modeling of voltage-gated ion channels453207/fulltext01.pdf · the recent determination of several...

Modeling of voltage-gated ion channels

Modeling of voltage-gated ionchannels

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. Coverof 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-gatedion channels has catalyzed computational efforts of studying these re-markable molecular machines that are able to conduct ions across bi-ological membranes at extremely high rates without compromising theion 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 thechannel were captured, manifested in a secondary-structure change inthe voltage-sensor. In a follow-up study, we found that the energetic costof translocating this 310-helix was significantly lower than in the origi-nal conformation. Thirdly, collaborators of ours identified new molecularconstraints for different states along the gating pathway. We used thoseto build new protein models that were evaluated by simulations. Allthese results point to a gating mechanism where the S4 helix undergoesa secondary structure transformation during gating.These simulations also provide information about how the protein in-

teracts with the surrounding membrane. In particular, we found thatlipid molecules close to the protein diffuse together with it, forming alarge dynamic lipid-protein cluster. This has important consequences forthe understanding of protein-membrane interactions and for the theoriesof lateral diffusion of membrane proteins.Further, simulations of the simple ion channel antiamoebin were per-

formed where different molecular models of the channel were evaluatedby calculating ion conduction rates, which were compared to experimen-tally measured values. One of the models had a conductance consistentwith the experimental data and was proposed to represent the biologicalactive state of the channel.Finally, the underlying methods for simulating molecular systems

were probed by implementing the CHARMM force field into theGROMACS simulation package. The implementation was verifiedand specific GROMACS-features were combined with CHARMM andevaluated on long timescales. The CHARMM interaction potential wasfound to sample relevant protein conformations indifferently of themodel of solvent used.

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List of Papers

This thesis is based on the following papers, which are referred to in thetext by their Roman numerals.

I Bjelkmar, P., Niemela, P. S., Vattulainen, I., Lindahl, E.(2009) Conformational changes and slow dynamics throughmicrosecond polarized atomistic molecular simulation of anintegral Kv1.2 ion channel. PLoS Computational Biology, 5(2)e1000289.

II Schwaiger, C. S., Bjelkmar, P., Hess, B., Lindahl, E. (2011)310-helix conformation facilitates the transition of a voltagesensor S4 segment toward the down state. Biophysical Journal,100(6):1446-54.

III Henrion, U., Renhorn, J., Börjesson, S. I., Nelson, E. M.,Schwaiger, C. S., Bjelkmar, P., Wallner, B., Lindahl, E., Elin-der, F. (2011) Tracking a complete voltage-sensor cycle withmetal-ion bridges. Submitted.

IV 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 withlipids. Journal of the American Chemical Society, 132(22):7574-5.

V Wilson, M. A., Wei, C., Bjelkmar, P., Wallace, B. A., Po-horille, A. (2011) Molecular dynamics simulation of the an-tiamoebin ion channel: linking structure and conductance.Biophysical Journal, 100(10):2394-402.

VI Bjelkmar, P., Larsson, P., Cuendet, M., Hess, B., Lindahl, E.(2010) Implementation of the CHARMM force field in GRO-MACS: Analysis of protein stability effects from correctionmaps, virtual interaction sites, and water models. Journal ofChemical Theory and Computation, 6:459-66.

Reprints were made with permission from the publishers.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Membranes and transport proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.1 Material transport across membranes . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.2 Cellular membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 The membrane potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2 Voltage-gated potassium channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Voltage-dependent gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.2 Slow inactivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.3 The role of 310-helices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Molecular modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 Basic molecular dynamics theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.2 Force fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.3 Statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.1.4 Modeling the membrane potential in simulations . . . . . . . . . . . . . . . . . 34

2.2 Modeling of protein structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2.1 Comparative modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2 Physics-based modeling and structural refinement . . . . . . . . . . . . . . . . 39

3 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1 Conformational changes in the KV 1.2 ion channel through polarized

microsecond molecular simulation (paper I) . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Secondary structure of the S4 helix in gating. (paper II) . . . . . . . . . . . . . . . . 423.3 Tracking the voltage-sensor gating motion with metal-ion bridges, computer

modeling and simulation (paper III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Lateral diffusion of membrane proteins (paper IV) . . . . . . . . . . . . . . . . . . . . 433.5 Linking structure and conductance in a simple ion channel (paper V) . . . . . . 443.6 Implementation of the CHARMM force field in GROMACS (paper VI) . . . . . . 45

4 Concluding remarks and future perspectives . . . . . . . . . . . . . . . . . . . . 475 Svensk populärvetenskaplig sammanfattning . . . . . . . . . . . . . . . . . . . 516 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

1. Introduction

The fact that life evolved out of nearly nothing, some10 billion years after the universe evolved out ofliterally nothing, is a fact so staggering that I wouldbe mad to attempt words to do it justice.

Richard Dawkins

This doctoral thesis describes the work that I have done during my timeas a PhD student at the Department of Biochemistry and Biophysics atStockholm University during the years of 2006 through 2011. The studieshave been published in peer-reviewed scientific journals and the papersare the core material of the thesis. They are presented as appendices inthe end of this book. To put this work in perspective and to provide abrief overview to the field, a preceding introductory and summarizingtext has been compiled. It starts with a background chapter introducingthe underlying biology and continues to describe the methodology andkey concepts in greater depth. The next two chapters contain a summaryof the main results and conclusions, and a technology walkthrough in-cluding some speculation of where we might head in the future. The textends with a popular science summary in Swedish, my acknowledgementsand 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 areincredible molecular machines incorporated in cellular membranes. Theyare precisely regulated by electrical changes in the local environment andcapable of highly selective conduction of ions at great speeds. The open-ing and closing of these proteins have been the main subject of my re-search. In my toolbox I have had models describing interactions betweenatoms, and cutting-edge software programs utilizing hundreds of com-puter processing units in parallel on the most modern computer clustersthere is. These models together with the data programs can be thoughtof as a "computational microscope" (term coined by Klaus Schulten),generating a great amount of atomic-level information on the dynamicsof the molecules studied. The data has been analyzed, complemented andcorroborated with experimental results derived from biological systemsin the wet lab.

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1.1 Membranes and transport proteinsAs soon as the information-bearing molecules were encapsulated by amembrane somewhere in the early evolution of life, material for the pro-cesses of life needed to be transported across this impermeable barrier.In truth, for small and uncharged molecules, the transport could in factbe unassisted diffusion across the membrane due to the relatively lowfree-energy cost of such transport, but this energy barrier is in practicetoo high for larger and charged, even though small, molecules. Addition-ally, unassisted diffusion is unspecific by nature, and what is required is ahighly efficient and regulated selective transport of a particular moleculeor group of molecules.For such a regulated transport of larger and charged molecular species,

dedicated molecular machines evolved. The birth of such remarkablemolecules was of course not a blink-of-an-eye event but, as is the casefor evolution in general, it was achieved by gradual improvement of themolecules involved by molecular evolution over millions of years. Thereare still traces of the gradual evolution of transport proteins in the liv-ing organisms today, both as simple molecules with primitive functions,as well as independently evolved protein domains found in the modularorganization of modern molecular machines.

1.1.1 Material transport across membranesCells are distinct entities separated from the surrounding environment bythe plasma membrane. Within one cell, membranes are also efficientlydividing it into different compartments, called organelles, for localizedand specialized functions (Figure 1.1). By incorporating specific proteinsinto the membranes the necessary flow of material between the organellesand the surrounding environment can be achieved. There are many dis-tinct types of membrane transport proteins and they can be regulatedin different ways.There are two mechanisms for transport of substances, substrates,

across a membrane; energy-independent facilitated diffusion andenergy-driven active transport. There are six characterized classes oftransporters [1]: channels and pores, electrochemical potential-driventransporters, primary active transporters, group translocators,transmembrane electron carriers, and accessory factors involved intransport. The first and second groups are proteins catalyzing facilitateddiffusion. The channel group opens up an aqueous transmembrane porefor substrate passage along its concentration gradient whereas in thesecond group the protein itself functions as a carrier. The other groupsof transporters include proteins that for example transport a soluteagainst its concentration gradient using a primary source of energy,

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Figure 1.1: The eukaryotic cell. The cell is separated from the outside by the plasmamembrane. Internally, it is divided into subcellular structures, called organelles. Thenuclear envelope separates the information-bearing molecules from the rest of thecell. The rough endoplasmic reticulum is a complex membrane structure holding theribosomes, which are the molecular factories where proteins are built. In the mito-chondria, 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 inwhich 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 duringthe transport, and proteins that facilitate transmembrane electrontransport.Membrane channels typically control the flux of materials across cellu-

lar membranes through high selectivity combined with high conductivityand through gating that is sensitive to essential environmental factors.For example, they transport water, ions, and other small substrates, asalcohols, as well as large proteins, and their opening and closing can becontrolled by ligand-binding, voltage, pH, or pressure changes.

1.1.2 Cellular membranesMembranes found in cells typically consist of phospholipids and aconsiderable amount of membrane proteins, cholesterol and lipids withsugar adornments, called glycolipids. Phospholipid molecules have apolar head-group and long non-polar tails. In a water environment these

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Figure 1.2: Membrane protein in bilayer. In this snapshot from paper IV, thetransmembrane part of the KV 1.2 channel (green) is embedded in a membrane bilayerconsisting of the phospholipid POPC (detailed in inset). The lipids have a polarend facing the surrounding water and two long apolar hydrocarbon chains. Thesehydrophobic tails aggregate and form the bilayer core (gray). The surrounding waterand lipids in front of the protein, in the line of sight, were omitted for clarity.

non-polar chains spontaneously aggregate so that their water-exposedsurface is minimized, shielded by the polar head-groups pointing towardthe water. One of the ways these lipids can organize themselves is asbilayers (Figure 1.2). These membranes are efficient barriers, in practiceimpermeable to all substances except small uncharged molecules. Thisis due to their stability and the associated energy cost to make alarge-enough hole for a molecule with considerable size to be able todiffuse through, and additionally, the cost of exposing charged andpolar 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 two-dimensional liquid where the molecules within it can diffuse laterally inthe plane of the membrane. The composition of the membranes can al-ter their properties significantly and can in fact also affect membraneprotein function. For example, membranes with less amount of choles-terol and/or a higher fraction of short-chained or unsaturated lipids willhave a lower melting temperature, i.e. a higher fluidity. Compared to awater environment the diffusion in the bilayer is two to three orders ofmagnitude slower [3, 4].

1.1.3 The membrane potentialVirtually all cells have a transmembrane potential across the plasmamembrane. The potential has two basic functions, first it provides a driv-ing force for operating the numerous molecular machines embedded inthe membrane, and secondly, for excitable cells such as neurons and mus-cle cells, it is used for transmitting signals in between cells. The potential

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is due to an ion concentration imbalance across the membrane establishedby the membrane protein Na+/K+-ATPase. These active transport pro-teins each continuously expel three sodium ions out of the cell and at thesame time import two potassium ions. Hence, both a charge separation,giving rise to an electrical potential, and concentration gradients areestablished. Membrane protein channels are typically only permeable tospecific ions and their collective interplay of conductances establishes theresting membrane potential as well as the action potential of excitablecells.Having open potassium channels in the membrane enable the potas-

sium ions to passively diffuse out of the cell, in the direction of theirconcentration gradient that is established by the Na+/K+-ATPase. How-ever, as potassium ions move out of the cell, the inside becomes even morenegative and the rising transmembrane electric potential starts to coun-teract this diffusion. These two opposite forces become equal and thenet flow of potassium ions is zero and a steady state has been reached[5]. It is not a thermodynamic equilibrium state since ions are continu-ously pumped across the membrane. The resulting resting potential dueto the charge separation can be calculated using the Nernst equation,see e.g. [6], given a concentration difference. If only potassium ions areconsidered the membrane potential would be approximately -100 mV at37 ◦C and typical ion concentrations. Performing the same calculationfor the other key ion involved, sodium, gives a transmembrane potentialof +55 mV. The fact that the true, measured, resting potential (around-70 mV) is close to the value for potassium ions is because the perme-ation of potassium ions is approximately an order of magnitude largerthan 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 (Figure1.3), is formed. In neurons this pulse is used for cell-to-cell communica-tion and in other cells its main function is to activate intracellular pro-cesses, such as muscle cell contraction and insulin secretion. The recentreview article by Barnett and Larkman [8] summarizes what is knownabout the action potential. Its origin is a small local depolarizing re-sponse initiated in some part of the cell membrane by external stimuli,either from specialized nerve endings in sensory neurons or from up-stream neuron signaling in the central nervous system. This local shifttowards less negative potential triggers fast voltage-gated sodium chan-nels to open if the local depolarization is larger than a threshold potentialof about -45 mV, increasing the permeation of this ion species consider-ably. Consequently, the extracellular sodium ions start to flow into thecell along their concentration gradient further depolarizing the mem-brane. Within one millisecond, the potential reverses and continues torise towards the steady state potential for sodium ions (+55 mV). How-

15

Undershoot

Resting potential

Threshold

Peak

Ris

ing

Phas

e Falling Phase

Overshoot

FailedInitiations

Stimulus

~ +40

0

~ -45

~ -70

Mem

bran

e Vo

ltage

(mV)

0 1 2 3 4 5Time (ms)

fredag den 11 november 2011

Figure 1.3: The action potential of excitable cells. Once a local depolarizationstimulus from an external source becomes larger than a threshold of approximately-45 mV, the opening of sodium channels depolarize the membrane voltage duringthe rising phase. The voltage reverses before the slower potassium ion channels havebeen activated in response to the depolarization. The sodium channels deactivateand the open potassium channels repolarize the voltage during the falling phase.Before the potassium channels have had time to close the normal resting potential ofapproximately -70 mV is undershoot somewhat.

ever, before this value is reached two opposing factors start to act. First,although voltage-gated sodium channels are activated fast they also havean inherent property of inactivation on the millisecond scale. The sec-ond factor is an increased flux of potassium ions due to the opening ofvoltage-gated potassium channels as the membrane potential becomesmore positive. Hence, potassium ions start to flow out of the cell at thesame time as sodium channels are inactivated and the depolarizationceases (peak) and starts dropping again causing the sodium channels toclose even faster. Consequently, the membrane is repolarized and returnsto the resting potential.

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1.2 Voltage-gated potassium channelsAll potassium channels are related members of a single family of proteinsand are found in all domains of life; bacteria, archea, and eukaryotes -both plants and animals. Their amino acid sequence is easily recognizedbecause they share a characteristic sequence of residues in the narrowestpart of the ion-conducting pore responsible for the molecular discrimina-tion of ions by a remarkably efficient mechanism [9, 10, 11]. The channelsare fascinating molecular machines capable of conducting K+ ions almostat the theoretical limit at a rate of ∼108 ions per second, while at the sametime keeping the permeation of potassium ions approximately 104 timeshigher compared to sodium ions [12, 13]. While their ion-permeationcharacteristics are common, diversity among different subfamilies of K+

channels is mainly manifested by the various ways the channel gate isopened.Potassium channels that are voltage-gated, abbreviated KV , are a

diverse and ubiquitous group of membrane proteins belonging to thevoltage-dependent cation channel family, which also includes the evolu-tionary related sodium (NaV ) and calcium (CaV ) channels [14]. Normally,the concentration of K+ outside the cell is more than an order of mag-nitude lower compared to that in the intracellular fluid and hence, uponopening these channels, an outward current of positively charged cations,called the alpha current, is established. Accordingly, the KV channelslimit cell excitability by returning the membrane potential to its restingstate as explained in the preceding section. They are therefor key play-ers in a range of cellular processes [15], such as dictating the durationof the action potential in excitable cells, as well as cell proliferation [16]and volume regulation [17] in non-excitable cells. They are also suscep-tible to disease-causing mutations [18, 19] and are the targets of drugsused against pain, epilepsy, cardiac arrhythmias, hypertension and hy-perglycemia for example. Understandably, the pharmaceutical industryhas 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 uncon-trolled shaking motion of the fly’s legs when anesthetized with ether[21]. Much of what is known about voltage-gated K+ channels has beenderived from electrophysiological studies of this channel in addition tothe pioneering work of Hodgkin, Huxley and Katz on the giant squidaxon (e.g. [7, 22, 23]). Since the first isolation, hundreds of genes codingfor diverse KV channels in different organisms have been found. In thehuman genome, some 40 KV genes have been identified and character-ized, and they have been assigned to 12 subfamilies (KV 1 to KV 12) basedon sequence similarities. The KV 1 subfamily is called the Shaker-related

17

subfamily since the Shaker-homologous human gene KCNA3 is codingfor 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 hasturned out to be very tricky to determine the three-dimensional high-resolution structure of membrane proteins, and it was not until 1998that the first (potassium) ion channel structure, KcsA, was solved bythe MacKinnon lab [12]. It is however not voltage-gated, rather pH-sensitive, but only a few years later the same group was able to crystallizethe KVAP channel [24] from the thermophilic archaebacteria Aeropyrumpernix (for which, together with previous work, Roderick MacKinnon wasawarded the Nobel Prize in Chemistry 2003). Subsequently, the partlyincomplete structures of the mammalian KV 1.2 channel [25, 26] weresolved and, most recently, a complete structure of a chimeric structureof a KV 1.2/KV 2.1 channel from rat was published [27]. Although thesecrystal structures have been extremely important for the description ofthe atomic workings of these proteins, the KV structures all caught thechannel in the open state and to date, the closed state, or intermediatestates along the opening pathway, have not been properly characterized.Most of the work in this thesis is an attempt to bridge this knowledge gapby studying the Shaker and KV 1.2 channels by computational and com-plementary experimental methods. An alignment of the transmembranesection of these two sequences can be found in Figure 1.4.The KV channels are made of four subunits, each containing six trans-

membrane helices, called S1-S6 (Figure 1.5). S1 through S4 form thevoltage-sensing domain (VSD) that responds to changes in the mem-brane potential. Helices S5 and S6 from the four subunits come togetherinto the centrally located pore domain (PD) where the ion-conductingpore is present. The structural alterations within the VSDs upon gatingare mediated to the PD by a linker helix, called the S4-S5 linker, inducinga conformational change that alters the kink angles of the pore-lining S6helices which, in turn, control the blocking of the ion-conducting path-way at the intracellular side [28, 24]. In contrast to the distinct subunits(i.e. separate peptide chains) of the KV channels, the evolutionary re-lated 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, calledthe tetramerization domain, or T1, aiding the assembly of the trans-membrane part as well as providing a binding platform for an accessoryβ-subunit whose function is not entirely understood but it is related toa 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 stableon their own. Interestingly, recently gene orthologs of the VSD in voltage-

18

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!"#$%&'()*+,,,,,,-..,+#%//!%01'$/22$334*443565'*4135325!035%7"$0$53740$,,,,-89,,,,,,,,,,,,,,,,,,,,,:;<:<:<<<<<<<<<<<;<<<<<<<<<:<::;:<<<<<<<<<<<<<<<<<!"#$.&($4,,,,,,,,=>-,('%?/1%01'$/22$33%*443565'*3144055!035%7"$0$53740$,,,,-@=

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233

R1/294S5

S2

R2 R3 R4

K5 R6

S4

S3

Linker

FilterPore helix S6

S1

183

226 236 259

fredag den 7 oktober 2011

Figure 1.4: Alignment of Shaker and KV 1.2 sequences. Sequence alignment ofthe transmembrane section of the Shaker gene KCNAS from Drosophila melanogasterand the KV 1.2 gene KCNA2 from Rattus norvegicus (common rat) as given by theEMBOSS implementation of the Smith-Waterman algorithm [32] with default pa-rameters. Key residues are highlighted (KV 1.2 enumeration) in red, blue and gray fornegative, positive and neutral, respectively. The characteristic sequence signature inthe selectivity filter of the potassium channels is shown in yellow. Secondary structureis indicated above the alignment in gray; S1-S6 transmembrane helices, the interfacialS4-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 andbeautiful result since, in an evolutionary perspective, one could think ofthe VSD as a distinct functional entity that evolved by itself and lateron was coupled to other functional domains, such as ion channels andphosphatases.

19

VSD PD

S1 S2 S3 S4 S5 S6

Linker

PH

SF

+++

måndag den 26 september 2011

Figure 1.5: Structure of the voltage-gated potassium channel. The topologyof 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 voltage-sensing domain (VSD) while the darker-green part forms the pore domain (PD) withits preceding interfacial S4-S5 linker helix, re-entrant pore helix (PH) and selectivityfilter (SF). In the S4 helix, there are several positively charged amino acids that areaffected by changes in the electric potential across the membrane. Side and top viewsof 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 thatis accessible to the ions once they are in the open conformation. At the extracellularend of the channel the selectivity filter narrows the channel for efficient ion filtering.

1.2.1 Voltage-dependent gatingThe four-helix bundle of the VSD has a shape reminiscent of an hour-glass, with helical ends that splay outwards at the intracellular and extra-cellular sides, forming two water-filled cavities and a central constrictedregion. The protein surfaces lining the two water-filled cavities are rel-atively polar whereas the environment around the domain center is hy-drophobic. Most of the polarity comes from several charged and evolu-tionary conserved residues, in particular arginines, in the S4 helix. Thesepositive charges are located at every third position of the helix makingit a rather peculiar transmembrane helix. Generally, it is very unfavor-able to have charges buried into the non-polar membrane bilayer coreand this would hence be evolutionarily selected against. However, bothbiophysical experiments [33] and computer simulations [34] have shownthat the S4 helix can be stable in isolation in the membrane bilayer al-though other studies report a considerably higher energy cost [35, 36].In any case, this helix has been conserved throughout evolution becauseit has a very special purpose, namely to function as a voltage-sensor;an electromechanical converter transforming the energy of the change inthe transmembrane electrical potential into a mechanical response of theprotein 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

[O] [C1] [C2] [C3]

Membrane coreVSD hydrophobic region

Extracell. neg. cluster

Intracell. neg. cluster

Hyperpolarization

Depolarization

måndag den 10 oktober 2011

Figure 1.6: Model of gating. Side view of a voltage-gated ion channel, where onlyone voltage-sensing domain (VSD) is shown. Extracellular side is up. In the bottompanel, a cartoon of the model of the S4 gating motion is shown. In the open activatedstate of the channel (left), the S4 helix (in cyan ribbon representation) is locatedtowards the extracellular side due to the depolarization of the membrane. All fourarginine residues (blue spheres) on the extracellular half of the helix are locatedabove the central hydrophobic region of the VSD. Upon gating, the S4 helix movestowards the intracellular side, due to repolarization of the membrane. This is a step-by-step process involving several meta-stable states (here represented by the C1 andC2 states in analogy with the terminology used in paper III), where the arginineresidues change their interaction partners along the way (as depicted in the modelcartoon and described more in the text). At the most closed state C3 (right), theS4 helix has moved approximately 10 Å vertically towards the intracellular side, andall arginine residues except the most extracellular one have passed the hydrophobicregion of the VSD. The open state is the homology model of Shaker from paper IIIand the closed model is based on the most closed state, C3, in paper III and the PDfrom [41].

is at rest and the ion channel is closed, the transmembrane potential isapproximately -70 mV (negative at the intracellular side). Consequently,the voltage-sensor S4 helix is attracted and positioned towards the in-tracellular side of the membrane. Once the depolarization occurs, thetransmembrane potential reverses, causing the helix to move toward theextracellular side (Figure 1.6). This movement of arginine charges upongating is producing a gating current that is readily detected experimen-tally [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 alarge barrier, which would explain their slow activation within a fewmilliseconds. Once the random thermal motion of the system overcomesthe barrier the transition is considerably faster, perhaps as rapid asten microseconds [42]. The barrier has been proposed to be due to thehydrophobic region in the VSD core, centered around the evolutionaryconserved F233 in Figure 1.4, through which charged amino acidsfrom the voltage-sensor have to pass to open and close the channel[43, 27, 44]. The gating charges’ movement is catalyzed by sequential ionpairing with negatively charged residues in two clusters on neighboringhelices, one cluster located at the extracellular side of the hydrophobicregion (corresponding to E183 and E226 in Figure 1.4) and the otheron the intracellular side (E236 and D259) [38, 45]. Since experimentalstructures of the intermediate and the closed states are missing, theatomic details of the gating movement of the voltage-sensor and thesubsequent conformational changes leading to opening and closing ofthe gate is still not understood properly. Probing this gating pathwayby computer simulations and modeling we have tried to shed light onthe understanding of this mechanism on an atomic level.

1.2.2 Slow inactivationA striking and somewhat unexpected feature of the voltage-gated chan-nels 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 theprotein can adopt at least three distinct conformations; a closed state,where the channel is non-conducting, which is the predominant restingstate at hyperpolarized membrane potentials; an active conducting stateoccupied transiently during depolarization; and a non-conducting relaxedstate reached at prolonged depolarization [46]. Since crystal growth oc-curs under depolarized conditions it has been assumed that the stateof the proteins reflect this relaxed state even though the central pore isopen [47].

1.2.3 The role of 310-helicesThere is a general belief in the protein structural community that 310-helices are relatively rare and short in protein structures. The idea has itsorigins in the classical studies of the helical conformations of polypeptidesby Pauling, Corey and Branson [48]. The conformation is energetically

22

586 310 helices in membrane proteins

Data Bank accession nos. and residues in helices are listed in the legend of Fig. 2). We found two modes of packing. Mode A was observed in eight cases. In this mode, a side-chain ridge lies in a groove formed by other regions of the protein (example in Fig. 2 A). Mode B was present in four helices. In this arrange-ment, one or two side chains from another region of the protein pack against the 310 helix face formed between the side-chain ridges (example in Fig. 2 B). A single case was observed where the helix is completely encased by the rest of the protein; this helix has a very speci!c sequence with four glycines and one proline out of eight residues. More quantitatively, the different modes are distinguishable by the fraction of solvent-exposed surface area, such that helices lying in a groove (mode A) tend to be less solvent exposed than in the other pack-ing arrangement (mode B; Fig. 2 C). Overall, this simple analysis supports the idea that packing of long 310 heli-ces has speci!c requirements. It also highlights the need for more extensive studies of the structural con-text in which 310 helices are found. We suggest that long 310 helices are rarely observed in protein structures, not just because they might be intrinsically less stable than

helices but also because they have packing require-ments that are rarely met over long extensions, result-ing in the unraveling of the 310 conformation or in the variability of the helical stereochemical parameters de-tected in many studies.

Another important characteristic of 310 helices is dy-namics. This property has been well documented in iso-lated peptides, which are important model systems for the study of helices. In the case of 310 helices, one of the best-studied systems involves peptides rich in methyl-alanine, a non-natural amino acid with an extra methyl group on the carbon (Karle and Balaram, 1990; Karle et al., 1994; Crisma et al., 2006; Bellanda et al., 2007). Because of the stereochemical constraints imposed by the presence of two large substituents in the C of the amino acid, these peptides have a strong tendency to adopt a helical conformation, 310 or helical. In these systems, it has been demonstrated that a peptide can adopt both conformations and that it is possible to shift the equilibrium between one conformation and the other by altering the polarity of the solvent or tempera-ture. In some cases, these two conformations appear to coexist in the same peptide so that one region adopts a 310-helical conformation and another region adopts an

-helical conformation. A few similar observations have been documented for peptides formed by natural amino acids; the two conformations appear to be present in different segments of the peptide, and this distribution is altered by small sequence changes (Fiori et al., 1994; Dike and Cowsik, 2006; Mikhonin and Asher, 2006). Molecular dynamics studies of helices in peptides have also proposed that 310 and helices coexist along the same peptide and that interconversion between the two

proteins, it has been shown that there is a maximization of the interaction surface between helices (Bowie, 1997). In an helix, the 3.6 residues per turn positions side chains every 100° around the helical axis, stagger-ing the side chains and offering an extensive set of modes of interaction (Fig. 1). This arrangement there-fore imposes few structural restrictions on the inter-acting partners and increases the probability of the stabilization of long helices in a protein. In con-trast, as a result of the 3.2 residues per helical turn in 310 helices, side chains are disposed in a more restrictive fashion; they form ridges along the helix (Fig. 1). Intui-tively, this disposition imposes speci!c structural re-quirements for the packing of long 310 helices with other protein regions. In an effort to verify this idea, we performed a visual inspection of long 310 helices con-taining at least two turns (seven or more residues) in 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 struc-tures included in their studies. As a result, the !nal pool contained 12 proteins for a total of 13 helices (Protein

Figure 1. Canonical helical conformations. Side views and views along the axes of a helix in a canonical -helical conformation (left) and canonical 310 conformation (right). Dotted lines in-dicate hydrogen bonds between atoms in the backbone of poly-peptide. Helices were generated using PepBuild. All !gures were prepared with PYMOL.

on August 1, 2011jgp.rupress.org

Dow

nloaded from

Published November 29, 2010

måndag den 10 oktober 2011

Figure 1.7: The structure of α- and 310-helices. In the idealized α-helix (left),residue i is interacting with residue i+4 forming a helical structure having 3.6 residuesper turn so that consecutive residues make an angle of 100◦ around the helical axis.The helical pitch is 5.4 Å and the rise along the helix axis is 1.5 Å per residue (5.4/3.6).In the 310-helix however (right), residue i interacts with residue i + 3 giving three(aligned) residues per turn of the helix and an angle of 120◦ between them. The namecomes from the observation that there are ten atoms in the ring formed by makingthe hydrogen bond. The pitch is consequently greater at around 6 Å, and the riseper residue is 1.93-2 Å. A more intuitive description; the 310-helix is a tighter-woundα-helix, being longer (given the same number of residues) and more narrow. Figureadapted from [52].

unfavorable and the cost of transforming a decapeptide in a protein-likeenvironment 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 structuresnow available has shown that 310-helices in proteins are quite common.Although they are most often only 1-2 turns long a significant number oflonger 310-helices (>7 residues) has been found [50, 51]. However, theyseem 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 hasbeen noticed that they have a wide variability in main-chain torsion an-gle distribution and this irregularity increases rapidly with helix length.

23

Another important point in explaining the difference between the twohelix types is the way helices are packed against each other. The peri-odic alignment of the residues in the 310-helix imposes more constraintsin 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 thepresence of a 310 conformation in the S4 helix [53, 54]. Moreover, re-cent experimental and computational studies support the idea that asignificant 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 isunclear how the helix behaves during gating. Our studies suggest thatit undergoes a conformational change such that there is a stationary 310

region, located around the phenylalanine residue in the VSD core, slidingalong the helix during its translation. The 310 conformation of S4 offersan apparently simple explanation for some of the issues previously raisedby the properties of the voltage-sensor helix. First, it would explain theconserved sequence pattern of positively charged residues at every thirdposition, because the 310 conformation places them on the same helicalface, promoting interactions with the polar environment in the VSD coreand hence shield the charges from unfavorable exposure to the non-polarmembrane interior. Second, the conformation would limit the necessaryS4 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 ofbiomolecules, but in practice only two give the complete high-resolutioninformation of the relative positions of all atoms in a protein, namelynuclear magnetic resonance (NMR) spectroscopy and X-ray crystallog-raphy. Whereas the structures of water-soluble proteins are often rela-tively easy to determine, membrane proteins have for a long time eludedstructural biologist. Their structural determination has proven very hardand expensive. Consequently, the Protein Data Bank (PDB), where allbiomolecular structures are deposited, only consist of a couple of hundredmembrane protein structures, in contrast to tens of thousands structuresof water-soluble proteins. Even when a structure of a particular proteinis not known there is a fair chance of successful creation of a relevantmodel of it based on known structures using computational molecularmodeling techniques. This is because they are evolutionary related andproteins with similar function can often safely be assumed to have similarstructures.Once a protein structure has been obtained, either by experiments

or modeling, one would typically like to study its function. This toocan be done using molecular modeling methods. For example, moleculardocking protocols try to estimate the binding free energy of a ligand tothe protein by evaluating the energies of different ligand conformationsin different positions relative the protein. Another common applicationis the elucidation of reaction mechanisms in the active sites of enzymesby quantum mechanical methods. In both these applications, the proteinstructure is typically fixed or at least restricted to only a few degrees offreedom. This can be a rather severe limitation in some cases because theprotein’s flexibility could be very important for the binding of the ligand.To study the dynamics in the systems one has to turn to simulationtechniques that sample the conformational flexibility of the molecules.

25

Methods of modeling and simulation of molecular systems can becrudely divided into two distinct classes based on how the interactionsof the atoms are described. In the first class, usually called quantumchemistry or simply quantum mechanics (QM), electrons are treated ex-plicitly and the energy of the system is calculated using the Schrödingerequation. In the other class, called empirical force field methods or molec-ular mechanics (MM), only the positions of the atom nuclei are consid-ered and they move and interact according to classical physics. Quan-tum mechanical methods rely on the Born-Oppenheimer approximation(as does molecular mechanics) stating that the higher velocities of theelectrons relative the nuclei allows these two motions to be treated inde-pendently and the underlying Schrödinger equation can take a simplerform. The explicit treatment of electrons in these methods allow for thestudy of chemical processes of permanent changes in electronic struc-ture, e.g. chemical reactions, at the price of an increase in computationalcost. Even though both computer software and hardware have developedenormously in recent decades, quantum chemistry is still in practice tooexpensive for studying biological macromolecules on relevant timescales.Additionally, the treatment of explicit water is troublesome in quan-tum chemistry methods and they are typically not able to capture thedynamics of biologically relevant temperatures. Treating the molecularsystem classically using molecular mechanics methods rectifies some ofthese shortcomings and allow increased simulation timescales of severalorders of magnitude. Although these methods generally can not be usedto study chemical reactions there is still a wealth of interesting systemswhere they can be used.

2.1 Molecular dynamicsMolecular dynamics (MD) is a computer simulation of the physical move-ments of atoms, or particles in general. The atoms are allowed to interactduring a period of time and their positions are calculated by solving New-ton’s equations of motion numerically. The method assumes a classicaldescription of the system where the atoms are point charges that in-teract through a molecular mechanics interaction potential. The partialcharges on the nuclei are assigned depending on their chemical envi-ronment. Hence, this method can in principle not be used in the studyof processes where changes in the electronic structure matters althoughhybrid QM/MM methods have been developed trying to overcome thislimitation. On the other hand, the benefit of molecular dynamics is that itcan simulate large molecular systems on biologically relevant timescales.Molecular dynamics simulations generate a sequence of states, a trajec-

26

tory, of the molecular system. The states are time-dependent and corre-spond to a possible trajectory of motion of the particles. The energiesof the states in the trajectory generated by solving Newton’s equationsfollow the thermodynamic equilibrium distribution (the Boltzmann dis-tribution, see section 2.1.3). Another way of exploring conformationalspace is by Monte Carlo (MC) sampling. Here, new conformations aregenerated randomly and by evaluating the energy of the new state withthe underlying MM potential the random move is either accepted or re-jected in a way such that the set of accepted conformations also followthe equilibrium distribution.To start a MD or MC simulation three things are needed:

1. A starting structure2. An interaction potential3. An algorithm for generating new states

The first requirement will be dealt with in section 2.2 and the secondand third ones will be described in more detail in the following sec-tions. The simulation package used throughout the work in this thesisis GROMACS [57]. Other commonly used packages are CHARMM [58]and NAMD [59].Another important point regarding the simulation of molecular sys-

tems is that it consists of a finite number of particles and hence there isa boundary between the outer-most located particles of the simulationunit cell (often also called the simulation box) and the surroundings. Thisis usually coped with by using periodic boundary conditions. One couldthink of this as cloning the entire system in all directions and lettingparticles on the boundary of the unit cell interact with particles fromthe neighboring clone(s). Also, particles moving out of the box will, inthe next step, appear on the opposite side of the system. This is a neattrick to circumvent the finite size effects of the simulations but one has tomake sure the simulation box is large enough to prevent particles frominteracting with their periodic copies because such interactions wouldpropagate throughout the collection of systems giving severe artifacts.

2.1.1 Basic molecular dynamics theoryIn MD simulations, the time evolution of the positions of the N atomsof the system is calculated by solving Newton’s second law of motionstating that the acceleration of atom i times its mass equals the netforce exerted on it by the remainder of the system:

~Fi = mi · ~ai = mi ·d2~ridt2

. (2.1)

27

The force on atom i is, in turn, calculated by differentiating the totalpotential energy function V (see next section) with respect to the atom’sposition

~Fi = −∇iV (~r1, ..., ~rN ). (2.2)

Given the position of atom i at a specific time ~ri(t), the position aftera short finite time interval, ~ri(t+∆t), is given by a standard Taylor seriesexpansion around ~ri(t):

~ri(t+ ∆t) = ~ri(t) +d~ri(t)dt

∆t+d2~ri(t)dt2

∆t2

2+ ... (2.3)

Numerous algorithms exist for integrating Eq. 2.1 and many are fi-nite difference methods based on the Taylor series expansion above. Thesimple Verlet integration algorithm [60] is deduced by adding the Taylorexpansion above with the corresponding one for ~ri(t−∆t) and truncatingthe expansion after the third order, yielding

~ri(t+ ∆t) = 2~ri(t)− ~ri(t−∆t) +~Fi(t)mi

∆t2. (2.4)

Note that the expression for the second derivative of the position withrespect to time in Eq. 2.3 is the acceleration and hence can be substi-tuted 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 ofall atoms in the system. Other popular, and related, algorithms are forexample the leapfrog (default in GROMACS) and the Velocity-Verletintegrators [61, 62]. All these methods are time reversible, meaning thatinversing the velocities of all atoms would run the simulation in exactlythe opposite direction. The computational expense of any particular in-tegration scheme is insignificant compared to the cost of calculating theforces acting within the system but the length of the time step, ∆t, ison the other had very important. The maximum time step length de-pends on the integration method but it is has to be significantly smallerthan the period of the fastest movements. The fastest modes of motionin atomistic systems is bond vibration involving hydrogens, limiting thetime step to 0.5− 2 · 10−15 s (femtoseconds, or fs). This small time step isone of the main drawbacks of MD simulations since it limits the availabletimescale.

2.1.2 Force fieldsThe underlying potential energy function describing the interactions be-tween atoms (or particles in general) is usually called a force field. Thefundamental ideas behind it were conceived in the late sixties by Shneior

28

Lifson, Arieh Warshel and Michael Levitt. It is a mathematical functionrelating the total potential energy of the system to the positions of itsparticles. In this sense the name is somewhat misleading because it is anexpression of the energy, not the force. However, since differentiation ofthe energy function gives the force and it is the force that governs theparticles’ motion (see previous section), the name is understandable. Thepotential consists of several terms of elementary mathematical functionseach describing an intra- or intermolecular interaction. There are numer-ous force fields used in the molecular simulation community but most ofthem have the same basic structure. They typically contain three or fourterms describing the bonded interactions, mediated by covalent bondsbetween atoms, and two terms for non-bonded interactions between allpairs of atoms. Since the number of pairs in a system is quadratic in thenumber of particles, the way in which the non-bonded interactions arecalculated 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 be-tween 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 ofthe covalent bond is the Morse potential, an asymmetric function giv-ing a zero force at infinite distance, not infinite as is the case for theharmonic function. The bond vibrations involving light atoms are thefastest modes of motion and they are therefor the limiting factor of thetime step length in the integration scheme. Constraining bonds to theirequilibrium bond lengths allows for time steps of ∼2 fs and the rationalfor doing this is that it is in fact a better approximation to a quantummechanical ground-state oscillator than the classical harmonic oscillator.Constraining bonds in this way is common in MD simulations today andhas 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 bet-ter optimize the fit to vibrational spectra two additional terms can beadded, the Urey-Bradley function for the angle bending of certain tripletsand an improper dihedral potential term for preserving the planarity ofplanar groups. These two terms are typical for the CHARMM force field[63] for example. In analogy to the constraining of bonds to their equilib-

29

rium bond lengths, the next-fastest degree of motion is angle-vibrationsinvolving hydrogens and removing those vibrations would motivate inte-gration time steps of 4-5 fs. This can be done by introducing so calledvirtual interaction sites, replacing the fastest-moving atoms. In contrastto standard atoms, the positions of virtual interaction sites are calcu-lated in a predefined manner (based on the geometry of the groups asdefined by the force field) from neighboring heavy atoms. This is a fea-ture implemented in GROMACS and has been studied in paper V andused in all papers except papers II and VI.The energy of torsional rotation around the middle bond of four con-

secutive atoms, A-B-C-D, is modeled by a periodic (proper) dihedralfunction

Etorsion = Kω(1 + cos(nω + δ)) (2.7)

where the rotational angle, ω, by convention is measured with respectto the cis conformation (A and D on the same side of the bond). Theenergy maximum is set by Kω and the number of minima is representedby the n parameter, and δ determines their location. For more compli-cated functional forms one could either sum several such functions forthe same quadruple of atoms or express it as an expansion in powers ofcos ω, the so called Ryckaert-Bellemans potential.The treatment of the torsional rotation around bonds in the pro-

tein backbone is crucial for the correct sampling of its conformations.Rather recently, an extension to the CHARMM force field was pub-lished [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 ex-pressed as a two-dimensional grid-based energy that is added to thedihedral energy of the pair. The authors found that the inclusion of theCMAP term gave a better conformational sampling of the protein whencomparing to crystallographic data. In paper V, the implementation ofCHARMM in GROMACS study the behavior of this correction term onlonger timescales.So far all interactions have been between particles connected by cova-

lent bonds. Now, let us turn to the non-bonded interactions. The van derWaals interaction is a balance between two forces, the nuclei-nuclei repul-sion at short distances and the attraction of atoms at moderate distancesdue to the dispersive force (London force). Typically this interaction isrepresented by the Lennard-Jones potential:

EvdW = ε

[(Rminrij

)12

− 2(Rminrij

)6]. (2.8)

30

An exponential function would represent the repulsion better but thedifference is small and the choice is motivated by the greater speed ofthe calculation of r−12

ij from r−6ij . The energy minimum is given by ε

and is located at distance Rmin between the atoms. Instead of havingone parameter pair for each pair of atom types in the force field, the εand Rmin parameters for a pair are calculated by averaging the individualatom’s respective parameter value. The total van der Waals energy of thesystem is a sum over all pairs separated by more than two bonds sincethe corresponding interaction between close neighbors already has beenaccounted for by the bonded interactions. The van der Waals interactionsbetween pairs separated by exactly three bonds are usually too large aswell, because of its partial treatment in the torsion term, and have tobe scaled down. Since the interaction decays fast at longer distances, inpractice 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 =qiqj

4πε0εrrij(2.9)

where qi and qj are the charges, ε0 the vacuum permittivity and εr isthe relative permittivity of the medium. Since the electrostatic interac-tion decays slowly its calculation is pivotal for the computational load ofthe simulation. The simplest approach to reduce the cost, is to truncatethe interaction after a certain cut-off distance as for the van der Waalsinteraction. Although well-motivated in that case, here the energy is stillconsiderable for cut-offs that make a difference, leading to discontinu-ous forces at the boundary. One could adjust the potential entirely tocircumvent this but nowadays the best practice is to use more sophisti-cated approaches, such as the particle-mesh Ewald (PME) method [65].Briefly, it adds a neutralizing potential to the point charges in the sys-tem so that this new sum of electrostatic interactions converges fast. Itis calculated in real space up to a cut-off of around 1 nm from the atomof interest, while the contribution of a counteracting potential, oppositeto the neutralizing one, is solved on a grid in reciprocal space by the fastFourier method. In principle, the PME algorithm account for all electro-static interactions, also between particles in the periodic copies of thesimulated 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 individualsums of all the above-mentioned terms. However, how all the parametersin the model are set hast hitherto been left out. In principle, all parame-ters in the expression for the potential energy (Kb, b0,Kθ, θ,Kω, n, δ, ε andRmin) 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 chem-ical environments, giving them different partial charges, the number ofatoms types are quite large and consequently, the number of parametersneeded to be set is even greater. The parameterization of a force-fieldis the process of fitting numerical values for all these parameters, typi-cally by comparing energies to quantum chemical results and carefullymeasured experimental data. The functional forms together with all theparameters in the model define the force field. Even though one mightget the impression that this is a huge simplification, simulations basedon MM-type force fields have proven to be exceptionally good in manycases, see e.g. review articles [66, 67].

2.1.3 Statistical mechanicsWhen trying to model and study real macroscopical systems and theirproperties it does not make sense, conceptually, to simply consider onesingle molecule of the substance of interest since, in reality, the behaviordepends on a enormous number of such molecules. On the other hand,assuming the force field is able to capture the physics of the system satis-factorily, MD simulations give the complete information of the system inatomic detail for the simulated period of time, something which in gen-eral is not accessible to conventional molecular biology experiments. Inorder to investigate how a microscopical system in a simulation behaveson the macroscopic scale and hence correlate the simulation results toexperimentally measurable thermodynamic properties, one has to turnto statistical mechanics. The experimentally measured value of an arbi-trary property, denoted A here, of a system with N particles is a timeaverage over the time-course of the measurement:

Aave = limτ→∞

1τ

∫ τ

t=0

A(~pN (t), ~rN (t))dt. (2.10)

The integrand A(~pN (t), ~rN(t)) is the instantaneous value of A at timet, depending on the momenta ~p, and position ~r of all atoms. To calcu-late such a macroscopical time average in a simulation, a system of theorder of 1023 particles has to be followed for a time-period comparableto the experiment. This is of course an impossible task. Boltzmann andGibbs recognized this problem and developed the theory of statisticalmechanics, in which a single system evolving in time is replaced by alarge number of replications of the system, called an ensemble, that areconsidered simultaneously. The ergodic hypotesis is one of the axioms ofstatistical mechanics and it states that the time average is equal to theensemble average:

32

〈A〉 =∫∫

A(~pN , ~rN )ρ(~pN , ~rN ) d~pNd~rN . (2.11)

Note that the time-dependence has disappeared and the ensemble av-erage is an average over all replications of the system generated by thesimulation. Each member of the ensemble has an energy, and for systemsof constant number of particles N , volume V and temperature T , i.e. thecanonical NV T ensemble, the distribution of systems in the ensemble,ρ(~pN , ~rN ), follow the famous Boltzmann distribution:

ρ(~pN , ~rN ) = exp

(−E(~pN , ~rN )

kBT

)/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 frominteractions between particles described by the potential energy functionV (~rN ). kB is the Boltzmann constant and Q is the canonical partitionfunction:

QNV T =1N !

1h3N

∫∫exp

(−E(~pN , ~rN )

kBT

)d~pNd~rN . (2.13)

N ! comes from the indistinguishability of the particles and 1/h3N isa normalizing factor. The partition function carries all thermodynamicinformation about a system, and it is hence possible to calculate allmacroscopic properties either directly from it or from its derivative. Sincethe simulation method sample conformations from the Boltzmann dis-tribution, the partition function never has to be calculated explicitly.In fact, to calculate it in a MD simulation one would need to sam-ple all possible conformations of the system and sum up the values ofexp(−E(~pN , ~rN )/kBT ). This number of conformations is in practice infi-nite. However, for simple systems the partition function can be calculatedfrom quantum chemistry. This is actually quite a beautiful result, sincestatistical mechanics hence provide us with the awestruck tool to tietogether quantum chemistry, developed during the 20th century, withthermodynamics that was described centuries earlier.The molecular dynamics method and the as-famous Monte

Carlo method, generate system configurations from the Boltzmanndistribution and the ensemble average of the property at study cantherefor 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 confor-mations in a particular way so that the acquired ensemble of structuresare Boltzmann-distributed. The random nature of the sampling has theconsequence that consecutive states have no true connection in time. The

33

MC technique is in practice a more effective sampling strategy, since itdoes not require the calculation of forces nor the numerical integrationof the equations of motion, but the problem is that there is no generalalgorithm for efficient generation of new test conformations. Moleculardynamics on the other hand, guarantees sampling from the Boltzmanndistribution for arbitrarily complex systems and, additionally, generatesa trajectory of conformations reflecting realistic dynamics of the system.

2.1.4 Modeling the membrane potential in simulationsAs described previously in Section 1.1.3, the membrane potential in livingcells is established by an unequal distribution of ions between the watersolutions on each side of the membrane. In a typical atomistic simu-lation set-up, this situation is not as easy to accomplish as one mightexpect. This is because the standard method of using periodic bound-ary conditions (see Section 2.1.2) to overcome the finite-size effects ofthe simulation, in fact connects the two water baths so that they arein physical contact with each other. Hence, they are not two separatedwater 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 simulationunit cell [68]. The two parallel membranes divide the simulation systeminto two truly separated water solutions. This method has been shown toreproduce transmembrane potentials in a realistic way but the compu-tational overload is significant. Another complication is that to establisha biological transmembrane potential on the order of hundred millivolts,an imbalance of only one or a few ions is needed for the small systemsthat are practical to simulate. Approximately one access ion/104 Å2 ofbilayer is enough, requiring an order of 105 atoms in the simulated sys-tem. This means that if one ion permeation event occurs (for examplethrough passing an ion channel), the membrane potential changes dras-tically and could even be reversed. Similarly, if the embedded protein,carrying charged residues, changes conformation significantly during thesimulation the transmembrane potential varies considerably. However, ascomputational power and software efficiency increases this method willbecome 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, thewater solutions on each side of a single membrane are separated fromeach other even though periodic boundary conditions are applied. Thismethod has been successfully used in several studies [70, 71, 72] but italso suffers from the problem of membrane potential variability men-tioned above.

34

0 2 4 6 8 10

Φ (V

)E

(V/n

m)

ρ (q

/nm

)

0 2 4 6 8 10

0

0

0

-0.1

0.1

0.20.4

-0.2-0.4-0.6

0.20.40.6

-0.2-0.4

System coordinate z (nm) (=L )z

3

-0.2

0.2

onsdag den 12 oktober 2011

Figure 2.1: The transmembrane potential. The charge density (top) in a simula-tion is calculated by summing the charges in slices along the direction of the appliedelectric field. Integrating it once gives the electric field (middle) and twice the electro-static potential (bottom), see text and Eq. 2.14. Figure produced from data of paperI. The dashed line represent the simulation without an applied field and the solid linea simulation with an applied constant field of 0.052 V/nm. With a box dimensionof 10 nm in z-direction this would correspond to a potential of approximately 0.5 V(φ = EzLz). This is also the potential given by the integration.

In contrast to the all-atom simulation techniques mentioned so far,continuum electrostatic models have previously been used to study pro-teins in general and ion permeation of channels in particular [73, 74, 75].In these methods, the polar solvent (water and lipids in the case ofa membrane-protein system) is represented as a structureless dielectricmedium leaving out the necessity of explicit solvent particles. In such arepresentation, the effect of the membrane potential can easily be incor-porated by modifying the underlying Poisson-Boltzmann equation [73].It has been shown that this representation can capture the dominantfeatures of energetics of ion solvation in bulk but not in confined en-vironments such as aqueous pores (e.g. in the selectivity filter of ionchannels).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 poten-

35

tial by an added constant external electric field along the z-direction,Ez, perpendicular to the membrane surface. The resulting electrostaticpotential is given by φ = EzLz, where Lz is the simulation box lengthin the z-direction. In practice this is achieved by adding a force alongz, Fi = qiEz, to each particle i with charge qi in the system. It is arather unphysiological set-up, in particular because the potential dropsinstantaneously from one end of the simulation box to the other (i.e. adiscontinuous hop over the periodic boundary), but it has been shownto be a sound representation [80]. The Poisson equation gives the rela-tion between the equilibrium charge distribution and the electrostaticpotential:

d2φ(z)dz2

= − 1ε0ρ(z) (2.14)

where φ is the potential, ε0 is the vacuum permittivity, and ρ is thecharge density. The membrane potential can therefor be calculated froma simulation (giving the charge density) by integrating this equationtwice. The reference point of the potential is arbitrarily set to z = 0 andthe boundary conditions used are, φ(0) = 0 and Ez(0) = dφ(0)/dz = 0. Anexample of the integration from paper I is shown in Figure 2.1.

2.2 Modeling of protein structuresAnalysis of three-dimensional high-resolution protein structures hasproven very fruitful for biological and medical discovery in the lastcouple of decades. Experimental determination of such structures istime-consuming and expensive, and solving all human protein structuresusing high-resolution methods, such as X-ray crystallography and NMRspectroscopy, is an immense task and it is questionable if it is a realisticand well-motivated goal at all given the cost. However, computationalmodeling of protein structures could very well bridge this gap because,as the structural information in the databases accumulates, suchmethods are increasingly successful in predicting biologically relevantstructures for proteins with unknown structure.Comparative modeling is the means by which models of protein struc-

tures are built based on the information in the sequence and structuredatabases. Its foundations lie in the discovery of strong correlative pat-terns between sequence similarity on one hand and an analogous functionon the other. In other words, as the volume of protein sequences startedto accumulate in the 1950s, the data showed that similar sequences codedfor proteins with identical or similar functions. This lead to the classi-fication of proteins (based on sequence) into families and superfamilies

36

1972 0 0 0 0 0

1973 0 0 0 0 0

1974 0 0 0 0 0

1975 0 0 0 0 0

1976 13 13 10 10 0

1977 24 37 7 17 0

1978 6 43 2 19 0

1979 11 54 0 19 0

1980 16 70 1 20 0

1981 16 86 5 25 0

1982 32 118 14 39 0

1983 35 153 0 39 0

1984 22 175 5 44 0

1985 20 195 3 47 1

1986 18 213 3 50 1

1987 25 238 8 58 2

1988 53 291 7 65 3

1989 74 365 12 77 5

1990 143 508 12 89 6

1991 187 695 20 109 7

1992 192 887 18 127 7

1993 695 1582 41 168 9

1994 1289 2871 61 229 10

1995 947 3818 59 288 10

1996 1172 4990 78 366 14

1997 1565 6555 84 450 20

1998 2057 8612 86 536 26

1999 2360 10972 86 622 31

2000 2627 13599 101 723 40

2001 2831 16430 107 830 43

2002 3002 19432 86 916 59

2003 4167 23599 92 1008 75

2004 5183 28782 146 1154 86

2005 5364 34146 86 1240 110

2006 6478 40624 101 1341 135

2007 7209 47833 46 1387 159

2008 6974 54807 6 1393 197

2009 7400 62207 0 1393 233

2010 7929 70136 0 1393 263

2011 6152 76288 0 1393 302

Note: searchable structures vary over time as some become obsolete and are removed from the database.Note: searchable structures vary over time as some become obsolete and are removed from the database.Note: searchable structures vary over time as some become obsolete and are removed from the database.Note: searchable structures vary over time as some become obsolete and are removed from the database.Note: searchable structures vary over time as some become obsolete and are removed from the database.Note: searchable structures vary over time as some become obsolete and are removed from the database.

Number of searchable structures per year.Number of searchable structures per year.Number of searchable structures per year.Number of searchable structures per year.Number of searchable structures per year.Number of searchable structures per year.

Year Yearly Total Yearly Total

0

20000

40000

60000

80000

1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 2011

0

375

750

1125

1500

Num

ber

of

stru

ctu

res

Num

ber

of

fold

s

0

16!000

32!000

48!000

64!000

80!000

1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 2011

0

300

600

900

1!200

1!500

Namnlös 1 Namnlös 2 Namnlös 3Figure 2.2: Protein Data Bank statistics. Yearly growth of total structures (bluediamonds, left axis), of unique membrane protein structures (green triangles, rightaxis) 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 familyare all evolutionary related, or homologous. This fact can be exploitedin comparative modeling because a new protein sequence with unknownstructure, the target structure, can be assigned to these families and itsstructure can be modeled using the structures of the proteins in thatparticular 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 wasconfirmed: similar protein sequences had similar structures and there-for similar functions as well. It was also realized that the number ofunique ways a protein can fold, the structural space, is limited, whilethe sequence space in practice is unlimited. As of today, the StructuralClassification of Proteins (SCOP) database [83] contains some 1400 dis-tinct protein folds and judging from the plot of accumulated folds foundin Figure 2.2, the true number is not much larger. Therefor, for new se-quences found today, there is a high probability of finding a homologousprotein with a known structure that can be used as a template for struc-tural modeling. The situation is a bit more complicated for membraneproteins however. The first high-resolution structure was not publisheduntil 1985 [84] and as can be seen in the figure there is only a coupleof hundred structures known today. The structural space of membraneproteins is hence sparsely explored and the number of folds for this classof proteins is unknown although it probably is considerably less than thefolds of globular proteins because of the environmental restraint of themembrane.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 simulationtechniques and the underlying molecular mechanics description treatedpreviously in this chapter. A branch of this research area is the refine-ment of protein structures. Here, protein structures of moderate to poorquality are refined with the objective to improve the model by doing localadjustments, typically achieved by structural sampling and minimizationof the interaction energy function.

2.2.1 Comparative modelingThe first step in comparative modeling is to align the target proteinsequence to proteins with known structure. By aligning sequences, evo-lutionary conserved, and hence important, residues are identified. Theyare later pivotal in the model building. The search of homologous se-quences is typically done using PSI-BLAST [85, 86]. It creates a multiplesequence alignment by an iterative process, were it finds new sequencehits in each step that are incorporated in the next query to the sequencedatabase. In this fashion it is quite successful in finding distantly relatedsequences in a relatively short time. The alignment is extremely impor-tant for the outcome of the modeling. A poor alignment will result ina bad model. Once homologous sequences have been found, the target-template alignment and the protein structures are sent to the modelingtools.Briefly, models are built by optimizing spatial constraints derived from

the alignment. The constraints are typically distances between backboneatoms of different residues. The structural model is generated by mini-mizing the violations of these constraints. When several template struc-tures are used for the model building, local sequence similarities guidethe selection of the appropriate choice of template structure for thissequence section. That is, different sections of the target structure aremodeled from different template structures based on local sequence sim-ilarity.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 onthe template structure of the KV 1.2/2.1 chimera protein [27]. A sequencealignment of the two sequences was performed using HHalign [87] andsince the sequence identity is more than 40 % their alignment is quitestraightforward. The model was built using one of the most commonlyused homology modeling tools; Modeller [88].

38

2.2.2 Physics-based modeling and structural refinementThe origin of de novo structure prediction lies in the experiments onprotein folding performed by Anfinsen in the middle of the 20th century(for which he was awarded the Nobel Prize in Chemistry 1972) [89]. Hewas able to observe folding of the denatured ribonuclease protein intoits biologically active conformation, the native state, in a plain solution.This means that, at least for small globular proteins, all the informationfor the folding of the protein lies in the amino acid sequence. Hence, ifthe physical interactions are described accurately enough, the modelingmethod should be able to find the native state. These methods are com-putationally demanding and have been used mostly on smaller modelproteins. It is safe to say they have not been as successful as compara-tive modeling but as computational power is increasing they will becomemore and more useful.These methods generally have a molecular mechanics-type of interac-

tion potential under the hood. To decrease the computational cost it isnot uncommon that the methods are extended in their description of themolecular system and the way conformations are sampled. For example,coarse-graining the atomistic model by grouping several atoms in func-tional groups, each represented by a single particle, reduces the numberof particles considerably. This approach has been used for the model ofthe lipid molecules in most simulations in this thesis, where the hydrogenatoms in the long non-polar chains are represented together with the car-bon they are attached to as a united atom. Another trick frequently usedis to model the surrounding medium by an analytic function describingits average properties without treating the atoms explicitly. This is calledimplicit solvent methods and was studied in paper V. Analogous effortstreating the lipid bilayer implicitly have been made and can be used inthe interaction potential of the Rosetta molecular modeling tool [90] thatwe used for the modeling of the Shaker VSD states in paper III.For successful structural refinement of protein models the modeler of-

ten has to extend the sampling technique with methods exploring theconformational space more efficiently. This is due to the fact that therefinement process readily gets stuck in local energy minima of the en-ergetic landscape. MD and MC methods typically sample conformationsfrom low-energy regions and hence these methods are very inefficient insurmounting energy barriers. Such barriers could be due to an subopti-mal packing of amino acid side chains for example. To circumvent thisproblem one could use simulated annealing techniques that temporarilyincrease the temperature in the simulation, hence providing additionalenergy for passing barriers. Another way is to completely rebuild partsof the protein from a library of protein structure fragments, similar tothe methodology used in homology modeling based on several template

39

structures, or less dramatic, to rebuild the s from libraries of torsionalangles 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 partof the domain for each state using the corresponding constraints fromthe metal-bridges in the experiments. The S3-S4 loop and S4 helix wasrebuilt completely from fragments in Rosetta, allowing this part to adoptdifferent positions relative the rest of the protein and the implicitly repre-sented water and membrane. During conformational sampling the back-bone and s of the remodeled part were allowed to move whereas thenon-remodeled residues were kept fixed.

40

3. Summary of papers

- So Dr. Bjelkmar, in what area are you doingresearch?- Well Professor Knowitall,I’m answering biological questionsbuilt up by chemical entitiesmodeled by physical principlesdescribed by mathematicsusing computational methods.

Pär Bjelkmar

3.1 Conformational changes in the KV1.2 ion channelthrough polarized microsecond molecular simulation(paper I)The opening of the voltage-gated ion channel and initiation of the al-pha current have been shown to occur in the millisecond range [13] butan early component of the gating current has been reported in themicrosecond-range [42]. With recently developed highly efficient and par-allel molecular dynamics code (e.g. GROMACS [57]) and current state-of-the-art computer clusters, it is now practically feasible to performsimulations in the microsecond regime for large membrane-protein sys-tems like the KV channels and early stages of gating can therefor, inprinciple, be studied.The crystal structure of the open KV 1.2 channel [25], complemented

with modeled regions where electron density was missing, was insertedinto a lipid bilayer membrane and simulated with an applied electricfield - mimicking the effect of the resting potential - for more than amicrosecond with the goal of capturing early stages of VSD gating mo-tion. To our knowledge, this work was the first simulation reaching thislong sought-after timescale for a comparable system and in the wake ofit 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 isslow and beyond the scope of this study. Both in the hyperpolarized and

41

a control simulation without an applied electric field, a small rotationof the S4 helix was found, indicating that the original crystal structuremight 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 theprotein in the hyperpolarized simulation, a more pronounced rotation ofthe extracellular half of the helix converted it to a 310-helix during thetime-course of hundreds of nanoseconds. This is a key result support-ing the original idea of 310 helix content in voltage-gated ion channels[53, 54] and has also been confirmed by experiments [46] and recentcomputational studies [55, 56], paper II and III, as well by the S4 struc-ture 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 pro-tein conformations in the second half of the simulation are overall moresimilar to the new chimera structure compared to the starting KV 1.2structure. Another interesting, and rather unexpected result, is the factthat lipid molecules in the simulation matched the positions of resolvedlipid molecules in the chimera crystal structure, successfully predictinga bilayer thinning of the same magnitude and at the same membranelocation.

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 quan-titative approach of assessing the likelihood of S4 adopting a 310 con-formation during gating. The VSD from the KV 1.2/2.1 crystal structure[27], in two conformations, was simulated in a POPC membrane. Thefirst being the conformation in the crystal structure, with a mix of α and310 content in the S4 helix, while the other had an all-310 S4 conforma-tion. By measuring the work of translating these two helices along theiraxes by steered molecular dynamics simulations, the cost of moving the310 S4 conformation one helix turn towards the closed state was foundto be significantly lower compared to the original α-helical conforma-tion. The barrier seemed to be caused by the breaking and reformingof electrostatic interactions between the arginine residues in S4 and thenegative residues on neighboring helices, together with the passing ofthe most intracellular arginine residue through the hydrophobic regionformed by the phenylalanine residue F233 (see Figure 1.4). The 310 con-formation appeared to reduce the work by limiting the amount of S4rotation needed to reform salt-bridge interactions and its smaller size(see Figure 1.7) reduced the distortion in the rest of the VSD whenmoved.

42

3.3 Tracking the voltage-sensor gating motion withmetal-ion bridges, computer modeling and simulation(paper III)In an attempt to study the conformations along the gating pathway of theKV channels, a collaboration with an experimental group in Linköpingwas initiated. By substituting residues in the Shaker channel to Cd2+-binding cysteines, they were able to collect a set of 17 newly describedinteractions between the VSD segments in different states along the gat-ing 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 suc-cessively more closed states were constructed using these experimentalconstraints by incorporating them in the rebuilding the voltage-sensorS4 helix and its preceding loop (using methods described in section 2.2).These models were evaluated by molecular dynamics simulations andrepresentative conformations were chosen based on structural clusteringof the simulation conformations. The three selected models of succes-sively more closed states suggest a gating mechanism where a station-ary region of the S4 helix, opposite the central hydrophobic region ofthe VSD (centered at the phenylalanine residue 290 in Shaker, F233in KV 1.2 enumeration), is in a 310 conformation as the helix translatesalong its axis. Additionally, two interactions in the experimental set ofconstraints distinguish the open active and open inactive states of thechannel confirming that the KV 1.2/2.1 crystal structure is indeed in theopen active state despite the long depolarization of the membrane in thecrystallization experiment.

3.4 Lateral diffusion of membrane proteins (paper IV)While diffusion in pure lipid membranes has been given much attentionand is rather well understood, the lateral diffusion of membrane proteinsis still an unclear process in membrane dynamics [94, 95]. It is importantfor understanding protein sorting, membrane protein complex formationand binding of ligands, such as toxins and pharmaceuticals, to membraneproteins.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 witha dynamic complex of 50-100 lipids surrounding it. Lateral diffusion co-efficients were calculated for neighboring lipids versus the more distantones and indeed, neighbors were an order of magnitude slower in theirdiffusion, 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 cel-lular membranes are crowded with membrane proteins occupying some30% of the total surface area this suggest that there are no free lipidswith bulk properties in them. These results also have implications for thetheories of lateral diffusion of membrane proteins. The equations typi-cally include the protein radius and membrane viscosity as parameters,and as this study shows, it is effectively the radius of the whole lipid-protein complex that matters and the protein senses a viscosity differentfrom a protein-free membrane.

3.5 Linking structure and conductance in a simple ionchannel (paper V)Eukaryotic membrane proteins such as the KV channels are complexesof several protein domains, often making them too large for molecularsimulations on relevant timescales, even though this shortcoming is grad-ually diminished due to the new computational capabilities. Fortunately,more simple channels are common in nature and these can be used asmodel systems, still giving insight into the workings of complex ion chan-nels [96]. One such channel is formed by the 16-residue antiamoebin-1peptide (AAM-1) [97], secreted by fungi as a weapon in the microbialwarfare among organisms. It is a member of the nongenomic family offungal peptides called peptaibols [96, 98]; 15-20 residues long peptidesrich in helix-promoting nonstandard amino acids. They are not encodedby the genome but rather synthesized by a special set of enzymes. Pep-taibols are able to assemble into helical bundles that form lethal poresin 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 byAAM-1 channels have shown that it only has one conductance level in-dicating one single conducting structure [99]. Measuring the current as afunction of protein concentration has not unambiguously revealed its na-ture but suggested that the channel is a tetramer, hexamer or octamer,depending on the underlying theory of helix aggregation [100, 99]. Modelsof these different channel models were constructed from the monomericpeptide structure [101, 102, 103], and molecular dynamics simulationswere used to calculate their conductance.The tetrameric channel did not conduct ions at all because of a too nar-

row pore and the octameric channel had a conductance that was an orderof magnitude too large. The hexameric channel on the other hand hada conductance consistent with the experimentally measured value. Thecalculated conductances were derived both from explicit counting of ion

44

crossings in the simulation as well as from traditional continuum methodsusing the Nernst-Planck equation [13, 104]. Moreover, the channel waslarge enough for Fickian, rather than single-file, diffusion and the ionstransversed the channel having their solvation shells of water moleculesintact. Consequently, the free-energy barrier of ion transport was only acouple of kcal/mol and the channel was slightly cation selective.

3.6 Implementation of the CHARMM force field inGROMACS (paper VI)If the underlying model of particle interactions in the force fields used inmolecular dynamics simulations is accurate enough, it should, in princi-ple, be able to find the native state of the protein (given enough computa-tional time). The development of force fields is therefor a very importantand continuously current topic of research. As described in chapter 2, aforce field is constituted by a set of functions and their accompanyingparameters describing the interactions between atoms. One of the mostcommonly used force fields is CHARMM [63, 64] and this paper describesthe implementation of it into the GROMACS program [57]. CHARMMhas a special term, called CMAP (for correction map), modulating thetorsion potential of pairs of neighboring residues (see section 2.1.2). Thetorsion energy of protein backbone residues is a key factor of the energylandscape determining protein structure and it has been shown that ithas 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 withdifferent water models and integration time steps. The inclusion of theCMAP did indeed improve the structural sampling of the protein byexploring backbone torsional angle ensembles more similar to a crystalstructure. Interestingly, this was true for all water models, including theimplicit ones, and time steps. In particular, the most computationally ef-ficient combination of settings - CHARMM with CMAP, implicit solventand long time steps - did not compromise the overall structural qualityand is therefor promising for protein structure refinement. Using thesesettings in a small test case, it was possible to improve RMSD-values ofdecoy structures of Protein G from 3 Å down to 2 Å, and starting from2 Å down to 1 Å.

45

4. Concluding remarks and futureperspectives

If we have learned one thing from the history ofinvention and discovery, it is that in the long run,and often in the short run, the most daringprophecies seem laughably conservative.

Arthur C. Clark

The function of a protein is to a large extent determined by its struc-ture. As the techniques for determining high-resolution structures im-prove and comparative modeling of protein structures become more ac-curate, we will rather soon find ourselves in a situation where most pro-teins will have a reliable high-resolution structure, deduced either fromexperiments 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. Insteadof being seen as a single static conformation, a better representation ofthe native state of a protein seems to be an ensemble of conformationsthat all are important for function. This could for example be differentside chain conformations in the active site of an enzyme, all contributingto the binding of the substrate. Additionally, large-scale conformationalchanges often occur within the protein as it performs its function, as wehave seen for the voltage-gated ion channels in this thesis. In other words,dynamics is important. Molecular dynamics simulation is a well estab-lished method for modeling the conformational flexibility of biomoleculesand it provides spatial and temporal information on scales that are dif-ficult to access with experimental techniques. Historically, the compu-tational demands of MD have been too great to be able to reach thetimescales of these functionally important motions that typically takeplace in microseconds to seconds. This is starting to change.The first all-atom MD simulation of a small protein in vacuo was per-

formed in the 70’s and reached less than 10 ps. Thirteen years ago, thefirst microsecond all-atom simulation was performed. It was an immensetask involving several months of supercomputer time even though the

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system consisted of only some 10,000 atoms. The technology has de-veloped considerably since then and we are now on the verge of reach-ing the region where biological interesting phenomena occur. The mostlarge-scale all-atom MD simulations nowadays consist of hundreds ofthousands atoms and reach the microsecond timescale, with millisecondson the horizon.The increase in simulation speed is both due to development of software

and hardware. Algorithms for performing the MD calculations on a singleprocessing unit have improved but the main difference the last few yearshas been the parallelization of the problem over many computing units.New national computer centers have been established to provide thesesupercomputing capabilities to the scientific community, now enabling usto perform calculations on thousands of processors simultaneously. An-other approach to enhance the capabilities of MD simulations (and othertechniques) is to use distributed computing. In contrast to the computercenters, the computational capabilities are spread around the world andare connected via the internet. The very successful folding@home ini-tiative is an example of this. Here, volunteering users around the globelend their computer power to the project, for example through screen-savers, and the folding@home servers send the simulation information tothe node and collect the results. This infrastructure limits the way thesecalculations are performed. Because of the temporary availability of thecompute nodes and the slow interconnection between them, only smallsimulation tasks are sent out to the users but the immense number ofthem 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 referto the traditional central processing units (CPUs). In particular, thegraphics processing units (GPUs) that are common in current desktopcomputers, typically for the demanding graphics of computer games, havean immense capacity of floating-point arithmetics that can readily beused for MD calculations. Implementation of MD code on GPUs is stillan area under development but the technology is promising, not leastbecause these chips are very cost-effective. Another type of hardwareused for MD simulations is the Cell processor in Sony’s Playstation 3.The folding@home project, for example, has quite an impressive numberof PS3s in its arsenal. These are all general-purpose hardware, but thereare also efforts in developing custom-built hardware specialized for MDsimulations. For example, DE Shaw Research in New York has built theimpressive supercomputer Anton for molecular dynamics simulations,showing unprecedented simulation speeds.Since the calculation of the underlying interactions in molecular sim-

ulations has become faster it promotes improvements in the force fieldsas well. In particular, considerable effort has been given to represent

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the polarization in chemical systems in a more realistic way. The cur-rent standard is to assign fixed partial charges on all the particles inthe simulation but that, of course, prohibits the change in the electronicstructure in the simulation. On the other end of the spectrum, modelreduction through coarse-graining will enable yet another boost in sam-pling efficiency even though the accuracy is reduced somewhat. In thelong run, I believe it is likely that the development of more sophisti-cated coarse-graining strategies will bridge the gap between atomisticand macroscopic representations through a hierarchical structure of sim-ulations 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 inhow the biomolecular conformational space is explored, exemplified byenhanced sampling techniques and new methods for free-energy calcu-lations through nonequilibrium work methods. Additionally, new effortsusing Markov-state modeling is trying to represent dynamical evolutionover long timescales from short MD simulations by automatic decom-position of conformational space in disjoint but connected metastablestates with Markovian transitions between them. In these examples thestrategy is to make the conformational search more efficient, with re-spect to the process of interest, rather than relying on one or a few long,brute-force, equilibrium simulations. These strategies will for sure pavethe way for the extended applicability of molecular simulations in thefuture, for example in the areas of computer-aided drug design, proteinengineering, 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 experi-mentalists. More experimental collaborations should be initiated to com-pare, complement, and validate our simulations to a higher degree thanwhat 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 experimentaldata in our work, the potential of molecular modeling and simulationwill be enhanced and become a standard tool throughout the molecularsciences. Further increases in simulation speed in combination with im-provements in force fields and sampling techniques will provide us withunprecedented insight into the relationship between biomolecular struc-ture, dynamics, and function.

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5. Svensk populärvetenskapligsammanfattning

Ju mer man tänker, desto mer inser man att det intefinns något enkelt svar.

Nalle Puh

En cell avgränsar sig från sin omgivning genom att omge sig av ettmembran. På så sätt kan den koncentrera de livsnödvändiga molekylernatill en liten volym för att därmed kunna utföra all den kemi som är karak-täristiskt för livet. Även inuti en cell finns det membran som särskiljerdelar av den i distinkta enheter, kallade organeller, med specialiseradfunktion. Så fort de informationsbärande molekylerna som utgör livetskärna omgav sig av ett membran i begynnelsen av livets utveckling upp-kom bohovet att kunna transportera materiel över dessa annars ogenom-släppliga barriärer. Därför utvecklades speciella membranbundna trans-portmolekyler vars kemiska sammansättning finns lagrad, tillsammansmed ritningarna för alla andra proteiner som cellen kan producera, i deinformationsbärande molekyler som idag utgörs av DNA.Molekylerna som sitter insprängda i membranen kallas membranpro-

teiner och de är en mycket viktig klass av proteiner som i princip sköterall interaktion mellan cellen och dess omgivning samt mellan de membra-nomgivna organellerna inuti cellen. DNA-molekylerna specificerar endastproteinernas kemiska sammansättning men för att förstå hur proteinerfungerar måste man studera deras tre-dimensionella struktur. Dennabestäms av de fysikaliska interaktionerna mellan proteinet och dess om-givning och är en häpnadsväckande process som under normala om-ständigheter låter proteinet vecka sig till en helt unik struktur som ärden biologiskt aktiva. Felveckning av proteiner kan ha livsavgörande kon-sekvenser i och med att de kan framkalla diverse sjukdomstillstånd. Veck-ningsprocessen är inte förstådd i sin helhelt men efter mycket forskninghar 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 gällerspeciellt gruppen av membranproteiner. Idag finns endast några hundra

51

högupplösta strukturer av membranproteiner jämfört med tio-tusentalsstrukturer av icke membranbundna proteiner, så kallade globulära pro-teiner, som rör sig fritt i vattenblandningen i cellen. I och med kostnadenassocierad med att bestämma proteiners struktur har datorbaserade al-ternativ utvecklats. De är väldigt mycket mer tidseffektiva och mindrekostsamma och är därmed viktiga metoder inom forskningen nuförtiden.Datormodellering av proteiner på atomär nivå kan göras på många olikasätt, t.ex. med hjälp av den evolutionära informationen från alla DNA-sekvenser av proteiner från olika organismer som finns tillgängliga, till-sammans med de redan bestämda strukturerna. Ett annat alternativ äratt försöka beskriva de fysikaliska interaktionerna mellan atomer och pådenna väg simulera proteiners beteende.I denna avhandling har jag, tillsammans med flertalet andra forskare,

främst studerat en viss typ av membranproteiner som har den fantastiskaegenskapen att selektivt transportera kaliumjoner med väldig effektivitet.Regleringen av öppnandet och stängandet av kanalen sker genom spän-ningsförändringar över membranet. Dessa så kallade spänningskänsligakaliumkanaler har studerats med hjälp av datorbaserad modellering,både utifrån evolutionär information och med fysikbaserade metoder somkompletterats med nya data från ett samarbete med en experimentellforskargrupp. Våra studier har visat att den spänningskänsliga delen avproteinet förändrar sin struktur på ett speciellt sätt under öppnandetoch stängandet av kanalen. Proteinets interaktioner med det omgivandemembranet har också studerats och vi har visat att proteinet påverkarmembranet i stor utsträckning. Framförallt så rör sig proteinet tillsam-mans med ett kluster av lipider runt omkring sig och dessa lipider hardärmed helt andra egenskaper än de opåverkade. Detta har givit ny kun-skap för förståelsen av interaktionerna mellan proteiner och membranoch 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 genomatt, utifrån olika modeller av kanalen, beräkna dess genomsläpplighetför joner och jämföra resultaten med experimentellt uppmätta värden.Endast en av modellerna stämde överens med vad som uppmätts ilabbet och vi bekräftade dessutom kanalens moderata selektivitet förpositivt laddade joner.Det sista arbetet som jag utfört är att överföra ett kraftfält som

beskriver de atomära interaktionerna till den programvara som vianvänder i forskargruppen. Specifika egenskaper hos modellerings-programmet kunde därför kombineras med kraftfältet och dess beteendekunde studeras på långa tidsskalor. Vi fann att kraftfältet beskriverinteraktionerna på ett godtagbart sätt och kombinationen med den nyaprogramvaran uppvisade lovande egenskaper när det gäller att förbättrastrukturella proteinmodeller.

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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 youdeserve.

Bilbo Baggins

The thesis you hold in your hand had not been possible to writeif not for a certain number of people. Some of you might not havemade a direct contribution to it but you kept me happy and mo-tivated during these years, something that should not be underestimated.

Foremost, I would like to thank you Erik, my advisor andpartner in crime, for being such a generous person both when it comesto 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 neverlooked twice when I have gone for a workout session in the middle ofthe day, although you’ve quite commonly commented on my sanityfor so doing. Besides this, one of the greatest things you’ve done is toalways believe in me, from the day you hired me after one Skype-calluntil today, when most of our communication still is electronic... Nooffense! You are a busy professor with an average altitude of at least3 000 meters. You have an storehouse worth of knowledge in the mostsurprising 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 whatyou know in half the subjects, I am half again as knowledgeable as therest on this planet.

I have been surrounded by more very inspirational people,Arne, with all your ideas and enthusiasm, as well as Martin andGunnar, for the same reasons. Berk, I really appreciate your abilityto explain complicated stuff in an understandable way and taking thetime for so doing. You have that curiosity and drive to understand thenitty-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 thewell-guarded Moffett Field, and for your and Joanna’s hospitality. Mikeand Karl, I’ll never forget our daily trips to Dana street for the greatcompany and the best coffee I’ve ever had.

I’d like to acknowledge all my collaborators and co-authors forthe great work we’ve put together. The Ames people of course, andIlpo, for the Finnish hospitality and computer power. Fredrik for beingsuch a ninja in the electrophysiology of the voltage-gated ion channels,and a ninja master when it comes to running! You introduced me to theultras 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 aroundanother round of Vetligan. Thank you Jenny for those times, and Aron,my longest-lasting room-mate, for all the good times we’ve had duringthese years.

My past and current colleagues in the Lindahl group, youhave been great to work and hang out with!

During my exile at KTH, I have missed the C4 corridor andthe group of social and positive people that inhabits it. I’llremember the dart sessions with Håkan and Andreas, supersweetFriday cake clubs, nice cold(-room) after work beers with theoccasional Rock Band jamming, and great discussions over coffee, of-ten involving Charlotta, Karin, Catrine, Dan, Daniel, Katta, and Linnea.

In fact, it’s not only the C4 corridor that’s a great place towork in, but the whole DBB. I’m impressed by the deeply rootedfamiliar spirit in the department. You are a handful of people thatnourishes it, respect to you!

A special thanks goes to Henrik and Johan, and the others inPuppy TS, for keeping my body tired and my mind sound.

A curious thing that might be fit to mention here is that Istarted to log all the songs I listen to in the office in September2006 when I began my PhD studies. Since then I have listened to91,389 songs. Assuming each song is 4 minutes long, and that I haveworked all days of the year (yeah right!), I have listened to inspira-tional 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 rela-tion to this, for teaching me to think critically and see the importanceof education.

The best thing I did during these years had nothing to dowith science, or wait, of course it had! My genetic information wasfused with that of Lina’s and our wonderful biochemical experiment,Alma, was born. You are my treasures and I’m the world’s happiestman. I love you!

fredag den 4 november 2011

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