Modeling of voltage-gated ion channels
Modeling of voltage-gated ion
channels
Pär Bjelkmar
c Pär Bjelkmar, Stockholm 2011, pages 1-65
Cover picture: Produced by Pär Bjelkmar and Jyrki Hokkanen at CSC - IT Center for Science, Finland. Cover
of PLoS Computational Biology February 2009 issue.
ISBN 978-91-7447-336-0
Printed in Sweden by US-AB, Stockholm 2011
Distributor: Department of Biochemistry and Biophysics, Stockholm University
Abstract
The recent determination of several crystal structures of voltage-gated
ion channels has catalyzed computational efforts of studying these remarkable molecular machines that are able to conduct ions across biological membranes at extremely high rates without compromising the
ion selectivity.
Starting from the open crystal structures, we have studied the gating
mechanism of these channels by molecular modeling techniques. Firstly,
by applying a membrane potential, initial stages of the closing of the
channel were captured, manifested in a secondary-structure change in
the voltage-sensor. In a follow-up study, we found that the energetic cost
of translocating this 310 -helix was significantly lower than in the original conformation. Thirdly, collaborators of ours identified new molecular
constraints for different states along the gating pathway. We used those
to build new protein models that were evaluated by simulations. All
these results point to a gating mechanism where the S4 helix undergoes
a secondary structure transformation during gating.
These simulations also provide information about how the protein interacts with the surrounding membrane. In particular, we found that
lipid molecules close to the protein diffuse together with it, forming a
large dynamic lipid-protein cluster. This has important consequences for
the understanding of protein-membrane interactions and for the theories
of lateral diffusion of membrane proteins.
Further, simulations of the simple ion channel antiamoebin were performed where different molecular models of the channel were evaluated
by calculating ion conduction rates, which were compared to experimentally measured values. One of the models had a conductance consistent
with the experimental data and was proposed to represent the biological
active state of the channel.
Finally, the underlying methods for simulating molecular systems
were probed by implementing the CHARMM force field into the
GROMACS simulation package. The implementation was verified
and specific GROMACS-features were combined with CHARMM and
evaluated on long timescales. The CHARMM interaction potential was
found to sample relevant protein conformations indifferently of the
model of solvent used.
5
List of Papers
This thesis is based on the following papers, which are referred to in the
text by their Roman numerals.
I
II
III
IV
V
VI
Bjelkmar, P., Niemela, P. S., Vattulainen, I., Lindahl, E.
(2009) Conformational changes and slow dynamics through
microsecond polarized atomistic molecular simulation of an
integral Kv1.2 ion channel. PLoS Computational Biology, 5(2)
e1000289.
Schwaiger, C. S., Bjelkmar, P., Hess, B., Lindahl, E. (2011)
310 -helix conformation facilitates the transition of a voltage
sensor S4 segment toward the down state. Biophysical Journal,
100(6):1446-54.
Henrion, U., Renhorn, J., Börjesson, S. I., Nelson, E. M.,
Schwaiger, C. S., Bjelkmar, P., Wallner, B., Lindahl, E., Elinder, F. (2011) Tracking a complete voltage-sensor cycle with
metal-ion bridges. Submitted.
Niemela, P. S., Miettinen, M. S., Monticelli, L., Hammaren,
H., Bjelkmar, P., Murtola, T., Lindahl, E., Vattulainen, I.
(2010) Membrane proteins diffuse as dynamic complexes with
lipids. Journal of the American Chemical Society, 132(22):75745.
Wilson, M. A., Wei, C., Bjelkmar, P., Wallace, B. A., Pohorille, A. (2011) Molecular dynamics simulation of the antiamoebin ion channel: linking structure and conductance.
Biophysical Journal, 100(10):2394-402.
Bjelkmar, P., Larsson, P., Cuendet, M., Hess, B., Lindahl, E.
(2010) Implementation of the CHARMM force field in GROMACS: Analysis of protein stability effects from correction
maps, virtual interaction sites, and water models. Journal of
Chemical Theory and Computation, 6:459-66.
Reprints were made with permission from the publishers.
7
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . . . . . . 12
. . . . . . . . . . . . . . . . . . . . . . . . . 12
. . . . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . 14
. . . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . 20
. . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Molecular modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.1 Basic molecular dynamics theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.2 Force fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.3 Statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.4 Modeling the membrane potential in simulations . . . . . . . . . . . . . . . . . 34
2.2 Modeling of protein structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Comparative modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 Physics-based modeling and structural refinement . . . . . . . . . . . . . . . . 39
3 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.1 Membranes and transport proteins . . . . . .
1.1.1 Material transport across membranes
1.1.2 Cellular membranes . . . . . . . . . . . .
1.1.3 The membrane potential . . . . . . . . .
1.2 Voltage-gated potassium channels . . . . . .
1.2.1 Voltage-dependent gating . . . . . . . .
1.2.2 Slow inactivation . . . . . . . . . . . . . . .
1.2.3 The role of 310 -helices . . . . . . . . . . .
3.1 Conformational changes in the KV 1.2 ion channel through polarized
microsecond molecular simulation (paper I) . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Secondary structure of the S4 helix in gating. (paper II) . . . . . . . . . . . . . . . .
3.3 Tracking the voltage-sensor gating motion with metal-ion bridges, computer
modeling and simulation (paper III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Lateral diffusion of membrane proteins (paper IV) . . . . . . . . . . . . . . . . . . . .
3.5 Linking structure and conductance in a simple ion channel (paper V) . . . . . .
3.6 Implementation of the CHARMM force field in GROMACS (paper VI) . . . . . .
4 Concluding remarks and future perspectives . . . . . . . . . . . . . . . . . . . .
5 Svensk populärvetenskaplig sammanfattning . . . . . . . . . . . . . . . . . . .
6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
42
43
43
44
45
47
51
53
57
1. Introduction
The fact that life evolved out of nearly nothing, some
10 billion years after the universe evolved out of
literally nothing, is a fact so staggering that I would
be mad to attempt words to do it justice.
Richard Dawkins
T
his doctoral thesis describes the work that I have done during my time
as a PhD student at the Department of Biochemistry and Biophysics at
Stockholm University during the years of 2006 through 2011. The studies
have been published in peer-reviewed scientific journals and the papers
are the core material of the thesis. They are presented as appendices in
the end of this book. To put this work in perspective and to provide a
brief overview to the field, a preceding introductory and summarizing
text has been compiled. It starts with a background chapter introducing
the underlying biology and continues to describe the methodology and
key concepts in greater depth. The next two chapters contain a summary
of the main results and conclusions, and a technology walkthrough including some speculation of where we might head in the future. The text
ends with a popular science summary in Swedish, my acknowledgements
and the bibliography.
The work has been a great journey in the interdisciplinary field I prefer
to call computational biology. The proteins that have been studied are
incredible molecular machines incorporated in cellular membranes. They
are precisely regulated by electrical changes in the local environment and
capable of highly selective conduction of ions at great speeds. The opening and closing of these proteins have been the main subject of my research. In my toolbox I have had models describing interactions between
atoms, and cutting-edge software programs utilizing hundreds of computer processing units in parallel on the most modern computer clusters
there is. These models together with the data programs can be thought
of as a "computational microscope" (term coined by Klaus Schulten),
generating a great amount of atomic-level information on the dynamics
of the molecules studied. The data has been analyzed, complemented and
corroborated with experimental results derived from biological systems
in the wet lab.
11
1.1
Membranes and transport proteins
As soon as the information-bearing molecules were encapsulated by a
membrane somewhere in the early evolution of life, material for the processes of life needed to be transported across this impermeable barrier.
In truth, for small and uncharged molecules, the transport could in fact
be unassisted diffusion across the membrane due to the relatively low
free-energy cost of such transport, but this energy barrier is in practice
too high for larger and charged, even though small, molecules. Additionally, unassisted diffusion is unspecific by nature, and what is required is a
highly efficient and regulated selective transport of a particular molecule
or group of molecules.
For such a regulated transport of larger and charged molecular species,
dedicated molecular machines evolved. The birth of such remarkable
molecules was of course not a blink-of-an-eye event but, as is the case
for evolution in general, it was achieved by gradual improvement of the
molecules involved by molecular evolution over millions of years. There
are still traces of the gradual evolution of transport proteins in the living organisms today, both as simple molecules with primitive functions,
as well as independently evolved protein domains found in the modular
organization of modern molecular machines.
1.1.1
Material transport across membranes
Cells are distinct entities separated from the surrounding environment by
the plasma membrane. Within one cell, membranes are also efficiently
dividing it into different compartments, called organelles, for localized
and specialized functions (Figure 1.1). By incorporating specific proteins
into the membranes the necessary flow of material between the organelles
and the surrounding environment can be achieved. There are many distinct types of membrane transport proteins and they can be regulated
in different ways.
There are two mechanisms for transport of substances, substrates,
across a membrane; energy-independent facilitated diffusion and
energy-driven active transport. There are six characterized classes of
transporters [1]: channels and pores, electrochemical potential-driven
transporters, primary active transporters, group translocators,
transmembrane electron carriers, and accessory factors involved in
transport. The first and second groups are proteins catalyzing facilitated
diffusion. The channel group opens up an aqueous transmembrane pore
for substrate passage along its concentration gradient whereas in the
second group the protein itself functions as a carrier. The other groups
of transporters include proteins that for example transport a solute
against its concentration gradient using a primary source of energy,
12
Figure 1.1: The eukaryotic cell. The cell is separated from the outside by the plasma
membrane. Internally, it is divided into subcellular structures, called organelles. The
nuclear envelope separates the information-bearing molecules from the rest of the
cell. The rough endoplasmic reticulum is a complex membrane structure holding the
ribosomes, which are the molecular factories where proteins are built. In the mitochondria, the proteins of the cellular respiratory system produces most of the ATP.
The Golgi apparatus is used for protein sorting and the vesicles are the particles in
which the material is transported or, in the case of the specialized lysosome, degraded.
Figure reproduced from the On-line Biology Book [2].
such as hydrolysis of ATP, proteins that modify the substrate during
the transport, and proteins that facilitate transmembrane electron
transport.
Membrane channels typically control the flux of materials across cellular membranes through high selectivity combined with high conductivity
and through gating that is sensitive to essential environmental factors.
For example, they transport water, ions, and other small substrates, as
alcohols, as well as large proteins, and their opening and closing can be
controlled by ligand-binding, voltage, pH, or pressure changes.
1.1.2
Cellular membranes
Membranes found in cells typically consist of phospholipids and a
considerable amount of membrane proteins, cholesterol and lipids with
sugar adornments, called glycolipids. Phospholipid molecules have a
polar head-group and long non-polar tails. In a water environment these
13
Figure 1.2: Membrane protein in bilayer. In this snapshot from paper IV, the
transmembrane part of the KV 1.2 channel (green) is embedded in a membrane bilayer
consisting of the phospholipid POPC (detailed in inset). The lipids have a polar
end facing the surrounding water and two long apolar hydrocarbon chains. These
hydrophobic tails aggregate and form the bilayer core (gray). The surrounding water
and lipids in front of the protein, in the line of sight, were omitted for clarity.
non-polar chains spontaneously aggregate so that their water-exposed
surface is minimized, shielded by the polar head-groups pointing toward
the water. One of the ways these lipids can organize themselves is as
bilayers (Figure 1.2). These membranes are efficient barriers, in practice
impermeable to all substances except small uncharged molecules. This
is due to their stability and the associated energy cost to make a
large-enough hole for a molecule with considerable size to be able to
diffuse through, and additionally, the cost of exposing charged and
polar particles to the hydrophobic core of the membrane.
Under physiological conditions phospholipids in the membrane are
in the liquid crystalline phase, which can be considered to be a twodimensional liquid where the molecules within it can diffuse laterally in
the plane of the membrane. The composition of the membranes can alter their properties significantly and can in fact also affect membrane
protein function. For example, membranes with less amount of cholesterol and/or a higher fraction of short-chained or unsaturated lipids will
have a lower melting temperature, i.e. a higher fluidity. Compared to a
water environment the diffusion in the bilayer is two to three orders of
magnitude slower [3, 4].
1.1.3
The membrane potential
Virtually all cells have a transmembrane potential across the plasma
membrane. The potential has two basic functions, first it provides a driving force for operating the numerous molecular machines embedded in
the membrane, and secondly, for excitable cells such as neurons and muscle cells, it is used for transmitting signals in between cells. The potential
14
is due to an ion concentration imbalance across the membrane established
by the membrane protein Na+ /K+ -ATPase. These active transport proteins each continuously expel three sodium ions out of the cell and at the
same time import two potassium ions. Hence, both a charge separation,
giving rise to an electrical potential, and concentration gradients are
established. Membrane protein channels are typically only permeable to
specific ions and their collective interplay of conductances establishes the
resting membrane potential as well as the action potential of excitable
cells.
Having open potassium channels in the membrane enable the potassium ions to passively diffuse out of the cell, in the direction of their
concentration gradient that is established by the Na+ /K+ -ATPase. However, as potassium ions move out of the cell, the inside becomes even more
negative and the rising transmembrane electric potential starts to counteract this diffusion. These two opposite forces become equal and the
net flow of potassium ions is zero and a steady state has been reached
[5]. It is not a thermodynamic equilibrium state since ions are continuously pumped across the membrane. The resulting resting potential due
to the charge separation can be calculated using the Nernst equation,
see e.g. [6], given a concentration difference. If only potassium ions are
considered the membrane potential would be approximately -100 mV at
37 ◦ C and typical ion concentrations. Performing the same calculation
for the other key ion involved, sodium, gives a transmembrane potential
of +55 mV. The fact that the true, measured, resting potential (around
-70 mV) is close to the value for potassium ions is because the permeation of potassium ions is approximately an order of magnitude larger
than sodium ions under normal circumstances [7].
In excitable cells, the membrane potential can be perturbed from its
resting state so that a pulse-like signal, called the action potential (Figure
1.3), is formed. In neurons this pulse is used for cell-to-cell communication and in other cells its main function is to activate intracellular processes, such as muscle cell contraction and insulin secretion. The recent
review article by Barnett and Larkman [8] summarizes what is known
about the action potential. Its origin is a small local depolarizing response initiated in some part of the cell membrane by external stimuli,
either from specialized nerve endings in sensory neurons or from upstream neuron signaling in the central nervous system. This local shift
towards less negative potential triggers fast voltage-gated sodium channels to open if the local depolarization is larger than a threshold potential
of about -45 mV, increasing the permeation of this ion species considerably. Consequently, the extracellular sodium ions start to flow into the
cell along their concentration gradient further depolarizing the membrane. Within one millisecond, the potential reverses and continues to
rise towards the steady state potential for sodium ions (+55 mV). How15
Peak
Phase
Rising
~ -45
Threshold
Failed
Initiations
~ -70
Overshoot
0
Falling Phase
Membrane Voltage (mV)
~ +40
Resting potential
Undershoot
Stimulus
0
1
2
3
Time (ms)
4
5
Figure 1.3: The action potential of excitable cells. Once a local depolarization
stimulus from an external source becomes larger than a threshold of approximately
-45 mV, the opening of sodium channels depolarize the membrane voltage during
the rising phase. The voltage reverses before the slower potassium ion channels have
been activated in response to the depolarization. The sodium channels deactivate
and the open potassium channels repolarize the voltage during the falling phase.
Before the potassium channels have had time to close the normal resting potential of
approximately -70 mV is undershoot somewhat.
fredag den 11 november 2011
ever, before this value is reached two opposing factors start to act. First,
although voltage-gated sodium channels are activated fast they also have
an inherent property of inactivation on the millisecond scale. The second factor is an increased flux of potassium ions due to the opening of
voltage-gated potassium channels as the membrane potential becomes
more positive. Hence, potassium ions start to flow out of the cell at the
same time as sodium channels are inactivated and the depolarization
ceases (peak) and starts dropping again causing the sodium channels to
close even faster. Consequently, the membrane is repolarized and returns
to the resting potential.
16
1.2
Voltage-gated potassium channels
All potassium channels are related members of a single family of proteins
and are found in all domains of life; bacteria, archea, and eukaryotes both plants and animals. Their amino acid sequence is easily recognized
because they share a characteristic sequence of residues in the narrowest
part of the ion-conducting pore responsible for the molecular discrimination of ions by a remarkably efficient mechanism [9, 10, 11]. The channels
are fascinating molecular machines capable of conducting K+ ions almost
at the theoretical limit at a rate of ∼108 ions per second, while at the same
time keeping the permeation of potassium ions approximately 104 times
higher compared to sodium ions [12, 13]. While their ion-permeation
characteristics are common, diversity among different subfamilies of K+
channels is mainly manifested by the various ways the channel gate is
opened.
Potassium channels that are voltage-gated, abbreviated KV , are a
diverse and ubiquitous group of membrane proteins belonging to the
voltage-dependent cation channel family, which also includes the evolutionary related sodium (NaV ) and calcium (CaV ) channels [14]. Normally,
the concentration of K+ outside the cell is more than an order of magnitude lower compared to that in the intracellular fluid and hence, upon
opening these channels, an outward current of positively charged cations,
called the alpha current, is established. Accordingly, the KV channels
limit cell excitability by returning the membrane potential to its resting
state as explained in the preceding section. They are therefor key players in a range of cellular processes [15], such as dictating the duration
of the action potential in excitable cells, as well as cell proliferation [16]
and volume regulation [17] in non-excitable cells. They are also susceptible to disease-causing mutations [18, 19] and are the targets of drugs
used against pain, epilepsy, cardiac arrhythmias, hypertension and hyperglycemia for example. Understandably, the pharmaceutical industry
has given these channels considerable attention.
The first voltage-gated K+ channel that was isolated, called Shaker,
came from the fruit fly Drosophila melanogaster [20]. Somewhat amusing,
the name comes from the coupling between the gene and the uncontrolled shaking motion of the fly’s legs when anesthetized with ether
[21]. Much of what is known about voltage-gated K+ channels has been
derived from electrophysiological studies of this channel in addition to
the pioneering work of Hodgkin, Huxley and Katz on the giant squid
axon (e.g. [7, 22, 23]). Since the first isolation, hundreds of genes coding
for diverse KV channels in different organisms have been found. In the
human genome, some 40 KV genes have been identified and characterized, and they have been assigned to 12 subfamilies (KV 1 to KV 12) based
on sequence similarities. The KV 1 subfamily is called the Shaker-related
17
subfamily since the Shaker-homologous human gene KCNA3 is coding
for the KV 1.3 channel.
Structural knowledge of the ion channels have been sparse and mostly
based on biochemical studies on bacterial proteins. Unfortunately, it has
turned out to be very tricky to determine the three-dimensional highresolution structure of membrane proteins, and it was not until 1998
that the first (potassium) ion channel structure, KcsA, was solved by
the MacKinnon lab [12]. It is however not voltage-gated, rather pHsensitive, but only a few years later the same group was able to crystallize
the KV AP channel [24] from the thermophilic archaebacteria Aeropyrum
pernix (for which, together with previous work, Roderick MacKinnon was
awarded the Nobel Prize in Chemistry 2003). Subsequently, the partly
incomplete structures of the mammalian KV 1.2 channel [25, 26] were
solved and, most recently, a complete structure of a chimeric structure
of a KV 1.2/KV 2.1 channel from rat was published [27]. Although these
crystal structures have been extremely important for the description of
the atomic workings of these proteins, the KV structures all caught the
channel in the open state and to date, the closed state, or intermediate
states along the opening pathway, have not been properly characterized.
Most of the work in this thesis is an attempt to bridge this knowledge gap
by studying the Shaker and KV 1.2 channels by computational and complementary experimental methods. An alignment of the transmembrane
section of these two sequences can be found in Figure 1.4.
The KV channels are made of four subunits, each containing six transmembrane helices, called S1-S6 (Figure 1.5). S1 through S4 form the
voltage-sensing domain (VSD) that responds to changes in the membrane potential. Helices S5 and S6 from the four subunits come together
into the centrally located pore domain (PD) where the ion-conducting
pore is present. The structural alterations within the VSDs upon gating
are mediated to the PD by a linker helix, called the S4-S5 linker, inducing
a conformational change that alters the kink angles of the pore-lining S6
helices which, in turn, control the blocking of the ion-conducting pathway at the intracellular side [28, 24]. In contrast to the distinct subunits
(i.e. separate peptide chains) of the KV channels, the evolutionary related NaV and CaV proteins, consist of four covalently bonded domains,
each one with a similar structure to a single subunit of the KV channels.
Eukaryotic KV channels have an additional intracellular domain, called
the tetramerization domain, or T1, aiding the assembly of the transmembrane part as well as providing a binding platform for an accessory
β -subunit whose function is not entirely understood but it is related to
a particular class of enzymes [25].
The crystal structures of voltage-gated ion channels have also shown
that the VSDs are loosely attached to the pore domain, seemingly stable
on their own. Interestingly, recently gene orthologs of the VSD in voltage18
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S:T42B-4V"""""""""O8"-b_U#K-TIb__U-T-#M_U4CKbbb]MWW-K--#`T`#KUC_MIIC-_b""""66O
"""""""""""""""""""""aaaaaaaaaaaaaa)aaaaaaaaaaaaaaaaaaaaaaaaaa)aaaaa)aa
S:T4MBU-L1I""""""696"-b_U#K-TIb__U-M-#M_U4CKbbb]MWW-K--#`T`#KU`_MIICS_b""""6D5
"""""""""""""""""""""aaa))a)GGaaaaaa)aaaaaaaaa)aaGaa)aaaaaaaaaaaGGaa))a
S:T4MBU-L1I""""""6D6"IKWU]4CTS_-IUIW_CSIII-#K#UTIS]-S`cKK_Ib#IMM]44-``4""""295
S:T42B-4V"""""""""O8"-b_U#K-TIb__U-T-#M_U4CKbbb]MWW-K--#`T`#KUC_MIIC-_b""""66O
"""""""""""""""""""""aaaaaaaaaaaaaa)aaaaaaaaaaaaaaaaaaaaaaaaaa)aaaaa)aa
S:T42B-4V""""""""668"IKWII41I1_-IUIWbCSIII-#K#ITI_]-]`cKK_Ib#IMMW#4-CC4""""6OO
"""""""""""""""""""""aaa))a)GGaaaaaa)aaaaaaaaa)aaGaa)aaaaaaaaaaaGGaa))a
S:T4MBU-L1I""""""6D6"IKWU]4CTS_-IUIW_CSIII-#K#UTIS]-S`cKK_Ib#IMM]44-``4""""295
S:T42B-4V"""""""""O8"-b_U#K-TIb__U-T-#M_U4CKbbb]MWW-K--#`T`#KUC_MIIC-_b""""66O
S:T42B-4V""""""""668"IKWII41I1_-IUIWbCSIII-#K#ITI_]-]`cKK_Ib#IMMW#4-CC4""""6OO
"""""""""""""""""""""aaa))a)GGaaaaaa)aaaaaaaaa)aaGaa)aaaaaaaaaaaGGaa))a
S:T4MBU-L1I""""""6D6"IKWU]4CTS_-IUIW_CSIII-#K#UTIS]-S`cKK_Ib#IMM]44-``4""""295
S1
S:T4MBU-L1I""""""296"CCM`_`CKKMC`C_:KIVK#I_S^bS========`_TVVVTWVSCIIUI`""""282
S:T42B-4V""""""""668"IKWII41I1_-IUIWbCSIII-#K#ITI_]-]`cKK_Ib#IMMW#4-CC4""""6OO
"""""""""""""""""""""aaa))a)GGaaaaaa)aaaaaaaaa)aaGaa)aaaaaaaaaaaGGaa))a
S:T4MBU-L1I""""""6D6"IKWU]4CTS_-IUIW_CSIII-#K#UTIS]-S`cKK_Ib#IMM]44-``4""""295
"""""""""""""""""""""a)aaGaaa)aaaGaaaaaaaGa)GGG""""""""Ga)aG)aGa"aGGG)G
S:T4MBU-L1I""""""296"CCM`_`CKKMC`C_:KIVK#I_S^bS========`_TVVVTWVSCIIUI`""""282
S:T42B-4V""""""""668"IKWII41I1_-IUIWbCSIII-#K#ITI_]-]`cKK_Ib#IMMW#4-CC4""""6OO
"""""""""""""""""""""aaa))a)GGaaaaaa)aaaaaaaaa)aaGaa)aaaaaaaaaaaGGaa))a
S:T42B-4V""""""""6O8"C`M`1`CKCMC`M_:KIVK#C_-UITIU1^WWW`V_^VbMTMV=CWb]]M""""267
"""""""""""""""""""""a)aaGaaa)aaaGaaaaaaaGa)GGG""""""""Ga)aG)aGa"aGGG)G
S:T4MBU-L1I""""""296"CCM`_`CKKMC`C_:KIVK#I_S^bS========`_TVVVTWVSCIIUI`""""282
S:T42B-4V""""""""668"IKWII41I1_-IUIWbCSIII-#K#ITI_]-]`cKK_Ib#IMMW#4-CC4""""6OO
S:T42B-4V""""""""6O8"C`M`1`CKCMC`M_:KIVK#C_-UITIU1^WWW`V_^VbMTMV=CWb]]M""""267
"""""""""""""""""""""a)aaGaaa)aaaGaaaaaaaGa)GGG""""""""Ga)aG)aGa"aGGG)G
S:T4MBU-L1I""""""296"CCM`_`CKKMC`C_:KIVK#I_S^bS========`_TVVVTWVSCIIUI`""""282
S:T4MBU-L1I""""""289"#UCVU#__KCIVK:CCc_V_IKV`-_K4:#TSKT_:-U`1T`CUCC4CC#""""922
183
S:T42B-4V""""""""6O8"C`M`1`CKCMC`M_:KIVK#C_-UITIU1^WWW`V_^VbMTMV=CWb]]M""""267
"""""""""""""""""""""a)aaGaaa)aaaGaaaaaaaGa)GGG""""""""Ga)aG)aGa"aGGG)G
S:T4MBU-L1I""""""296"CCM`_`CKKMC`C_:KIVK#I_S^bS========`_TVVVTWVSCIIUI`""""282
"""""""""""""""""""""GGGaaaaa))aaaaaaaa)aaGGaaaGaaa)aGGaGG))aa)aaa)aaaa
S2
S3
S:T4MBU-L1I""""""289"#UCVU#__KCIVK:CCc_V_IKV`-_K4:#TSKT_:-U`1T`CUCC4CC#""""922
S:T42B-4V""""""""6O8"C`M`1`CKCMC`M_:KIVK#C_-UITIU1^WWW`V_^VbMTMV=CWb]]M""""267
"""""""""""""""""""""a)aaGaaa)aaaGaaaaaaaGa)GGG""""""""Ga)aG)aGa"aGGG)G
S:T42B-4V""""""""26O"VM_VU#__C`IVK:CCc_M_I_K`-__4:#MS4W__VTC1TCCUC`4CC#""""2O7
"""""""""""""""""""""GGGaaaaa))aaaaaaaa)aaGGaaaGaaa)aGGaGG))aa)aaa)aaaa
S:T4MBU-L1I""""""289"#UCVU#__KCIVK:CCc_V_IKV`-_K4:#TSKT_:-U`1T`CUCC4CC#""""922
S:T42B-4V""""""""6O8"C`M`1`CKCMC`M_:KIVK#C_-UITIU1^WWW`V_^VbMTMV=CWb]]M""""267
S:T42B-4V""""""""26O"VM_VU#__C`IVK:CCc_M_I_K`-__4:#MS4W__VTC1TCCUC`4CC#""""2O7
"""""""""""""""""""""GGGaaaaa))aaaaaaaa)aaGGaaaGaaa)aGGaGG))aa)aaa)aaaa
S:T4MBU-L1I""""""289"#UCVU#__KCIVK:CCc_V_IKV`-_K4:#TSKT_:-U`1T`CUCC4CC#""""922
S:T4MBU-L1I""""""929"b_CVK4V``4IIIUVKTK#S4#`M#]U=SMMT]41MK4CK-`C-K`-`_-""""986
S:T42B-4V""""""""26O"VM_VU#__C`IVK:CCc_M_I_K`-__4:#MS4W__VTC1TCCUC`4CC#""""2O7
"""""""""""""""""""""GGGaaaaa))aaaaaaaa)aaGGaaaGaaa)aGGaGG))aa)aaa)aaaa
S:T4MBU-L1I""""""289"#UCVU#__KCIVK:CCc_V_IKV`-_K4:#TSKT_:-U`1T`CUCC4CC#""""922
"""""""""""""""""""""aaaaaGaG)aa)""""""""""""a)a")GGGaaaaaaaaaaaaaaaaaa
S:T4MBU-L1I""""""929"b_CVK4V``4IIIUVKTK#S4#`M#]U=SMMT]41MK4CK-`C-K`-`_-""""986
226
233 236
259
S:T42B-4V""""""""26O"VM_VU#__C`IVK:CCc_M_I_K`-__4:#MS4W__VTC1TCCUC`4CC#""""2O7
"""""""""""""""""""""GGGaaaaa))aaaaaaaa)aaGGaaaGaaa)aGGaGG))aa)aaa)aaaa
S:T42B-4V""""""""2OO"b_CVKWVIK4IS============#IU4]]W]]41MK4CK-`C-K`-`_-""""959
"""""""""""""""""""""aaaaaGaG)aa)""""""""""""a)a")GGGaaaaaaaaaaaaaaaaaa
S:T4MBU-L1I""""""929"b_CVK4V``4IIIUVKTK#S4#`M#]U=SMMT]41MK4CK-`C-K`-`_-""""986
S:T42B-4V""""""""26O"VM_VU#__C`IVK:CCc_M_I_K`-__4:#MS4W__VTC1TCCUC`4CC#""""2O7
S4
S:T42B-4V""""""""2OO"b_CVKWVIK4IS============#IU4]]W]]41MK4CK-`C-K`-`_-""""959
"""""""""""""""""""""aaaaaGaG)aa)""""""""""""a)a")GGGaaaaaaaaaaaaaaaaaa
S:T4MBU-L1I""""""929"b_CVK4V``4IIIUVKTK#S4#`M#]U=SMMT]41MK4CK-`C-K`-`_-""""986
S:T4MBU-L1I""""""982"C_SKM-^MSWK]CKW-VKS4M1-IKWKKC__K_CW``K_MM4`b_4I4WM""""E26
S:T42B-4V""""""""2OO"b_CVKWVIK4IS============#IU4]]W]]41MK4CK-`C-K`-`_-""""959
"""""""""""""""""""""aaaaaGaG)aa)""""""""""""a)a")GGGaaaaaaaaaaaaaaaaaa
S:T4MBU-L1I""""""929"b_CVK4V``4IIIUVKTK#S4#`M#]U=SMMT]41MK4CK-`C-K`-`_-""""986
"""""""""""""""""""""aaaaaaaaaaaaaaa)aaaaaaaaaaaaaaaaaaaa)aaaaaaaaaaaGG
S:T4MBU-L1I""""""982"C_SKM-^MSWK]CKW-VKS4M1-IKWKKC__K_CW``K_MM4`b_4I4WM""""E26
S:T42B-4V""""""""2OO"b_CVKWVIK4IS============#IU4]]W]]41MK4CK-`C-K`-`_-""""959
"""""""""""""""""""""aaaaaGaG)aa)""""""""""""a)a")GGGaaaaaaaaaaaaaaaaaa
S:T42B-4V""""""""95E"C_SKM-^MSWK]CKW]VKS4M1-IKWKKC__K_CW`CK_MM4`b_4I4UI""""979
"""""""""""""""""""""aaaaaaaaaaaaaaa)aaaaaaaaaaaaaaaaaaaa)aaaaaaaaaaaGG
S:T4MBU-L1I""""""982"C_SKM-^MSWK]CKW-VKS4M1-IKWKKC__K_CW``K_MM4`b_4I4WM""""E26
R1/294 R2 R3 R4
S:T42B-4V""""""""2OO"b_CVKWVIK4IS============#IU4]]W]]41MK4CK-`C-K`-`_-""""959
S:T42B-4V""""""""95E"C_SKM-^MSWK]CKW]VKS4M1-IKWKKC__K_CW`CK_MM4`b_4I4UI""""979
"""""""""""""""""""""aaaaaaaaaaaaaaa)aaaaaaaaaaaaaaaaaaaa)aaaaaaaaaaaGG
S:T4MBU-L1I""""""982"C_SKM-^MSWK]CKW-VKS4M1-IKWKKC__K_CW``K_MM4`b_4I4WM""""E26
Linker
S5
S:T42B-4V""""""""95E"C_SKM-^MSWK]CKW]VKS4M1-IKWKKC__K_CW`CK_MM4`b_4I4UI""""979
"""""""""""""""""""""aaaaaaaaaaaaaaa)aaaaaaaaaaaaaaaaaaaa)aaaaaaaaaaaGG
S:T4MBU-L1I""""""982"C_SKM-^MSWK]CKW-VKS4M1-IKWKKC__K_CW``K_MM4`b_4I4WM""""E26
S:T42B-4V""""""""95E"C_SKM-^MSWK]CKW]VKS4M1-IKWKKC__K_CW`CK_MM4`b_4I4UI""""979
"""""""""""""""""""""aaaaaaaaaaaaaaa)aaaaaaaaaaaaaaaaaaaa)aaaaaaaaaaaGG
S:T42B-4V""""""""95E"C_SKM-^MSWK]CKW]VKS4M1-IKWKKC__K_CW`CK_MM4`b_4I4UI""""979
K5 R6
Pore helix
Filter
S6
!"#$%&'()*+,,,,,,-..,+#%//!%01'$/22$334*443565'*4135325!035%7"$0$53740$,,,,-89
,,,,,,,,,,,,,,,,,,,,,:;<:<:<<<<<<<<<<<;<<<<<<<<<:<::;:<<<<<<<<<<<<<<<<<
!"#$%&'()*+,,,,,,-..,+#%//!%01'$/22$334*443565'*4135325!035%7"$0$53740$,,,,-89
!"#$.&($4,,,,,,,,=>-,('%?/1%01'$/22$33%*443565'*3144055!035%7"$0$53740$,,,,-@=
,,,,,,,,,,,,,,,,,,,,,:;<:<:<<<<<<<<<<<;<<<<<<<<<:<::;:<<<<<<<<<<<<<<<<<
!"#$.&($4,,,,,,,,=>-,('%?/1%01'$/22$33%*443565'*3144055!035%7"$0$53740$,,,,-@=
!"#$%&'()*+,,,,,,-8.,7131303%#/#6/6A(+4'?++*?%?#/#A34%"16715475?A*!!%%7,,,,>.9
,,,,,,,,,,,,,,,,,,,,,<<<<<<<<<<<<<<<<<<;:<<,<;<,;::<<<<<:;<:;::::::;<;:
!"#$%&'()*+,,,,,,-8.,7131303%#/#6/6A(+4'?++*?%?#/#A34%"16715475?A*!!%%7,,,,>.9
!"#$.&($4,,,,,,,,-@-,7131303%#/#6/6A(+4+5++B?$?B67?34%"1!01%%1'7!!%(%$%,,,,->9
,,,,,,,,,,,,,,,,,,,,,<<<<<<<<<<<<<<<<<<;:<<,<;<,;::<<<<<:;<:;::::::;<;:
!"#$.&($4,,,,,,,,-@-,7131303%#/#6/6A(+4+5++B?$?B67?34%"1!01%%1'7!!%(%$%,,,,->9
!"#$%&'()*+,,,,,,>..,%+%%%'**'7''53+%4,,,,>=C
,,,,,,,,,,,,,,,,,,,,,;:<:<<:<;;:;<<:;;
!"#$%&'()*+,,,,,,>..,%+%%%'**'7''53+%4,,,,>=C
Figure
1.4: Alignment of Shaker and KV 1.2 sequences. Sequence alignment of
!"#$.&($4,,,,,,,,->.,40%!%'6*+0?+53##%,,,,-DC
,,,,,,,,,,,,,,,,,,,,,;:<:<<:<;;:;<<:;;
the
transmembrane section of the Shaker gene KCNAS from Drosophila melanogaster
!"#$.&($4,,,,,,,,->.,40%!%'6*+0?+53##%,,,,-DC
and
the KV 1.2 gene KCNA2 from Rattus norvegicus (common rat) as given by the
EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
EMBOSS
implementation of the Smith-Waterman algorithm [32] with default paEBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
rameters.
Key residues are highlighted (KV 1.2 enumeration) in red, blue and gray for
negative, positive and neutral, respectively. The characteristic sequence signature in
the selectivity filter of the potassium channels is shown in yellow. Secondary structure
is indicated above the alignment in gray; S1-S6 transmembrane helices, the interfacial
S4-S5 helix linker, the re-entrant pore helix and the selectivity filter.
gated ion channels have been found, functioning as proton pores [29, 30]
or regulators of phosphatase activity [31]. This is a very important and
beautiful result since, in an evolutionary perspective, one could think of
the VSD as a distinct functional entity that evolved by itself and later
on was coupled to other functional domains, such as ion channels and
phosphatases.
19
SF
S1 S2 S3 S4
+
+
+
S5
PH
S6
Linker
PD
VSD
Figure 1.5: Structure of the voltage-gated potassium channel. The topology
of one single subunit of a voltage-gated potassium channel is schematically drawn
(left). Extracellular side is up. The four helices in green correspond to the voltagesensing domain (VSD) while the darker-green part forms the pore domain (PD) with
its preceding interfacial S4-S5 linker helix, re-entrant pore helix (PH) and selectivity
filter (SF). In the S4 helix, there are several positively charged amino acids that are
affected by changes in the electric potential across the membrane. Side and top views
of the transmembrane part of the crystal structure of the open KV 1.2/2.1 protein [27]
(middle, right). The entire PD is shown but for clarity only the VSD from one subunit.
The S6 helices from the four subunits form the surface of a water-filled channel that
is accessible to the ions once they are in the open conformation. At the extracellular
end of the channel the selectivity filter narrows the channel for efficient ion filtering.
måndag den 26 september 2011
1.2.1
Voltage-dependent gating
The four-helix bundle of the VSD has a shape reminiscent of an hourglass, with helical ends that splay outwards at the intracellular and extracellular sides, forming two water-filled cavities and a central constricted
region. The protein surfaces lining the two water-filled cavities are relatively polar whereas the environment around the domain center is hydrophobic. Most of the polarity comes from several charged and evolutionary conserved residues, in particular arginines, in the S4 helix. These
positive charges are located at every third position of the helix making
it a rather peculiar transmembrane helix. Generally, it is very unfavorable to have charges buried into the non-polar membrane bilayer core
and this would hence be evolutionarily selected against. However, both
biophysical experiments [33] and computer simulations [34] have shown
that the S4 helix can be stable in isolation in the membrane bilayer although other studies report a considerably higher energy cost [35, 36].
In any case, this helix has been conserved throughout evolution because
it has a very special purpose, namely to function as a voltage-sensor;
an electromechanical converter transforming the energy of the change in
the transmembrane electrical potential into a mechanical response of the
protein that controls the opening and closing of the ion channel gate.
Charges are affected by electrical fields, and so are the positively
charged arginine residues in the S4 helix of the VSD. When the cell
20
Hyperpolarization
Depolarization
[O]
Membrane core
[C1] [C2]
Extracell.
neg. cluster
[C3]
VSD hydrophobic region
Intracell.
neg. cluster
Figure 1.6: Model of gating. Side view of a voltage-gated ion channel, where only
one voltage-sensing domain (VSD) is shown. Extracellular side is up. In the bottom
panel, a cartoon of the model of the S4 gating motion is shown. In the open activated
state of the channel (left), the S4 helix (in cyan ribbon representation) is located
towards the extracellular side due to the depolarization of the membrane. All four
måndag den 10 oktober 2011
arginine
residues (blue spheres) on the extracellular half of the helix are located
above the central hydrophobic region of the VSD. Upon gating, the S4 helix moves
towards the intracellular side, due to repolarization of the membrane. This is a stepby-step process involving several meta-stable states (here represented by the C1 and
C2 states in analogy with the terminology used in paper III), where the arginine
residues change their interaction partners along the way (as depicted in the model
cartoon and described more in the text). At the most closed state C3 (right), the
S4 helix has moved approximately 10 Å vertically towards the intracellular side, and
all arginine residues except the most extracellular one have passed the hydrophobic
region of the VSD. The open state is the homology model of Shaker from paper III
and the closed model is based on the most closed state, C3, in paper III and the PD
from [41].
is at rest and the ion channel is closed, the transmembrane potential is
approximately -70 mV (negative at the intracellular side). Consequently,
the voltage-sensor S4 helix is attracted and positioned towards the intracellular side of the membrane. Once the depolarization occurs, the
transmembrane potential reverses, causing the helix to move toward the
extracellular side (Figure 1.6). This movement of arginine charges upon
gating is producing a gating current that is readily detected experimentally [37, 38]. At least three unit charges per VSD have to be translated
21
through the electric field of the membrane to open or close the channel
[39, 40].
The free energy difference between the open and closed states is
thought to be fairly small, but they are thermally separated by a
large barrier, which would explain their slow activation within a few
milliseconds. Once the random thermal motion of the system overcomes
the barrier the transition is considerably faster, perhaps as rapid as
ten microseconds [42]. The barrier has been proposed to be due to the
hydrophobic region in the VSD core, centered around the evolutionary
conserved F233 in Figure 1.4, through which charged amino acids
from the voltage-sensor have to pass to open and close the channel
[43, 27, 44]. The gating charges’ movement is catalyzed by sequential ion
pairing with negatively charged residues in two clusters on neighboring
helices, one cluster located at the extracellular side of the hydrophobic
region (corresponding to E183 and E226 in Figure 1.4) and the other
on the intracellular side (E236 and D259) [38, 45]. Since experimental
structures of the intermediate and the closed states are missing, the
atomic details of the gating movement of the voltage-sensor and the
subsequent conformational changes leading to opening and closing of
the gate is still not understood properly. Probing this gating pathway
by computer simulations and modeling we have tried to shed light on
the understanding of this mechanism on an atomic level.
1.2.2
Slow inactivation
A striking and somewhat unexpected feature of the voltage-gated channels is that the ion current is transient when depolarization is maintained.
The current decays with a time constant of a few ms. This means that the
protein can adopt at least three distinct conformations; a closed state,
where the channel is non-conducting, which is the predominant resting
state at hyperpolarized membrane potentials; an active conducting state
occupied transiently during depolarization; and a non-conducting relaxed
state reached at prolonged depolarization [46]. Since crystal growth occurs under depolarized conditions it has been assumed that the state
of the proteins reflect this relaxed state even though the central pore is
open [47].
1.2.3
The role of 310 -helices
There is a general belief in the protein structural community that 310 helices are relatively rare and short in protein structures. The idea has its
origins in the classical studies of the helical conformations of polypeptides
by Pauling, Corey and Branson [48]. The conformation is energetically
22
protein structures listed in three different papers (Peters
et al., 1996; Pal and Basu, 1999; Enkhbayar et al., 2006).
These reports only name a small fraction of the structures included in their studies. As a result, the final pool
contained 12 proteins for a total of 13 helices (Protein
analysis supports the idea that p
ces has specific requirements.
need for more extensive studies
text in which 310 helices are foun
310 helices are rarely observed in
just because they might be intrin
A helices but also because they
ments that are rarely met over l
ing in the unraveling of the 310 c
variability of the helical stereoch
tected in many studies.
Another important characteri
namics. This property has been w
lated peptides, which are impor
the study of helices. In the case o
best-studied systems involves pep
alanine, a non-natural amino aci
group on the A carbon (Karle an
et al., 1994; Crisma et al., 2006;
Because of the stereochemical c
the presence of two large substit
amino acid, these peptides have
adopt a helical conformation, 31
systems, it has been demonstrat
adopt both conformations and th
the equilibrium between one c
other by altering the polarity of t
ture. In some cases, these two co
coexist in the same peptide so th
310-helical conformation and ano
A-helical conformation. A few sim
been documented for peptides fo
Figure 1.7: The structure of α- and 310 -helices. In the idealized α-helix (left),
acids; the two conformations ap
residue i is interacting with residue i+4 forming a helical structure having
3.6 residues
different
segments of the peptid
måndag den 10 oktober 2011
◦
per turn so that Figure
consecutive
residues
an angleSide
of views
100 and
around
helical
1. Canonical
helicalmake
conformations.
views the
is altered axis.
by small sequence cha
axes the
of a rise
helixalong
in a canonical
A-helical
The helical pitch along
is 5.4the
Å and
the helix
axis isconformation
1.5 Å per residue
Dike(5.4/3.6).
and Cowsik, 2006; Mikho
(left) and canonical 310 conformation (right). Dotted lines inIn the 310 -helix however (right), residue i interacts with residue i + 3Molecular
giving three
dynamics studies of h
dicate hydrogen bonds between atoms in the backbone
of poly◦
(aligned) residuespeptide.
per turn
of
the
helix
and
an
angle
of
120
between
them.
The
name that 310 and A he
also
proposed
Helices were generated using PepBuild. All figures were
prepared with PYMOL.
same
comes from the observation
that there are ten atoms in the ring formed
bypeptide
makingand that interconv
the hydrogen bond. The pitch is consequently greater at around 6 Å, and the rise
586
membrane proteins
per residue is 1.93-2
Å. 3A10 helices
more inintuitive
description; the 310 -helix is a tighter-wound
α-helix, being longer (given the same number of residues) and more narrow. Figure
adapted from [52].
unfavorable and the cost of transforming a decapeptide in a protein-like
environment from α- to 310 -helix has been estimated to 24 kJ/mol, i.e.
2.4 kJ/mol per residue [49]. Despite this, the wealth of protein structures
now available has shown that 310 -helices in proteins are quite common.
Although they are most often only 1-2 turns long a significant number of
longer 310 -helices (>7 residues) has been found [50, 51]. However, they
seem to differ from the idealized conformation (depicted in Figure 1.7)
by having an average of 3.2 residues per turn instead of three, and it has
been noticed that they have a wide variability in main-chain torsion angle distribution and this irregularity increases rapidly with helix length.
23
Another important point in explaining the difference between the two
helix types is the way helices are packed against each other. The periodic alignment of the residues in the 310 -helix imposes more constraints
in helix packing, resulting in the destabilization of the 310 conformation
[52].
Long 310 -helices have been found in the recent crystal structure of
the KV 1.2/2.1 chimera channel, confirming the early suggestions of the
presence of a 310 conformation in the S4 helix [53, 54]. Moreover, recent experimental and computational studies support the idea that a
significant part of the voltage-sensor S4 helix is in a 310 conformation
[46, 55, 56]. This is also what we found in papers I-III. However, it is
unclear how the helix behaves during gating. Our studies suggest that
it undergoes a conformational change such that there is a stationary 310
region, located around the phenylalanine residue in the VSD core, sliding
along the helix during its translation. The 310 conformation of S4 offers
an apparently simple explanation for some of the issues previously raised
by the properties of the voltage-sensor helix. First, it would explain the
conserved sequence pattern of positively charged residues at every third
position, because the 310 conformation places them on the same helical
face, promoting interactions with the polar environment in the VSD core
and hence shield the charges from unfavorable exposure to the non-polar
membrane interior. Second, the conformation would limit the necessary
S4 rotation when translated along the gating pathway.
24
2. Molecular modeling
Simplify as much as possible, but no further.
Albert Einstein
A
protein’s function is to a large extent determined by its structure.
Numerous methods are available to deduce structural information of
biomolecules, but in practice only two give the complete high-resolution
information of the relative positions of all atoms in a protein, namely
nuclear magnetic resonance (NMR) spectroscopy and X-ray crystallography. Whereas the structures of water-soluble proteins are often relatively easy to determine, membrane proteins have for a long time eluded
structural biologist. Their structural determination has proven very hard
and expensive. Consequently, the Protein Data Bank (PDB), where all
biomolecular structures are deposited, only consist of a couple of hundred
membrane protein structures, in contrast to tens of thousands structures
of water-soluble proteins. Even when a structure of a particular protein
is not known there is a fair chance of successful creation of a relevant
model of it based on known structures using computational molecular
modeling techniques. This is because they are evolutionary related and
proteins with similar function can often safely be assumed to have similar
structures.
Once a protein structure has been obtained, either by experiments
or modeling, one would typically like to study its function. This too
can be done using molecular modeling methods. For example, molecular
docking protocols try to estimate the binding free energy of a ligand to
the protein by evaluating the energies of different ligand conformations
in different positions relative the protein. Another common application
is the elucidation of reaction mechanisms in the active sites of enzymes
by quantum mechanical methods. In both these applications, the protein
structure is typically fixed or at least restricted to only a few degrees of
freedom. This can be a rather severe limitation in some cases because the
protein’s flexibility could be very important for the binding of the ligand.
To study the dynamics in the systems one has to turn to simulation
techniques that sample the conformational flexibility of the molecules.
25
Methods of modeling and simulation of molecular systems can be
crudely divided into two distinct classes based on how the interactions
of the atoms are described. In the first class, usually called quantum
chemistry or simply quantum mechanics (QM), electrons are treated explicitly and the energy of the system is calculated using the Schrödinger
equation. In the other class, called empirical force field methods or molecular mechanics (MM), only the positions of the atom nuclei are considered and they move and interact according to classical physics. Quantum mechanical methods rely on the Born-Oppenheimer approximation
(as does molecular mechanics) stating that the higher velocities of the
electrons relative the nuclei allows these two motions to be treated independently and the underlying Schrödinger equation can take a simpler
form. The explicit treatment of electrons in these methods allow for the
study of chemical processes of permanent changes in electronic structure, e.g. chemical reactions, at the price of an increase in computational
cost. Even though both computer software and hardware have developed
enormously in recent decades, quantum chemistry is still in practice too
expensive for studying biological macromolecules on relevant timescales.
Additionally, the treatment of explicit water is troublesome in quantum chemistry methods and they are typically not able to capture the
dynamics of biologically relevant temperatures. Treating the molecular
system classically using molecular mechanics methods rectifies some of
these shortcomings and allow increased simulation timescales of several
orders of magnitude. Although these methods generally can not be used
to study chemical reactions there is still a wealth of interesting systems
where they can be used.
2.1
Molecular dynamics
Molecular dynamics (MD) is a computer simulation of the physical movements of atoms, or particles in general. The atoms are allowed to interact
during a period of time and their positions are calculated by solving Newton’s equations of motion numerically. The method assumes a classical
description of the system where the atoms are point charges that interact through a molecular mechanics interaction potential. The partial
charges on the nuclei are assigned depending on their chemical environment. Hence, this method can in principle not be used in the study
of processes where changes in the electronic structure matters although
hybrid QM/MM methods have been developed trying to overcome this
limitation. On the other hand, the benefit of molecular dynamics is that it
can simulate large molecular systems on biologically relevant timescales.
Molecular dynamics simulations generate a sequence of states, a trajec26
tory, of the molecular system. The states are time-dependent and correspond to a possible trajectory of motion of the particles. The energies
of the states in the trajectory generated by solving Newton’s equations
follow the thermodynamic equilibrium distribution (the Boltzmann distribution, see section 2.1.3). Another way of exploring conformational
space is by Monte Carlo (MC) sampling. Here, new conformations are
generated randomly and by evaluating the energy of the new state with
the underlying MM potential the random move is either accepted or rejected in a way such that the set of accepted conformations also follow
the equilibrium distribution.
To start a MD or MC simulation three things are needed:
1. A starting structure
2. An interaction potential
3. An algorithm for generating new states
The first requirement will be dealt with in section 2.2 and the second
and third ones will be described in more detail in the following sections. The simulation package used throughout the work in this thesis
is GROMACS [57]. Other commonly used packages are CHARMM [58]
and NAMD [59].
Another important point regarding the simulation of molecular systems is that it consists of a finite number of particles and hence there is
a boundary between the outer-most located particles of the simulation
unit cell (often also called the simulation box) and the surroundings. This
is usually coped with by using periodic boundary conditions. One could
think of this as cloning the entire system in all directions and letting
particles on the boundary of the unit cell interact with particles from
the neighboring clone(s). Also, particles moving out of the box will, in
the next step, appear on the opposite side of the system. This is a neat
trick to circumvent the finite size effects of the simulations but one has to
make sure the simulation box is large enough to prevent particles from
interacting with their periodic copies because such interactions would
propagate throughout the collection of systems giving severe artifacts.
2.1.1
Basic molecular dynamics theory
In MD simulations, the time evolution of the positions of the N atoms
of the system is calculated by solving Newton’s second law of motion
stating that the acceleration of atom i times its mass equals the net
force exerted on it by the remainder of the system:
d2~ri
F~i = mi · ~ai = mi · 2 .
dt
(2.1)
27
The force on atom i is, in turn, calculated by differentiating the total
potential energy function V (see next section) with respect to the atom’s
position
F~i = −∇i V (r~1 , ..., r~N ).
(2.2)
Given the position of atom i at a specific time ~ri (t), the position after
a short finite time interval, ~ri (t + ∆t), is given by a standard Taylor series
expansion around ~ri (t):
~ri (t + ∆t) = ~ri (t) +
d2~ri (t) ∆t2
d~ri (t)
∆t +
+ ...
dt
dt2
2
(2.3)
Numerous algorithms exist for integrating Eq. 2.1 and many are finite difference methods based on the Taylor series expansion above. The
simple Verlet integration algorithm [60] is deduced by adding the Taylor
expansion above with the corresponding one for ~ri (t − ∆t) and truncating
the expansion after the third order, yielding
~ri (t + ∆t) = 2~ri (t) − ~ri (t − ∆t) +
F~i (t) 2
∆t .
mi
(2.4)
Note that the expression for the second derivative of the position with
respect to time in Eq. 2.3 is the acceleration and hence can be substituted with the force and the mass of the particle according to Eq. 2.1.
The procedure is iterated and produces the trajectory of the motions of
all atoms in the system. Other popular, and related, algorithms are for
example the leapfrog (default in GROMACS) and the Velocity-Verlet
integrators [61, 62]. All these methods are time reversible, meaning that
inversing the velocities of all atoms would run the simulation in exactly
the opposite direction. The computational expense of any particular integration scheme is insignificant compared to the cost of calculating the
forces acting within the system but the length of the time step, ∆t, is
on the other had very important. The maximum time step length depends on the integration method but it is has to be significantly smaller
than the period of the fastest movements. The fastest modes of motion
in atomistic systems is bond vibration involving hydrogens, limiting the
time step to 0.5 − 2 · 10−15 s (femtoseconds, or fs). This small time step is
one of the main drawbacks of MD simulations since it limits the available
timescale.
2.1.2
Force fields
The underlying potential energy function describing the interactions between atoms (or particles in general) is usually called a force field. The
fundamental ideas behind it were conceived in the late sixties by Shneior
28
Lifson, Arieh Warshel and Michael Levitt. It is a mathematical function
relating the total potential energy of the system to the positions of its
particles. In this sense the name is somewhat misleading because it is an
expression of the energy, not the force. However, since differentiation of
the energy function gives the force and it is the force that governs the
particles’ motion (see previous section), the name is understandable. The
potential consists of several terms of elementary mathematical functions
each describing an intra- or intermolecular interaction. There are numerous force fields used in the molecular simulation community but most of
them have the same basic structure. They typically contain three or four
terms describing the bonded interactions, mediated by covalent bonds
between atoms, and two terms for non-bonded interactions between all
pairs of atoms. Since the number of pairs in a system is quadratic in the
number of particles, the way in which the non-bonded interactions are
calculated is critical for the computational cost of the force calculation.
Next, a walk-through of the most common terms is presented.
The potential energy of covalent bond vibration is typically modeled
by a harmonic oscillator, analogous to a spring connecting the two atoms:
Ebond = Kb (b − b0 )2 .
(2.5)
The energy is dependent on the difference between the distance between the atom pair in the bond, b, and the reference bond length, b0 .
The total bonded energy is then a sum over all bonded pairs. The Kb
parameter defines the stiffness of the bond. A more realistic model of
the covalent bond is the Morse potential, an asymmetric function giving a zero force at infinite distance, not infinite as is the case for the
harmonic function. The bond vibrations involving light atoms are the
fastest modes of motion and they are therefor the limiting factor of the
time step length in the integration scheme. Constraining bonds to their
equilibrium bond lengths allows for time steps of ∼2 fs and the rational
for doing this is that it is in fact a better approximation to a quantum
mechanical ground-state oscillator than the classical harmonic oscillator.
Constraining bonds in this way is common in MD simulations today and
has been used in all simulations in this work.
Angle bending is also represented by a harmonic potential
Eangle = Kθ (θ − θ0 )2
(2.6)
where θ is the angle between covalently bonded atoms A-B-C. To better optimize the fit to vibrational spectra two additional terms can be
added, the Urey-Bradley function for the angle bending of certain triplets
and an improper dihedral potential term for preserving the planarity of
planar groups. These two terms are typical for the CHARMM force field
[63] for example. In analogy to the constraining of bonds to their equilib29
rium bond lengths, the next-fastest degree of motion is angle-vibrations
involving hydrogens and removing those vibrations would motivate integration time steps of 4-5 fs. This can be done by introducing so called
virtual interaction sites, replacing the fastest-moving atoms. In contrast
to standard atoms, the positions of virtual interaction sites are calculated in a predefined manner (based on the geometry of the groups as
defined by the force field) from neighboring heavy atoms. This is a feature implemented in GROMACS and has been studied in paper V and
used in all papers except papers II and VI.
The energy of torsional rotation around the middle bond of four consecutive atoms, A-B-C-D, is modeled by a periodic (proper) dihedral
function
Etorsion = Kω (1 + cos(nω + δ))
(2.7)
where the rotational angle, ω , by convention is measured with respect
to the cis conformation (A and D on the same side of the bond). The
energy maximum is set by Kω and the number of minima is represented
by the n parameter, and δ determines their location. For more complicated functional forms one could either sum several such functions for
the same quadruple of atoms or express it as an expansion in powers of
cos ω , the so called Ryckaert-Bellemans potential.
The treatment of the torsional rotation around bonds in the protein backbone is crucial for the correct sampling of its conformations.
Rather recently, an extension to the CHARMM force field was published [64], trying to improve this property by adding a correction term,
called CMAP (for correction map), to the protein backbone torsions.
The CMAP term is a function of two neighboring dihedral quadruples
(the Ramachandran dihedral pair, φ and ψ , of each residue) and is expressed as a two-dimensional grid-based energy that is added to the
dihedral energy of the pair. The authors found that the inclusion of the
CMAP term gave a better conformational sampling of the protein when
comparing to crystallographic data. In paper V, the implementation of
CHARMM in GROMACS study the behavior of this correction term on
longer timescales.
So far all interactions have been between particles connected by covalent bonds. Now, let us turn to the non-bonded interactions. The van der
Waals interaction is a balance between two forces, the nuclei-nuclei repulsion at short distances and the attraction of atoms at moderate distances
due to the dispersive force (London force). Typically this interaction is
represented by the Lennard-Jones potential:
"
EvdW = 30
Rmin
rij
12
−2
Rmin
rij
6 #
.
(2.8)
An exponential function would represent the repulsion better but the
difference is small and the choice is motivated by the greater speed of
−12
−6
the calculation of rij
from rij
. The energy minimum is given by and is located at distance Rmin between the atoms. Instead of having
one parameter pair for each pair of atom types in the force field, the and Rmin parameters for a pair are calculated by averaging the individual
atom’s respective parameter value. The total van der Waals energy of the
system is a sum over all pairs separated by more than two bonds since
the corresponding interaction between close neighbors already has been
accounted for by the bonded interactions. The van der Waals interactions
between pairs separated by exactly three bonds are usually too large as
well, because of its partial treatment in the torsion term, and have to
be scaled down. Since the interaction decays fast at longer distances, in
practice only the pairs within a certain cut-off, of approximately 1 nm,
are considered.
The electrostatic interaction between two charged particles is given by
the Coulomb equation:
ECoulomb =
qi qj
4π0 r rij
(2.9)
where qi and qj are the charges, 0 the vacuum permittivity and r is
the relative permittivity of the medium. Since the electrostatic interaction decays slowly its calculation is pivotal for the computational load of
the simulation. The simplest approach to reduce the cost, is to truncate
the interaction after a certain cut-off distance as for the van der Waals
interaction. Although well-motivated in that case, here the energy is still
considerable for cut-offs that make a difference, leading to discontinuous forces at the boundary. One could adjust the potential entirely to
circumvent this but nowadays the best practice is to use more sophisticated approaches, such as the particle-mesh Ewald (PME) method [65].
Briefly, it adds a neutralizing potential to the point charges in the system so that this new sum of electrostatic interactions converges fast. It
is calculated in real space up to a cut-off of around 1 nm from the atom
of interest, while the contribution of a counteracting potential, opposite
to the neutralizing one, is solved on a grid in reciprocal space by the fast
Fourier method. In principle, the PME algorithm account for all electrostatic interactions, also between particles in the periodic copies of the
simulated system.
The total potential energy of the system with respect to the positions
of the N atoms in it, V (~rN ), is now given by summing up the individual
sums of all the above-mentioned terms. However, how all the parameters
in the model are set hast hitherto been left out. In principle, all parameters in the expression for the potential energy (Kb , b0 , Kθ , θ, Kω , n, δ, and
Rmin ) must be specified for all different atom types one would like to
31
be able to study. Since the elements may appear in a variety of chemical environments, giving them different partial charges, the number of
atoms types are quite large and consequently, the number of parameters
needed to be set is even greater. The parameterization of a force-field
is the process of fitting numerical values for all these parameters, typically by comparing energies to quantum chemical results and carefully
measured experimental data. The functional forms together with all the
parameters in the model define the force field. Even though one might
get the impression that this is a huge simplification, simulations based
on MM-type force fields have proven to be exceptionally good in many
cases, see e.g. review articles [66, 67].
2.1.3
Statistical mechanics
When trying to model and study real macroscopical systems and their
properties it does not make sense, conceptually, to simply consider one
single molecule of the substance of interest since, in reality, the behavior
depends on a enormous number of such molecules. On the other hand,
assuming the force field is able to capture the physics of the system satisfactorily, MD simulations give the complete information of the system in
atomic detail for the simulated period of time, something which in general is not accessible to conventional molecular biology experiments. In
order to investigate how a microscopical system in a simulation behaves
on the macroscopic scale and hence correlate the simulation results to
experimentally measurable thermodynamic properties, one has to turn
to statistical mechanics. The experimentally measured value of an arbitrary property, denoted A here, of a system with N particles is a time
average over the time-course of the measurement:
Aave
1
= lim
τ →∞ τ
Z
τ
A(~
pN (t), ~rN (t))dt.
(2.10)
t=0
The integrand A(p~N (t), ~rN (t)) is the instantaneous value of A at time
t, depending on the momenta p~, and position ~r of all atoms. To calculate such a macroscopical time average in a simulation, a system of the
order of 1023 particles has to be followed for a time-period comparable
to the experiment. This is of course an impossible task. Boltzmann and
Gibbs recognized this problem and developed the theory of statistical
mechanics, in which a single system evolving in time is replaced by a
large number of replications of the system, called an ensemble, that are
considered simultaneously. The ergodic hypotesis is one of the axioms of
statistical mechanics and it states that the time average is equal to the
ensemble average:
32
ZZ
hAi =
A(~
pN , ~rN )ρ(~
pN , ~rN ) d~
pN d~rN .
(2.11)
Note that the time-dependence has disappeared and the ensemble average is an average over all replications of the system generated by the
simulation. Each member of the ensemble has an energy, and for systems
of constant number of particles N , volume V and temperature T , i.e. the
canonical N V T ensemble, the distribution of systems in the ensemble,
ρ(~
pN , ~rN ), follow the famous Boltzmann distribution:
N
N
ρ(~
p , ~r ) = exp
−E(~
pN , ~rN )
kB T
/Q.
(2.12)
The exponent E is the total energy, i.e. both the kinetic contribution,
K(~
pN ), due to the momentum of the particles, and the contribution from
interactions between particles described by the potential energy function
V (~rN ). kB is the Boltzmann constant and Q is the canonical partition
function:
QN V T
1 1
=
N ! h3N
ZZ
exp
−E(~
pN , ~rN )
kB T
d~
pN d~rN .
(2.13)
N ! comes from the indistinguishability of the particles and 1/h3N is
a normalizing factor. The partition function carries all thermodynamic
information about a system, and it is hence possible to calculate all
macroscopic properties either directly from it or from its derivative. Since
the simulation method sample conformations from the Boltzmann distribution, the partition function never has to be calculated explicitly.
In fact, to calculate it in a MD simulation one would need to sample all possible conformations of the system and sum up the values of
exp(−E(~
pN , ~rN )/kB T ). This number of conformations is in practice infinite. However, for simple systems the partition function can be calculated
from quantum chemistry. This is actually quite a beautiful result, since
statistical mechanics hence provide us with the awestruck tool to tie
together quantum chemistry, developed during the 20th century, with
thermodynamics that was described centuries earlier.
The molecular dynamics method and the as-famous Monte
Carlo method, generate system configurations from the Boltzmann
distribution and the ensemble average of the property at study can
therefor be calculated by numerical integration of Eq. 2.11.
Briefly, the difference between the two methods is that MC performs
a random walk through conformational space and accepts new conformations in a particular way so that the acquired ensemble of structures
are Boltzmann-distributed. The random nature of the sampling has the
consequence that consecutive states have no true connection in time. The
33
MC technique is in practice a more effective sampling strategy, since it
does not require the calculation of forces nor the numerical integration
of the equations of motion, but the problem is that there is no general
algorithm for efficient generation of new test conformations. Molecular
dynamics on the other hand, guarantees sampling from the Boltzmann
distribution for arbitrarily complex systems and, additionally, generates
a trajectory of conformations reflecting realistic dynamics of the system.
2.1.4
Modeling the membrane potential in simulations
As described previously in Section 1.1.3, the membrane potential in living
cells is established by an unequal distribution of ions between the water
solutions on each side of the membrane. In a typical atomistic simulation set-up, this situation is not as easy to accomplish as one might
expect. This is because the standard method of using periodic boundary conditions (see Section 2.1.2) to overcome the finite-size effects of
the simulation, in fact connects the two water baths so that they are
in physical contact with each other. Hence, they are not two separated
water solutions but one and the same.
Rather recently, the lab of Woolf has developed a method to circumvent
this problem by introducing a double-bilayer system in the simulation
unit cell [68]. The two parallel membranes divide the simulation system
into two truly separated water solutions. This method has been shown to
reproduce transmembrane potentials in a realistic way but the computational overload is significant. Another complication is that to establish
a biological transmembrane potential on the order of hundred millivolts,
an imbalance of only one or a few ions is needed for the small systems
that are practical to simulate. Approximately one access ion/104 Å2 of
bilayer is enough, requiring an order of 105 atoms in the simulated system. This means that if one ion permeation event occurs (for example
through passing an ion channel), the membrane potential changes drastically and could even be reversed. Similarly, if the embedded protein,
carrying charged residues, changes conformation significantly during the
simulation the transmembrane potential varies considerably. However, as
computational power and software efficiency increases this method will
become more feasible.
Another way to achieve charge imbalance is to introduce an air-water
interface at the outer edges of the water baths [69]. In such a way, the
water solutions on each side of a single membrane are separated from
each other even though periodic boundary conditions are applied. This
method has been successfully used in several studies [70, 71, 72] but it
also suffers from the problem of membrane potential variability mentioned above.
34
ρ (q/nm 3 )
0.2
0.1
0
-0.1
E (V/nm)
0.4
0.2
0
-0.2
-0.4
-0.6
Φ (V)
-0.2
0.6
0.4
0.2
0
-0.2
-0.4
00
2
4
6
System coordinate z (nm)
8
10
(=Lz )
Figure 2.1: The transmembrane potential. The charge density (top) in a simulation is calculated by summing the charges in slices along the direction of the applied
electric field. Integrating it once gives the electric field (middle) and twice the electrostatic potential (bottom), see text and Eq. 2.14. Figure produced from data of paper
I. The dashed line represent the simulation without an applied field and the solid line
a simulation with an applied constant field of 0.052 V/nm. With a box dimension
of 10 nm in z-direction this would correspond to a potential of approximately 0.5 V
(φ = Ez Lz ). This is also the potential given by the integration.
onsdag den 12 oktober 2011
In contrast to the all-atom simulation techniques mentioned so far,
continuum electrostatic models have previously been used to study proteins in general and ion permeation of channels in particular [73, 74, 75].
In these methods, the polar solvent (water and lipids in the case of
a membrane-protein system) is represented as a structureless dielectric
medium leaving out the necessity of explicit solvent particles. In such a
representation, the effect of the membrane potential can easily be incorporated by modifying the underlying Poisson-Boltzmann equation [73].
It has been shown that this representation can capture the dominant
features of energetics of ion solvation in bulk but not in confined environments such as aqueous pores (e.g. in the selectivity filter of ion
channels).
The approach we took in paper I, and that has been used in numerous
other studies, e.g. [76, 77, 78, 79], is to represent the membrane poten35
tial by an added constant external electric field along the z-direction,
Ez , perpendicular to the membrane surface. The resulting electrostatic
potential is given by φ = Ez Lz , where Lz is the simulation box length
in the z-direction. In practice this is achieved by adding a force along
z, Fi = qi Ez , to each particle i with charge qi in the system. It is a
rather unphysiological set-up, in particular because the potential drops
instantaneously from one end of the simulation box to the other (i.e. a
discontinuous hop over the periodic boundary), but it has been shown
to be a sound representation [80]. The Poisson equation gives the relation between the equilibrium charge distribution and the electrostatic
potential:
1
d2 φ(z)
= − ρ(z)
dz 2
0
(2.14)
where φ is the potential, 0 is the vacuum permittivity, and ρ is the
charge density. The membrane potential can therefor be calculated from
a simulation (giving the charge density) by integrating this equation
twice. The reference point of the potential is arbitrarily set to z = 0 and
the boundary conditions used are, φ(0) = 0 and Ez (0) = dφ(0)/dz = 0. An
example of the integration from paper I is shown in Figure 2.1.
2.2
Modeling of protein structures
Analysis of three-dimensional high-resolution protein structures has
proven very fruitful for biological and medical discovery in the last
couple of decades. Experimental determination of such structures is
time-consuming and expensive, and solving all human protein structures
using high-resolution methods, such as X-ray crystallography and NMR
spectroscopy, is an immense task and it is questionable if it is a realistic
and well-motivated goal at all given the cost. However, computational
modeling of protein structures could very well bridge this gap because,
as the structural information in the databases accumulates, such
methods are increasingly successful in predicting biologically relevant
structures for proteins with unknown structure.
Comparative modeling is the means by which models of protein structures are built based on the information in the sequence and structure
databases. Its foundations lie in the discovery of strong correlative patterns between sequence similarity on one hand and an analogous function
on the other. In other words, as the volume of protein sequences started
to accumulate in the 1950s, the data showed that similar sequences coded
for proteins with identical or similar functions. This lead to the classification of proteins (based on sequence) into families and superfamilies
36
7
9
10
0
10
1972 1975 1978 1981 1984 1987 1990 1993
14
20
26
31
40
80!000
43
59
64!000
75
86
110
48!000
135
159
197
32!000
233
263
16!000
302
come
structures
obsolete
varyand
over
are
time
removed
as some
from
become
the database.
obsolete and are removed from the database.
hable structures per year.
0
1996
1999
2002
2005
2008
2011
1!500
1!200
0
900
600
300
0
1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 2011
Figure
2.2:1 Protein Namnlös
Data Bank
statistics.
growth of total structures (blue
Namnlös
2
NamnlösYearly
3
diamonds, left axis), of unique membrane protein structures (green triangles, right
axis) and unique protein folds as defined by SCOP v1.75 (red squares, right axis).
[81]. The underlying idea is that members of the same protein family
are all evolutionary related, or homologous. This fact can be exploited
in comparative modeling because a new protein sequence with unknown
structure, the target structure, can be assigned to these families and its
structure can be modeled using the structures of the proteins in that
particular family, the template structures.
The first protein crystal structure was solved in 1958 [82] and as more
and more structures were solved the above-mentioned observation was
confirmed: similar protein sequences had similar structures and therefor similar functions as well. It was also realized that the number of
unique ways a protein can fold, the structural space, is limited, while
the sequence space in practice is unlimited. As of today, the Structural
Classification of Proteins (SCOP) database [83] contains some 1400 distinct protein folds and judging from the plot of accumulated folds found
in Figure 2.2, the true number is not much larger. Therefor, for new sequences found today, there is a high probability of finding a homologous
protein with a known structure that can be used as a template for structural modeling. The situation is a bit more complicated for membrane
proteins however. The first high-resolution structure was not published
until 1985 [84] and as can be seen in the figure there is only a couple
of hundred structures known today. The structural space of membrane
proteins is hence sparsely explored and the number of folds for this class
of proteins is unknown although it probably is considerably less than the
folds of globular proteins because of the environmental restraint of the
membrane.
Besides the comparative modeling approach, protein structures can be
predicted using physics-based methods, also called de novo or ab initio
37
modeling. These methods are in principle based on molecular simulation
techniques and the underlying molecular mechanics description treated
previously in this chapter. A branch of this research area is the refinement of protein structures. Here, protein structures of moderate to poor
quality are refined with the objective to improve the model by doing local
adjustments, typically achieved by structural sampling and minimization
of the interaction energy function.
2.2.1
Comparative modeling
The first step in comparative modeling is to align the target protein
sequence to proteins with known structure. By aligning sequences, evolutionary conserved, and hence important, residues are identified. They
are later pivotal in the model building. The search of homologous sequences is typically done using PSI-BLAST [85, 86]. It creates a multiple
sequence alignment by an iterative process, were it finds new sequence
hits in each step that are incorporated in the next query to the sequence
database. In this fashion it is quite successful in finding distantly related
sequences in a relatively short time. The alignment is extremely important for the outcome of the modeling. A poor alignment will result in
a bad model. Once homologous sequences have been found, the targettemplate alignment and the protein structures are sent to the modeling
tools.
Briefly, models are built by optimizing spatial constraints derived from
the alignment. The constraints are typically distances between backbone
atoms of different residues. The structural model is generated by minimizing the violations of these constraints. When several template structures are used for the model building, local sequence similarities guide
the selection of the appropriate choice of template structure for this
sequence section. That is, different sections of the target structure are
modeled from different template structures based on local sequence similarity.
In paper III, the starting point of the modeling of different states of
the VSD was to build a homology model of the Shaker VSD based on
the template structure of the KV 1.2/2.1 chimera protein [27]. A sequence
alignment of the two sequences was performed using HHalign [87] and
since the sequence identity is more than 40 % their alignment is quite
straightforward. The model was built using one of the most commonly
used homology modeling tools; Modeller [88].
38
2.2.2
Physics-based modeling and structural refinement
The origin of de novo structure prediction lies in the experiments on
protein folding performed by Anfinsen in the middle of the 20th century
(for which he was awarded the Nobel Prize in Chemistry 1972) [89]. He
was able to observe folding of the denatured ribonuclease protein into
its biologically active conformation, the native state, in a plain solution.
This means that, at least for small globular proteins, all the information
for the folding of the protein lies in the amino acid sequence. Hence, if
the physical interactions are described accurately enough, the modeling
method should be able to find the native state. These methods are computationally demanding and have been used mostly on smaller model
proteins. It is safe to say they have not been as successful as comparative modeling but as computational power is increasing they will become
more and more useful.
These methods generally have a molecular mechanics-type of interaction potential under the hood. To decrease the computational cost it is
not uncommon that the methods are extended in their description of the
molecular system and the way conformations are sampled. For example,
coarse-graining the atomistic model by grouping several atoms in functional groups, each represented by a single particle, reduces the number
of particles considerably. This approach has been used for the model of
the lipid molecules in most simulations in this thesis, where the hydrogen
atoms in the long non-polar chains are represented together with the carbon they are attached to as a united atom. Another trick frequently used
is to model the surrounding medium by an analytic function describing
its average properties without treating the atoms explicitly. This is called
implicit solvent methods and was studied in paper V. Analogous efforts
treating the lipid bilayer implicitly have been made and can be used in
the interaction potential of the Rosetta molecular modeling tool [90] that
we used for the modeling of the Shaker VSD states in paper III.
For successful structural refinement of protein models the modeler often has to extend the sampling technique with methods exploring the
conformational space more efficiently. This is due to the fact that the
refinement process readily gets stuck in local energy minima of the energetic landscape. MD and MC methods typically sample conformations
from low-energy regions and hence these methods are very inefficient in
surmounting energy barriers. Such barriers could be due to an suboptimal packing of amino acid side chains for example. To circumvent this
problem one could use simulated annealing techniques that temporarily
increase the temperature in the simulation, hence providing additional
energy for passing barriers. Another way is to completely rebuild parts
of the protein from a library of protein structure fragments, similar to
the methodology used in homology modeling based on several template
39
structures, or less dramatic, to rebuild the s from libraries of torsional
angles typical for the different amino acids.
The modeling in paper III was done along these lines. Starting from
the homology model of the Shaker VSD, we rebuilt the C-terminal part
of the domain for each state using the corresponding constraints from
the metal-bridges in the experiments. The S3-S4 loop and S4 helix was
rebuilt completely from fragments in Rosetta, allowing this part to adopt
different positions relative the rest of the protein and the implicitly represented water and membrane. During conformational sampling the backbone and s of the remodeled part were allowed to move whereas the
non-remodeled residues were kept fixed.
40
3. Summary of papers
- So Dr. Bjelkmar, in what area are you doing
research?
- Well Professor Knowitall,
I’m answering biological questions
built up by chemical entities
modeled by physical principles
described by mathematics
using computational methods.
Pär Bjelkmar
3.1 Conformational changes in the KV 1.2 ion channel
through polarized microsecond molecular simulation
(paper I)
The opening of the voltage-gated ion channel and initiation of the alpha current have been shown to occur in the millisecond range [13] but
an early component of the gating current has been reported in the
microsecond-range [42]. With recently developed highly efficient and parallel molecular dynamics code (e.g. GROMACS [57]) and current stateof-the-art computer clusters, it is now practically feasible to perform
simulations in the microsecond regime for large membrane-protein systems like the KV channels and early stages of gating can therefor, in
principle, be studied.
The crystal structure of the open KV 1.2 channel [25], complemented
with modeled regions where electron density was missing, was inserted
into a lipid bilayer membrane and simulated with an applied electric
field - mimicking the effect of the resting potential - for more than a
microsecond with the goal of capturing early stages of VSD gating motion. To our knowledge, this work was the first simulation reaching this
long sought-after timescale for a comparable system and in the wake of
it similar large-scale simulations have been performed, e.g. [91, 92].
Since gating is a reversible process the hyperpolarization is expected to
return the protein to the closed, resting state although this full closure is
slow and beyond the scope of this study. Both in the hyperpolarized and
41
a control simulation without an applied electric field, a small rotation
of the S4 helix was found, indicating that the original crystal structure
might be slightly different from the natural state of the voltage-sensor.
This was also the conclusion of Lewis et al. [93]. In one subunit of the
protein in the hyperpolarized simulation, a more pronounced rotation of
the extracellular half of the helix converted it to a 310 -helix during the
time-course of hundreds of nanoseconds. This is a key result supporting the original idea of 310 helix content in voltage-gated ion channels
[53, 54] and has also been confirmed by experiments [46] and recent
computational studies [55, 56], paper II and III, as well by the S4 structure in the more recent crystal structure of the KV 1.2/2.1 chimera [27]
that was unpublished at the initiation of the project. In fact, the protein conformations in the second half of the simulation are overall more
similar to the new chimera structure compared to the starting KV 1.2
structure. Another interesting, and rather unexpected result, is the fact
that lipid molecules in the simulation matched the positions of resolved
lipid molecules in the chimera crystal structure, successfully predicting
a bilayer thinning of the same magnitude and at the same membrane
location.
3.2 Secondary structure of the S4 helix in gating.
(paper II)
As a continuation of the work in paper I, this study had a more quantitative approach of assessing the likelihood of S4 adopting a 310 conformation during gating. The VSD from the KV 1.2/2.1 crystal structure
[27], in two conformations, was simulated in a POPC membrane. The
first being the conformation in the crystal structure, with a mix of α and
310 content in the S4 helix, while the other had an all-310 S4 conformation. By measuring the work of translating these two helices along their
axes by steered molecular dynamics simulations, the cost of moving the
310 S4 conformation one helix turn towards the closed state was found
to be significantly lower compared to the original α-helical conformation. The barrier seemed to be caused by the breaking and reforming
of electrostatic interactions between the arginine residues in S4 and the
negative residues on neighboring helices, together with the passing of
the most intracellular arginine residue through the hydrophobic region
formed by the phenylalanine residue F233 (see Figure 1.4). The 310 conformation appeared to reduce the work by limiting the amount of S4
rotation needed to reform salt-bridge interactions and its smaller size
(see Figure 1.7) reduced the distortion in the rest of the VSD when
moved.
42
3.3 Tracking the voltage-sensor gating motion with
metal-ion bridges, computer modeling and simulation
(paper III)
In an attempt to study the conformations along the gating pathway of the
KV channels, a collaboration with an experimental group in Linköping
was initiated. By substituting residues in the Shaker channel to Cd2+ binding cysteines, they were able to collect a set of 17 newly described
interactions between the VSD segments in different states along the gating pathway.
Starting from a homology model of the open Shaker VSD built on the
template structure of the KV 1.2/2.1 protein [27], models of three successively more closed states were constructed using these experimental
constraints by incorporating them in the rebuilding the voltage-sensor
S4 helix and its preceding loop (using methods described in section 2.2).
These models were evaluated by molecular dynamics simulations and
representative conformations were chosen based on structural clustering
of the simulation conformations. The three selected models of successively more closed states suggest a gating mechanism where a stationary region of the S4 helix, opposite the central hydrophobic region of
the VSD (centered at the phenylalanine residue 290 in Shaker, F233
in KV 1.2 enumeration), is in a 310 conformation as the helix translates
along its axis. Additionally, two interactions in the experimental set of
constraints distinguish the open active and open inactive states of the
channel confirming that the KV 1.2/2.1 crystal structure is indeed in the
open active state despite the long depolarization of the membrane in the
crystallization experiment.
3.4
Lateral diffusion of membrane proteins (paper IV)
While diffusion in pure lipid membranes has been given much attention
and is rather well understood, the lateral diffusion of membrane proteins
is still an unclear process in membrane dynamics [94, 95]. It is important
for understanding protein sorting, membrane protein complex formation
and binding of ligands, such as toxins and pharmaceuticals, to membrane
proteins.
An enlarged version of the KV 1.2 system described in paper I was used
to study this phenomenon. The protein was found to diffuse together with
a dynamic complex of 50-100 lipids surrounding it. Lateral diffusion coefficients were calculated for neighboring lipids versus the more distant
ones and indeed, neighbors were an order of magnitude slower in their
diffusion, having approximately the same value as the protein. Lipids up
43
to a distance of 1-2 nm from the protein surface were affected. Since cellular membranes are crowded with membrane proteins occupying some
30% of the total surface area this suggest that there are no free lipids
with bulk properties in them. These results also have implications for the
theories of lateral diffusion of membrane proteins. The equations typically include the protein radius and membrane viscosity as parameters,
and as this study shows, it is effectively the radius of the whole lipidprotein complex that matters and the protein senses a viscosity different
from a protein-free membrane.
3.5 Linking structure and conductance in a simple ion
channel (paper V)
Eukaryotic membrane proteins such as the KV channels are complexes
of several protein domains, often making them too large for molecular
simulations on relevant timescales, even though this shortcoming is gradually diminished due to the new computational capabilities. Fortunately,
more simple channels are common in nature and these can be used as
model systems, still giving insight into the workings of complex ion channels [96]. One such channel is formed by the 16-residue antiamoebin-1
peptide (AAM-1) [97], secreted by fungi as a weapon in the microbial
warfare among organisms. It is a member of the nongenomic family of
fungal peptides called peptaibols [96, 98]; 15-20 residues long peptides
rich in helix-promoting nonstandard amino acids. They are not encoded
by the genome but rather synthesized by a special set of enzymes. Peptaibols are able to assemble into helical bundles that form lethal pores
in the membrane of the target cell.
The ion-conducting multimeric state of the AAM-1 channel is not
known but electrophysiological measurements of currents generated by
AAM-1 channels have shown that it only has one conductance level indicating one single conducting structure [99]. Measuring the current as a
function of protein concentration has not unambiguously revealed its nature but suggested that the channel is a tetramer, hexamer or octamer,
depending on the underlying theory of helix aggregation [100, 99]. Models
of these different channel models were constructed from the monomeric
peptide structure [101, 102, 103], and molecular dynamics simulations
were used to calculate their conductance.
The tetrameric channel did not conduct ions at all because of a too narrow pore and the octameric channel had a conductance that was an order
of magnitude too large. The hexameric channel on the other hand had
a conductance consistent with the experimentally measured value. The
calculated conductances were derived both from explicit counting of ion
44
crossings in the simulation as well as from traditional continuum methods
using the Nernst-Planck equation [13, 104]. Moreover, the channel was
large enough for Fickian, rather than single-file, diffusion and the ions
transversed the channel having their solvation shells of water molecules
intact. Consequently, the free-energy barrier of ion transport was only a
couple of kcal/mol and the channel was slightly cation selective.
3.6 Implementation of the CHARMM force field in
GROMACS (paper VI)
If the underlying model of particle interactions in the force fields used in
molecular dynamics simulations is accurate enough, it should, in principle, be able to find the native state of the protein (given enough computational time). The development of force fields is therefor a very important
and continuously current topic of research. As described in chapter 2, a
force field is constituted by a set of functions and their accompanying
parameters describing the interactions between atoms. One of the most
commonly used force fields is CHARMM [63, 64] and this paper describes
the implementation of it into the GROMACS program [57]. CHARMM
has a special term, called CMAP (for correction map), modulating the
torsion potential of pairs of neighboring residues (see section 2.1.2). The
torsion energy of protein backbone residues is a key factor of the energy
landscape determining protein structure and it has been shown that it
has not been satisfactorily modeled before.
After verification of the implementation in GROMACS, it was used
to study the effect of the CMAP term on long timescales together with
different water models and integration time steps. The inclusion of the
CMAP did indeed improve the structural sampling of the protein by
exploring backbone torsional angle ensembles more similar to a crystal
structure. Interestingly, this was true for all water models, including the
implicit ones, and time steps. In particular, the most computationally efficient combination of settings - CHARMM with CMAP, implicit solvent
and long time steps - did not compromise the overall structural quality
and is therefor promising for protein structure refinement. Using these
settings in a small test case, it was possible to improve RMSD-values of
decoy structures of Protein G from 3 Å down to 2 Å, and starting from
2 Å down to 1 Å.
45
4. Concluding remarks and future
perspectives
If we have learned one thing from the history of
invention and discovery, it is that in the long run,
and often in the short run, the most daring
prophecies seem laughably conservative.
Arthur C. Clark
T
he function of a protein is to a large extent determined by its structure. As the techniques for determining high-resolution structures improve and comparative modeling of protein structures become more accurate, we will rather soon find ourselves in a situation where most proteins will have a reliable high-resolution structure, deduced either from
experiments or modeling. For globular proteins this time will soon here,
but the research on membrane proteins is a few decades behind.
To fully understand protein function however, one or a few structures
of a protein is not enough even though it can take you quite far. Instead
of being seen as a single static conformation, a better representation of
the native state of a protein seems to be an ensemble of conformations
that all are important for function. This could for example be different
side chain conformations in the active site of an enzyme, all contributing
to the binding of the substrate. Additionally, large-scale conformational
changes often occur within the protein as it performs its function, as we
have seen for the voltage-gated ion channels in this thesis. In other words,
dynamics is important. Molecular dynamics simulation is a well established method for modeling the conformational flexibility of biomolecules
and it provides spatial and temporal information on scales that are difficult to access with experimental techniques. Historically, the computational demands of MD have been too great to be able to reach the
timescales of these functionally important motions that typically take
place in microseconds to seconds. This is starting to change.
The first all-atom MD simulation of a small protein in vacuo was performed in the 70’s and reached less than 10 ps. Thirteen years ago, the
first microsecond all-atom simulation was performed. It was an immense
task involving several months of supercomputer time even though the
47
system consisted of only some 10,000 atoms. The technology has developed considerably since then and we are now on the verge of reaching the region where biological interesting phenomena occur. The most
large-scale all-atom MD simulations nowadays consist of hundreds of
thousands atoms and reach the microsecond timescale, with milliseconds
on the horizon.
The increase in simulation speed is both due to development of software
and hardware. Algorithms for performing the MD calculations on a single
processing unit have improved but the main difference the last few years
has been the parallelization of the problem over many computing units.
New national computer centers have been established to provide these
supercomputing capabilities to the scientific community, now enabling us
to perform calculations on thousands of processors simultaneously. Another approach to enhance the capabilities of MD simulations (and other
techniques) is to use distributed computing. In contrast to the computer
centers, the computational capabilities are spread around the world and
are connected via the internet. The very successful folding@home initiative is an example of this. Here, volunteering users around the globe
lend their computer power to the project, for example through screensavers, and the folding@home servers send the simulation information to
the node and collect the results. This infrastructure limits the way these
calculations are performed. Because of the temporary availability of the
compute nodes and the slow interconnection between them, only small
simulation tasks are sent out to the users but the immense number of
them enables an incredible sampling of molecular conformations.
Interestingly, new types of hardware have also emerged on the market
so the computing units mentioned above do not necessarily have to refer
to the traditional central processing units (CPUs). In particular, the
graphics processing units (GPUs) that are common in current desktop
computers, typically for the demanding graphics of computer games, have
an immense capacity of floating-point arithmetics that can readily be
used for MD calculations. Implementation of MD code on GPUs is still
an area under development but the technology is promising, not least
because these chips are very cost-effective. Another type of hardware
used for MD simulations is the Cell processor in Sony’s Playstation 3.
The folding@home project, for example, has quite an impressive number
of PS3s in its arsenal. These are all general-purpose hardware, but there
are also efforts in developing custom-built hardware specialized for MD
simulations. For example, DE Shaw Research in New York has built the
impressive supercomputer Anton for molecular dynamics simulations,
showing unprecedented simulation speeds.
Since the calculation of the underlying interactions in molecular simulations has become faster it promotes improvements in the force fields
as well. In particular, considerable effort has been given to represent
48
the polarization in chemical systems in a more realistic way. The current standard is to assign fixed partial charges on all the particles in
the simulation but that, of course, prohibits the change in the electronic
structure in the simulation. On the other end of the spectrum, model
reduction through coarse-graining will enable yet another boost in sampling efficiency even though the accuracy is reduced somewhat. In the
long run, I believe it is likely that the development of more sophisticated coarse-graining strategies will bridge the gap between atomistic
and macroscopic representations through a hierarchical structure of simulations with different coarseness.
The future of molecular dynamics simulations is not simply a matter
of longer and faster simulations. The field has also seen a development in
how the biomolecular conformational space is explored, exemplified by
enhanced sampling techniques and new methods for free-energy calculations through nonequilibrium work methods. Additionally, new efforts
using Markov-state modeling is trying to represent dynamical evolution
over long timescales from short MD simulations by automatic decomposition of conformational space in disjoint but connected metastable
states with Markovian transitions between them. In these examples the
strategy is to make the conformational search more efficient, with respect to the process of interest, rather than relying on one or a few long,
brute-force, equilibrium simulations. These strategies will for sure pave
the way for the extended applicability of molecular simulations in the
future, for example in the areas of computer-aided drug design, protein
engineering, and the study of macromolecular conformational dynamics.
Does molecular modeling have a future? As you have seen above I
am quite optimistic, but I think we have to interact more with experimentalists. More experimental collaborations should be initiated to compare, complement, and validate our simulations to a higher degree than
what is done today. The era of pure-MD papers is coming to an end,
but on the other hand, as we incorporate more and more experimental
data in our work, the potential of molecular modeling and simulation
will be enhanced and become a standard tool throughout the molecular
sciences. Further increases in simulation speed in combination with improvements in force fields and sampling techniques will provide us with
unprecedented insight into the relationship between biomolecular structure, dynamics, and function.
49
5. Svensk populärvetenskaplig
sammanfattning
Ju mer man tänker, desto mer inser man att det inte
finns något enkelt svar.
Nalle Puh
E
n cell avgränsar sig från sin omgivning genom att omge sig av ett
membran. På så sätt kan den koncentrera de livsnödvändiga molekylerna
till en liten volym för att därmed kunna utföra all den kemi som är karaktäristiskt för livet. Även inuti en cell finns det membran som särskiljer
delar av den i distinkta enheter, kallade organeller, med specialiserad
funktion. Så fort de informationsbärande molekylerna som utgör livets
kärna omgav sig av ett membran i begynnelsen av livets utveckling uppkom bohovet att kunna transportera materiel över dessa annars ogenomsläppliga barriärer. Därför utvecklades speciella membranbundna transportmolekyler vars kemiska sammansättning finns lagrad, tillsammans
med ritningarna för alla andra proteiner som cellen kan producera, i de
informationsbärande molekyler som idag utgörs av DNA.
Molekylerna som sitter insprängda i membranen kallas membranproteiner och de är en mycket viktig klass av proteiner som i princip sköter
all interaktion mellan cellen och dess omgivning samt mellan de membranomgivna organellerna inuti cellen. DNA-molekylerna specificerar endast
proteinernas kemiska sammansättning men för att förstå hur proteiner
fungerar måste man studera deras tre-dimensionella struktur. Denna
bestäms av de fysikaliska interaktionerna mellan proteinet och dess omgivning och är en häpnadsväckande process som under normala omständigheter låter proteinet vecka sig till en helt unik struktur som är
den biologiskt aktiva. Felveckning av proteiner kan ha livsavgörande konsekvenser i och med att de kan framkalla diverse sjukdomstillstånd. Veckningsprocessen är inte förstådd i sin helhelt men efter mycket forskning
har de grundläggande principerna kartlagts.
Att bestämma ett proteins DNA-sekvens är relativt enkelt men att
fastställa dess struktur har visat sig mycket svårt och dyrt. Det gäller
speciellt gruppen av membranproteiner. Idag finns endast några hundra
51
högupplösta strukturer av membranproteiner jämfört med tio-tusentals
strukturer av icke membranbundna proteiner, så kallade globulära proteiner, som rör sig fritt i vattenblandningen i cellen. I och med kostnaden
associerad med att bestämma proteiners struktur har datorbaserade alternativ utvecklats. De är väldigt mycket mer tidseffektiva och mindre
kostsamma och är därmed viktiga metoder inom forskningen nuförtiden.
Datormodellering av proteiner på atomär nivå kan göras på många olika
sätt, t.ex. med hjälp av den evolutionära informationen från alla DNAsekvenser av proteiner från olika organismer som finns tillgängliga, tillsammans med de redan bestämda strukturerna. Ett annat alternativ är
att försöka beskriva de fysikaliska interaktionerna mellan atomer och på
denna väg simulera proteiners beteende.
I denna avhandling har jag, tillsammans med flertalet andra forskare,
främst studerat en viss typ av membranproteiner som har den fantastiska
egenskapen att selektivt transportera kaliumjoner med väldig effektivitet.
Regleringen av öppnandet och stängandet av kanalen sker genom spänningsförändringar över membranet. Dessa så kallade spänningskänsliga
kaliumkanaler har studerats med hjälp av datorbaserad modellering,
både utifrån evolutionär information och med fysikbaserade metoder som
kompletterats med nya data från ett samarbete med en experimentell
forskargrupp. Våra studier har visat att den spänningskänsliga delen av
proteinet förändrar sin struktur på ett speciellt sätt under öppnandet
och stängandet av kanalen. Proteinets interaktioner med det omgivande
membranet har också studerats och vi har visat att proteinet påverkar
membranet i stor utsträckning. Framförallt så rör sig proteinet tillsammans med ett kluster av lipider runt omkring sig och dessa lipider har
därmed helt andra egenskaper än de opåverkade. Detta har givit ny kunskap för förståelsen av interaktionerna mellan proteiner och membran
och för teorierna som beskriver molekylrörelserna i ett membran.
Vidare har datormodellering av en liten och primitiv antibiotisk
jonkanal använts för att bestämma dess biologiskt aktiva form genom
att, utifrån olika modeller av kanalen, beräkna dess genomsläpplighet
för joner och jämföra resultaten med experimentellt uppmätta värden.
Endast en av modellerna stämde överens med vad som uppmätts i
labbet och vi bekräftade dessutom kanalens moderata selektivitet för
positivt laddade joner.
Det sista arbetet som jag utfört är att överföra ett kraftfält som
beskriver de atomära interaktionerna till den programvara som vi
använder i forskargruppen. Specifika egenskaper hos modelleringsprogrammet kunde därför kombineras med kraftfältet och dess beteende
kunde studeras på långa tidsskalor. Vi fann att kraftfältet beskriver
interaktionerna på ett godtagbart sätt och kombinationen med den nya
programvaran uppvisade lovande egenskaper när det gäller att förbättra
strukturella proteinmodeller.
52
6. Acknowledgements
I don’t know half of you half as well as I should like,
and I like less than half of you half as well as you
deserve.
Bilbo Baggins
T
he thesis you hold in your hand had not been possible to write
if not for a certain number of people. Some of you might not have
made a direct contribution to it but you kept me happy and motivated during these years, something that should not be underestimated.
Foremost, I would like to thank you Erik, my advisor and
partner in crime, for being such a generous person both when it comes
to giving me the opportunity to use the most current toys to play with,
as well as giving me the freedom to plan my own time. You have never
looked twice when I have gone for a workout session in the middle of
the day, although you’ve quite commonly commented on my sanity
for so doing. Besides this, one of the greatest things you’ve done is to
always believe in me, from the day you hired me after one Skype-call
until today, when most of our communication still is electronic... No
offense! You are a busy professor with an average altitude of at least
3 000 meters. You have an storehouse worth of knowledge in the most
surprising subjects and I’m trying to steal as many boxes as I can.
Bilbo would perhaps put it in this way: if I would know half of what
you know in half the subjects, I am half again as knowledgeable as the
rest on this planet.
I have been surrounded by more very inspirational people,
Arne, with all your ideas and enthusiasm, as well as Martin and
Gunnar, for the same reasons. Berk, I really appreciate your ability
to explain complicated stuff in an understandable way and taking the
time for so doing. You have that curiosity and drive to understand the
nitty-gritty details. I would like to be the same.
One could say that my scientific journey started during my
53
master’s thesis work when you, Andrew, took me under your wings.
Thanks for all the effort you underwent to grant me access to the
well-guarded Moffett Field, and for your and Joanna’s hospitality. Mike
and Karl, I’ll never forget our daily trips to Dana street for the great
company and the best coffee I’ve ever had.
I’d like to acknowledge all my collaborators and co-authors for
the great work we’ve put together. The Ames people of course, and
Ilpo, for the Finnish hospitality and computer power. Fredrik for being
such a ninja in the electrophysiology of the voltage-gated ion channels,
and a ninja master when it comes to running! You introduced me to the
ultras and hopefully we can run and compete together many more times.
The start of my PhD studentship could not have been better,
with extensive discussions on the Dragon and weekly gatherings around
another round of Vetligan. Thank you Jenny for those times, and Aron,
my longest-lasting room-mate, for all the good times we’ve had during
these years.
My past and current colleagues in the
have been great to work and hang out with!
Lindahl
group,
you
During my exile at KTH, I have missed the C4 corridor and
the group of social and positive people that inhabits it. I’ll
remember the dart sessions with Håkan and Andreas, supersweet
Friday cake clubs, nice cold(-room) after work beers with the
occasional Rock Band jamming, and great discussions over coffee, often involving Charlotta, Karin, Catrine, Dan, Daniel, Katta, and Linnea.
In fact, it’s not only the C4 corridor that’s a great place to
work in, but the whole DBB. I’m impressed by the deeply rooted
familiar spirit in the department. You are a handful of people that
nourishes it, respect to you!
A special thanks goes to Henrik and Johan, and the others in
Puppy TS, for keeping my body tired and my mind sound.
A curious thing that might be fit to mention here is that I
started to log all the songs I listen to in the office in September
2006 when I began my PhD studies. Since then I have listened to
91,389 songs. Assuming each song is 4 minutes long, and that I have
worked all days of the year (yeah right!), I have listened to inspirational music for 3.34 hours on average each day. Thank you for the music!
54
Thank you mom and dad for many, many things, but in relation to this, for teaching me to think critically and see the importance
of education.
The best thing I did during these years had nothing to do
with science, or wait, of course it had! My genetic information was
fused with that of Lina’s and our wonderful biochemical experiment,
Alma, was born. You are my treasures and I’m the world’s happiest
man. I love you!
dag den 4 november 2011
55
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