84] (see Fig. 2.3). The side chain of Glu-242, points
down towards the D-channel, away from the propionates of heme a3 , which are
suggested to be good candidates for the pump site [136, 162]. It has therefore
been suggested that the side chain of Glu-242 must rotate to a conformation in
which it can form a proton wire by the water molecules above this residue, to the
BNC and the propionates of heme a3 [163, 164]. Wikström et al. suggested that
the fluctuation of the Arg-438 - heme a3 D-propionate ion pair forms the path for
proton exit to the P-side of the membrane [165]. EPR spectroscopy has indicated
that the Mg2+ site have a rapid exchange rate of water with the bulk (see section
2.4.5) [166]. It may therefore be possible that protons follow the same exit path as
water due to the fast proton conduction in water. Computational studies suggest
that protons might pass Lys-171 and Asp-173 of su II, which are in close vicinity
of the Mg2+ site [85]. The uni-directionality of proton-pumping is one of the major unsolved questions in the pumping mechanism of CcO. This is discussed in
papers III and V.
The K-channel is not as well defined as the D-channel but certain conserved and
functionally important residues have been found. The opening of the K-channel
is probably near Ser-255 or Glu-62 (su II). Glu-62 might connect to Lys-319 by a
water molecule, from which proton transfer takes place through water molecules
via Thr-316, to Tyr-244 of the CuB center [167]. Electrostatic calculations show
that the pKa of Lys-319 and Glu-62 are strongly coupled [168]. In addition, re-
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 23
cent FTIR experiments show that Tyr-244 titrates with a pKa of 6.6 in the O state,
whereas a Hill factor 0.5 indicates a strong electrostatic coupling which might be
due to the Lys-319/Glu-62 pair [83].
It has also been suggested that the mitochondrial oxidases have an additional
H-channel [79, 153]. This hypothesis is based on structural differences of Asp-51
between reduced and oxidized bovine structures. Mutation of Asp-51, which is
located above the propionates of heme a, lead to loss of proton-pumping activity
while oxygen reduction activity is not altered [25]. No homologous amino acid
has been found in the bacterial enzymes which suggests that Asp-51 might have
a different function in the mitochondrial oxidases. In addition, the mutational experiments were done in human HeLa-cells with 20% remaining wt activity. Another possible explanation might be the structure of the sodium/calcium binding
site which is relatively close to Asp-51. Binding of a second sodium ion might influence the conformation of Asp-51, which is the basis of the H-channel hypothesis. Recent studies indicate that two sodium ions bind to this site, although only
one sodium can be observed in the X-ray structure [88]. The H-pathway would
involve a pT through a peptide bond, which consist of a keto-enol tautomerization reaction. This reaction was theoretically shown to have an energy barrier of
13.5 kcal/mol [169].
2.4.4 Oxygen channels
Non-polar molecules such as oxygen can diffuse through lipid membranes. However, for oxygen to diffuse into the buried active center of CcO, a special oxygen
diffusion channel has evolved. It has been shown by combined structural studies
and mutational analysis that certain amino acids which are important in forming
the channel [62, 103]. In addition, the location of xenon atoms in X-ray structures
has helped in identifying the channel [56]. MD simulations using the Locally Enhanced Sampling (LES) method6 indicate that oxygen diffusion to the BNC takes
place via specific trajectories [164].
2.4.5 Water exit channel
As the maximum turnover of CcO is ca. 1000 e− /s, 500 water molecules are
produced every second. MD-simulations have indicated that if the non-polar
cavity around the hemes are filled with more than four water molecules, excess
water can leave the cavity by opening of the Arg-438 - D-propionate ion pair of
heme a3 [165]. This leads the water molecules to the Mg2+ /Mn2+ center. EPR
studies have indicated that this site exchange water with the surrounding bulk
on a ms-time regime [166].
2.5 The catalytic cycle
The catalytic cycle of CcO can be divided into an oxidative and a reductive part
(Fig. 2.8). In the former, the oxidizing power of oxygen is transferred in one
6
Several replicas of oxygen molecules which cannot interact with each other, but normally with
surrounding groups, are used in this simulation method to enhance the sampling.
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 24
step to the enzyme, whereas in the latter, the fully reduced state of the BNC is
stepwise re-generated. The oxidative phase starts by the binding of molecular
oxygen to heme a3 , when the enzyme is in a fully reduced (ferrous/cuprous) Rstate. Although the rates of binding and dissociation of oxygen are almost equal,
oxygen is kinetically trapped to heme a3 by a rapid electron transfer [170]. The
bound intermediate is known as species A, from the early studies of Chance [171].
Recent multiconfigurational calculations have indicated that this state is a linear
combination of many configurations but the Fe3+ -O•−
2 might be of dominant importance [172]. In the next step of the reaction cycle, the O-O bond is cleaved
in one step; two electrons are taken from heme a3 , one electron from the copper
of CuB , and one electron and proton is probably derived from Tyr-244 of CuB ,
which makes it a neutral tyrosyl radical [4, 69, 80]. Splitting of the O-O bond in
one step prevents formation of hazardous oxygen intermediates, and transfers all
of the oxidizing power of oxygen to the enzyme. The formed species is called PM
due to historical reasons from the time it was though that this state was a "peroxy" intermediate7 . The structure of the PM state is ferryl iron, FeIV =O, a cupric
hydroxo-copper CuII -OH, and a neutral tyrosine radical Tyr-O• . Reduction of
PM yields the PR , in which the neutral radical is reduced to a tyrosinate, Tyr-O− .
Although the CuB center is in general invisible for spectroscopic detection, PR
has a specific EPR spectra which is characterized by four peaks at g=2.24 [82].
Recent EPR studies have challenged that Tyr-244 would be the primary donor
of the fourth electron. These studies suggest that a conserved tryptophan, Trp236, which forms a π-stacking interaction with His-291 of CuB could donate this
electron [173]. However, the study suggested that the Trp-radical relaxes into a
neutral Tyr radical. Protonation of PR from the D-channel forms the F state, in
which the heme a3 iron still remains as a ferryl (FeIV =O) but a water molecule is
most probably ligated to the copper of CuB , as the optical spectra of the ferryl
is different in P and F states. de Vries has recently suggested that there might
also exist an FW • state with Trp-236 in a radical form [174]. The role of Trp-236 in
the reaction cycle of CcO is still poorly understood. Reduction and protonation
of heme a3 yields a ferric FeIII -OH state with CuB as CuII -H2 O YO− or CuII OH YOH. This state is known as OH ("oxidized"). Reduction and protonation of
OH yields EH in which CuB is either as CuI - OH YOH or CuI - H2 O YO− and
a water molecule is bound to heme a3 . Reduction and protonation yields the
ferrous/cuprous R state with no bound ligands. Recent FTIR experiments have
indicated that Tyr-244 would be anionic in states PM , F, and depending on pH
also in OH [83].
2.6 Proton translocation
In 1977, Wikström showed for the first time that CcO in rat-liver mitochondria
pumps protons in response to the addition of a reductant in a reaction which can
be inhibited by addition of carbon monoxide [52]. Pumping was also confirmed
to take place when CcO is reconstituted in phospholipid vesicles. The first indications of the stochiometry of proton-pumping in different stages of the catalytic
7
The subindex "M" refers to mixed valence, a state from which it has been prepared; reduced
BNC, and oxidized CuA and heme a.
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 25
Figure 2.8. Reaction cycle of CcO and the structure of possible intermediates. The oxidative phase; P → O, and reductive phase; O → R. States XR indicate the structure of
a reduced state X, which has not yet been protonated. Green protons are taken from
the K-channel, red protons from the D-channel, and blue protons are pumped across the
membrane. Reproduced from paper [IV].
cycle came from reverting the reaction cycle of CcO, with a high pH concentration and ATP. These studies indicated that two protons are pumped in the P →
F and F → O transitions, respectively [175]. Reanalysis led to a suggestion that
the P → F transition couples to pumping of two protons, whereas the F → O,
and the reductive phase, possibly E → R couple each to transfer of one proton
across the membrane [176]. New experiments, by optical flow-flash combined
with electrochemical methods, suggested that both P → F, and F → O are linked
to pumping of one proton each [177]. However, it was known that the reduction of a prepared fully-oxidized O state is not coupled to proton translocation
[177], but that reduction and re-oxidation would prepare the enzyme in a state in
which two protons are translocated in the reductive phase. This suggested that
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 26
there might be two different reaction paths in the reductive phase; high-energy
OH and EH states which couple to proton-pumping and "low energy" O- and E
states which are not linked to proton-pumping [177, 178, 179]. So far, no spectroscopic differences have been found between these states [180]. It has been known
for long that there are two different forms of the oxidized enzyme; a "slow" (also
known as "resting") and a "fast" (or "pulsed") form [181, 182], where the names
refer to cyanide bindning kinetics. In contrast to the states O and OH , the "slow"
and "fast" forms have different spectral features, the former has a Soret band with
λmax =417 nm and the latter with λmax =424 nm [183]. As both O and OH have
spectral features corresponding to the "fast" form, this indicates that there might
be two distinct "pulsed" forms of the enzyme. Two hypotheses of the possible
difference between state OH and O are suggested in papers IV, and V.
2.7 Mechanism of proton translocation
Each of the four electron transfer steps, after binding and splitting of the O-O
bond, are coupled to transfer of one proton across the membrane, and another
proton for the oxygen reduction chemistry. Ever since it was discovered that CcO
is a proton pump, several mechanisms for proton translocation have been suggested. Today it is known that the pumped protons derive from the D-channel
and are transferred via Glu-242 to a pump site located above the heme groups.
Good candidates are the propionates of heme a3 . The chemical protons derive
in the oxidative phase also from Glu-242, whereas in the reductive phase, they
are abstracted from the K-channel [155, 156, 157]. From combined time-resolved
spectroscopic and electrochemical studies of the O → E transition [184], it is suggested that the sequence of proton transfer is the following:
• Electron transfer from cyt c to the CuA center. The electron is distributed in
a 30/70 proportion on CuA and heme a, respectively.
• Reduction of heme a is coupled to proton transfer from Glu-242 to the pump
site located "above" the hemes, perpendicular to the membrane plane, within
150 µs.
• Electron transfer from heme a to heme a3 in 1.2 ns.
• Proton transfer to the BNC and internal electron transfer from heme a3 to
CuB within 800 µs.
• Ejections of the proton at the pump site to the P-side of the membrane within 2.6 ms.
In the "Histidine cycle"-model from mid 1990s, His-291 was suggested to be the
pump site of CcO [185]. Depending on the redox state of the enzyme, His-291 was
suggested either to be protonated and bound to CuB , or deprotonated and dissociated from CuB . Proton-pumping would thus involve a conformational change
of the CuB site. This model was abandoned due to the lack of experimental evidence. Colbran and Paddon-Row found by quantum chemical DFT-studies that
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 27
there might be a redox state dependent thermodynamic favoring of dissociating
one His-ligand of the CuB site [186].
In the Coulomb pump model suggested by Popovic et al., a redox state dependent pKa variation of His-291 was found [187, 188, 189]. This was however not
supported in the studies by Fadda et al. [190], nor Song et al. [191]. The drawback
with these computational models (including the one by Popovic et al.), is that
Tyr-244 is left out from the QM-model. This might have a strong effect on the
imidazol pKa since a larger charge delocalization would be expected to diminish
large proton affinity variations of His-291.
In most pumping models the pumped proton is transferred first, and the chemical proton second. However, Larsson and Brzezinski proposed a model in which
the chemical proton is transferred first [192]. According to their model, the anionic Glu would cause a conformational change around the region of the pumpsite (D-propionate of heme a3 ), which drastically increases its pKa and leads to
protonation of the pump site. This model is based on structural studies of the
Glu278Gln mutant (Glu-242 in Bos taurus), where structural changes are found in
the regions around the D-propionate of heme a3 [56]. This pumping-model is also
based on experimental result of the P → F transition in which it was suggested
that the internal eT is not linked to proton transfer over the membrane [193]. The
pH of the N- and P-side was monitored in the measurements by a pH-indicator.
However, it was later shown that the electron transfer links to an internal proton transfer event inside the enzyme [194]. The pumping model by Larsson and
Brzezinski is based on purely thermodynamic arguments.
The water-gated mechanism by Wikström et al., suggested based on MD simulations that proton transfer to the pump site or the binuclear site is controlled
by the orientation of water molecules in the non-polar cavity around the active
site of CcO [195]. More specifically, it was suggested that if heme a is reduced
and the BNC oxidized, the water molecules have an orientation such that proton transfer can take place to the pump site but not to the BNC. Furthermore,
when heme a is oxidized and the BNC reduced, the water chain orientates to
conduct protons to the binuclear site. Wikström et al. further suggested that it
is the electric field between the hemes that interact with the dipole moment of
the water molecules, which is responsible for the orientating effect. This theory
has received much criticism [196], partially due to a similar suggestion of the
water orientation preventing proton conductance by the aquaporin water channel [197, 198]. However, the electrostatic field in the non-polar cavity next to the
hemes and CuB might be much stronger than the one caused by the helix dipoles
in aquaporins. If the water-gated model is a feasible mechanism for transferring
the protons to the right site at the right time (or maybe rather for preventing the
protons from being transferred to the wrong site at the wrong time), there should
be a high kinetic barrier of orientating the water molecules in the "wrong" direction. In addition, the connection between a pre-formed water chain and the
barrier of transferring the proton itself should be investigated. These issues are
studied in paper VI.
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 28
Siegbahn and Blomberg explained the pumping mechanism by thermodynamic
arguments based on quantum chemical DFT-studies of the binuclear site [199].
In this model the protons are "guided" rather than "gated" to the correct site
in the correct reaction step by a redox state dependent pKa -variation of the Apropionate of heme a3 and the metal ligands. Their calculations suggest that as
the BNC is reduced, the A-propionate of heme a3 works as an electron sink, making it more favorable to transfer a proton to this site rather than to the BNC. The
order of eT and pT in their model is contradictory with the results obtained later
by time-resolved experiments by Belevich et al. However, as the BNC is neutralized after the reduction, the pKa of the protonated pump site is expected to
drop, which explains why the proton is ejected to the P-side of the membrane.
Generally, quantum chemical studies often favor thermodynamic explanations
as kinetic barriers are difficult to estimate, due to difficulty in optimizing true
transition states. However, pKa - variation on the A-propionate might be important, as this group is a good candidate for the pump site. Based on thermodynamic modeling, Siegbahn and Blomberg suggested in a very recent study that
unidirectionality of proton-pumping might be obtained by forming a positively
charged transition state for the pumped proton, and a neutral transition state for
the chemical proton [200]. The former was suggested to prevent protons from
leaking back from the pump site, whereas the latter could allow proton transfer
with a low barrier to the binuclear site.
The Warshel-group studied proton transfer from the D-channel to the D-propionate of heme a3 and to the BNC by combined Kinetic Monte Carlo and EVB simulations [201]. The simulation time was extended impressively to the microsecond
time scale by considering only the "dynamics" of the proton wire. Based on this
study, it was suggested that Glu-242 cannot deprotonate due to a high pKa value
obtained. Furthermore, an electrostatic gate was suggested which could control
the transfer of the proton either to the pump site or to the BNC. The high pKa
of Glu might have been an artifact of the additional proton, which was placed in
the cavity. It was later shown by FTIR experiments in a slow D-channel mutant
that Glu actually can deprotonate [202]. In a more recent study, Pisliakov et al.
studied the deprotonation of Glu-242, in which it was found that the conformational rotation of Glu might facilitate proton transfer to the D-propionate of heme
a3 [196]. It was also suggested that proton transfer might occur directly from Glu
to the D-propionate without any or just via one water molecule in between. The
kinetic barriers were obtained by protonating this water molecule, by forming a
E− W1 H+ P− transition state (where E is Glu-242, W1 H+ is a protonated water
molecule, and P is the D-propionate of heme a3 ). However, proton transfer might
follow a concerted reaction in which the charge is delocalized to many water molecules. This issue is studied further in paper VI.
The Voth-group studied proton transfer of an excess proton from Glu-242 to the
D-propionate of heme a3 with MS-EVB2 simulations [203]. Similarly to the study
by Pisliakov et al. it was found that the D-propionate of heme a3 might bend toward the solvated H3 O+ . However, as the proton is forced to the non-polar cavity
this might introduce some artificial structural variations. As Glu-242 and the Dpropionate of heme a3 were not included in the MS-EVB-Hamiltonian the charge
CHAPTER 2. STRUCTURE AND FUNCTION OF CYTOCHROME C OXIDASE 29
of the proton in not allowed to relax and the deprotonation reaction of Glu-242
could not either be studied. In agreement with early MD-studies by Wikström et
al. a redox-state dependent dynamical behavior was found to prefer transfer of a
proton from Glu-242 to the D-propionate of heme a3 when heme a is reduced.
Chapter 3
Foundations of theoretical
biochemistry
This chapter reviews the theory behind the computational methods used in biomolecular simulations. The first sections cover basic quantum chemical methods, after which the classical methods are discussed. In the last section combined
quantum and classical methods are reviewed.
3.1 The Schrödinger equation
The fundamental equation of quantum mechanics is the Schrödinger equation
(S.E.) [204]. This eigenvalue equation establishes a connection between the energy and the wave-function of the system, which can be written as,
(3.1)
ĤΨ = EΨ.
The Hamilton operator Ĥ for a molecular system, contains kinetic and potential
energy terms for the nuclei (α, β) and electrons (i, j),
Ĥ = −
X h̄2
X e2 Zα X e2
X Zα Zβ
X h̄2
∇2i −
∇2α −
−
+
2me
2mα
ri,α
ri,j
rα,β
α
i
i,α
i>j
(3.2)
α>β
So far no approximations have been made. However, for solving the S.E. for a
system with more than two particles, approximations are needed.
3.2 The Born-Oppenheimer approximation
In 1927 Born and Oppenheimer [205], and later generalized by Born and Huang
[206], suggested an ansatz in which the S.E. is split into an electronic and nuclear part. Due to the large ratio between the mass of the nuclei compared to the
electrons, electrons can be treated as moving in a fixed nuclear framework. The
interactions between the electrons and the nuclei are treated as a static external
31
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
32
potential (VN ). In the Born-Oppenheimer approximation (BOA) the electronic S.E.
of Eqn. 3.1 can be written as,
(Ĥel + VN )Ψel (ri ; Ri ) = Eel Ψel (ri ; Ri ),
(3.3)
where the coordinates of the electrons, ri depend only parametrically on the nuclear coordinates Ri . The BOA is an excellent approximation to the molecular
electronic S.E. In addition, it provides practically the only way to define a molecular potential energy surface and molecular structure. There are, however, special cases where the BOA does not hold; for example, in case of proton tunneling
and excited state dynamics [207], which play an important role in many biochemical reactions. In this case, nuclear dynamics must be taken into account by e.g. a
discretized Feynman path integral approach [208].
3.3 The Hartree-Fock method
The electronic Hamiltonian can be described by a sum of one-electron Hamiltonians,
Ĥ =
N
X
ĥi ,
(3.4)
i
if the electron-electron repulsion is neglected or included in some average way.
The one-electron Hamiltonians hi would then satisfy the corresponding oneelectron S.E.,
ĥi ψi = ǫi ψi .
(3.5)
Inserting these conditions in the S.E. gives for the wave-functions,
Φ(x1 , x2 , ..., xN ) = Φ(x1 )Φ(x2 )...Φ(xN ).
(3.6)
This many-electron wave-function is referred to as the Hartree product [209]. In
the Hartree method [209], the energy is obtained by variationally minimizing Φ,
R ∗
Φ ĤΦdx
E= R ∗
.
(3.7)
Φ Φdx
The drawback of the Hartree product is that it violates a fundamental property of
fermionic particles; as shown by Pauli, the overall wave-function of fermions is
antisymmetric with respect to interchange of two electrons, a principle known as
the Pauli exclusion principle [210]. J.C. Slater found in 1929 that the antisymmetric nature of the wave-function is satisfied by a determinant relation (the Slater
determinant) [211],
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
χ1 (x1 ) χ2 (x1 ) ... χN (x1 )
χ1 (x2 ) χ2 (x2 ) ... χN (x2 )
.
.
.
.
1
Φ(x1 , x2 , ..., xN ) = √N ! .
.
.
.
.
.
.
.
χ (x ) χ (x ) ... χ (x )
1 N
2 N
N N
.
33
(3.8)
The electrons are described by the spin-orbitals χi , which are products of space
and spin parts of the electrons, and represented by the columns of the determinant. The coordinates of a specific electron appear on the rows. Thus, interchange
of two electrons corresponds to the interchange of two rows of the determinant,
which changes the sign of the wave-function.
The Hartree-Fock method is obtained by using a Slater determinant in the Hartree
method [212, 213]. The equations are solved iteratively, in a self-consistent way,
until the energy converges to a pre-set threshold (the self-consistent field, SCF
procedure). The initial guess orbitals can be calculated e.g. semi-empirically. The
Hartree-Fock equation can be written as,
F (1)Φi (1) = ǫi Φi (1),
(3.9)
in which the effective one-electron Fock operator is,
F̂ (1) = ĥ(1) +
N/2
X
i
[2Jˆi (1) − K̂i (1)],
(3.10)
where,
X ZM
1
ĥ(1) = − ∇21 −
2
r1M
M
R
dx2 χ∗b (2)χb (2)
ˆ
J(1)χ
χa (1)
a (1) =
r12
R
dx2 χ∗b (2)χa (2)
χb (1).
K̂(1)χa (1) =
r12
(3.11)
(3.12)
(3.13)
Jˆ and K̂ are the Coulomb and exchange operators, respectively. The former is a
local operator which describes the Coulombic interaction between electrons; the
latter is a non-local operator considering the exchange of two particles, the action
of which has no classical analogue.
Hartree-Fock theory provides 99% of the total energy. However, the remaining
1% is important for obtaining chemical accuracy (1 kcal/mol). It is caused by
the fact that the HF-method takes into account only the average interaction of
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
34
the electrons with each other (the mean-field approximation). More specifically,
the HF-method fails to recover the Coulomb cusp of the wave-function, i.e. the
reduced probability of finding one electron in the neighborhood of another electron. The energy difference between the HF-method and the exact energy is, due
to Per-Olof Löwdin, referred to as the correlation energy [214]. There are two
types of electron correlation: dynamic, which is due to the correlated movement
of electrons, and non-dynamic (sometimes also called static), which is due to the
failure of the single determinant description of the wave-function.
The anti-symmetric nature of the wave-function can instead of using a Slater
determinant, also be expressed by the formalism of second quantization. This
formalism is based on the properties of two operators; the creation operator a†p ,
and the annihilation operator ap . When a†p operates on empty space, it creates an
electron in the spin orbital χp . Analogously, ap removes an electron from an occupied spin orbital χp . Furthermore, the creation and annihilation operators are
connected with each other through specific anti-commutation relations1 . A oneelectron excitation from spin orbital p to q is thus obtained by operating on a
configuration, |...1p ...0q ...i, where 1 refers to an occupied orbital and 0 to an unoccupied orbital, first by ap and then by a†q , i.e. a†q ap |...1p ...0q ...i = |...0p ...1q ...i.
Including these kinds of electron excitations in the derivation provides the most
intuitive way of recovering the correlation energy.
3.4 Beyond HF-theory
The simplest way to include electron excitations in the calculation is to introduce
them perturbatively, if it is assumed that the HF wave-function is relatively close
to the exact wave-function. The second-order energy correction can according to
Møller-Plesset perturbation theory be written as [215],
(2)
EM P 2 =
|hab||iji|2
1 X
,
4
ǫi + ǫj − ǫa − ǫb
(3.14)
i,j,a,b
where a, b refer to unoccupied orbitals and i, j to occupied orbitals. hab||iji equals
hab|iji − hab|jii, i.e. the effect of Coulomb and exchange operators. In the MP2theory only two-electron excitations contribute and the computational scaling
can further be reduced by considering only excitations in the valence orbitals - the
core excitations seldom change in chemical reactions. Although it is also possible
to construct higher order MPn series, this introduces a higher computational cost,
and in fact, a perturbation series which might diverge [216]. The MP2 method has
been widely used in parametrizing biomolecular force fields [217, 218]. In addition, some properties such as dispersion forces, which are important in van der
Waals interaction, are described reasonably well with MP2-theory but not with
the widely popular DFT-methods (see section 3.5).
A more accurate way to recover the electron correlation is to include singly, doubly, triply, ..., etc. excited determinants in the calculation. However, the size of
1
[a†p , aq ]+ = a†p aq + aq a†p = δpq , [a†p , a†q ]+ = 0, [ap , aq ]+ = 0 , (δpq is a Kronecker delta).
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
35
the determinant increases extremely fast together with the computational scaling. There are two methods; Configuration Interaction (CI) [219], which is based
on a linear combination of the excited determinants, and Coupled Cluster (CC)
[220], which is based on an exponential ansatz. The latter is more widely used
as it is size-consistent2 [221, 222]. The CCSD(T) level, where triple excitations
are treated perturbatively [223], gives with a quite large certainty chemical accuracy if the system does not have a significant multi-configurational character.
There are special diagnostic tests for this purpose [224]. There are also approximate methods for determining the cluster equations, known as the CCn-methods
(n=2,3,...) [225]. The application of the CC-methods to study biomolecular systems is limited, as the computational scaling is N 6−8 . However, the methods
have been successfully applied on model systems. For example, the rotational
barriers for the ground and first excited state of retinal have been studied at the
CC2-level [226] as well as spin and charge distribution in simple porphyrin models [227, 228].
The methods described above fail to describe a system with a significant multiconfigurational character, i.e. when the molecular system is a combination of
several comparably important electronic configurations. In this case, the Multiconfigurational Self-Consistent Field (MCSCF) approach must be employed. In
addition to the CI coefficients, the orbital coefficients are optimized simultaneously. The most important MCSCF method is CASSCF (Complete Active Space
SCF), in which the space is divided into inactive orbitals, which are fully occupied, virtual orbitals, which are fully unoccupied, and active orbitals, in which
the occupation can vary and a full CI-calculation is performed [229]. Dynamical electron correlation can be recovered in CASSCF by introducing it perturbatively. This is done in the CAS second order perturbation theory (CASPT2)
method [230], which is analogous to the MP2 perturbation for the HF-wave function, but in which a CASSCF wave function is used instead of the HF-wave function. The CASPT2 approach has been applied to study the excited states in many
biochemical systems [231, 232, 233]. In addition, the electronic structure of porphyrin model systems [228, 234, 235, 236] and the oxoheme [172] has recently
been treated with the CASPT2 methodology. The difficulty in the CASPT2 calculation is to choose the active space, which is currently limited to 15 electrons in 15
orbitals due to the size of the CI. As there are already 10 d-electrons in transition
metals, treating bioinorganic complexes with this methods is very challenging.
3.5 Density Functional Theory
Instead of the wave-function, which depends on 3n spatial coordinates in a nelectron system, density functional theory describes the system by its electron
density, which depends only on 3 spatial coordinates. The one-particle electron
density is defined as,
Z
Z
ρ(r1 ) = N ... |Ψ(x1 , x2 , ...xN )|2 ds1 dx2 ...dxN ,
(3.15)
2
the sum of the energy of two non-interacting fragments at very long distance should equal the
sum of the energy for the separate fragments.
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
36
where the space-spin coordinate dxi is composed of a spin (dsi ) and space (dri )
part.
3.5.1 Early models
Already before the development of Hartree-Fock theory and any rigorous proof
that the energy actually can be written in terms of the density, the Thomas-Fermi
(TF) model was published in 1927 [122, 237], where the density has a functional
dependence of the energy,
E[ρ] = T [ρ] + Vne [ρ] + J[ρ].
(3.16)
Both the electron-nuclear Vne [ρ] and electron-electron repulsion J[ρ] energies can
be written as functionals of ρ,
X Z Zα ρ(r)
Vne [ρ] = −
dr
(3.17)
|Rα − r|
α
Z Z
1
ρ(r)ρ(r ′ )
J[ρ] =
drdr ′ .
(3.18)
2
|r − r ′ |
Index α refers to atomic nucleus α, Rα to its position, and Zα to its charge. The
factor 1/2 corrects for double counting. The kinetic energy expression in the TFmodel is given by,
Z
3
2 2/3
T [ρ] = (3π )
ρ5/3 dr.
(3.19)
10
The Thomas-Fermi theory was shown to be useful in describing e.g. trends of
atomic total energies. However, the model fails to account for many properties
important in chemistry, e.g. to predict a covalent chemical bond [238].
P.A.M. Dirac tried to correct for these effects in 1930 by introducing an exchange
functional [121],
Z
3 3
(3.20)
Ex [ρ] = − ( )1/3 ρ4/3 dr.
4 π
This correction did not lead to any significant improvement compared to the TFtheory. However, the theories by Thomas-Fermi, and Dirac led to the work by
Walter Kohn, where he showed that the electron density ρ is a complete and exact
description of the ground state electronic structure. This was the HohenbergKohn formulation of density functional theory.
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
37
3.5.2 The Hohenberg-Kohn formulation of DFT
The basic lemma of the Hohenberg-Kohn theorem states that the non-degenerate
ground state in some external potential v(r) is uniquely3 determined by the density ρ [239]. The proof of this mathematically rigorous theorem is very simple.
Two potentials v1 (r) and v2 (r) are considered. The potentials are defined to differ
by more than an additive constant, and it is asked if they can arise from the same
density ρ. Thus, there would be two Hamiltonians Ĥ1 and Ĥ2 describing the
same ground state density ρ, which would arise from different wave-functions
Φ1 and Φ2 . By employing the Rayleigh-Ritz minimal principle,
E10 < hΦ2 |Ĥ1 |Φ2 i = hΦ2 |Ĥ2 |Φ2 i + hΦ2 |Ĥ1 − Ĥ2 |Φ2 i
Z
0
= E2 + [v1 (r) − v2 (r)]ρ(r)dr.
(3.21)
Similarly,
E20
<
E10
+
Z
[v2 (r) − v1 (r)]ρ(r)dr.
(3.22)
Adding 3.21 and 3.22 together gives,
E10 + E20 < E20 + E10 .
(3.23)
Reductio ad absurdum, there cannot exist another potential v2 (r) different from
v1 (r) described by the same density ρ. Therefore, the external potential v(r) is
uniquely determined by ρ(r), quod erat demonstrandum. This proves that all information there is to know about the ground state system is determined by the
density.
The second Hohenberg-Kohn theorem introduces a variational principle for determining the ground state energy. The theorem was first derived for non-degenerate ground states, but Levy extended it to degenerate ground states as well [240].
From the proof presented above, the energy can be written as a functional of
the density,
E[ρ] = Vne [ρ] + T [ρ] + Vee [ρ] =
Z
v(r)ρ(r)dr + F [ρ],
(3.24)
where Vee [ρ] is the electron-electron interaction, Vne [ρ] and T [ρ] have the same
meaning as in Eqn. 3.16, and F [ρ] is a new functional. The Euler-Lagrange
equaR
4
tion for minimizing E[ρ] against ρ, subjected to the constraint ρ(r)dr = N
3
This means up to an uninteresting trivial additive constant.
A method for minimizing a functional F [x] vs. x(y) subjected to a constraint G[x] = a gives,
δ
[δF [x] − µ(G[x] − a)] = 0.
δx(y)
4
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
38
gives for the Lagrange undetermined multiplier,
µ = v(r) +
δF [ρ]
δρ(r)
(3.25)
where,
F [ρ] = T [ρ] + Vee [ρ].
(3.26)
However, without knowledge of the exact form of T [ρ] and Vee [ρ], the theorem is
useless.
3.5.3 The Kohn-Sham theorem
The central idea behind the Kohn-Sham theorem from 1965 is to solve the ground
state density of a fictious system with non-interacting electrons but with the exact same ground state density as the real system [241]. The advantage of this
procedure is that the ground state density of a non-interacting system is known.
However, there are some physical differences between the two systems; the interacting electrons of the real system are subjected to an exchange-correlation energy which accounts for the difference between the classical and quantum mechanical repulsion energy between the electrons, as well as the difference in the
kinetic energy of the two systems. Kohn and Sham defined,
F [ρ] = Ts [ρ] + J[ρ] + Exc [ρ],
(3.27)
in which Ts [ρ] is the kinetic energy of the non-interacting system, and J[ρ] is
the classical electron-electron repulsion energy defined as in Eqn. 3.18. If one
assumes that for the real system, F [ρ] is given by Eqn. 3.26, the exchange correlation energy Exc [ρ] is given by,
Exc [ρ] = (T [ρ] − Ts [ρ]) + Vee [ρ] − J[ρ].
(3.28)
According to the second Hohenberg-Kohn theorem [239], the Kohn-Sham energy
expression can be minimized according to the variational principle which gives
the Euler-Lagrange equations. The Lagrangian undetermined multiplier is then,
µ = vef f (r) +
δTs [ρ]
,
δρ(r)
(3.29)
where,
vef f (r) = v(r) +
δJ[ρ] δExc [ρ]
+
.
δρ(r)
δρ(r)
(3.30)
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
39
The density of the real system is therefore exactly the same as for the non-interacting system (where T [ρ] = Ts [ρ], and Vee [ρ] = 0), moving in an effective external
potential vef f . The Kohn-Sham Hamiltonian for the fictious system is,
Ĥ
KS
N
N
i
i
1X 2 X
=−
vef f (ri ).
∇i +
2
(3.31)
Solving the Schrödinger equation using the Kohn-Sham operator gives,
N
N
i
i
1X 2 X
[−
vef f (ri )]φi (r) = ǫi φi (r),
∇i +
2
(3.32)
in which the density and kinetic energy of the fictious system is,
ρ(r) =
N
X
i
|φi (r)|2 ,
(3.33)
and,
N
X
1
hφi | − ∇2i |φi i,
Ts [r] =
2
(3.34)
i
respectively. The total energy can be evaluated from these as,
Z
E[ρ] = v(r)ρ(r)dr + Ts [ρ] + J[ρ] + Exc [ρ].
(3.35)
The Kohn-Sham (KS) orbitals φi (r) and their eigenvalues (i.e. orbital energies), ǫi
can thus be determined in a similar way as in the Hartree-Fock theory. The KohnSham orbitals are expressed in terms of basis functions of which the orbital coefficients are determined by solving the same secular equations as in HF-theory.
The exchange term of the HF-theory (Eqn. 3.13) is however replaced in KohnSham theory by the exchange correlation term. The most prominent difference
between the two theories is that the Hartree-Fock theory is approximate whereas
DFT is exact, although approximations must be introduced due to the unknown
exchange correlation term (see below).
The KS-orbitals were in the first place constructed as a mathematical machinery
for solving the density of the real systems, as they represent the non-interacting
electrons. Therefore, it was thought that they lack physical meaning. Interestingly, it has been noted that the KS-orbitals themselves resemble strikingly much
the Dyson orbitals [242, 243].
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
40
3.5.4 Exchange-correlation functionals
Local Density Approximation
In the Local Density Approximation (LDA) it is assumed that the exchange correlation energy in a certain point in space depend on the density in that same point
[241],
Z
Exc = F (ρ)dr.
(3.36)
For a uniform electron gas, analytical expressions for the kinetic and exchange
correlation energy are given by Eqn. 3.19 and 3.20, respectively. For spin polarized systems the Local Spin Density Approximation (LSDA) has the form,
Z
486 1/3
4/3
LSDA
Exc
= −(
)
(3.37)
(ρ4/3
α + ρβ )dr.
64π 2
There are analytical solutions for the exchange correlation energy of an electron
gas but not for more complicated electronic system. Ceperly and Alder performed quantum Monte Carlo simulations on an electron gas for several densities [244]. By subtracting the total energy from the analytical expressions, the
contribution of the exchange correlation energy could be estimated. The numerical fit of this data by Vosko, Wilk, and Nusair (VWN functional) is given as [245],
ǫ(rs ) =
r√s
A
2 [ln rs +b rs +c
+
√
4c−b2
−1 ( √
√ 2b
tan
2 rs +b )
4c−b2
√
√
( rs − x0 )2
4c − b2
bx0
2(b + 2x0 )
−1
− 2
)]],
tan ( √
[ln[
]+ √
√
rs + b rs + c
2 rs + b
x0 + bx0 + c
4c − b2
(3.38)
in which several empirical fitting parameters (A, b, c, x0 ) are used. Vosko, Wilk,
and Nusair proposed several fitting schemes of which the VWN and VWN-5 are
most commonly used. Perdew and Wang have also their own LDA-functional
(the PW-functional) [246, 247]. LDAs perform surprisingly well in describing e.g.
molecular structure, but predict bond dissociation energies which are far too high
[248].
Generalized Gradient Approximation
The reason for the failure of the LDA-functionals is the assumption of a uniform electron density. In the Generalized Gradient Approximation (GGA), the
exchange correlation energy depends in addition to the local density, also on its
gradient [249],
Z
Exc = F (ρ, ∇ρ)dr.
(3.39)
GGAs were initially, but incorrectly, referred to as non-local DFT, although the
gradient of a function in a point in space is mathematically also a local property.
The initial formulation of the gradient-correction of the Thomas-Fermi kinetic
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
41
energy expression was actually derived in 1935 by von Weizsäcker [250], but the
breakthrough of GGAs came in the 1980s. Most GGA exchange functionals have
the form,
Z
3 3 1/3
GGA
Exc = − ( )
ρ4/3 F (ρ, ∇ρ)dr,
(3.40)
4 π
which have the ρ4/3 -dependence derived by Dirac (cf. Eqn. 3.20). In 1986, Becke
defined the B86-functional for which [251],
B86
Fxc
= βx2 ,
(3.41)
|∇ρ(r)|
.
ρ4/3
(3.42)
where,
x=
The drawback with B86 is that the exchange potential diverges. This was corrected in the "Becke 1988"-functional (B88X) [252] for which,
B88X
Fxc
=
βx2
.
1 + 6sinh−1 x
(3.43)
The empirical parameter β was used to fit the exchange functional to known
exchange energies of six noble gas atoms, He-Rn. The alternative GGA functional
by Perdew and Wang, PW86, has the following form,
P W 86
Fxc
= (1 +
0.0864s2
+ bs4 + cs6 )m ,
m
(3.44)
where,
s=
1
|∇ρ(r)|
.
1/3
(24π)
ρ4/3
(3.45)
Another widely used functional from 1988 by Lee, Yang, and Parr (LYP) does
not use the LDA expression for the exchange correlation energy [43], but is instead derived by fitting four parameters to the Colle-Salvetti formula for the He
atom [253]. The combination of B88X and LYP gives the BLYP functional, which
has had a major importance in chemistry. The GGA functionals discussed above
contain semi-empirical fitting parameters but there are many GGA functionals
derived from purely theoretical assumptions e.g. PW91 [254] and PBE [255].
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
42
The GGA functionals describe dynamical correlation by converting the non-localized exchange-hole into a localized one. Energetics, molecular structure and valence excitations are described reasonably well by the GGAs. Kinetic energy barriers are often underestimated, whereas Rydberg excitations and charge transfer
are seriously underestimated.
Meta-GGAs
The following step on Jacob’s ladder [256]5 is to introduce the Laplacian dependence in the exchange correlation energy,
Z
Exc = F (ρ, ∇ρ, ∇2 ρ)dr.
(3.46)
Evaluation of the Laplacian is, however, numerically challenging. This led to the
introduction of a kinetic energy density,
1
τ=
2
occupied
X
i
|∇φi |2 .
(3.47)
The meta-GGAs perform somewhat better than the GGAs. Examples of metaGGAs are B95 [257], PKZB [258], and TPSS [259]. It has recently been suggested
that the TPSS-functional performs surprisingly well in describing transition metals [260] and biomolecular systems [261, 262].
Adiabatic connection and hybrid functionals
In the adiabatic connection method, the extent of electron-electron interaction is
smoothly switched on, thus converting the non-interacting system into an interacting one. For the real (1) and fictious system (0),
F1 [ρ] = T [ρ] + Vee [ρ],
(3.48)
F0 [ρ] = Ts [ρ].
(3.49)
Exc = F1 − F0 − J[ρ].
(3.50)
and,
Thus,
Using the Hellman-Feynman theorem one can write,
Exc =
Z
1
0
δFλ [ρ]
dλ − J[ρ] =
δλ
Z
0
1
hΦλ |V̂ee |Φλ idλ − J[ρ],
(3.51)
5
Jacob’s ladder is a biblical analogue of a ladder to heaven where each improved functional is
one step closer to the exact functional.
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
43
in which the extent of λ is varied from 0 to 1. If the notation is changed to,
Exc =
Z
1
Wλ dλ,
(3.52)
0
this can be thought of as a complex curve in which a certain amount of exact
exchange is introduced from Hartree-Fock theory. According to Becke, Wλ =
a + bλ from which the exchange functional is determined as,
LDA
Exc = (1 − a)Exc
− aExHF .
(3.53)
a = 1/2 defines the half-and-half functional (BHLYP), which is historically the
first hybrid functional [263]. Later, Becke introduced three semi-empirical fit parameters in a functional called "B3". The B3PW91-functional is written as [264],
B3P W 91
Exc
= (1 − a)ExLSDA + aExHF + b∆ExB88X + EcLSDA + cEcP W 91 ,
(3.54)
where a=0.20, b=0.72, and c=0.81. Stevens et al. replaced the PW91 [254] with
LYP, because LYP was developed for describing the total correlation energy. This
introduced the functional that has had the largest impact on chemistry and modeling of enzymatic reactions using quantum chemistry methods. The exchange
correlation energy of the B3LYP functional can be written as [265],
B3LY P
Exc
= (1 − a)ExLSDA + aExHF + b∆ExB88X + (1 − c)EcLSDA + cEcLY P . (3.55)
The hybrid functionals, and in particular B3LYP describe the energetics better
than the GGAs. Kinetic barriers are somewhat underestimated in B3LYP but
there are special functionals such as the MPW1K designed to describe them more
accurately [266].
Other functionals
Coulomb-attenuated functionals
While hybrid functionals describe the short range exchange correlation energy
reasonably well, the long-range electron-electron interaction energy is poorly described. Adding more exact exchange would fix the long-range electron interaction but at the same time, the short-range interaction would become worse. One
way of avoiding this problem is to have different short- and long-range expressions for the 1/r-term. An example of such functionals is the Coulomb attenuated, CAM-B3LYP, which performs better than conventional B3LYP in describing
charge transfer and Rydberg states [267].
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
44
Dispersion corrected functionals
DFT is poor in describing dispersion forces which play an important role in e.g.
π-π stacking of aromatic residues and π-cation interactions. One way to recover
the dispersion forces is to introduce an empirical 1/r 6 -interaction between the
atoms in interacting fragments [268]. The dispersion corrected DFT (DFT-D) has
been used e.g. to study aromatic stacking in DNA base pairs and aromatic amino
acids [269].
Semi-empirical functionals, SCC-DFTB
When atomic partial charges are used instead of the quantum mechanical electron density in the DFT formulation, a "truly" semi-empirical DFT method known
as the self-consistent charge density functional tight binding (SCC-DFTB) theory is obtained [270]. This method is computationally as efficient as other semiempirical models, and it has recently been shown to describe proton transfer in
carbonic anhydrase and alcohol dehydrogenase reasonably well [271, 272]. Like
conventional DFT, SCC-DFTB has difficulties in describing dispersion forces. Inclusion of a 1/r 6 -term is required to obtain agreement with experimental findings
[273].
3.5.5 Performance of DFT in modeling biomolecular systems
DFT methods became of major importance in theoretical biochemistry by the development of gradient corrected and hybrid functionals in the late 1980s and
the early 1990s. Until this, the 1-2 eV error of LDAs was too large to describe
chemical reactions. G2 and G3 benchmarking tests have shown that the average error of GGAs is 6-8 kcal/mol in describing standard small organic compounds [274, 275]. For the hybrid functionals, in particular B3LYP, the average
error is 2-4 kcal/mol, although it has been argued that the G2 average error of
1.5 kcal/mol should be taken as a general reference point. Biomolecular systems
contain however often transition metals, for which the average error is somewhat
larger [276, 277, 278, 279, 280, 281]. Although the DFT methods have been shown
to perform reasonably well, there are some worst-case scenarios in which DFT
has predicted also qualitatively wrong results [228, 234, 235, 236]. This may take
place if the system has a considerable multiconfigurational character. The results
obtained by DFT calculations should therefore always be compared with experimental results (as any theoretical method). In addition to the intrinsic error of
the DFT methods, it is important to use a large enough basis set; the error of a too
small basis set can be as large as 5-10 kcal/mol.
3.6 The basis set approximation
The electronic wave-function is in quantum chemical calculations represented
by a basis set. The molecular orbitals are obtained by Linear Combinations of
Atomic Orbitals, χp (LCAO-MO),
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
Φi =
X
45
(3.56)
cip χp
p
Often, Gaussian type orbitals (GTOs) are used as a basis set,
2
χlp (r, θ, φ) = Yl,m (θ, φ)r l e−γr ,
(3.57)
where the spherical harmonics Yl,m define the angular momentum for quantum
numbers l and m, and r is the distance of the electrons from the nuclei. Slater
type orbitals (STO, χlp ∝ e−γr ), which have the correct nuclear cusp and long
term behavior, are also used in some electronic structure packages (e.g. ADF).
However, the popularity of the GTOs is due to analytical expressions for all fourcenter two-electron Coulomb integrals.
The minimal basis set, STO-nG, is formed by fitting n Gaussians to a Slater type
function to describe each orbital by one function [282]. However, this does not
give a very satisfying accuracy. Instead the number of functions per electron are
usually increased to a double-ζ, triple-ζ, etc. quality. The split valence basis set
(SV) is obtained by adding functions only to the valence shell. For calculation
of properties, and for describing anions, it might be important to add diffuse or
polarization functions.
The split-valence basis set by Pople et al. [282] is usually designated X-YZg,
where X is the number of primitive Gaussians describing the core orbitals and,
Y and Z are representing the valence orbitals. More than two numbers indicate
triple, quadruple etc. −ζs, a ∗ indicate polarization functions, and a + indicate
diffuse functions.
The correlation consistent basis sets by Dunning et al. are especially useful for
correlation calculations [283]. The basis sets are designated aug-cc-pCVxZ where
x=D,T,Q,..., aug implies diffuse functions, and C implies augmentation with steep
functions. The Atomic Natural Orbital basis (ANO) are examples of basis sets in
which natural orbital coefficients are optimized by optimizing atomization energies [284, 285]. In the effective core potential (ECP) basis sets, the core electrons
are described implicitly, which reduces the computational cost [286]. In addition,
relativistic effects can implicitly be included in these. The Karlsruhe basis defxZVP [287, 288], where x stands for S,D,T,..., and where P stands for polarization
functions, is implemented in the program package TURBOMOLE [289], and are
extensively used in the QM-studies of this thesis.
A sufficiently large basis set should always be used in electronic structure calculations. For DFT calculations, the basis set convergence is reached usually quite
fast. Interestingly, "optimal" results can often be found at a TZVP level basis due
to error cancellation. Correlated calculations such as MP2 or CC, often require a
much larger basis set for convergence. QZVP or even a higher level are recommended.
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
46
The basis set superposition error (BSSE) should be taken into account when interaction energies are calculated [290]. In the BSSE, the basis set of one fragment
improves the basis set of the other fragment, thus biasing the energies. The extent
of this effect can be estimated by calculating a "counterpoise" energy correction.
In this calculation, the energies for fragments in presence of only the basis set
of the other fragments are evaluated and subtracted from the total interaction
energy.
3.7 Classical methods
Although a quantum mechanical description of a molecular system can be very
accurate, it often becomes unpractical or even impossible to solve the Schrödinger
equation (or its approximations) for large systems, such as biomolecular systems.
It is known that macroscopic particles obey the classical laws of mechanics. However, can the limit between classical and quantum mechanics easily be drawn?
The wave-particle duality, introduced in the 1920s by Louis de Broglie, enlightens this limit [291, 292]. The thermal "de Broglie" wave-length is defined as,
Λ= √
h
,
2πmkB T
(3.58)
in which m is the mass of the particle, and T is the temperature of the system.
The translational partition function can be expressed as,
Qtrans =
1 V N
( ) ,
N ! Λ3
(3.59)
in which V is the volume of the system containing N particles. For an ideal
gas the inter-particle distance can be assumed to be rinter ≈ (V /N )1/3 . Whenever rinter is much smaller than the thermal wavelength, the system obeys quantum laws and thus either Fermi-Dirac or Bose-Einstein statistics, depending on
whether the particles are fermions or bosons. Otherwise, if this distance is comparable or much larger than Λ, classical effects are dominant and the system
obeys classical Maxwell-Boltzmann statistics. For example, for an electron at
T=273 K the thermal wavelength is 45 Å, for a proton 1 Å, and for a water molecule 0.1 Å.
3.7.1 Classical equations of motion
The classical equations of motion were first derived by Sir Isaac Newton in his
work Philosophiae Naturalis Principa Mathematica from 1687 [293]. In modern notation the second law of Newton can be written as,
mi
∂Vi
d2 ri
=−
, i = 1, 2, ..., N .
2
dt
∂ri
(3.60)
This well known equation states that if the position of N particles ri , are known
together with their masses mi and the forces −∂Vi /∂ri affecting them at some
time point t, their positions can be calculated at any time t′ by integrating Eqn.
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
47
3.60. Stating it otherwise, a force affecting a particle with a certain mass, causes
it to accelerate.
The botanist Robert Brown discovered in 1827, that a microscope preparate of
pollen grains immensed in water is in a constant jitterying motion [294]. He
also found that dust grains behaved in a similar manner. This proved that the
phenomena was a property of the microscopic particle rather than due to a mysterious vis vitae, force of life.
George Stokes suggested based on his studies on fluid viscosity that frictional
force is proportional to the velocity of the particle times a frictional coefficient ζ.
Inserting this to the Newtonian equations of motion gives,
mr̈ = −ζ ṙ,
(3.61)
v(t) = v(0)e−ζt/m .
(3.62)
with the solution,
In the time limit when t → ∞, v → 0. This contradicts the equipartition theorem,
which states that for a system in equilibrium,
hv 2 ieq = kB T /m.
(3.63)
Paul Langevin therefore added a random (stochastic) force for keeping the system "alive",
mr̈ = −ζ ṙ + δF (t),
(3.64)
δF (t) = F (t) − hF i.
(3.65)
where,
The stochastic, fluctuating force satisfies the relation,
hδF (t)i = 0,
(3.66)
hδF (t)δF (t′ )i = 2Bδ(t − t′ ).
(3.67)
and,
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
48
B is the strength of the stochastic force (and equals ζkB T according to the fluctuation-dissipation theorem). The Langevin equation of motion is widely used in
classical molecular dynamics simulations to simulate the temperature effect.
3.7.2 Classical force-fields
The central idea behind classical molecular dynamics is to parametrize intra- and
intermolecular interactions in a simple analytic function known as the force field.
Usually only pair-wise terms are treated to secure low scaling. Of central importance is also the transferability of the force field parameters. The analytical form
of the force fields used in biomolecular simulations is usually [295],
Vf f =
X
bonds
kr (r − ri )2 +
−
X
angles
kθ (θ − θi )2 +
X
kω [1 + cos(nω + φ)]
dihedrals
X qi qj
X
σi,j 12
σi,j 6
−
4ǫi,j [(
) −(
) ].
4πǫrij
rij
rij
i,j
(3.68)
i,j
The force field can be divided into bonded (upper row) and non-bonded terms
(lower row). Bond-, angle- and dihedral terms are treated using simple harmonic
potentials which is a quite reasonable approximation. Some force fields employ
a Morse potential for describing a chemical bond,
VM orse = −
X
bond
De [1 − ea(r−ri ) ]2 ,
in which De is the dissociation energy and a = ω
quency of the vibration and µ the reduced mass.
p
(3.69)
(µ/2De ), where ω is the fre-
The non-bonded terms consist of electrostatic Coulombic interactions between
point charges and weak Lennard-Jones interactions. The dielectric constant ǫ in
the Coulombic term is usually put equal to 1 in explicit condense phase simulations (all-atom simulations). The Lennard-Jones term is based on an attractive
1/r 6 -term which can be derived for systems interacting through van der Waals
forces. The 1/r 12 -term is a strong repulsive term, which prevents atoms from
colliding if they come too close to each other. σ is the collision diameter since
it defines the distance where the interaction energy is zero. ǫ is the depth of the
Lennard-Jones potential. An alternative way of modeling van der Waals interactions is by using a Buckingham potential where α is a fitting parameter,
VBuckingham = −
X
ǫ[
α σ 6
6
e−α(r/σ−1) −
( ) ].
α−6
α−6 r
(3.70)
Because there are N (N − 1)/2 pair-interactions in a molecular system with N
interacting particles, the number of pair-interactions might become unacceptably
large especially in a biomolecular simulation. Therefore, a cut-off value and a
down-switching function S (which is multiplied with the interaction term) are
usually employed to lower the computational cost,
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
(3.71)
S = 1.0, rij < rswitch
S=
49
rcut−of f − rij
, rswitch ≤ rij ≤ rcut−of f
rcut−of f − rswitch
S = 0.0, rij > rcut−of f .
(3.72)
(3.73)
Cutting-off the electrostatic interactions must be done with care because the electrostatic interactions are long-range. Typical cut-off values used in large biomolecular simulations are 10-14 Å. Too short cut-off values might affect the results
of the simulation, especially properties which depend on the distribution of molecules.
Long-range interactions can also be treated by the particle mesh Ewald method
[296]. In this method, interactions between particles close in space are calculated in direct space and particles far away, are calculated in reciprocal space and
Fourier transformed to direct space. This method is computationally somewhat
more expensive, scaling as N log(N ).
3.7.3 Velocity Verlet
For integrating the classical equations of motion in computer simulations, the
"continuous" time is divided into small time-steps, ∆t. The "Verlet" expression is
obtained by adding together the Taylor expansions for r(t + ∆t) and r(t − ∆t)
[297],
r(t + ∆t) ≈ 2r(t) − r(t − ∆t) +
f (t) 2
∆t + O(∆t4 ).
m
(3.74)
The error of the approximation is O(∆t4 ) since the third order terms cancel out.
Choosing a sufficient time-step is of central importance in MD-studies. A rule of
thumb is to choose a time-step which is approximately 10 times faster than the
fastest vibration in the system. For example, in a system with C-H vibrations,
which are typically found at 3000 cm−1 (equal to a period of 11 fs), would require
a time-step of 1 fs to be used in a MD-simulation. If the time step is chosen too
large, some molecular collisions might be avoided which produces unphysical
results. The velocity of a particle is obtained from v(t) = [r(t + ∆t) − r(t −
∆t)]/2∆t, and the initial particle velocities are usually randomly assigned from a
Maxwell-Boltzmann velocity distribution at a given simulation temperature.
3.7.4 The Langevin equations of motion in computer simulations
The stochastic Langevin equation of motion (Eqn. 3.64) is combined with the
velocity Verlet in the Brügner-Brooks-Karplus (BBK) method [298],
rn+1
1 − γ∆t/2
1
F (rn )
= rn +
(rn − rn−1 ) +
∆t2 [
+
1 + γ∆t/2
1 + γ∆t/2
m
r
2γkB T
Rn ], (3.75)
m
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
50
where Rn is a Gaussian random number with a standard deviation of 1 and a
mean of zero, and n is the time-step. To obtain a true stochastic temperature
behavior, it is of central importance that the random number generator is of good
quality.
3.7.5 Simulation ensembles
Especially in biomolecular simulations, one is often interested in simulating other
ensembles than a microcanonical NVE ensemble, for example NVT or NPT ensembles. A constant temperature can be simulated by scaling the acceleration of
each particle as,
ai (t) =
Fi
pi (t) T0
+
[
− 1],
mi
mi τ T (t)
(3.76)
where T (t) is an instantaneous temperature and τ is a coupling parameter. This is
a typical Baerends coupling scheme [299], but also other methods are commonly
used [300, 301, 302]. The pressure can be controlled in an MD-simulation by
scaling the volume [303, 304]. The instantaneous pressure can be computed as,
P (t) =
1X
1
Fij rij ],
[N kB T (t) +
V (t)
3
(3.77)
i>j
where V (t) is the instantaneous volume, which can be scaled similarly to the
temperature in Eqn. 3.76.
3.7.6 Dynamical properties
The trajectories obtained from MD simulations, can be used to calculate several
dynamical properties. For example, the root-mean-square deviation (RMSD),
which is a measure of the overall structural changes taking place in time during a simulation is defined as,
v
u Nα
uP
u
[r (t )− < rα >]2
t α=1 α j
(3.78)
RM SDα (ti ) =
Nα
where,
Nt
1 X
hrα i =
rα (tj )
Nt
(3.79)
j=1
Auto-correlation functions correspond to many chemical and physical properties. For example, the diffusion coefficient can be measured from the velocity
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
51
auto-correlation function, and IR-spectra from the dipole moment auto-correlation
function. For a property a, the auto-correlation function is defined as,
Z
1 t
C(t) =< a(t0 )a(t0 + t) >t0 = lim
a(t0 )a(t0 + t)dt.
(3.80)
t→∞ t 0
The radial distribution function is of special importance, for example in determining the properties of a liquid. It is defined as,
g(r)AB =
< NB (r, r + ∆r) >
,
4πρB r 2 ∆r
(3.81)
where NB is the average number of B atoms found within r and r + ∆r, and
ρB is the average density of B. When g(r) is integrated from 0 to r, it defines the
average number of neighboring B atoms at a distance r.
3.7.7 Free energy calculations
The free energy associated with a change in the state of a system from an initial
state i to a final state f with corresponding potentials Vi and Vf , is defined as,
∆Gf,i = Gf − Gi = −kB T ln(
Qf
) = −kB T ln[< e−β∆V >i ],
Qi
(3.82)
where ∆V = Vf − Vi , and the ensemble average with respect to state i is defined
as,
Z
Qf
= d(∆V )e−β∆V ρi (∆V ).
(3.83)
Qi
ρi (∆V ) is the density of different configurations of ∆V , and β = 1/kB T . If the
densities of states ρi (∆V ) have a relatively broad distribution with a large number of states outside the mean value of ∆V , the integrand in Eqn. 3.83 (the product of ρi (∆V ) and exp(−β∆V )), will not be properly sampled in an equilibrium
MD simulation, since neither very high nor low values of ∆V are obtained from
the simulation. The main aim of free energy calculations is to obtain accurate
sampling of the phase-space to get a good estimate of the free energy. If the free
energy surface of a process, e.g. a chemical reaction, is sampled along a reaction
coordinate, which can be e.g. a bond length, it is often referred to as the potential
of mean force (PMF) defined as,
W (q) = −kB T ln[p(q)],
(3.84)
where,
1
p(q) =
Q
Z Z
δ[q′ − q]e−βE(p,q) dqdp.
(3.85)
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
52
Q is the partition function, δ[q′ − q] is a Dirac-delta function, and q and p are
generalized coordinates and momenta, respectively.
Umbrella sampling
A biasing force can be introduced in an MD simulation, which drives the system
to sample configurations of the phase-space with extreme values of ∆V . However, this drives the system into a non-Boltzmann distribution. Torrie and Valleu
showed that the effect of the biasing potential can be removed, which leads back
to Boltzmann statistics [40]. If the biasing potential is w(r), then the unbiased
probability can be calculated from,
R
drw(r)e−βVf /w(r)
p= R
.
(3.86)
drw(r)e−β∆Vi /w(r)
Free energy perturbation
The free energy can also be evaluated by slowly varying a perturbation parameter
λ from 0 to 1 which takes the system from an initial to a final state,
∆V (λ) = λVf + (1 − λ)Vi .
(3.87)
The free energy calculation is thus broken into small sampling windows in which
λ varies "slow enough" to keep the system close to thermodynamic equilibrium.
There are several variants of how the free energy is obtained. In the FEP-approach
the free energy is calculated as [39],
∆Gf,i =
λ=1
X
kB T ln < eβ(Vλ+dλ −Vλ ) >λ .
(3.88)
λ=0
FEP is usually performed in both forward and backward directions. If the FEP
calculation involves a chemical reaction, the topology of the molecule must also
change along with the perturbation parameter λ. Alternatively, the free energy
can be calculated by the slow growth-method,
∆Gf,i = lim
dλ→0
λ=1
X
(Vλ+dλ − Vλ )
(3.89)
∂V
iλ ∆λ.
∂λ
(3.90)
λ=0
or by thermodynamic integration,
∆Gf,i ≈
λ=1
X
λ=0
h
Practical issues usually determine the best suitable method for the free energy
calculation.
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
53
Weighted Histogram Analysis Method
The free energy surface can seldom be sampled in a single perturbation run, but
rather needs a combination of the PMF profiles generated from many perturbation runs. The PMF profiles must have overlapping regions because the absolute free energy of the profiles is not known. The Weighted Histogram Analysis
Method (WHAM) is an iterative method for combining different PMF-profiles
from different runs in an optimal way [305]. The 2D-WHAM method is illustrated below as an example. Let l = 1, 2, .., L be data points of the reactioncoordinate q, and m = 1, 2, ..., M the corresponding data points on the reaction
coordinate s. Thus the reaction coordinates can be written as ql = q0 + l∆q, and
similarly for s. The discretized probability p(q, s) can then be normalized as,
M
L X
X
plm = 1.
(3.91)
l=1 m=1
Furthermore, there are K perturbation runs (k = 1, 2, ..., K) with different biasing potentials uk (s) from which histograms are collected from each run separately. The total number of points for each simulation is Nk , which is obtained by
summing over nk,lm,
Nk =
L X
M
X
nk,lm.
(3.92)
l=1 m=1
The WHAM-equation for the probability p(q, s) can be written as,
PK
k=1 nk,lm
,
−β(uk,m −Gk )
k=1 Nk e
plm = PK
(3.93)
where uk,m is the potential caused by the perturbation in bin m for run k i.e.
uk (sm ). Everything except the normalization coefficients Gk are known, which
gives the normalization conditions,
e−βGk =
L X
M
X
plm e−βuk,m .
(3.94)
l=1 m=1
Equations 3.93 and 3.94 must be solved self-consistently. Before the development of WHAM in the mid-90s, separate PMF profiles were combined "by hand"
to give the full PMF profile.
3.7.8 Electrostatic calculations
Evaluation of amino acid pKa -values in proteins is a challenging task. This is
due to the highly anisotropic and dynamic protein medium, and the interactions
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
54
between the protonation states between all protanatable groups. The pKa of an
amino acid residue can be considerably up- or down-shifted inside the protein
compared to its pKa in solution. For example, a glutamic acid has a pKa of 4.7
in water whereas Glu-242 in CcO has been suggested to have a pKa close to 10
[202, 306], which means that it is thermodynamically favorable to protonate it
from pH=7. The pKa -value inside a protein must be evaluated by considering
the thermodynamic cycle in Fig 3.1, which gives,
Figure 3.1. A thermodynamic cycle used to calculate protein pKa -values. The ∆Gw is
obtained from the pKa of the residue in solution, and ∆Gw→p is calculated for both the
protonated and deprotonated residues. From this, the ∆Gp and the pKa of the residue
inside the protein can be estimated.
pKa =
=
1
ln(10)RT
∆Gp (AHp → Ap + Hw+ )
1
[∆Gw→p (A− ) − ∆Gw→p (AH) + ∆Gw (AHw → Aw + Hw+ )]. (3.95)
ln(10)RT
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
55
This equation can be written as,
pKa = pKaw −
δi
w→p
∆∆Gsol
(AH → A− ),
ln(10)RT
(3.96)
w→p
where δi is -1 for acids and +1 for bases. The ∆∆Gsol
-term has three contributions,
w→p
∆∆Gsol
= ∆GBorn + ∆Gback +
X
∆Gpij ,
(3.97)
i6=j
where ∆GBorn is the desolvation energy of an ionizable residue when it is moved
from water to protein environment, and ∆Gback derives from the background
charges of all other residues in their neutral form.
P The sum of these two terms
forms the intrinsic pKa,int -value. The last term, i6=j ∆Gpij is the interaction between the ith and jth ionizable residue. In pKa -calculations, e.g. the classical
Poisson-Boltzmann equation (PB) is solved numerically,
∇ǫ(r) · ∇φ(r) − ǫ(r)λ(r)κ2
qφ(r)
kB T
sinh[
] = −4πρ(r),
q
kB T
(3.98)
where the Debye-Hückel parameter is defined as,
κ2 =
8πq 2 I
,
ǫkB T
(3.99)
where I is the ionic strength of the solution. If no ions are present in the solution, the PB equation simplifies to the Poisson equation,
∇2 φ(r) = −
4πρ(r)
.
ǫ(r)
(3.100)
The Born-term in Eqn. 3.97 can be written as,
GBorn = −
1
q2
(1 − ),
2a
ǫ
(3.101)
where a is the effective radius of the solute, and ǫ is the dielectric constant of
the medium. In the Generalized-Born (GB) treatment Eqn. 3.101 is replaced by,
1
1 X qi qj
GBorn = − (1 − )
,
2
ǫ
fGB
i,j
(3.102)
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
56
where,
fGB =
q
2 + R R exp(−r 2 /4R R ),
rij
i j
i j
ij
(3.103)
in which Ri is the effective atomic Born radius. The goal of the GB theory is
to find simple analytical solutions to Eqn. 3.98 which approximate real molecular geometries.
Implicit solvation models can also be used in QM-calculations. For example, in
the COnductor Like Screening Model (COSMO) [48] interactions are screened by
the Kirkwood-Onsager factor,
f (ǫ) =
2(ǫ − 1)
,
2ǫ + 1
(3.104)
which is used in the evaluation of solvation energies.
The pKa -values depend on the local environment and therefore it is adviceable to
perform classical electrostatic calculations on more than one structure, and weigh
the separate pKa -values by the relative energy of the structures. Different conformations can be obtained from MD-simulations [307], or from a rotational library
of amino acid conformations [308, 309].
3.8 Combined QM/MM calculations
Kinetics and energetics of bond-breaking and bond-making are of high importance in enzyme catalysis. This can accurately be described only by quantum
chemical methods. However, the complete protein environment, which plays
an important role in the process, cannot be treated at the QM-level of theory
due to the high computational cost. It is therefore more practical to describe
the surrounding protein environment by classical methods. This view led to
development of combined QM/MM methods in which the active "interesting"
part is modeled by quantum chemical methods and the surrounding protein
is described by classical biomolecular force fields [46, 310]. The very first protein QM/MM simulation was done very early after the development of the biomolecular force fields in 1976 by Warshel and Levitt, who studied the hexa-Nacetylglucoseamine cleavage reaction of lysozyme [34].
The most difficult task in QM/MM modeling is to choose the partitioning between the QM and MM regions to avoid over-polarization. Three popular approaches exist: 1) the simple fragment method [311, 312], 2) the link atom method
[311, 313, 314, 315], and 3) the frozen orbital method [316, 317, 318, 319]. In the
first approach, the QM-region is capped with hydrogen atoms and the energy is
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
57
calculated for the simple fragment both on QM and MM levels, and the energy
of these is subtracted from the MM energy of the total system,
Etot = Eactive+surr.,M M + [Eactive,QM − Eactive,M M ].
(3.105)
The total classical energy is in this approach "corrected" with quantum effects.
The drawback with this method is that classical parameters for the complete system are required.
In the link-atom approach, the active region is similarly to the previous method
truncated with hydrogen atoms, but the QM and MM-fragments are explicitly
coupled to each other via dummy atoms. The coupling between the two regions
is calculated as,
ĤQM/M M =
QM
M
X
XM
i
j
{
qi,j
Zi qj
σi,j 12
σi,j 6
+
+ 4ǫi,j [(
) −(
) ]}.
ri,j
ri,j
ri,j
ri,j
(3.106)
In the frozen orbital method, frozen localized orbitals are introduced between
the QM and MM regions. These orbitals are obtained from simple model compounds, and connected with standard bonding interactions to the MM-region.
Hybrid DFT-methods are most commonly used to describe the QM-region in
QM/MM simulation due to their good overall performance. It has been demonstrated that the error done in the QM/MM coupling can be quite small, in the
order of a couple of kcal/mol, which is comparable to the intrinsic error of the hybrid functionals. However, if the active region alters its total charge in a reaction
step, it has been demonstrated that the error is not always negligible, especially
in the region close to the link atoms [310].
Empirical Valence Bond approach
An alternative approach to describe the QM-region in QM/MM simulations is to
use semi-empirical methods. One method, which is actually more of a classical
method than a semi-empirical QM-method (cf. [320]) but has had a major impact
in modeling enzymatic reactions, is the Empirical Valence Bond method (EVB),
pioneered by Warshel and Weiss [321]. For example, for proton transfer reactions
the VB wave function can be written as [322],
φ1 = [RX − H Y R]
φ2 = [RX − H − Y + R]
CHAPTER 3. FOUNDATIONS OF THEORETICAL BIOCHEMISTRY
58
These wave functions are used as a basis for the EVB Hamiltonian,
0
ǫ1 H12 Ĥ = H21 ǫ01 (3.107)
where,
1
(1)
0
1
ǫ01 = H11
= ∆M (b1 ) + Unb + k(θ1 − θ10 )2 + Uinact
2
1
(1)
2
0
+ α02
ǫ02 = H22
= ∆M (b2 ) + Unb + k(θ2 − θ20 )2 + Uinact
2
0
H12
= Aexp(−µ(r3 − r30 )).
(3.108)
(3.109)
(3.110)
b1 and b2 are the covalent bond distance between X-H and H-Y, and r3 the nonbonded X · · · Y distance. θ1 and θ2 are the harmonic angle terms, and ∆M is a
i is the nonbonded electrostatic and van der Waals interacMorse potential. Unb
0
tions. α2 is the energy difference between Ψ1 and Ψ2 when the fragments are at
infinite separation. The off-diagonal elements are calibrated with gas-phase experimental data or accurate quantum chemical calculations. The interaction be(i)
tween the active and inactive electrons are described by Uinact , which is modeled
according to a general force field potential. The protein environment is added as
diagonal terms in the Hamiltonian,
0
(1)
ǫS1 = H11 = H11
+ Vrs
+ Vss
(3.111)
ǫS2
(3.112)
= H22 =
0
H22
+
(2)
Vrs
+ Vss ,
(i)
where Vss corresponds to interaction within solvent (s), and Vrs to the potential of configuration i in the active part and the environment. The ground-state
energy is obtained by a SCF procedure by solving the secular equation,
ĤEV B C = Eg C.
(3.113)
After obtaining the EVB-potential energy surface, molecular dynamics can be
performed on this surface by employing the Hellman-Feynman theorem,
dH
dE
= hΦ|
|Φi.
dλ
dλ
(3.114)
When λ is the atomic position, dE/dλ equals the force, which can be inserted
in the velocity Verlet.
Chapter 4
Aims of the Study
The mechanism of proton-pumping in CcO is one of the most important unsolved
problems in bioenergetics. Ever since the first high resolution X-ray structures of
CcO appeared over ten years ago, the pumping mechanism has been tackled by
theoretical methods. It is very difficult to deduce molecular scale interpretations
from the experimental results, although many elegant experiments have brought
detailed understanding of the system. This makes it feasible to tackle the problem further by computational modeling. The aims of this thesis were:
• To understand what prevents the pumped protons from leaking.
• To understand how the protons are gated. That is, how the redox chemistry
dictates that the protons are transferred to the right site at the right time.
• Why two proton conducting channels are needed.
In addition, there were some specific aims linked to these major questions:
• How can protons be transferred further from Glu-242 with no connectivity
to the pump site or the binuclear site.
• To understand the difference between the non-pumping states O and E, and
the pumping states OH and EH .
• To derive reliable simulation parameters for the redox active metal centers
in cytochrome c oxidase.
The last aim was purely technical. At the time when the work started, there
were no published parameters of the metal centers of CcO. In addition, there
were no consistent way of deriving charge parameters for complex biological
metal centers.
59
Chapter 5
Models and Methods
The mechanism of proton-pumping in CcO has been studied in this thesis by
a combination of several theoretical methods. In this section the models and
methods used in the publications are briefly reviewed.
5.1 Quantum chemical models
Quantum chemical model systems were built for all four metal centers in CcO.
The aim of these models were to derive charge parameters for classical simulations of the centers (paper I), to study how different redox and ligand states of
the CuB system are possibly linked to the activation of the K-channel (paper IV),
and to study the importance of the His-Tyr crosslink (paper IV). The QM-models
included amino acid residues from the first solvation sphere and conserved interactions around the metal centers. All models were built based on the Bos taurus
X-ray structure, PDB entry 1V54 [25]. The full backbone of each amino acid was
included in the model, with N-terminal N-methyl and C-terminal acetyl groups
capping each residue, in order to mimic the continuation of the peptide chain.
The terminal carbons of the capping residues were frozen during the optimization procedure, which was done using a DFT/B3LYP-method [43, 264, 265] and
a SVP-basis set [287] on all atoms, except for the metals for which a TZVP basis set was used [288]. The dielectric medium of the protein was described by
the COnductor like Screening MOdel (COSMO) [48] with ǫ=4. The QM-package
TURBOMOLE (v 5.8 and v 5.9) [289] was used for the QM-optimization. Before
final analysis of the fully relaxed systems, single point calculations were done
in the QM-package NWChem [323], using a TZVP-basis set on all atoms. For
the point charge derivation scheme, "CHarge from Electrostatic Potential using a
Grid" method (CHELPG) [324] was used for deriving the ESP point charges for
each atom in the metal sites. In this procedure the atomic partial charges of the
nuclei α, qα , are fitted to produce the quantum mechanical electrostatic potential,
VESP =
X
α
qα
|r − rα |
(5.1)
The charges were hyperbolically restrained to be close to zero using a Restrained
ESP procedure (RESP). In the fitting methodology the system was divided into
61
CHAPTER 5. MODELS AND METHODS
62
two regions, an inner and outer (see Fig. 5.1). The former includes the redox active
metal site, and the latter a region which provides the accurate interaction field for
the metal site. The total charge of the outer region was constrained equal to the
force field charge. In addition, border atoms between the inner and outer parts
were constrained to have charges equal to standard CHARMM [217] or AMBER
[218] point charges. Without these constraints, an extra charge was found to accumulate on the Cβ atoms of the inner part of the system. Figure 5.1 exemplifies
the partitioning between different regions in the QM-system. In the RESP fitting,
the dielectric constant was set equal to 1 due to the explicit outer region of the
system. The Mulliken charge analysis method was also employed for further
analysis [325]. The Mulliken charge is defined as,
qA = ZA − NA
(5.2)
where ZA is the nuclear charge of nuclei A, and
NA =
AO
AO X
X
Dαβ Sαβ ,
(5.3)
α∈A β
where D, and S are the density and overlap matrices, respectively.
Figure 5.1. Charge fitting scheme exemplified. Black boxes indicate fixed atoms during
structure optimization, asterisks indicate atoms which are assigned their unconstrained
RESP charge, and orange atoms (Cα , Hα ) are constrained to have a charge equal to the
force field charge. Thick lines indicate inner region atoms, the outer region is indicated
by thin lines. Reproduced from paper [I].
CHAPTER 5. MODELS AND METHODS
63
In paper IV, the charge and spin densities of the CuB system were calculated together with the gross spin density, defined as the sum of α and β spin densities.
Energy level diagrams were constructed for the different redox and ligand states.
The energy of an electron reducing the CuB site was taken to be -109.5 kcal/mol,
which is equal to the Em of heme a [326]. The energy of a proton was taken to
be -270.6 kcal/mol, from gas-phase experiments [327]. The results were analyzed
in the molecular visualization programs VMD [328] and gOpenMol [329, 330].
In addition, environmental polarization effects were studied on the electronic
structure of the CuB site, by performing single point calculations with a scalable charge located approximately 2.5 Å from the copper of CuB in the direction
of heme a3 , simulating the effect of the negative oxygenous ligand of heme a3 .
The effect of a positive Lys-319 in the K-channel was also simulated by placing a
positive unit charge approximately 10 Å away from Tyr-244 down towards the
K-channel.
In paper VI, the barrier of proton transfer was studied by optimizing formic acid,
hydrogen bonded to formiate by three water molecules. The optimization was
done at DFT/B3LYP and MP2 levels of theory using a def2-TZVP basis set. The
O-H bond length of the formic acid was scanned whilst the rest of the system
was optimized. Stabilizing effects of the water molecules in the proton transfer
process were studied by single-point calculations.
5.2 Classical molecular dynamics
Classical molecular dynamics simulations were used in papers II, III, V, and VI.
In these studies, MD simulations were performed on subunits I and II of Bos
taurus enzyme, PDB entry 1V54. In most studies, two states were considered,
•
red /heme aIV =O/CuII -OH Tyr
the so-called PM with Cuox
244 -O , and PR
3
A /heme a
II
−
ox
IV
with Cuox
A /heme a /heme a3 =O/Cu -OH Tyr244 -O . Standard CHARMM27force field parameters [217] were used to describe the amino acids together with
DFT-based charge parameters discussed above for the metal centers [Paper I]. In
most simulations four water molecules were modeled above Glu-242, according
to results obtained from PMF-calculations [84]. In paper II the effect of the water
potential was explicitly studied and simulations were performed with 0-4 water
molecules above Glu-242. In paper III, in addition to the four water molecules
above Glu-242, four water molecules were found by MD-simulations to reside
also below this residue. These molecules are not present in the Bos taurus structure but are resolved in the Rhodobacter sphaeroides structure [97]. The simulations
were performed in MD-package NAMD [331], with typical simulation times of
1-10 ns, and a time-step of 1 fs. The structures were subjected to initial minimization and equilibration procedures. The temperature was controlled using
Langevin dynamics with T=310 K. A cut-off of 20 Å was used for the electrostatic interactions. The trajectories were analyzed in the molecular visualization
program VMD and the mathematical program package MATLAB.
CHAPTER 5. MODELS AND METHODS
64
5.3 Free energy calculations
In paper VI the free energy of different redox-linked proton pathway formations
were studied. A reaction coordinate, dmax , was used to describe the proton pathways. dmax was defined as the longest distance between a donor and an acceptor
group within the shortest pT-pathway from the donor (Glu-242) to the acceptor
(the pump site or the oxygenous ligand of CuB ). Based on QM-calculations, it
was assumed that proton transfer can take place when two groups are hydrogen
bonded to each other, i.e. when dmax < 2 Å. To obtain sampling of the "chemistry" pathway in the "pump" step, a perturbation of the charge distribution was
performed,
Qperturb (λ) = λQ(heme ared /PM ) + (1 − λ)Q(heme aox /PR ),
(5.4)
where λ=0 is the charge distribution of the state with heme ared / PM , and λ=1
the charge distribution of the state with heme aox / PR . Optimal back-and-forth
sampling of the dmax parameter was obtained with λ ≈0.6 during a 15 ns simulation. The unbiased dmax was calculated according to perturbation theory,
P
δ[dmax − dmax,i ]exp(−β∆Ui )
P
p(dmax ) = i
,
(5.5)
i exp(−β∆Ui )
where δ[dmax − dmax,i ] is a Dirac delta function and the PMF is calculated as,
G(dmax ) = −kB T ln[p(dmax )].
(5.6)
To understand the molecular basis for the redox-linked proton pathway formations, the interaction energies between the water molecules and surrounding
groups were divided into their contributions from single groups.
5.4 Other theoretical methods
The results obtained in paper III were incorporated in paper V with the framework of the overall reaction mechanism based on time-resolved spectroscopy and
electrochemical measurements. In the model, pKa -variations of Glu-242 from the
surrounding metal centers were incorporated by a simple Coulombic perturbation,
∆pKa =
qi qj
1
( ),
4πǫǫ0 kB T ln(10) r
(5.7)
where r is the distance between two groups. For simplicity, the charge was assumed to localize on the metals of the redox sites or the propionate of heme a3 .
The ground-state pKa of Glu-242 was defined as 10 for the protonated Glu in
its "down" configuration with one charge uncompensated electron in the system
CHAPTER 5. MODELS AND METHODS
65
(heme a or BNC reduced). Effects of membrane potential were studied by assuming a linear electrostatic pKa -perturbation across the membrane dielectric,
d
∆pKa = ( )Vm ,
L
(5.8)
where d is the distance of a protonatable group from the P-side of the membrane,
L is the membrane thickness, and Vm is the membrane potential (200 mV with full
membrane potential). Transition rates were assumed to follow transition state
theory,
k=
kB T −∆G† /kB T
κe
,
h̄
(5.9)
where the transmission factor (κ) was assumed to equal one. Energy diagrams
were drawn for one proton-pumping step with and without a membrane potential. In addition, proton-pumping was studied for the case in which the Glu-valve
would not be present. The model was calibrated by optimizing the dielectric constants of the electrostatic interaction energies to satisfy two experimental boundary conditions [136]: 1) The increase of the pKa of the pump site from 5.3 to 9.0
upon reduction of heme a, and 2) the total energy of one pumping step to equal
-7.7 pK-units (10.9 kcal/mol) (see section 6.5).
Chapter 6
Results and Discussion
6.1 Charge parametrization of the metal centers in CcO
Accurate charge parameters are crucial for classical molecular dynamics simulations and electrostatic calculations, as they define electrostatic interaction energies. Electrostatic energy is of high importance e.g. in providing the catalytic
power of enzymes [31]. In addition to the use of "point charges" in classical simulations, they can give insight in the electronic structure of the system.
This study (paper I) resulted in QM-derived point charges for the four redox
active metal centers in CcO. CuA and heme a were studied in their reduced and
oxidized states, whereas heme a3 and CuB were studied in several different redox and ligand states. A few noteworthy points concerning the derived charges
are discussed below.
Interestingly, the formal charge of the metal in all metal centers is very delocalized; on average only 15-30% of this charge is localized on the metal itself. Spin
and charge densities of the two copper atoms in the dimetallic CuA center were
), the difference between spin
found to be nearly equal. As oxidized (Cu1.5+1.5
A
and charge densities between the two coppers, when integrating over a sphere
with half the Cu-Cu bond radius, was only 1%. In the reduced state, the charge
density differs by 10% between the copper atoms. This is why the dinuclear character of the CuA center was not found until two copper atoms were explicitly observed in the X-ray structure. Upon reduction, 7% of the charge localizes on the
coppers, while one fourth of the electron is spread to Cys-196 and another fourth
to Cys-200. Structurally His-161 and His-204 could be assumed to be chemically
equal. However, according to the RESP charges, His-204 becomes 11% more negative compared to His-161. This inhomogenity in charge distribution may be of
importance for the electron transfer between CuA and heme a.
The charge of the reduced and oxidized heme a iron is +0.45e and +0.78e, respectively. This implies that only 23-26% of the charge of the incoming electron
is localized on the iron, in agreement with earlier observations [66]. The reason
might be the six negatively charged nitrogens ligating to the central iron; four
from the porphyrin ring and two from His-161 and His-378. Interestingly, the
67
CHAPTER 6. RESULTS AND DISCUSSION
68
porphyrin nitrogens are chemically non-equal; the nitrogen of the C-ring has a
charge of +0.25e more positive than the nitrogen of the A-ring, and +0.14e more
positive than the nitrogen of the D-ring. This effect is due to the electron attracting vinyl group in the C-ring. The two propionates of heme a are likewise
non-equal; the carboxyl group of the A-propionate is +0.15e more positive than
corresponding group on the D-propionate. The carboxyl groups of both propionates have a 0.05e difference between reduced and oxidized states.
The charge of heme a3 is, similarly to heme a, spread out in the ring. However, there is a large difference in the charge of the heme a3 iron if the iron is
five or six coordinated. The charge of the iron is +0.34e and +0.41e in five coordinated ferric and ferrous states, respectively. This corresponds approximately
to 15% of the formal metal charge. As six coordinated, the degree of charge delocalization is large; the iron has a charge between +1.0e and +1.1e as ferric with
different ligands (H2 O, OH− , and O2− ). A water molecule was found to bind
more weakly to a ferrous state than the ferric state; the Fe-O bond length is 2.5 Å
in the former and 2.3 Å in the latter. The same effect is observed for the charge
of the iron, which is +0.56e for heme aII
3 -H2 O, resembling the charge of a five coordinated iron. Futhermore, there seems to be a connection between the bound
ligand of the iron and the charge of the A-propionate carboxyl group. The effect
might be due to the total charge of the heme a3 system rather than the character
of the ligand itself; the carboxyl group of the A-propionate has a charge of -0.90e,
-0.95e, and -0.83e, when the total charge of the heme a3 system is neutral, -e, and
+e, respectively. Similar variations in the carboxyl charge of the D-propionate
were not observed, which qualitatively supports that the A-propionate might be
a good candidate for the pump site [136, 162, 199].
Compared to the CuA center, the redox changes in the CuB system occur to a
much larger extent on the copper atom itself. As reduced, the charge of the copper varies between -0.07e and +0.15e, and as oxidized between +0.24e and +0.39e,
in different ligand states. In addition, charge variations of Tyr-244 seem to be
considerable. This is an important finding, since Tyr-244 has been simulated as a
methyl group in many previous theoretical studies [187, 188, 189, 190, 191]. For
example, the interpretation of His-291 as the pump-site might be influenced by
this [187, 188, 189]. However, also by including Tyr-244 in the structural model,
it is found that there is approximately a 0.1e variation of the Nǫ charge of His-291
between the reduced and oxidized CuB system when a water molecule or no ligand is bound to the copper and Tyr-244 is neutral. This is, however, not observed
when the neutral radical of Tyr-244 is reduced to a tyrosinate.
The point charge derivation methodology introduced here included a large part
of the protein environment to obtain a reasonable polarization of the redox active
group and to keep the charge of the redox system as close to neutral as possible.
For example, the charge of heme aIII
3 -O would be -3e without its environment.
Optimization of this system without the protein environment leads to the dissociation of the propionates from each other. The effect of environment on the RESP
charges is also considerable; in the present model, the total charge on the propionates and the farnesyl chain is -2.85e and -0.15e, respectively, whereas without
CHAPTER 6. RESULTS AND DISCUSSION
69
the environment, half of an electron charge shifts from the propionates to the farnesyl chain, yielding -2.36e and -0.64e for the propionates and farnesyl charges,
respectively. Heme a3 and CuB were studied separately, which gave a larger
modularity of the optimization scheme. This treatment was justified by studying
the difference in Mulliken charges between the combined and separate systems.
The maximum difference was found to be 0.043e. In the analyzed redox and ligII
and state, heme aIII
3 /Cu -OH YOH, the total charge of the subsystems should
equal +1e . It was found that the total charge of the subsystems were +0.95e and
+1.05e, respectively, which equals to a charge transfer of 1/20e. Although this
effect has been suggested to contribute to the 660 nm charge transfer band, this
effect might as well arise from an inaccuracy of the Mulliken charge method.
The drawback of the charge derivation scheme discussed above is the need of
very large QM-structures. For example, the heme a model system contains 278
atoms. It was found that the environmental effects are important for describing the correct charge distribution in the metal centers. A possible improvement
of this methodology, which would reduce the computational cost considerably,
could be obtained by introducing a QM/MM model, in which the environment is
described by classical point charges or preferably by point charges and LennardJones interactions. Preliminary tests showed that the charge distribution of the
propionates of heme a3 were described reasonably well with an MM environment. However, when representing the backbone of the CuB system with explicit point charges, the Cβ polarizes 0.118e too negative. Future work is therefore needed to obtain a consistent description of the border region in a QM/MMderivation scheme. The QM-derived point charges obtained from this study have
been used in all classical simulations discussed below.
6.2 Dynamics of the Glu-242 side chain in CcO
Glu-242, which is at the end of the D-channel, points "down" towards the Dchannel in all available X-ray structures [25, 55, 56]. It has therefore no apparent connectivity to the binuclear site or the propionates of heme a3 . Quite early
after the appearance of the first X-ray structures of CcO [104, 153] it was however predicted that the non-polar cavity may be occupied by 3-5 water molecules
[103], which could provide a pathway for proton conductance. Pomès found
that the rotational isomerization of Glu-242 has a relatively low barrier [163].
However, prior to the current study, this had not been observed in unbiased MDsimulations [332].
In this study (paper II), the dynamics of the protonated Glu-242 side chain was
found to depend on the number of water molecules in the non-polar cavity above
this residue (Fig. 6.1). The side chain isomerization was found to occur easily
when at least four water molecules were present in the non-polar cavity. Without
any water molecules in the non-polar cavity, a combination of maximum likelihood analysis and MD-simulations gave a rough estimate of an up/down equilibrium with a 1/20 preference for the "down" configuration, corresponding to
ca. 2 kcal/mol. In the "down" configuration Glu-242 was found to interact with
CHAPTER 6. RESULTS AND DISCUSSION
70
the sulfuric group of Met-71. By switching off the interaction between the four
water molecules and Glu-242, it was found that Glu-242 interacts by 6 kcal/mol
with the water molecules above it. Assuming that the interaction in the "down"
configuration is the same in an empty or a filled cavity, it was concluded that the
"up" configuration is preferred by 4 kcal/mol when the cavity is fully hydrated.
These findings could easily account for the observed X-ray structures of CcO,
where Glu-242 points "down". Theoretical predictions suggested that the water
molecules are relatively weakly bound to the enzyme [103, 333], and that water
molecules may escape the cavity through opening of the Arg-438-D-propionate of
heme a3 ion pair [165]. These findings may also be connected to the long known
"resting" and "pulsed" states of the enzyme [181, 182], of which the former state
has a retarded heme-heme eT-rate. Reduction and re-oxidation yield a "pulsed"
enzyme with a normal eT-rate. It is known that the heme-heme eT couples to
internal proton transfer, probably via water molecules in the active site of CcO.
As water molecules are synthesized during reduction and re-oxidation, the "resting" state of the enzyme might therefore correspond to a "dry" cavity, in which
the proton-linked eT is hindered. There are two possible explanations to why the
available X-ray structures of CcO are devoid of water molecules in the non-polar
cavity [25, 56, 55]; first, dynamic or unordered water molecules inside protein
cavities are not always resolved by X-ray crystallographic techniques [334], or
second, the enzyme might simply be devoid of water molecules and crystallized
in a "resting" state. The "down" conformation of Glu observed in these structures
suggest, in the light of the present results, the latter explanation.
Figure 6.1. The effect of water potential on the up-flip of protonated Glu-242. The figure shows that the isomerization to and "up" configuration of Glu-242 can easily take
place when at least four water molecules are present in the non-polar cavity whereas
the isomerization was not observed in 1 ns equilibrium MD-simulations with less water
molecules in the cavity. Reproduced from paper [II].
CHAPTER 6. RESULTS AND DISCUSSION
71
6.3 Glu-242 is a valve in the proton pump of CcO
For CcO to be able to pump protons, there must be a continuous proton pathway
from the N-side to the P-side of the membrane. However, if this pathway would
be open at any given time, protons would spontaneously leak back to the N-side
of the membrane. This poses a mechanistic question of a valve that prevents the
protons from leaking.
The results obtained from the previous study were further investigated by incorporating the dynamics of the deprotonated Glu-242 in the model. After fully
hydrating the initial X-ray structure by MD-simulations, it was found that the
deprotonated Glu-242 flips with a t1/2 ≈ 1 ps from an "up" to a "down" configuration, with a ≥ 104 preference for the latter (Fig. 6.2). In the X-ray structure
of CcO from Bos taurus [25], water molecules were lacking from the space below Glu-242, which strongly affected the obtained down-flip rate of the deprotonated Glu. The position of these water molecules, which were found by MDsimulations, are present in e.g. the X-ray structure of CcO from Rh. sphaeroides
[97]. It has been demonstrated the water molecules lacking from non-polar protein cavities in X-ray structures can effectively be found by MD-simulations [335].
The rate of the Glu down-flip was somewhat affected by the four negative hemepropionate charges together, but not by any of them individually, and as pointed
out above, the hydration state below Glu-242. However, when Glu-242 was protonated, both "up" and "down" states were found to be easily accessible, in accordance with the previous study [paper II]. A small variation of the equilibrium
constant between "up" and "down" was found between the PM and PR states,
which could explain the IR-shift of the peak at 1740 cm−1 depending on the redox state of heme a [306, 336, 337, 338, 339]. The "up"/"down"-dynamics for the
deprotonated and protonated Glu-242 were combined in a thermodynamic cycle
(Fig. 6.3a). It was found that the pKa of Gludown of 9 (based on experimental
[202, 306, 340] and theoretical [191, 332, 341] studies) is shifted to 13, and 11.7 for
the Gluup in PM and PR states, respectively. Most importantly, the dynamics of
the deprotonated Glu provides a valve which prevents protons from the proton
loading site (PLS) to leak back to Glu-242, because of the rapid and thermodynamically preferred "down" configuration with no connectivity to the PLS. The
derived energy diagrams are shown in Fig. 6.3 for the first proton transfer to the
PLS (Fig. 6.3b), and the second pT to the BNC (Fig. 6.3c). The proton pump
is most vulnerable to leaks when the PLS is protonated and the BNC has just
been reduced and protonated by Glu-242. However, the occupancy of the most
−6 before
vulnerable Glu−
up /BNCH compared to the lowest lying state is only 10
final slow release of the proton from the PLS (Fig. 6.3). This low occupancy is
suggested to provide a valve which can block protons from leaking back from
the PLS to the N-side. When the kinetic barrier for the protonation of Glu−
down
from the D-channel is increased, the population of the vulnerable Glu−
up state increases. This might explain the uncoupling mutants [159, 160] with mutations ca.
16 Å away from Glu-242 in the D-channel, by a kinetic rather than a thermodynamic basis [159, 160, 161] for the uncoupling effect.
CHAPTER 6. RESULTS AND DISCUSSION
72
It is interesting to compare the Glu valve to the mechanism by which bacteriorhodopsin (bR) prevents proton leaks. In the photocycle of bR, isomerization of
the all-trans to 13-cis Schiff base by absorption of light quanta (BR → K), couples
to proton transfer from the protonated Schiff base to Asp-85 (L → M1 ), located
close to the extracellular side of the membrane [342]. As the Schiff base deprotonates, its proton donating N-H group turns away from the protonated Asp-85
(in M2 ) and establishes a connection with Asp-96 near the cytoplasmic side of the
membrane [343]. Re-protonation of the the Schiff base (in M2 → N) restores the
all-trans configuration (in N’ → O) [343]. Both Glu-242 and the proton donating
group of the Schiff base turn away from the protonated PLS site (D-propionate
of heme a3 in CcO and Asp-85 in bR) to prevent leaks. In addition, Glu-242 and
the Schiff base have an equilibrium configuration as deprotonated towards the
N-side (cytoplasm/matrix) of the membrane, whereas as protonated they can be
in contact with both P- (extracellular/intermembrane space) and N-side of the
membrane. Interestingly, in bR there seems to be a stable water path between
the Schiff base and Asp-85 (the PLS), whereas the region between Asp-96 and the
Schiff base seems to be devoid of bound water molecules in the X-ray structures.
During the photocycle, this non-polar cavity is transiently filled with water to allow re-protonation of the Schiff base. As previously pointed out, the D-channel is
highly hydrated, whereas the non-polar cavity appears empty in the X-ray structures. However, transient emptying/filling of the D-channel is not likely to take
place. The proton is released from Asp-85 to the extracellular side approximately
100 µs after its protonation, whereas the proton at the PLS in CcO cannot be released to the P-side before a second pT reaction has taken place to the BNC. As
this leads to a slow proton release in CcO (2.6 ms) [184], it is important that the
re-protonation of Glu-242 takes place quite rapidly. Despite many similarities
[344], the main difference is that CcO takes up two protons per electron transfer
step, whereas bR takes up one proton per light quantum. This might put higher
requirements on a tight leak valve in systems resembling CcO.
6.4 The chemistry of the CuB site in CcO
The CuB site is spectroscopically invisible in most states of the reaction cycle.
However, this site is of major importance because it has proton connectivity to
both D- and K-channels [156, 157, 195] and its redox state might determine proton pathway formations in the non-polar cavity (see section 6.6). This motivated
to study the CuB site quantum chemically in different redox and ligand states of
the reaction cycle. This study (paper IV) had two main aims: first, to understand
the electronic structure of CuB system in different redox and ligand states and its
connection to the reaction cycle of CcO, in particular the connection to the activation of the K-channel, and second, to understand the importance of the unique
covalent His-Tyr bond.
In agreement with structural studies on simple inorganic compounds [345], and
EXAFS measurements [346] on the CuB site in CcO, the cupric and cuprous states
of the model system preferred a trigonal and tetragonal coordination of the copper, respectively. In CuI structures the bond between Cu and His-290 is on aver-
CHAPTER 6. RESULTS AND DISCUSSION
73
Figure 6.2. a) The contour of subunits I and II, and the location of redox active groups.
Dotted lines represent electron transfer pathways, and solid lines represent proton pathways. b) Glu-242 in its "up" and "down" configurations. Proton transfer to the pump
site (D-propionate of heme a3 ) is assumed to take place from the "up" conformation of
Glu-242. Reproduced from paper [III].
age 0.23 Å longer than the two other Cu-His bonds (2.18-2.26 Å vs. 1.96-2.01 Å),
whereas in CuII structures all three Cu-His bonds are equally long (1.96-2.01 Å).
It was found that the preferred structure in the F and O states is a formal CuII H2 O YO− state. This state is preferred by 2.1 kcal/mol over the isoelectronic
CuII -OH YOH form (Fig. 6.4a). This finding is in agreement with results obtained from the optical spectra of heme a3 , which suggest that a water molecule
might bind to CuB in state F [82]. Interestingly, the electronic structure of the former state was found to resemble more closely the CuI /Tyr-O• resonance form.
The proton affinity of the preferred F and O states was found to be low. However, when reduced to a CuI -H2 O YO− , which might take place in the OH,R state
(the reduced OH state), the proton affinity of Tyr-244 increases drastically (Fig.
6.4a), which might attract a proton from the K-channel. Protonation of the heme
a3 ligand by Tyr-244, in analogy to what has been proposed in the A → PM [347],
was suggested to trigger a second pT from the K-channel to Tyr-244. The radical character of CuII -H2 O YO− was found to depend on a positive charge in
the K-channel, simulating the effect of the protonated Lys-319. Recent IR-studies
suggest that Tyr-244 is anionic in PR , F, and partially in OH [83], which might
account for the Lys-319 perturbation effect. Interestingly, Lepp et al. found that
Lys-319 mutation to a methionine slows down the A → PR transition [348], which
might be explained by stabilization of an anionic tyrosinate resonance form of the
PM state in the PM → PR transition.
The His-Tyr bond was found to work as a wire for charge movement from Tyr-244
to His-240. Due to a higher charge delocalization, this increases the electron affinity of Tyr-244 and decreases its proton affinity. Perturbation of the proton and
electron affinity has also been observed previously in both experimental [81] and
computational [186, 349, 350] studies on simple model systems on a imidazol-
CHAPTER 6. RESULTS AND DISCUSSION
74
Figure 6.3. Thermodynamic cycle (a) and energy diagrams of the Glu-valve of the first (b)
and second (c) proton transfer step. a) The thermodynamic cycle predicts that the pKa of
Glu in its "up" configuration is 4.0 (PM ) and 2.7 (PR ) pKa units higher than in its "down"
configuration. b) and c) the down-flip of Gluup is kinetically fast and thermodynamically
favorable. Reproduced from paper [III].
phenol dimer. The preference of protonating the hydroxy ligand of the copper
over Tyr-244 was also found to be stronger when the His-Tyr bond is present
(Fig. 6.4b).
CHAPTER 6. RESULTS AND DISCUSSION
75
Figure 6.4. Energy level diagrams of the CuB system in the reaction cycle. Above the wild
type system, below the "His-Tyr mutant". The proton affinity of CuII -H2 O YO− is low,
whereas reduction to CuI -H2 O YO− increases the proton affinity of Tyr-244 drastically,
which might lead to proton uptake from the K-channel. Reproduced from paper [IV].
6.5 Mechanism and energetics for preventing leaks in CcO
The Glu-valve, which was proposed in paper III, was further analyzed in this
study (paper V) by incorporating the Glu dynamics in the general framework of
the pumping mechanism (Fig. 6.5). Glu cycles in this model between four different states; in protonated (GH/gH) and deprotonated (G-/g-), "up" (G-/GH)
and "down" (g-/gH) configurations, respectively (Fig. 6.5). The resulting energy
diagram is presented in Fig. 6.6, and the effect of a membrane potential on the
Glu-valve is shown Fig. 6.7. The total energy of -7.7 pK-units (10.8 kcal/mol) for
one pumping step is obtained by dividing the released energy of the cytochrome
c oxidation [326] and reduction of molecular oxygen, corrected by the physiological oxygen concentration (20-120 µM), equally between four proton pumping
CHAPTER 6. RESULTS AND DISCUSSION
76
steps, and correcting for the oxygen binding energy [136, 162, 347]. Another experimental requirement was the pKa perturbation of the PLS from 5.3 to 9 upon
reduction of heme a [136]. These ansätze led to a model with two ǫ values; one describing the environment between the hemes, and the other describing the dielectric constant of the medium between Glu-242 and its interacting groups (heme a,
PLS, BNC). These two ǫ values were found to satisfy a/ǫ′ + b/ǫ = c, where a,
b, and c, are contants determined by the energy constraints (see above). If the
heme-heme dielectric constant is chosen to be ǫ′ =3.8, this gives ǫ=33, and specifies the pKa (B−) to 11.2, which is lower than the value predicted by Wikström
and Verkhovsky [136]. These ǫ values represent also a limit, where the reprotonation of Glu in presence of a full membrane potential (200 mV) is nearly isergonic.
The choice of dielectric constants affects e.g. the last protonation step of g- as
the proton at the pump site lowers the pKa of g, which was set to have a ground
state pKa of 10 when one charge uncompensated electron is in the system. As
the charge perturbation causes the pK of g to approach 7, protonation from the
N-side becomes more difficult. There are also other possibilities for the choice of
a dielectric constant. For example, a distance dependent effective ǫ(r)1 has been
employed in some theoretical studies [352].
The membrane potential increases somewhat the probability of leaks, but the
high thermodynamic preference of the g- over G- effectively secures a low occupancy of the G- state. For the pump not to leak in the steady-state equilibrium
approximation, the kinetic barrier between states Va and VIIIa (see Figs. 6.5 and
6.6) must be considerably higher than the final release of protons from the proton
loading site to the P-side (Vc → VI, Fig. 6.5). If a minimum barrier for preventing
leaks is set to 16 kcal/mol or 11.3 pK-units (as in [162]), and the pump is required
to work also in 200 mV, where the relative steady-state occupancy of G- in state
Va is reduced from 6 to 3.8 pK-units, this kinetic barrier must be at least 11.3-3.8
pK-units=7.5 pK-units (shaded lines in Fig. 6.6).
If no Glu-valve would be present, it was found that the occupancy of G- without
a membrane gradient is ≈ 10−2 , which might prevent leaks to a certain extent.
However, in presence of a full membrane potential (200 mV) the occupation of
the vulnerable G- state is 100.5 without the valve, which is highly subjected to
leaks. This finding might suggest that enzymes which lack the Glu valve can
pump protons in zero pmf, but leaks are much more prominent in high pmf.
There are some known D-channel mutants which are unable to pump protons but
have a normal or even slightly increased oxygen reduction activity [159, 160] (see
section 2.4.3). It is possible that that Tyr-19 is a proton donor for Glu-242 [353]. If
re-protonation of Tyr-19 is slowed down in the uncoupling D-channel mutants,
it would increase the population of anionic Tyr-19. The question whether the
anionic Tyr-19 shifts the equilibrium constant between G- and g-, increasing the
population of the vulnerable G- state and thus increasing the amount of leaks, is
currently studied further. In addition, increase of the kinetic barrier of g- → gH
1
ǫ(r) = 1 + ǫ′ [1 − exp(−µr)] where ǫ′ is usually 20-80 and µ=0.18 [351].
CHAPTER 6. RESULTS AND DISCUSSION
77
also increases the probability for leaks. Therefore, it is possible that the uncoupling effect is due to an impaired Glu valve due to a retarded pT rate from the
D-channel.
It was also found that the occupancy of G- can increase due to a "dry" Gluenvironment and lead to leaks. It is plausible that during OH → O, some water molecules might escape the non-polar cavity, which would prevent the normal Glu dynamics. The water molecules in the non-polar cavity have theoretically been found to bind relatively weakly to the cavity [84, 333]. It has also
been observed in MD simulations that water molecules in the non-polar cavity
might escape through opening of the Arg-438-D-propionate ion pair of heme a3
[165]. Dehydration is further supported by experiments which show that reduction, and re-oxidation of the non-pumping O state, yields the pumping OH state
[178, 177, 180]. During this process, water molecules are formed in the BNC,
which might regain the fully hydrated state of the enzyme.
Figure 6.5. The Glu-valve inserted in the sequence of proton transfer in CcO, together
with the nomenclature of states. G and g, are Glu-242 in its "up" and "down" configurations, respectively. X- is the pump site, B is the binuclear site and the squares are heme a
(left) and heme a3 (right). The red color indicate that a group is reduced and a blue color
that it is protonated. The shaded states represent the assumed main leakage pathway,
where the proton at the pump site leaks back to G- instead of being transferred to the
P-side. Reproduced from paper [V].
CHAPTER 6. RESULTS AND DISCUSSION
78
Figure 6.6. Energy level diagram for one proton-pumping step in CcO. Energy levels and
kinetic barriers shown in red indicate rates obtained directly from experimental measurements or from experimental boundary conditions [184]. Nomenclature of states is the
same as in the previous figure. Shaded lines represent the leakage pathway. Reproduced
from paper [V].
Figure 6.7. Effect of membrane potential on the Glu-valve. Only the thermodynamic
energy levels are shown. Nomenclature of states is the same as in figure 6.5. Reproduced
from paper [V].
CHAPTER 6. RESULTS AND DISCUSSION
79
6.6 Formation of redox-linked proton pathways in CcO
As earlier discussed, CcO involves two types of proton transfer reactions; pumped
protons, which are transferred across the membrane, and chemical protons, which
are transferred for the oxygen chemistry. Ever since it was discovered that these
two proton transfer reactions are involved in the catalytic reaction of CcO [52],
several mechanisms have been proposed to explain the bifurcation of proton
transfer pathways (see section 2.7).
According to the water-gated mechanism by Wikström et al. [195] (see section
2.7), water molecules in the non-polar cavity above Glu-242 were suggested to
constitute the molecular switch. In this work (paper VI), gating of proton transfer by redox-state dependent orientation of water molecules was further investigated.
To understand gating of proton transfer in CcO it might be enlightening to discuss factors affecting proton transfer in simpler systems. As discussed in section 2.4.3, proton transfer in water involves solvation of Zundel (H5 O2 + ) [354]
and Eigen (H9 O4 + ) [355, 356] like species, where the fluctuation of the solvation
sphere dictates the ps-time scale proton transfer process [149, 150, 357, 358, 359].
In addition, intermediate species of the Zundel and Eigen cations have a significant population [150]. In contrast to this 3-dimensional structural diffusion of the
proton, proton transfer in biological systems often takes place in confined onedimensional hydrogen bonded water chains formed in non-polar protein cavities [360]. The proton conductance in these water chains is very rapid (about 40
times faster than in bulk, where the protons have a conductance of 36.23x10−8
m2 /Vs [361]). This is probably due to the reduced dimensionality and pT reaction, which takes place in a concerted reaction [362, 363, 364]. Although the conductance of protons can also be very rapid in these quasi-one dimensional water
arrays, the desolvation energy of the proton can be large, depending on the polarity and specific interactions in the pore. Dellago and Hummer estimated this
barrier in a water filled carbon nanotube (CNT) to be approximately 10 kcal/mol,
which would make the conductance of the CNT about 1 H+ /h [365]. Other systems, such as the gramicidin A channel compensate for this desolvation energy
by backbone carbonyl and amide groups, which stabilize the proton, and have
thus a relatively high proton conductance rate [366].
The blockage of proton transfer in the aquaporins (Aqp1) and glycerol (GlpF)
channels were first explained to be prevented by the formation of a wrongly oriented water chain [197, 198], very similarly to the idea in the water-gated mechanism in CcO. In equilibrium MD-simulations it was observed that 7-9 water molecules make a hydrogen bonded water array around three polar residues Arg-206
Asn-203, Asn-68, which are located between the M3 and M7 helices around the
NPA-motif of the channel [197, 198]. It was later shown that protons have a very
large desolvation energy for entering the channel, whereas the origin of this high
barrier has remained controversial. Chakrabati et al. argued that this barrier is
due to the helix dipole moment of M3 and M7 which culminates at the NPA-motif
[367]. Inserting a proton at this position would be influenced by a high repulsive
CHAPTER 6. RESULTS AND DISCUSSION
80
force from the helix dipole moments which could be removed by switching off
the helix dipoles, or compensated by introducing a negative charge in the channel
(N98D, N203D). Based on the earlier work by Burykin and Warshel [368], Kato et
al. argued that the origin of this barrier is neither the helix dipole moment nor the
NPA-motif [369]. By switching off these elements they found a reduction of the
barrier 15 kcal/mol (from 25 to 10 kcal/mol) by letting the protein environment
relax, whereas the total barrier of Chakrabati et al. was only 15 kcal/mol. Thus,
Kato et al. argued that the barrier is only due to the high desolvation energy of
the proton in the non-polar channel. In addition, they suggested that as this desolvation energy is very large, the orientational effect of water molecules must
be negligible in comparison to the desolvation energy. The same argument was
used in the study by Pisliakov et al. in the proton conductance of the proton from
Glu-242 to the pump site in CcO [196].
Proton transfer in CcO have some characteristic differences compared to the pT
in Aqp1 and CNTs. The donor of the proton is not an oxonium ion or its derivative as in bulk water, but rather a neutral group (Glu-242). As the "proton" is
already in the channel, the desolvation energy is therefore probably of less importance, whereas water orientation effects, and/or emptying/filling non-polar
cavities as in bR can have a more significant contribution. The issue in CcO is
choosing between to two plausible proton transfer pathways, rather than comparing the desolvation of the proton to the bulk.
The initial step in the pT process in the non-polar cavity of CcO is protonation
of the water molecule closest to Glu-242 and formation of a COO− H3 O+ pair.
As it has been found that proton transfer takes place through hydrogen bonded
water chains, the kinetic barrier of forming this ion pair in the presence and absence of the hydrogen bonded water chain was investigated by hybrid density
functional and ab initio calculations. It was found that the barrier of proton transfer is ca. 8 kcal/mol lower if the proton accepting water molecules are a part of a
pre-formed proton pathway compared to a situation where the proton pathway
is not pre-formed (Fig. 6.8). This finding indicates that proton transfer can be
controlled by forming proton pathways by hydrogen bonded water molecules.
This stabilization is due to a larger charge relaxation in the protonated water
chain. These findings are also in agreement with results obtained from combined
QM(quantum dynamics)/MM simulations on bacteriorhodopsin, where it was
found that pT from Asp-96 to the photoisomerized Schiff base takes place only
after a complete hydrogen bonded water chain has formed [370]. In fact, careful investigation of these QM-trajectories shows that the dissociation of the proton from Asp to the closest water molecule is very transient and does not fully
take place before the full proton pathways is formed to the Schiff base (QM/MM
movie of bR, personal communication with Dr. Yong-Sok Lee). This suggests
that formation of hydrogen bonded water-chains have a significant contribution
in determining the proton transfer process.
Formation of two proton pathways was found to strongly depend on the redox
state of the metal sites, in agreement with early studies by Wikström et al. [195].
More specifically, it was found that when heme a is reduced and the BNC is in
CHAPTER 6. RESULTS AND DISCUSSION
81
the PM state, a proton pathway is formed to the PLS, but never to the binuclear
site (Fig. 6.9). When the electron was transferred to the BNC, i.e. when heme a
was oxidized and the BNC in the PR state, a proton pathway could easily form
to the BNC (Fig. 6.9). The same trend can be observed in other transitions of
the catalytic cycle as well, which suggest the phenomenon is not limited to this
transition.
Figure 6.8. The effect of the proton pathway on the energetics of proton transfer. The
curves which are lower in energy correspond to protonating the first water molecule if a
proton pathway is fully formed to the acceptor. The curves which are higher in energy
correspond to a situation where the stabilizing effect of the two water molecules above
this protonated water have been removed. The profiles are calculated on B3LYP (blue)
and MP2 (red) levels of theory using a def2-TZVP basis set. At a def2-QZVPP basis set
the energies for protonating a fully formed water chain are 8.3 (B3LYP) and 7.4 kcal/mol
(MP2). Reproduced from paper [VI].
The most crucial point is to prevent formation of a proton pathway to the binuclear site in the "pump" step, because if the binuclear site could be protonated
before proton-pumping, it would compromise the pump. To study the "fidelity"
of this mechanism, the PMF-surface for forming a proton pathway to the BNC
in the "pump" step is shown in Fig. 6.10. The kinetic energy barrier for forming
a "chemistry pathway" in this redox state is ca. 14 kcal/mol, which provides a
reasonable barrier for preventing the "wrong" reaction. In addition, for creating
the pathway itself, the energy for transferring a proton to a pre-formed pathway
must be added, which gives a total barrier of ca. 22 kcal/mol, i.e. a 107 preference
for the correct pathway, as the pump step is experimentally known to take place
within 150 µs [184]. It must however be remembered that the 8 kcal/mol energy
of protonating a preformed water chain, is obtained from QM studies of a simple
model system, and provides only qualitatively the barrier for pT in the "wrong"
direction. The molecular basis for this high barrier can be understood from the
CHAPTER 6. RESULTS AND DISCUSSION
82
interaction energies. The four water-molecules interact very strongly with each
other (5-10 kcal/mol) in the non-polar cavity, and with the donor (Glu-242) and
acceptors (D-propionate of heme a3 or the oxygenous ligand of the CuB ). When
the redox state is relaxed from heme ared /PM to heme aox /PR and there is a proton pathway open to the BNC, the interaction energy relaxes by 10 kcal/mol of
which CuB provides 80% and heme a the remaining. This suggests that the redox state of the CuB site is of major importance in determining the formation of
proton pathways to the BNC. This is in contrast to the original proposal by Wikström et al. where it was suggested that it is the direction of electric field between
heme a and the BNC, which determines the orientation of water molecules. The
major importance of the redox state of the CuB site is also observed from equilibrium MD simulations, where both heme a and the BNC are reduced, which gives
a dominant "chemistry pathway". Fig. 6.10 shows two represenative structures
of typical PM (above, right) and PR (above, left) states. In the former, the space
between Glu-242 and CuB seems to be devoid of water molecules whereas in the
latter, it is filled by three water molecules, which form a proton conducting wire
to the same site. The behavior of water molecules can be understood from the
electric field present in the non-polar cavity; during PM → PM an electric field of
≈ 1.8 V/Å develops between Glu and CuB due to the reduction of the former.
The electric field causes a translational force to the water dipole moment, which
is proportial to ∇E · µ. Similar electric field induced hydration effects have perviously been observed in carbon nanotubes [371]. In addition, there is a striking
resemblance to other systems such as bR, cytochrome P450 (see above), and more
general phenomena such as membrane electroporation [372].
Figure 6.9. Above, probability distribution for formation of the "chemistry path" when
the electron is on heme a (blue curve) and CuB (red curve) obtained from 5 ns equilibrium
MD simulations. Below the probability distribution for the "pump path". The reaction
coordinate dmax measures the longest distance within the shortest pT route. When dmax
≤ 2 Å a Grotthuss proton transfer reaction can take place. Reproduced from paper [VI].
CHAPTER 6. RESULTS AND DISCUSSION
83
This study suggests that bifurcated proton transfer in biological systems can be
controlled by redox-state dependent proton pathway formations in "tight" nonpolar cavities. This finding can inspire the design of future biomimetic fuel cells
and enhance the understanding of biological proton transfer more generally.
Figure 6.10. Above, typical proton pathways in heme ared /PM state (to the right) and
heme aox /PR state (to the left). In the former, a proton pathway is formed between
Glu-242 and the D-propionate of heme a3 , but not to the CuB site. In the latter, proton
pathways are present to both sites. Below, the free energy surface for forming the wrong
proton transfer pathway. The reaction coordinate dmax describes the longest hydrogen
bonded distance between within the shortest pT route from Glu-242 to the oxygenous
ligand of CuB . Reproduced from paper [VI].
Chapter 7
Conclusions
In this thesis, methods of theoretical biochemistry were employed to explore the
coupled electron and proton transfer mechanism of cytochrome c oxidase. In
particular, three aspects of the pumping mechanism were studied: 1) what prevents the protons from leaking in the proton pump, 2) why two proton conducting channels are employed, and 3) how protons are gated to be pumped and
consumed for the oxygen reduction chemistry. In addition, a new parametrization scheme for deriving accurate charge distributions for classical simulations of
complicated metal active centers was introduced.
It was found that Glu-242, an amino acid in the end of the proton conducting
D-channel, can effectively work as a valve for preventing protons from leaking
in the pumping step. The conformation of the Glu-242 side chain was found to
depend strongly on the protonation state of the residue. As deprotonated, it both
kinetically and thermodynamically prefers the contact with the D-channel and
blocks the contact to the pump site, while as protonated it has roughly equal connectivity to both sides. It was found that especially under high proton motive
force, such as in the mitochondrial state 3, a valve of this kind effectively prevents leaks.
The activation of the K-channel might link to the chemistry of the CuB site. It
was found that in the O→E step, a state appears in which Tyr-244 has a high proton affinity. This state was suggested to attract a proton from the K-channel. An
interesting connection was found between the protonation state of the K-channel
and the electronic structure of the CuB site, more specifically between the internal electronic equilibrium between the copper and Tyr-244 of the CuB site. This
effect might be of importance in explaining recent experimental findings.
Formation of redox-state dependent proton pathways can work as a gate between
proton consumption and pumping in the active site of CcO. When heme a is reduced and the BNC is oxidized, the kinetic energy barrier for the formation of
a proton pathway to the binuclear site was found to be very high, but when the
electron is transferred to the binuclear site, this pathway opens up. Formation of
proton pathways together with the Glu valve secures a tight coupling in redoxlinked proton-pumping of cytochrome c oxidase.
85
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