Solvation properties ofproteins in membranes

Solvation properties ofproteins in membranes

Anna Johansson

Abstract

Knowledge about the insertion and stabilization of membrane proteins isa key step towards understanding their function and enabling membraneprotein design. Transmembrane helices are normally quite hydrophobic toinsert efficiently, but there are many exceptions with unfavorable polar ortitratable residues. Since evolutionary conserved, these amino acids are likelyof paramount functional importance, e.g. the four arginines in the S4 voltagesensor helix of voltage-gated ion channels. This has lead to vivid discussionsabout their conformation, protonation state and cost of insertion. To addresssuch questions, the main focus of this thesis has been membrane proteinsolvation in lipid bilayers, evaluated using molecular dynamics simulationsmethods.

A main result is that polar and charged amino acids tend to deform thebilayer by pulling water/headgroups into the hydrophobic core to keep theirhydrogen bonds paired, thus demonstrating the adaptiveness of the membraneto allow specific and quite complex solvation. In addition, this retainedhydration suggests that the solvation cost is mainly due to entropy, notenthalpy loss. To further quantify solvation properties, free energy profileswere calculated for all amino acids in pure bilayers, with shapes correlatingwell with experimental in vivo values but with higher magnitudes. Additionalprofiles were calculated for different protonation states of the titratable aminoacids, varying lipid composition and with transmembrane helices presentin the bilayer. While the two first both influence solvation properties, thelatter seems to be a critical aspect. When the protein fraction in the modelsresemble biological membranes, the solvation cost drops significantly - evento values compatible with experimental data.

In conclusion, by using simulation based methods I have been able toprovide atomic scale explanations to experimental results, and in particularpresent a hypothesis for how the solvation of charged groups occurs.

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List of Publications

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

I Anna CV. Johansson and Erik Lindahl. 2006. Amino-Acid Sol-vation Structure in Transmembrane Helices from Molecular Dy-namics Simulations. Biophys J. 91:4450-4463.

II Anna CV. Johansson and Erik Lindahl. 2008. Position-resolvedfree energy of solvation for amino acids in lipid membrane frommolecular dynamics simulations. Proteins. 70(4):1332-44.

III Anna CV. Johansson and Erik Lindahl. 2009. Titratable aminoacid solvation in lipid membranes as a function of protonationstate. J Phys Chem B. 113(1):245-53.

IV Anna CV. Johansson and Erik Lindahl. 2009. The role of lipidcomposition for insertion & stabilization of amino acids in mem-branes. J Chem Phys. In press.

V Anna CV. Johansson and Erik Lindahl. Protein contents in bi-ological membranes can explain abnormal solvation of chargedand polar residues. Submitted.

Reprints were made with permission from the publishers.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Life, cells and membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Water as a solvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 From sequence to structure to function . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 What this thesis is about . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Biological membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Composition of a biological membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Bilayer lipids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Influence of lipids on membrane protein function . . . . . . . . . . . . . . . . . . . . 172.3 Biological membranes vs membrane models . . . . . . . . . . . . . . . . . . . . . . . 18

3 Membrane proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 What do membrane proteins look like? . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 How are proteins inserted into the membrane? . . . . . . . . . . . . . . . . . . . . . 203.3 Amino acid composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Polar groups in transmembrane segments . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Structural features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 in silico chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Relation between statistical mechanics and thermodynamics . . . . . . . . . . . 284.3 The MD algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Force fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4.1 Bonded interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4.2 Non-bonded interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4.3 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.5 Technical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.1 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.2 Long range interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.5.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.5.4 Temperature and pressure coupling . . . . . . . . . . . . . . . . . . . . . . . . . 354.5.5 Parallelization approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.6 Applications to membrane proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Free energy calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1 Definition of free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Thermodynamic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Free energy perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4 Free energy along a reaction coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4.1 Umbrella sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4.2 Constraint free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.4.3 Adaptive biasing force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.5 Thermodynamic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.6 Applications to membrane proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.1 Solvation of transmembrane helices (paper I) . . . . . . . . . . . . . . . . . . . . . . 466.2 Solvation free energy of amino acid analogs (paper II) . . . . . . . . . . . . . . . . 476.3 Solvation free energy as function of protonation state (paper III) . . . . . . . . . 496.4 Solvation free energy as function of lipid type (paper IV) . . . . . . . . . . . . . . . 506.5 Solvation free energy as function of membrane protein content (paper V) . . . 51

7 Final thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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1. Introduction

1.1 Life, cells and membranesThe driving force behind all physical processes is an endeavor towards anequilibrium state where competing influences are balanced, although whensuch a state is obtained all life processes cease.

Some main factors are commonly stated as necessary for life to persist in-cluding compartmentalization, conversion of energy and reproduction, all ofwhich can be contributed to balance between a strive for equilibrium and themaintenance of non-equilibrium.

In living organisms there is a non-uniform distribution of matter maintainedby compartmentalization into different kinds of tissues, different types of cellsand varying concentrations of ions and molecules in different parts of the cell.According to the second law of thermodynamics the disorder in an isolatedsystem will increase until equilibrium is reached. This means that energy andmatter tend to spread out over the universe to form an equilibrated distribution.To concentrate energy/matter in one specific place, it is necessary to compen-sate for this by spreading out a greater amount of energy across the remainderof the universe. In living cells complex molecules are constructed from sim-pler building blocks and energy is stored in the form of energy rich molecules.This maintenance of non-equilibrium is so important for the cell that a largefraction of the energy absorbed as light or as chemical energy is released asheat to counterbalance the increased concentration of matter.

On a population scale natural selection attempts to accumulate traits impor-tant for survival, while the genetic drift introduces random changes. Togetherthese processes ensure the continuous development of life.

1.2 Water as a solventMany of the unique physical properties of water, including its solvation prop-erties and high boiling and melting points, are due to its ability to form hydro-gen bonds as illustrated in figure 1.1. According to thermodynamics, matterseeks to be in a low-energy state and bonding reduces chemical energy. Whensolvating a polar group in water it becomes part of this hydrogen bondingnetwork, but when solvating a non-polar group, water prefers to rearrange itshydrogen bond network and repels this hydrophobic solute. When attemptingthe opposite, solvating a polar group in a hydrophobic solvent, the solute will

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Figure 1.1: Hydrogen bonding network in liquid water.

be dehydrated without any possibility to pair its hydrogen bonds which is anenergetically very unfavorable state. A main theme of this thesis will be theinfluence of hydrophobicity and hydrophilicity for solvation properties.

1.3 From sequence to structure to functionMost processes in a cell are conducted by proteins acting either as enzymes,structural elements or by performing signal transduction. All informationneeded to synthesize the entire set of proteins in a cell (the proteome) iscaptured by a long sequence of four different sugar-bases denoted A, T, Cand G in the DNA-molecule within the cell. When a protein is synthesized,the part of the DNA-molecule that corresponds to this protein is transcribedto a RNA-molecule carrying the information to the ribosome machinery,which translates the sequence and puts individual amino acids togetheraccording to the coded sequence to form the actual polypeptide. Still how thenascent protein that emerges from the ribosome folds into its final structure isa somewhat open question, and a very interesting one since the structure cantell us how the protein performs its functions. As first proposed by Anfinsen[1], all the information required for folding is contained within the sequenceof amino acids. If the folding would occur by trial and error the number ofpossible conformations would according to the Levinthal paradox [2] bein the order of 10143, which implies that folding instead follows specificpathways.

The number of sequenced genomes is growing exponentially and eversince the first successful attempts to sequence DNA there has been a constantprogress in method development to analyze and manage this vast amountof information. The amount of structure and proteomics data is growing at

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a slower rate, making cheap computer based methods for predicting proteinproperties from sequence an appealing option. In addition it seems likenature reuses protein folds that have proven efficient. All living organisms arerelated but have diverged over evolution due to random mutations and naturalselection. If the sequences of two proteins are similar, they usually share thesame structural features as well. If the structure of a related protein is known,rather reliable conclusions can thus be drawn about the unknown structure. Ifno such related structures are known, structure prediction is however a veryhard problem without simple solutions.

Our understanding of molecular biology has increased tremendously overthe last couple of decades. It however seems as if for every rule that you try tostate in biology there is at least one exception and for every attempt to makesome kind of generalization, the picture grows more complex as more databecomes available. Nature is continuously evolving and finding new ways tosolve problems, which is what makes it such a challenging and interesting areaof research.

1.4 What this thesis is aboutThe main focus of this thesis is computational models that aim at making senseof the fundamental biological mechanisms that allow for proteins to occur inbiological membranes. This class of proteins is responsible for a vast arrayof different cellular functions but inherently difficult to characterize experi-mentally due to their non-water solubility, which makes them very interestingcandidates for computational studies.

Biological processes are usually studied with a top-down method startingfrom observations of a system and based on these trying to understand how itworks. Here an opposite approach is used; by starting with individual atomsfor which the basic properties have been determined, models can be createdand studied in action to gain understanding about the system as a whole. For-tunately the two approaches are by no means exclusive and rather complementeach other since theoretical models and predictions are just models and pre-dictions that need to be experimentally validated. In return they frequentlyprovide a much more detailed view of the system.

The first part of the thesis describes the biological problem at hand, howbiological membranes and the proteins within them are constructed, followedby a presentation of the computational methods used to study these systemsand finally a summary of the research papers whereupon this thesis is based.

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2. Biological membranes

The most fundamental role of a lipid bilayer is to function as a barrier be-tween living cells and their environment and to compartmentalize intracellu-lar organelles within eukaryotes. At the same time as being both mechanicallystrong and flexible, the membrane must also be impermeable to compoundsthat are unwanted in the cell and have mechanisms for passage of desiredcompounds into the cell. A composition based on lipids provides these de-sired structural properties and is impermeable to polar and charged groupswhile allowing the passage of small, hydrophobic molecules. The transportof larger and/or hydrophilic groups is managed by membrane bound proteins,which secure the optimal chemical composition inside the cell.

In 1972 Singer and Nicolson proposed a model where biological mem-branes were described as 2-dimensional liquids in which lipids and proteinsdiffuse more or less freely in the plane [3], offering a more dynamic view thanearlier hard cell models. This picture was prevalent for a number of years andcan still be found in most text books, although our view has evolved exten-sively over the last couple of decades. It is now generally acknowledged thatthe membrane is a complex entity compartmentalized into domains exhibitinglipid and protein composition differing from the bulk plasma membrane, asreviewed e.g. by Jacobson and Dietrich [4].

2.1 Composition of a biological membraneA biological membrane consists of variable proportions of lipids, proteins andcarbohydrates attached to either lipids or proteins. The mass fraction of pro-teins is often around half the mass of the membrane, which in most cases isroughly equivalent to 50 lipids per protein [5]. There are extreme cases withprotein content as low as 18% in neurons where the myelin membranes func-tion as insulation and as high as 75% in mitochondria with a large amount ofenzymatic activity in the membrane, for data see Guidotti [6].

2.1.1 Bilayer lipidsA lipid typically consists of a headgroup, either charged or zwitterionic withtwo charged groups with opposite sign and no overall charge, and one partwith one or more fatty acid hydrocarbon tail with a varying number of carbonsand double bonds. When placed in water these amphiphilic molecules have a

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Figure 2.1: A: A zwitterionic POPC lipid with a positive and a negative headgroupcharge and one double bond. B: Lipid shape determines the preferred lamellar struc-ture of the bilayer. C: A simulated membrane and the corresponding densities of dif-ferent groups. Red water, blue polar headgroups, green carbonyl groups and grey hy-drocarbon tails.

natural tendency to form bilayer structures to simultaneously expose the polargroups and shield the hydrophobic parts from the water.

The preferred lamellar structure depends on the geometry of headgroupsand tails where equal size promotes a flat bilayer and differences in size pro-mote curved bilayers. Figure 2.1 shows a typical lipid structure, different pos-sible lipid shapes and a phosphatidylcholin (PC) bilayer with varying compo-sition as function of bilayer depth.

A membrane typically contains hundreds to thousand of chemically dis-tinct species of lipid molecules, differing in the chemical composition andstructure. The most common lipids are zwitterionic phospholipids, while rarerbut important examples include cholesterol, saturated lipids and trans fats.

Cholesterol is essential in mammalian cell membranes to regulate fluidityover the range of physiological temperatures. This lipid is only slightly solu-ble in water and transported by different lipoproteins in the blood stream, onetype when going out to cells and one type when going back to the liver for

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excreation. High levels of the first type, known as "bad cholesterol", and lowlevels of the second, known as "good cholesterol", is associated with cardio-vascular disease.

Saturated lipids are made of the same building blocks as other lipids butarranged differently since they lack double bonds. This configuration givesstraight molecules that packs well resulting in high melting temperatures andhard membranes. The same effect can also be seen for special unsaturated fatsdenoted trans fats, which unlike most naturally occurring unsaturated fattyacids have the chains on opposite sides of the double bond. These lipids areusually a side effect from attempts by the food industry to hydrogenate lipidsand found in products containing processed fats. They are not essential andthey do not promote good health.

At the same time as being very diverse the exact lipid composition cannotbe critical since the fatty acyl chain composition of animal cell membranesvaries with changes in diet that, with a few exceptions as indicated above,have no obvious deleterious effect on cell function or on the health of theanimal [7]. The fatty acid composition is thus to some part determined bythe diet, although overall features of the lipid composition are neverthelessbelieved to be important and influence the bilayer properties. Factors suchas chain length, degree of chain saturation, headgroup geometry and chargedetermine the bilayer fluidity, curvature, charge distribution and thickness, amatter extensively reviewed e.g. by Anthony Lee [8, 9].

2.2 Influence of lipids on membrane protein functionThe direct effects of lipid composition on structure and function of membraneproteins have been quantified experimentally for a number of cases [10]. Astriking example is the voltage gated potassium channels that pump potassiumwhen placed in an mixed bilayer containing zwitterionic (POPE) and nega-tively charged phospholipids (POPG) but whose function is impaired whenplaced in a bilayer with positively charged lipids (DOTAP) [11]. Transportefficiency of the same channel has also been shown to be affected by alteringthe lipid composition enzymatically [12, 13].

Other examples include rhodopsin that does not fold properly without a spe-cific lipid composition [14], LacY that might change topology when the lipidcomposition is altered [15], the mechanosensitive channel MscL that dependson a specific curvature of the membranes for activation (reviewed by Hammiland and Martinac [16]) and Ca2+-ATPase which has lower activity in mem-brane with a high fraction of phosphoethanolamine (PE) lipid headgroups ascompared to phosphatidylcholine (PC) likely due to reduced ability to formhydrogen bonds in the interface region [17, 18, 19]. Varying lateral pressureprofiles and charge distributions associated with different bilayer lipids havebeen shown to effect membrane protein conformation and hence also function

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[20, 21, 22]. The importance of the lipid composition of the bilayer for cellfunction is also emphasized by the varying lipid composition between differ-ent organelles and by the uneven distribution of lipids between the two leafletsof the bilayer in the Golgi, plasma, and endosomal membranes that results indifferent environments on the two sides of the membrane [23].

2.3 Biological membranes vs membrane modelsA biological membrane is a very intricate medium composed of a large num-ber of molecules with differing properties. Due to this complexity any at-tempts to model the bilayer result in highly simplified models, where the aimof the modeling dictates the level of detail. If only the hydrophobicity of themembrane is needed it can be modeled very simplistically as an implicit sol-vent hydrophobic slab. If individual lipids are believed to contribute to thesolvation properties, an all-atom model where all lipids and the surroundingwater is modeled explicitly is needed.

Even the all-atom models are of course highly simplified since often onlyincluding a single lipid type, or possible a mixture of a few different types oflipids. In addition, the bilayer slab used is usually rather small excluding longrange effects like undulations from the model. Fortunately the rather simpledescriptions presently in use seem to be detailed enough for a large number ofapplications.

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3. Membrane proteins

A majority of articles about membrane proteins begin their introduction bystating the importance of membrane proteins based on their high abundance(about 30% of the genes in a typical genome codes for membrane relatedproteins according to Wallin et al. [24]), their importance as pharmaceuticaltargets [25] and the scarcity of experimentally determined structures (as ofMarch 2009, 184 structures had been determined [26], which amounts to lessthan 1% of all structures in the protein data bank (PDB) [27]). This is all trueand they are very important reasons to study membrane proteins. Beyond thatthe problem of understanding how this class of proteins function and behaveinside the membrane is a very fascinating and challenging problem in itself,and an area of research where computational and experimental methods havesuccessfully worked together. Although the number of structures [28, 29] andhence our knowledge is steadily increasing, there are still many aspects of themembrane protein life story that are poorly characterized.

3.1 What do membrane proteins look like?The definition of membrane proteins usually includes both peripheral mem-brane proteins associated with the membrane surface and integral or trans-membrane proteins traversing the membrane. This chapter will focus on thelatter.

The membrane puts a number of constraints on the possible structures thatcan be adopted by integral proteins but as more structures are solved, mem-brane protein structures seem to be just as complex as their water solublecounterparts [30].

Since no free hydrogen bonding acceptor or donors are possible inside thehydrophobic interior of the membrane, the only possible secondary structureelements that fulfill full hydrogen bonding potential of the protein backboneare α–helices and β–barrels, as illustrated in fig 3.1 for an α–helix–bundleand a β–barrel protein.

For the same reason that no unpaired hydrogen bond partners are availableinside the membrane, the thickness of the membrane puts length restrictionson the secondary structure elements. In addition, the amino acid compositionis dictated by varying chemical properties as function of membrane depth,ranging from pure water to pure hydrocarbons via the chemically complex

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Figure 3.1: A β–barrel, outer membrane protein (1QJA) (left) and an α–helical bundleprotein, rhodopsin (1UAZ) (right), together with small insets with the peptide bondhydrogen bonding patterns for these secondary structure elements.

interface region with ample possibilities for electrostatic, hydrogen bond andvan der Waals interactions.

β–barrel proteins have this far only been found in the outer membrane ofgram–negative bacteria and in organelles such as mitochondria and chloro-plast, which both are of bacterial origin. The β–strands are more difficult topredict based on sequence than the transmembrane α–helices due to morelong ranged–interactions and fewer residues. Genome analyses predict thatonly 2-3% of a bacterial genome codes for this class of proteins [31] and thereare also fewer structures of β–barrel proteins than of α–helical proteins.

A typical proteome consists of about 20-30% α–helical proteins [24], com-ing in all shapes and with a large variation in size ranging from single helicesto tight bundles formed from by a large number of helices. From here on theterm membrane protein will implicitly refer to α–helical membrane proteinssince they have been the focus of this thesis.

3.2 How are proteins inserted into the membrane?All proteins are transcribed by the ribosome where the genetic code in theRNA molecule is used to put together a polypeptide chain, which is subse-quently folded to form the three-dimensional protein. Membrane proteins are

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Figure 3.2: The translocon channel can either insert the helix into the membrane ortranslocate it to the other side of the membrane.

however not designed to survive in the aqueous cytosol, and if placed directlythere by the ribosome a multi-helical membrane protein would immediatelyaggregate into a non-native structure. Some small single-helix proteins can in-sert spontaneously into the membrane, but normally the insertion is aided byspecial channels called translocons [32, 33].

The first step in eucaryote membrane protein folding is the emergence of asignal peptide in the nascent polypeptide chain. This signal triggers the tightbinding of a signal recognition particle (SRP) to the ribosome, which tem-porarily halts the translocation and allows the ribosome-SRP complex to bindto a receptor on the ER-membrane [34]. Here the SRP is cleaved off and theribosome is placed on top of the translocon channel into which the polypep-tide chain is now synthesized. The translocon can either use a lateral gate todeliver transmembrane helices into the membrane or translocate the helix tothe other side of the membrane, as illustrated in figure 3.2.

This is in accordance with the classical two-state model proposed by Popotand Engelman [35], where transmembrane helices are first inserted into themembrane one by one and there aggregate to form the final three-dimensionalstructure.

The mechanism whereby the translocon recognizes sequences intended forthe membrane is not completely known, although the molecular features seemto be the same as seen to mediate protein-lipid interactions in known mem-brane protein structures implying that the translocon is designed to allow thenascent polypeptide chain to sample the surrounding bilayer [15]. Attemptshave been made to experimentally characterize the molecular code behindtranslocon recognition, resulting in position specific solvation profiles for allamino acids in a biological membrane [36, 37, 38, 39]. In paper I in this thesismolecular dynamics simulations were performed using the same type of he-

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lices as in these experimental studies to obtain a better understanding of aminoacid solvation properties within a membrane. In papers II to V molecular dy-namic simulation based methods were used to calculate solvation free energyprofiles using a range of different conditions.

3.3 Amino acid compositionTransmembrane helix sequences are dominated by highly hydrophobic aminoacids such as leucine, isoleucine, valine, alanine and phenylalanine, althoughsprinkled among these are polar or even charged residues having structuraland functional importance.

The otherwise rather rare aromatic residues tyrosine and tryptophan arecommonly found in the interface region [40] where they lock the helix inplace thanks to their shape and amphiphilic nature with a polar group and alarge apolar aromatic ring [41], as illustrated in fig 3.3 for tyrosine.

Another intriguing example of the interplay between amino acid compo-sition and function is the helix breaking residue proline that when found intransmembrane stretches is usually associated with helix kinks. It has beenshown that when substituting a kink-associated proline with an alanine, thekink remains [42], suggesting that the proline is not needed for the kink for-mation. A proposed explanation is that the introduction of proline generatesthe kink, but over time additional substitutions allow for the preservation ofthe bend in the helix.

When shifting the focus to the amino acid composition of parts outside themembrane, the loops on the inside of the cell are enriched in the positivelycharged amino acids lysine and arginine with a corresponding reduction onthe outside of the cell, a phenomenon known as the positive-inside rule [43]. Itseems to be valid over a large number of organisms [44] and it has been shownin protein engineering studies that it is a powerful determinant of membraneprotein topology and promotes helix insertion [45]. Evolution also seems tohave made use of this ability by creating homologous proteins with the samenumber of transmembrane helices but with opposite orientation of the topol-ogy, often referred to as dual topology proteins [46].

3.4 Polar groups in transmembrane segmentsSince it is energetically highly unfavorable to expose polar groups in trans-membrane stretches, their presence is a strong indicator of functional impor-tance in each individual case, which is also highlighted by a generally strongevolutionary conservation of these residues. Even though this is a well estab-lished fact, it is still unknown how they are stabilized inside the membrane[47]. In the final protein structure they are usually directed towards polar parts

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Figure 3.3: Left, tyrosine that intercalates with lipid tails at the same time as forminghydrogen bonds with interfacial groups, thus positioning the side chain in the interfaceregion. Right, lysine snorkeling towards the interface region where it pairs its hydrogenbonds.

in the membrane protein where they can pair their hydrogen bonds, whichis also an important driving force for helix dimerization [48, 49, 50]. Dur-ing protein folding when helices are inserted individually into the membranethey however still need to be exposed to the lipid environment. Experimentalstudies have shown that it is energetically possible to incorporate even signif-icantly hydrophilic residues into transmembrane helices as long as they arecounterbalanced by enough hydrophobic residues [36]. Molecular dynamicsstudies, including paper I in this thesis and others [51, 52, 53], have proposedthat this can be explained by the creation of water cavities inside the mem-brane allowing for polar groups to keep their hydrogen bonds paired.

Another option is used by the positively charged lysine and arginine, whichare both long and flexible making it possible for them to direct the polargroups towards the polar interface regions [41]. This behavior is often knownas snorkeling and is illustrated for lysine in figure 3.3.

3.5 Structural featuresThe first solved structure of a membrane protein was that of the photosyntheticreaction center published in the mid 80’s [54] that together with subsequentstructures represent highly regular examples both in terms of helix length andorientation, which was of course related to the fact that these could be crys-talized and have their structure determined in the first place. As the number ofavailable structures has increased, so has the number of encountered structuralpeculiarities. A typical helix-bundle structure was once believed to consistof a number of equally long interacting helices with slightly different angles

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Figure 3.4: The highly irregular structure of a glutamate receptor (1XFH). Aromaticresidues are pink to highlight the position of the aromatic belt. Lysine and arginineare shown in cyan, and as predicted by the positive inside rule more abundant on thecytosolic side (i.e. downwards) of the membrane.

through the membrane, but now this is extended to also include elements suchas very long or very short helices, kinked helices, interfacial helices, re-entrantloops or even irregular structure elements, all which contribute to give the pro-tein desired functional properties [55]. A number of these features can be seenin the irregular structure of the glutamate receptor shown in figure 3.4, wherethe aromatic belt and the positive inside rule is also highlighted by differentlycolored residues.

A number of studies have focused on dimerization patterns between inter-acting helices, and some early results showed that a specific motif based onglycine, GxxxG where x can be any other residue, was often found in the inter-face [56, 57]. This is natural as the small size of glycine allows the helices tocome in close contact. Attempts have been made to find additional dimeriza-tion targeting motifs but it appears as if the GxxxG-motif could be an excep-tion since no other similarly strong sequence signals have been found. Evenpolar groups such as serine and threonine are however commonly found indimerization interfaces, possibly due to their ability to form shared hydrogenbonds to the helix backbone [58, 59]. Even tough the task of predicting inter-acting helices seems to be a difficult problem, the information can be used theother way around. The DeGrado group has shown that it is possible to designpeptides specifically targeting transmembrane helices in a sequence specific

24

manner based on information about geometric constraints from known casesof helix interactions [60, 61, 62].

3.6 Computational methodsEven before the first 3-dimensional membrane protein structure was deter-mined, attempts were made to model the structure of membrane proteins basedsolely on sequence information. The main characteristic feature of transmem-brane helices is their hydrophobicity [63]. For easy cases the topology, posi-tion and orientation of helices can be determined very accurately based onlyon this hydrophobicity. Topology predictions can be made even more accurateby also including evolutionary information [64] or amino acid composition bi-ases such as the positive-inside rule or the aromatic belt, allowing for correcttopology predictions for close to 70% of all membrane proteins. This num-ber can be improved even further by addition of experimental informationabout the proteins at hand, e.g. the location of one of the peptide termini thatgives a constraint to the predication [65, 66]. Attempts have also been made toextend the 2-dimensional characterization by predicating features such as de-gree of lipid exposure, tilt angles relative the membrane [67], oligomerizationstates [68], the presence of proline induced kinks [42] and distance from thecenter of the membrane, a helpful parameter when characterizing structuralelements like re-entrant loops and interfacial helices [69]. In terms of fullthree-dimensional structure predictions, the same limitations hold for mem-brane proteins as for globular proteins; if a homolog with known structureexist it is a rather easy problem, otherwise it is a very hard problem.

Once the structure is known, other methods (frequently simulation based),can be used to further explore the dynamics of the protein, something that willbe the main focus in the following chapters.

25

4. Molecular dynamics

4.1 in silico chemistryHistorically there have been two main ways to gain understanding aboutchemical processes, either by developing theoretical models or by conductingexperimental work. During the last couple of decades another possibilitysomewhere in between has emerged with different computationallytechniques, making it possible to predict the behavior of complex chemicalsystems and perform theoretical experiments complementing laboratorywork. To emphasize the analogy to the latin expressions in vivo, which meansin a living organism and in vitro, in a controlled environment such as testtube, this approach is sometimes called in silico referring to the computersilicon microchips where calculations are conducted .

The first attempts to perform simulations of molecular systems were madeby Alder and Wainwright in the early 50’s [70] who simulated rigid spheres,and the research area soon expanded to more realistic systems, like argon [71]and even liquid water [72]. In the late 70’s the first simulation of a protein wasconducted for a total of 8.8 ps or 9000 steps by Martin Karplus and colleagues[73]. Three decades later this is common practice and the size and timescaleof systems possible to simulate are still growing rapidly and now extend intothe microsecond range.

A wide array of simulation techniques have been developed, all represent-ing different levels of trade-off between accuracy and efficient simulations.On the one extreme is quantum mechanical calculations, for which John A.Pope was awarded the chemistry Nobel prize in 1998, where the Schrodingerequation is solved for all atoms in the system allowing for calculations of e.g.partial charges and binding strengths. The only input needed is the participat-ing elements, although with the limitations that the calculations are very costlyand can usually only be applied to small systems under a set of somewhat un-realistic conditions such as vacuum and a temperature of zero degrees Kelvin.To model more complex systems for longer time spans approximations arenecessary. The other extreme is coarse grained simulations where individualparticles are groups of atoms allowing for calculations of concerted dynamicproperties for a large number of particles, but with lower accuracy. To conductsuch a simulation you need an extensive knowledge about the interactions be-tween particles in the system. Somewhere in between these extremes you findthe conventional molecular dynamics technique, which uses empirically de-

27

rived potentials and Newtonian dynamics to calculate interactions betweenatoms to study either dynamic or equilibrium properties.

Ideally one would of course prefer as accurate calculations as possible.Most interaction properties are however fortunately well described by clas-sical mechanics and except for the lightest atoms, such as hydrogen and he-lium, quantum effects can be neglected. This means that for a large numberof applications it is possible to settle for the classical view and solve Newtonsequations of motion for the interacting particles instead of the much more ex-pensive Schrodinger formulation. There are of course cases where molecularmechanics methods are too limited, e.g. when a very accurate energy esti-mate is needed or when studying bond-breaking and formation. In return fortreating interactions classically, these techniques also allow for more realisticconditions such as explicit solvent water and room temperature.

The main limitation for all simulation techniques, including the moleculardynamics technique, is the length of the simulations as determined by the sys-tem size and the available computer resources. The maximum time step in aatomistic simulation is in the order of a few femtoseconds, the exact value de-pending on the system and the integration and constraint algorithms, resultingin a maximum total length of a simulation in the order of microseconds thuslimiting the type of processes that can be studied successfully.

4.2 Relation between statistical mechanics andthermodynamicsExperiments usually measure macroscopic properties that are always averagesover a representative statistical ensemble of a molecular system. In a molec-ular dynamics simulation the position, velocity and force is known for eachparticle in every time step. The knowledge of a single structure is howevernot sufficient and a representative ensemble has to be generated in order tocompute the macroscopic properties of interest. Furthermore, detailed atomicdescriptions of structure and motions are often not relevant for macroscopicproperties and can be averaged by applying a statistical mechanics framework.This framework also offers the main justification of the MD method, that sta-tistical ensemble averages are equal to time averages of the system, known asthe ergodic hypothesis.

The phase space of a system with N atoms is defined as a virtual spacewhere all possible states of a system are represented. For a mechanical systemthe phase space usually consist of all possible values of position and mo-mentum variables (6N dimensions). A point in phase space is often denoteda micro-state while a macro-state is defined by properties like temperatureand pressure. The basic assumption in statistical mechanics is that all possibleconfigurations of the system that have the same total energy are equally likely,therefor the macro-state associated with the largest number of micro-states

28

is the one that is most likely to be observed. A partition function describesthe system at thermodynamical equilibrium and relates the probability dis-tribution of micro-states to the available macro-states, where the proportionof microscopic states for each macroscopic state is given by the Boltzmanndistribution. This partition function is defined as

Q = ∑i

e−Ei/kBT

where Ei is the energy associated with micro-state i and kB is the Boltzmannconstant that relates the units of temperature to the units of energy. Most ofthe aggregate thermodynamic variables of the system, such as the total energy,free energy, entropy, and pressure, can be expressed in terms of the partitionfunction or its derivatives.

4.3 The MD algorithmTo generate a representative equilibrium ensemble of a system the phase spaceas defined by the partition function must be efficiently sampled. Two mainmethods exist to do this, either by a biased random walk through phase spaceusing the Monte Carlo method or by solving Newton’s equations of motionfor the system and follow the time dependent path through phase space in themolecular dynamics method. For the study of dynamic events only the secondone is usable and since it usually produces comparable amounts of statisticsin a given amount of computational time, molecular dynamics simulationsare more commonly used. A third approach known as stochastic dynamicscombines the two by solving the equations of motions but adding a randompotential in each step to improve sampling.

The focus of this chapter is the molecular dynamics method, where theforces on every particle are integrated over each time–step according to New-tons second law F = ma to obtain new particle positions. The forces are cal-culated from interaction potentials between all particles as a function of theirrelative positions, which after a completed time step are updated allowing forthe next force calculation and integrattion to obtain the resulting positions af-ter yet another timestep. This procedure can be repeated as long as desired.The trajectory, the coordinates as function of time, is saved to disk and can beused to perform analysis of dynamic and equilibrium system properties afterthe completed simulation.

Since the force calculation is usually the most time consuming part, themain criterion on the integration algorithm is not calculation efficiency butthe ability to allow for long time steps. Another often desirable criterion istime-reversibility since the Newton equations are time reversible; the effectof adding a small time step ∆t should be equal to taking the same step back-wards. Good integrators are also area-preserving and leave the magnitude of

29

any volume element in phase space unchanged. If the latter is not fulfilled,the volume of the system will expand in phase space, which is not compatiblewith energy conservation.

The most commonly used class of integration algorithms is based on theVerlet formulation [74], which is derived by writing two Taylor expansions ofthe position vector x(t) in opposite time directions and combining these to anexpression for the new position at t = t +∆t.

r(t +∆t) = r(t)+ x(t)∆t +x(t)2!

∆t2 +...x (t)

3!∆t3 +O(∆t4)

r(t−∆t) = r(t)− x(t)∆t +x(t)2!

∆t2−...x (t)

3!∆t3 +O(∆t4)

r(t +∆t) ≈ 2r(t)− r(t−∆t)+ x(t)∆t2

The error in the updated positions will be in the order of O(∆t4). Variationsof this basic Verlet integrator exist, e.g. the Velocity Verlet algorithm, whichexplicitly incorporates velocities, and the Leap-frog algorithm that evaluatesthe velocities at half-integer time steps and uses these velocities to computethe new positions. The latter was originally an efficient choice using a mini-mum of memory since only the positions at a single step needed to be stored,and is still popular due to its combination of accuracy and efficiency in par-ticular for parallel simulations. More advanced schemes like the Beeman al-gorithm have also been developed, which produces identical positions as theVerlet integrator and gives more accurate velocities, but is more expensive andneeds significant storage.

4.4 Force fieldsThe forces acing on each individual particle are calculated using empiricallyderived potentials. This collection of potential terms is usually referred toas a potential function in physics and a force field in chemistry. The mostcommonly used force fields in our type of molecular dynamics simulationsare semi-empirical with parameters derived from both experimental work andhigh level quantum-calculations, optimized to reproduce experimental data asclose as possible. They have different levels of refinement, either only implic-itly including interaction effects such as polarization, or taking these effectsinto explicit consideration.

The basic form contains terms describing both bonded and non-bondedterms, where the specific decomposition depends on the force field. A gen-

30

k1(l-

l0)2k2(ξ-ξ0)2

Figure 4.1: Three bonded particles where harmonic constraint are used to model thebond length and the bond angle. The ideal length and angles are l0 and ξ0 and forceconstants are k1 and k2.

eral approach could be

Etot = Ebonded +Enonbonded

Ebonded = Ebond +Eangle +Edihedral

Enonbonded = Eelectrostatic +Evanderwaals

Force fields are usually either atomistic, describing the interactions betweenindividual atoms, or coarse grained describing interactions between groupsof atoms. Small groups as united atoms where methyl-groups are regardedas individual particles or larger groups where each particle may represent anumber of atoms.

4.4.1 Bonded interactionsThe potential due to covalent bond stretching and bending is usually modeledas harmonic oscillators, as sketched for three bonded particles in figure 4.1.The resulting potential is larger the further away from the ideal bond length orangle the molecule has moved. Dihedrals are angles involving more than threeatoms, either proper dihedrals describing torsion angles in chain molecules orimproper dihedrals to keep planar groups like aromatic rings planar.

4.4.2 Non-bonded interactionsThe non-bonded interactions are more expensive to calculate since they in-clude a much larger number of interacting neighbors for each particle. These

31

r

r

0

V(r) Repulsive (σ/r12)

Attractive(-σ/r6)εσ

Figure 4.2: The Lennard Jones potential between two particles as function of the sep-aration r.

potentials can either be calculated as pair– or many–body potentials, wherethe first is a simpler yet highly efficient and accurate choice, which will befurther described here.

A common way to jointly describe the van der Waals dispersion interac-tions due to induced dipoles and the short-ranged repulsion due to overlap-ping orbitals is to use the Lennard-Jones potential as function of the particleseparation r,

V (r) = 4ε

[(σ

r

)12−(

σ

r

)6]

where ε is the minimum of the potential and σ the particle separation forwhich the energy is zero, also illustrated in figure 4.2. The 1/r6 term describesthe dispersion interactions and the 1/r12 term is an approximative term de-scribing the repulsion. This repulsion force should depend exponentially onthe distance since the energy of the system increases abruptly once the elec-tronic clouds surrounding the atoms starts to overlap. This approximationhowever allows for more efficient calculation due to the ease of calculating1/r12 as the square of 1/r6 and in practice atoms will not get so close that itmatters.

The contribution from electrostatic interactions between two point chargesqi and q j is calculated using Coloumb’s law

32

V =qiq j

4πε0εrri j

where ε0 is the permittivity in vacuum, εr the relative dielectric constantand ri j the distance between the charges.

4.4.3 ParameterizationIn addition to the functional form of the potentials, a force field also definesthe values of parameters needed to calculate these potentials including partialcharges, equilibrium values for bond lengths and harmonic potentials associ-ated with these bonded interactions. Often several parameters are needed forthe same element when placed in different chemical environments. An ex-ample of this would be a carbon bound to two oxygens in a carbon dioxideversus a carbon part of a hydrocarbon chain. Force fields are parametrized toreproduce experimental properties like density, enthalpy of vaporization, var-ious spectroscopic parameters and sometimes even solvation free energy, andhence have differing capabilities of describing a certain systems.

4.5 Technical aspectsTo efficiently implement the molecular dynamical method, there are a numberof additional considerations;

4.5.1 Periodic boundary conditionsConsider a system with a finite number of particles placed in empty space. Thesmaller the system the larger the fraction of the system that will be located atthe edges. These particles will experience properties very different from therest of the particles and to minimize such edge effects a very large systemmust be used. Another option is to use periodic boundary conditions where aparticle on the left edge of the box is actually placed next to a particle on theright edge of the box and if this particle leaves the simulation box on the left,it will enter it again on the right. This effectively means that the simulated boxis surrounded by translated copies of itself, as sketched in two dimensions infigure 4.3.

When calculating short-ranged interactions between particles, only the clos-est replica of each molecule is considered, independent of its box location.Care must be taken to make the simulation box sufficiently large for moleculesto be embedded in enough solvent not to interact with copies in neighboringboxes.

33

Figure 4.3: Using period boundary conditions the simulation box is effectively sur-rounded by translated replicas, illustrated here in the x– and y direction.

4.5.2 Long range interactionsFor short range interactions that decay fast as a function of inter-group dis-tance, the periodic boundary conditions cause no additional complexity andonly the interactions between atoms within a cut-off have to be consideredwithout any substantial loss of information. For long ranged electrostatic in-teractions such a cut-off approach would result in errors since the potentialfunction depends on 1/r and does not converge when integrated over 3Dspace. Furthermore, the evaluation of a large number of pair-wise interac-tion is very costly. There fortunately exist options that combine reasonablecomputational cost with accuracy. The most widespread approaches are ei-ther based on reaction-field or different implementations of the Particle MeshEwald method (PME). In the reaction-field method pair-wise interactions areonly calculated within a cut-off. Beyond this a constant dielectric environmentis assumed, which works best for homogeneous systems. In PME the interac-tions are divided into two parts, the computationally cheap contribution fromshort ranged interactions between neighbors within a cut-off and the more ex-pensive long range interactions. The short range contribution is calculated inreal space and the long range part in reciprocal space allowing for accurateand efficient calculations.

4.5.3 ConstraintsTo allow for longer time steps and more efficient simulations the fastest mo-tions in the system, usually associated with bond vibrations for light atomslike hydrogens, can be removed by applying constraints to bond lengths andpossibly angles. Incidentally, modeling bond lengths as being constant is quitea good description of the quantum chemical ground state of a bond, at leastbetter than the harmonic oscillator bond description. Constraints offer an ef-

34

ficient way to increase the MD time step, but with the drawback of lost timereversibility of the integration step.

4.5.4 Temperature and pressure couplingA molecular dynamics simulation is per default performed in a NVE-ensemblem with constant number of particles (N), volume (V) and energy(E). There are however very few chemical processes that occur under theseconditions, a ordinary chemical experiment in a test tube is e.g. performedunder constant pressure and temperature, and there is hence frequently a needto couple other properties of the system to fixed values. The most commonlyused ensembles are either NVT or NPT, corresponding to constant number ofparticles (N), volume (V) or pressure (P) and temperature(T), which meansthat either the temperature have to be kept constant by a thermostat and/orthe pressure by a barostat.

4.5.5 Parallelization approachesThe development of more efficient computers has shifted focus from moreefficient single processors to the use of several processors in parallel. Whenrunning a simulation in parallel the calculation has to be divided on the avail-able nodes as efficiently as possible to minimize the need for communicationbetween the nodes. The simplest way is to divide the particles over differentnodes, perform the force calculations for these particles individually on eachnode and then communicate the result in an all-to-all communication step.This communication will limit how many nodes a system efficiently scales to,and more refined ways have been developed that allow for systems to scaleto a larger number of nodes by partitioning space instead [75]. Care must betaken since the part of the simulation–algorithm that scales least well willbe the bottle–neck that limits the parallelization performance. While a highlyparallelizable simulation software is desirable, good performance on a singleprocessor is still of paramount importance.

4.6 Applications to membrane proteinsMolecular dynamics is a commonly used tool when studying macromoleculesin general and, as emphasized by a number of recent reviews [76, 77, 78],membrane proteins are no exceptions. Since this class of protein is solvatedin a more complex environment than their water soluble counterpart, theirdynamics is a more intricate problem. A bilayer system requires longer equi-libration times due to slow lipid diffusion and relaxation and the processes ofinterest are often slow. This requires long calculations with simulation timesapproaching µs to sample simple protein-membrane interactions. Attempts

35

have been made to use implicit solvents to mimic the membrane dielectricproperties [79, 80] and make these inherently expensive simulations more eas-ily obtainable. As shown e.g. in paper I in this thesis, a lipid bilayer is how-ever a very adaptive solvent and by removing the atomic representation of thelipids these solvation properties are excluded. Another option is to use coarsegrained methods, where the number of particles in the system is reduced re-sulting in removal of the fastest motions allowing for much longer time steps.This approach has successfully been used e.g. by the Sansom group to studyprotein interactions with bilayers [81, 82, 83] and Treptow et al. [84, 85] tostudy ion channel gating mechanisms.

Even if the atomic details of the interactions are kept, the range of possi-ble process that can be studied can be extended by the use of non-equilibriumdynamics where instead of waiting for the system to evolve in a certain di-rection, it is forced there. One example is Gumbart’s studies of the translo-con channel being forced to open its lateral gate by pulling a helix out ofthe channel, allowing for increased understanding of the structural rearrange-ments [86, 87]. Another example is a recent microsecond simulation of thevoltage gated potassium channel with voltage applied over the membrane toinduce movement [88]. Thus, if simulations are used wisely by applying smalltricks, molecular dynamics can be used to study even rather slow processes onan atomic scale.

36

5. Free energy calculations

Every system seeks to achieve a minimum of free energy

5.1 Definition of free energyBy definition the thermodynamic free energy in a system is the amount of en-ergy available for doing thermodynamic work, which is minimized when thesystem is at equilibrium. A reaction that is associated with a negative free en-ergy difference is hence favored and will run spontaneously, which makes freeenergy calculations useful when evaluating e.g. the most likely conformationof a protein or which ligand that provides the best fit to a given receptor.

For a closed system with constant temperature and volume the free energyis described by the Helmholtz free energy F =U−T S, where U is the internalenergy of the system, T the absolute temperature and S the entropy. In a statis-tical mechanics context the partition function Q is, as discussed in section 4.2,the sum over all possible states i of a system in the ensemble and hence relatedto both the total energy and entropy of the system. This offers an alternativeway to define the Helmholtz free energy as F =−kBT lnQ.

In chemistry where reactions often take place in test tubes it is more com-mon with constant pressure than constant volume. Gibbs introduced an ad-ditional term to compensate for the work performed against the pressure tochange the volume and this Gibbs free energy is defined as G =U−T S+ pV =H−T S.

It is unfortunately impossible to calculate the free energy directly from asimulation since it is an ensemble property directly related to the volume inphase space associated with the system, not a simple average over phase spacecoordinates. There is however no way to measure absolute free energy experi-mentally either since it is defined as the work that one system does (or can do)on another. Only the changes in free energy associated with the transition of asystem from one state into another can thus be defined and measured. Exam-ples of this include the derivative of the free energy with respect to volume ortemperature, properties that can be measured directly in simulations as well:

(∂F∂V

)NT

=−P,(

∂F/T∂1/T

)V N

= E

37

5.2 Thermodynamic integrationThe free energy difference between two states can be calculated by first find-ing a reversible path that links them together and then integrating the deriva-tive of the free energy over this path, an approach denoted thermodynamicintegration.

The free energy is a state function that is only dependent on the states andnot the actual path travelled between them. In a simulation we are hence notrestricted to a physical path that can be followed experimentally, which im-plies that all the parameters in the potential energy function can be used asvariables for thermodynamical integration. To define the system along thispath a coupling parameter λ can be defined where the potential energy U0of the system corresponds to λ = 0 and potential energy U1 corresponds toλ = 1. The free energy difference between the states can be calculated by tak-ing the derivative with respect to λ of the free energy along the path betweenthe states, which can be written as an ensemble average, and integrating thisensemble average between λ = 0 and λ = 1. This is an important result, sinceensemble averages can be obtained from simulations. The coupling parametercan either be defined for systems with the same composition or for systemswith similar but distinct molecules where non-physical intermediate states areused, so called artificial thermodynamic integration.

The most common method to perform thermodynamic integration is to con-struct a set of configurations that are simulated for fixed values of the couplingconstant λ to obtain the derivative of the free energy expressed as an ensembleaverage from equilibrium simulations

dFdλ

=⟨

dHdλ

⟩

The ensemble averages from these simulations are numerically integratedfrom λ = 0 to λ = 1 to obtain the free energy difference

∆F =λ=1∫

λ=0

〈dH/dλ 〉dλ

An alternative is to use slow growth where the value of the coupling pa-rameter λ is changed slowly during a single simulation. For the latter there isnever a true average of any state since λ is constantly changing and the resultis thus often associated with a large degree of error due to hysteresis; i.e. thereis a risk of a systematic difference in the calculated free energy depending onwhether the calculation is performed forwards and backwards.

38

5.3 Free energy perturbationSuppose we are interested in the free energy difference between two systems,0 and 1, with partition functions Q0 and Q1 and free energy difference ∆F =F1 − F0 = −kBT ln(Q1/Q0). If a simulation is performed for system 0, thepotential energy can also be calculated for system 1 for each configurationvisited during this sampling. The potential energy difference, ∆U , can thus berecorded and used to construct a histogram as a measure of the probabilitydensity for the potential energy difference, which can be related to the freeenergy by F =−kBT lnP(∆U). This method to calculate free energy differencefrom a single simulation is called free energy perturbation and was presentedalready in 1954 by Robert Zwanzig [89]. If the phase space volumes betweenthe states of interest do not overlap sufficiently and the free energy differenceis not small enough to ensure convergence, additional simulations have to beperformed for intermediate systems.

5.4 Free energy along a reaction coordinateSometimes free energies correspond closely to physical processes acting alonga reaction coordinate, representing cases when not only the free energy differ-ence between two states is of interest but also the free energy along the paththe system travels between these states. The reaction coordinate can e.g. be asimple geometric coordinate representing bond length (as illustrated in figure5.1), bond order, or perhaps the distance between the active site of an enzymeand a ligand.

A naive approach to calculate the free energy profile along the reactioncoordinate would be to let the system sample the phase space and note thenumber of times the system is found in different locations along the coor-dinate to estimate the probability of the state being in that location. Thisprobability as function of position, P(ξ ), can be related to the free energy byF =−kBT lnP(ξ ), where kB is the Boltzmann constant and T the temperature.The problem with this approach is that only states associated with sufficientlylow energy will be sampled efficiently since the probability of finding the sys-tem at higher energy states will be very small. If there are barriers along thefree energy path, the system might even become stuck and some parts of thereaction coordinate pathway might not be sampled at all. To overcome thisproblem and ensure even sampling along the entire reaction coordinate sometricks can be used, which will be discussed in the following sections.

5.4.1 Umbrella samplingA frequently used approach to ensure sufficient sampling along the entire reac-tion coordinate is to add a potential to the system to cancel out the free energyprofile and to confine the system to a small region, where the free energy is

39

Free

ene

rgy

Separation

Figure 5.1: The free energy along a reaction coordinate represented by the separationbetween two bonded atoms.

40

F = −kBT lnP(ξ )+W (ξ ) and W (ξ ) a biasing potential. During simulation,the preferred position of the system is sampled and a histogram approach usedto evaluate the result and calculate the free energy. In the final step the bias-ing potential is subtracted again to recover the actual free energy. The addedpotentials often have the functional form of harmonic potentials, the shape ofwhich has given the method its name. This is a well known method with theadvantage of easy usage and system setup, but the disadvantage is that thebiasing potential has to be guessed beforehand. Since this cannot be done ex-actly, the normal solution is to partition the reaction coordinate into a numberof windows with sufficient overlap to ensure convergence resulting in a largenumber of simulations.

5.4.2 Constraint free energyAccording to the concept of potential of mean force (PMF), if a force

f (ξ ) =∂F∂ξ

depending on some reaction coordinate can be extracted, then the PMF canbe calculated according to

PMF =

ξ=1∫ξ=0

〈∂F/∂ξ 〉dξ

Constraint-based free energy calculation, which is the method used in pa-pers II to V presented in this thesis, is based on the same principle as umbrellasampling where a potential is added to the system to ensure uniform distri-bution of P(ξ ). In this case the added potential is infinite, which means thatthe system will stay in the exact same position. During simulation the forceneeded to keep the system in this position is recorded and used to calculatethe PMF, which can interpreted as the free energy along the reaction coordi-nate. In the most simple case the added constraint is one-dimensional and therecorded forces in this direction will only depend on this degree of freedomand hopefully all other degrees of freedom are sufficiently sampled during thesimulation. The advantages with this approach are that no assumptions haveto made beforehand about the system and each subsystem represents a pointon the reaction coordinate that can be simulated independently, making it em-barrassingly parallelizable. The disadvantage is that an advanced constraintalgorithm has to be used to keep the groups of interest at the defined distance,which is particularly complex in parallel runs.

41

Figure 5.2: The free energy between two states can be more easily calculated by con-structing a thermodynamic cycle with intermediate states.

5.4.3 Adaptive biasing forceAnother option is to calculate the biasing potential on the fly, as is the casein the adaptive biasing force method. The average mean force acting on thesystem is calculated and then removed from the system, which then performsa one-dimensional random walk with zero mean force along the reaction coor-dinate ξ . Only the fluctuating part of the force remains, ensuring uniform andefficient sampling. The calculated mean force is updated as more statistics iscollected, which results in the adaptive properties of the method. The methodgives better sampling than the aforementioned ones but at the cost of beingless suitable for simulation on multiple processors since it is not possible toperform independent simulations, as is the case e.g. for the constraint method.

5.5 Thermodynamic cyclesFor some applications it can be difficult to directly define a path betweenthe states of interest. However, since the free energy is a state function andthe path travelled is unimportant, intermediate states can be used to definethe sought difference. An example is shown in Fig 5.2 where the sought freeenergy difference is the hydration of a particle, ∆G4, a difference that is easierto calculate when using intermediate steps where the particle is transformedinto a dummy particle according to ∆G4 = ∆G1 +∆G2−∆G3.

5.6 Applications to membrane proteinsThe methods above can be used to perform the same types of free energy cal-culations on membrane proteins that are commonly performed on all types ofproteins, e.g. ligand binding to a protein surface or assessing different folding

42

options, but also to more specifically evaluate membrane protein structure andfunction. An interesting problem well suited for such calculations is the per-meability of molecules through channels of different types. Striking examplesinclude work by Allen et al. [90] who used umbrella sampling to study modelgramicidin channels in reasonable agreement with experimental results andmodeling of the permeation through the voltage gated potassium channel us-ing the adaptive biasing force method [84]. Other applications include recog-nition and association of transmembrane helical domains [91] and solvationprofiles of individual amino acids as function of position in the membrane, aspresented in papers II - V of this thesis but also by the Tieleman [92, 53] andAllen groups [51, 93, 94, 95] who both used umbrella sampling with aminoacids analogs and entire helices respectively, to accomplish this.

Although the development of faster and more accurate methods for freeenergy calculations have enabled applications that were not feasible only acouple of years ago, the methods are still not possible to use as black boxapplications. A number of different careful choices have to be made, bothwhen designing the free energy path between the states of interest and whenperforming the simulations to ensure efficient and sufficient sampling. A freeenergy calculation in a complex system thus offers great methodological chal-lenges, but can also substantially contribute to understanding the problem athand.

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6. Summary of papers

Knowledge about the insertion and stabilization of membrane proteins is akey step towards understanding their function and enabling membrane proteindesign. Transmembrane helices are normally quite hydrophobic to insert effi-ciently into membranes but there are many exceptions with polar or titratableresidues. An obvious example is the S4 helices of voltage-gated ion channelswith up to 4 arginines, leading to vivid discussions about whether such helicescan insert spontaneously and if so what their conformation, protonation stateand cost of insertion really are [36, 96, 97, 98, 99]. To address this question,the main focus of this thesis has been membrane protein solvation in lipidbilayers.

In paper I, we used single transmembrane helices with a systematically var-ied sequence inserted into a lipid bilayer to study how the solvation propertieschanged as function of amino acid type and position in the helix.

In the next paper, paper II, we tried to reproduce the experimental in vivoamino acid solvation profiles in a bilayer presented by Hessa et al. [36, 37]by using a molecular dynamics based method. Our results show good corre-lation with the experimental values, but the magnitude of the in vivo valuesare considerably smaller and the scale interestingly appears as a compressedversion of the calculated values. At the same time other groups also presentedcalculated values that lead to the same conclusion [51, 53], which made us be-lieve that there was something about the process of helix insertion that we stilldo not understand and that is hence not included in theoretical models. Thefollowing three papers, paper III to V, are all focused on possible explanationsfor the discrepancies between the two scales. Paper III evaluates the influenceof the protonation state for titratable amino acid on solvation properties, paperIV the corresponding influence from different types of bilayer lipids and thelast paper, paper V, the influence of transmembrane helices already present inthe bilayer to mimic a biological membrane. Both the protonation state andlipid types do have an influence on solvation, although not large enough tosingle handedly offer an explanation, while the presence of transmembranehelices seems to be able to account for virtually the entire discrepancy. Theanswer is however most likely a combination of different aspects since thebilayer used in models, no matter how realistic you try to make it, will differextensively from a complex biological membrane.

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6.1 Solvation of transmembrane helices (paper I)Inspired by experimental work by Hessa et al. [36] a set of transmembranetest helices was constructed to evaluate membrane solvation properties foramino acids in a transmembrane helix, both as function of amino acid type andposition in the membrane. Starting from a 19-residue long poly-alanine helixwith interfacial glycine and proline anchors on either side, nine test heliceswere constructed for each of the remaining amino acids by systematicallysubstituting two alanines at a time, starting next to each other in the middle ofthe membrane and stepwise moving outwards until placed at the ends of thehelix. These test helices were studied inside a DMPC bilayer using moleculardynamics simulations.

The most important result was that charged and, to some extent polar, aminoacids retain hydration by pulling in water and polar lipid headgroups, evenwhen placed in the middle of the membrane. The amount of water surround-ing each tested amino acid was quantified by calculating the cumulative radialdistribution and, interestingly enough, these values correspond well to exper-imental in vivo amino acid solvation profiles [36] indicating that this retainedsolvation could play an important role for the solvation cost. This also impliesthat the cost is mostly due to entropic contributions from molecules in thehydration shell, not loss of enthalpy from bond breaking.

Other interesting observations include a number of atomic scale explana-tions to already observed properties of membrane protein interactions with abilayer. Polar amino acids, and particular long flexible residues like lysine,tend to snorkel towards the interface region to pair their hydrogen bonds. Inour simulations we were able to quantify this behavior by calculating averagesnorkeling angles and, due to geometrical constraints of the peptide bonds,the angles are larger in the N-terminal than the C-terminal end of the helix.The size of the hydration shell for polar groups is also frequently larger in theN-terminal than the C-terminal end, which is consistent with earlier studies onamino acid composition of transmembrane helices indicating that polar groupsare more common in the N-terminal than in the C-terminal end. When furthercomparing the relative occurrence of amino acids, basic amino acids seem tooccur further into the membrane than acidic ones. This is caused by their abil-ity to form hydrogen bonds to lipid carbonyl groups, something acidic aminoacids cannot.

Tryptophan and tyrosine are known to be enriched in the interface regionand to function as interfacial anchors keeping helices in place in the mem-brane. In our simulations we could see how the aromatic ring orients itselfparallel to the lipid chains and intercalates at the same time as forming hy-drogen bonds with the interface regions, thus efficiently locking the helix inplace. Serine and threonine are furthermore known to be important for helixdimerization, which agrees well with our observations that they form sharedhydrogen bonds to the helix backbone as illustrated in figure 6.1, making their

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Figure 6.1: Hydrogen bonding network for serine (left) and threonine (right).

membrane insertion relatively cheap at the same as it is more advantageous tolater form real hydrogen bonds to other groups, e.g. to another helix.

In conclusion, the result from this paper showed how adaptive the mem-brane environment is as a solvent and that the solvation is both specific andquite complex.

6.2 Solvation free energy of amino acid analogs(paper II)In the second paper we wanted to further quantify the solvation propertiesof amino acids in a membrane environment by calculating the solvation freeenergy as a function of position. Sampling in an explicit membrane is veryslow, and as we in paper I showed that polar groups tend to retain hydration,accurate free energy calculations would necessitate extremely long simulationtimes. Here, we instead chose to use amino acid analogs where the backbonepart was removed to allow for accurate calculations using shorter simulationssince individual side chains cause much less disturbance to the membrane thanan entire helix.

To systematically study the effect individual amino acids have on the bilayerenvironment 100 conformations were constructed for each amino acid analog,except for glycine (no side chain), placed at different positions along the mem-brane z-axis. Simulations were performed for each of these conformationswith the position of the analog relative the lipid bilayer constrained. The direc-tion and size of the resulting force, denoted Fconstr, indicates the direction theanalog wants to move and to what extent. By integrating this set of position-dependent forces over the box-length in the z-direction, a likewise positiondependent potential of mean force (PMF) was obtained, V (z) =

∫z Fconstr(z)dz.

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Figure 6.2: Arginine side chain analogs in different positions along the membranenormal. The relative z-coordinate is constrained and the resulting forces acting on theanalogs are shown with arrows.

This PMF can be interpreted as the free energy of solvating amino acid sidechains at different positions in the bilayer. Figure 6.2 illustrates this with anarginine analog placed at three different locations in the membrane togetherwith the resulting forces from simulations.

Similar trends as in paper I were seen here as well, with polar groups retain-ing solvation and directing them selves to facilitate the pairing of hydrogenbonds, indicating that the solvation cost is to a large extent entropy driven.When comparing the results with the in vivo apparent free energy scale [36],there was initially a strikingly good correlation and we seemed to be able tocalculate the solvation free energy of entire helices with result correspond-ing very well to experimental results. However, as shown in the erratum topaper II, this turned out to be an artifact due to a software error and afterrecalculation the net values where clearly higher than the experimental. Thisagrees well with other simulation studies [53, 51] and is somewhat perplexing:The relative correlation is quite good despite the higher magnitude and the invivo scale almost seems like a compressed version of the calculated scales.Altogether, this implies that the membrane protein insertion is still not wellunderstood and that some aspect was not included in the models used for cal-culating the solvation free energies. Suggested explanations for this discrep-ancy include different protonation states of the titratable amino acids, differenttypes of bilayer lipids, the absence of protein in the pure lipid bilayers usedfor calculations, as well as different aspects of the still not fully understandhelix recognition by the translocon machinery. The three first suggestions arethe theme of the remaining three papers of this thesis, while the latter is stillan interesting question for further studies.

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6.3 Solvation free energy as function of protonationstate (paper III)The main objective of this paper was to evaluate the solvation properties oftitratable amino acids inside a lipid bilayer as function of protonation state.These amino acids are rare inside bilayers, and their presence in transmem-brane stretches implies importance for structure and/or function, most likelyassociated with its charge. By calculating solvation free energy profiles forboth the charged and the de-charged states we were able compare solvationproperties and thus evaluating if it is possible to have the charged forms in thebilayer or if insertion would lead to de-charging.

In paper II the solvation free energy was calculated for the most likely pro-tonation states in water, hence the profiles for the charged state were calculatedfor the analogs to arginine, lysine, histidine, aspartate and glutamate. In paperIII profiles were also calculated for the corresponding de-charged states usingthe same method as presented in paper II.

As expected, the cost of solvating the previously de-charged analogs is con-siderably lower than for the charged states, corresponding well to profiles forother polar amino acids. The non-charged protonation states induce little dis-tortion of the membrane even when placed in the hydrophobic core whereasthe charged form cause local distortion to maintain hydration, as illustrated infigure 6.3

Our results show that the differences between protonation states are notlarge enough to rule out the possibility that charged amino acids can be in-serted into a transmembrane helix if necessary for structure or function, es-pecially not if the free energy cost of de-charging the residue is taken intoaccount, being in the same order as the differences. Both the charged and de-charged solvation free energy show good correlation with experimental val-ues, which can thus not be used to discriminate between the two sets. It ishowever important to keep in mind that there is a cost associated with the sol-vation of polar groups in the membrane, smaller or larger depending on theprotonation state but still a cost.

The geometric aspects of solvation were also extensively evaluated andcharged states have a larger fraction of paired hydrogen bonds inside the mem-brane and a higher degree of order since consistently directing their polargroups towards the closest interface region. This almost perfect conservationof hydration for charged amino acids also results in a local thinning of themembrane around the analog, an effect that is not as strong for the unchargedprotonation states.

Even though the difference between protonation states is significant it is notlarge enough to explain the discrepancy between calculated and experimentalvalues presented in paper II. Furthermore for some proteins it is most likelynecessary to have the charged state of the titratable amino acid to maintainfunction.

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Figure 6.3: Systems containing arginine and aspartic acid at different positions andprotonation states.

6.4 Solvation free energy as function of lipid type(paper IV)A number of experimental observations for individual membrane proteinshave emphasized the importance of the membrane composition for proteinsto retain proper function. The typical membrane model used in molecular dy-namics simulations includes only a single type of lipids, while a biologicalmembrane consists of a mixture of different types of lipids as well as proteinsand carbohydrates. Here the aim was to systematically evaluate the influencedifferent types of bilayer lipids have on amino acids solvation properties.

To evaluate bilayers with different characteristics a set of six different lipidtypes was used, including positively and negatively charged lipids as well aszwitterionic lipids with different headgroup geometry and tail lengths. Thesolvation properties was determined for eight different amino acids compris-ing a representative test set of all twenty amino acids. We used the same basicmethod as presented in paper II to calculate solvation profiles for all 48 con-figurations.

A number of perhaps expected but nevertheless intriguing findings weremade that explain some observed lipid preferences for different membraneproteins. The maximum solvation cost for polar amino acids is lower in athinner membrane since the effort to retain hydration and snorkeling is facil-

50

DMPCDOPCPOPC

-20 0 20-20 0 20

SER PHE

LYS TRP

-5

0

5

10

15

-5

0

5

10

15

Box (Å)

ΔG

(kca

l/mol

)

Figure 6.4: Free energy profiles for analogs in bilayers with differing thickness andpacking properties.

itated by the shorter distance to the interface region. Figure 6.4 shows sol-vation profiles for amino acids in bilayers with the same headgroup (PC) butwith varying tail composition resulting in bilayers with different thickness andpacking properties.

Positively charged amino acids are more advantageous to solvate in a neg-atively charged membrane than in a positively charged, and vice versa fornegatively charged analogs. Lipids with smaller headgroups allow for moreadvantageous solvation in the interface region due to decreased pressure inthis region.

The presented differences in solvation free energy between lipids are con-siderable, although not large enough to account for the entire discrepancy be-tween experimental and calculated values as described in paper II.

6.5 Solvation free energy as function of membraneprotein content (paper V)On average a biological membrane has a protein fraction generally accountingfor around half the total mass [6]. To mimic this in a test system, we insertedtransmembrane helices into a lipid bilayer and used the same approach as pre-sented in paper II to calculate solvation free energy profiles. Two types of testhelices were used to evaluate the influence of amino acid composition, eitherhydrophobic or slightly polar helices, inserted in different numbers to createmembranes with varying mass fractions of protein. The solvation profiles were

51

10

0

10 10-10-10-20 -2020 2000Box(Å)

Solv

atio

n fr

ee e

nerg

y (k

cal/m

ol)

0%20.7%34.2%40.2%52.5%

0%20.4%33.5%39.7%49.9%

Polar helicesHydrophobic helices

Figure 6.5: Solvation free energy profiles for arginine as a function of position in themembrane and in bilayers with varying mass fractions of proteins, left hydrophobichelices, right polar helices.

calculated for arginine and leucine analogs representing a highly polar and ahydrophobic example of amino acids.

For arginine there is very good correlation between increasing mass frac-tion of protein in the bilayer and decreasing solvation costs, with a saturationtendency for high mass fractions of polar helices. In figure 6.5 solvation freeenergy profiles for arginine is plotted in bilayers with varying mass fractionsof hydrophobic and polar helices.

The chemical surroundings for the solvated analogs were evaluated by a cu-mulative radial distribution of solvation molecules. As expected, the numberof lipids surrounding the amino acids is higher for low mass fractions of pro-teins and the number of proteins higher for cases with high mass fractions. Thetotal number of proteins and lipids is however almost constant for all testedsetups while the amount of water in the hydration shell increases for higherprotein content. Furthermore the total number of paired hydrogen bonds forarginine is almost constant for all mass fractions and all positions in the mem-brane. Altogether this indicates that arginine always retains hydration but thepresence of a larger number of helices facilitates for water to be present in themembrane, thus explaining the much lower solvation costs.

The solvation cost for leucine also shows a decreasing trend with increasedprotein content, albeit not as strong as for arginine. In this case, the decreaseis most likely due to the additional space created inside the membrane as aresult of inserted helices and less efficient packing.

Interestingly enough, the presence of transmembrane helices seems to beable to account for virtually the entire discrepancy between the calculatedand experimental values. Most likely there is a variety of interacting causes,although the result presented in this paper offers at least one important partof the explanation for solvation properties of polar and charged residues inmembranes.

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7. Final thoughts

Molecular dynamics approaches do take a lot of computer power but at thesame time give us possibilities to study molecular interactions at a level notpossible to achieve in experimental systems, e.g. hydrogen bond dynamics andother atomic-scale solvation properties. To be useful computational methodsneed to be used as a complement to experimental methods, where both sidecan learn from each other and the results and understanding can be iterativelyimproved. The time when you could simply take the structure of a protein, putit into your simulation software of choice, press the button and watch it wiggleand giggle for a very brief period of time and then write an article about it, haspassed. Simulation approaches can however give you a vast array of answersas long as you are able to ask the right questions.

I have more specifically tried to use simulation techniques to quantify anobservable, in this case solvation free energy for individual amino acids in alipid bilayer, that can also be determined experimentally and hence the resultscan be justified. My initial results did not correspond to available experimentalresults; there was a strong correlation but a difference in the absolute magni-tude of the calculated solvation free energies. Since this discrepancy was seenby others as well, the problem was likely not my method per se, rather thatsome aspect of the transmembrane helix insertion machinery was poorly un-derstood and missing from the model, i.e. a simplistic membrane-as-a-bilayermodel. My further studies have been aimed at improving our understandingof this process and in the last paper, paper V, solvation free energies surpris-ingly close to experimental values were obtained by using bilayers incorporat-ing transmembrane protein segments to mimic biological membranes. By nomeans does this explain the whole story behind membrane protein insertionand a lot of work still lies ahead to fully understand this intricate problem. Animportant aspect well suited for further computational studies is the role ofthe translocon machinery and how this complex channel actually determineswhich helices are destined for the membrane.

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8. Acknowledgment

A number of people have supported and encourage me durning the workthat finally resulted in this book, and I would like to take this opportunity toacknowledge a few of these.

Erik Lindahl, my supervisor, for always having our best in mind and foralways being the most enthusiastic, encouraging and supportive supervisoryou can imagine.

Diana for being a very good friend and understanding that some problemsneed to be dwelt upon, Per for being able to see the humorous aspect ofeverything, no matter how bad it might seem. Both the office and my time asa PhD-student would have been much worse off without the two of you!

Other past and present office-mates, Samuel who once upon a time helpedme submit my first cluster-job, then I learnt how to do it by myself, Karin forkeeping us and the office organized, Sophie for filling the empty place afterKarin and starting every work day with a great smile and Sam for introducingthe concept of working in shifts to the office.

Johannes for your life coaching and for always being an optimist whosees possibilities and is ready for the the next adventure, Pär for alwayswanting to discuss both science and sports and for managing to have largerlunch-boxes than me, Sander for your invaluable help when I tried tolearn everything there is to know about MD, Åsa for carefully guidingme through the biochemistry teaching, Arne for creating such an openand creative research environment at the CBR, Aron for making surethat no party ends to soon, Kristoffer the proud captain of Swedmyra I,Andreas and Håkan V for good times at work, wherever work happenedto be, Erik S for your patience when everything goes wrong, Katarinafor never hesitating to squeeze in another social event, Maria for nevergiving up on me as a soccer player, Håkan L for helping me to survivehigh school teaching and for reminding me that there is a another worldjust across the building, Jenny for teaching me everything I know aboutwii, Björn for all the user-friendly scripts, Erik G for your never endinginterest during presentations, no matter how boring the subject might be,Rossen for organizing the cheer-leading team, Örjan for the reminder that

55

winning really is everything, whether it is in poker or pärk and Ali forchoosing the right part of the day for your dissertation. I would also like tothank all other colleagues at SBC/CBR and in the C4 corridor for all thecoffee clubs, lunch-breaks, unemptied dishwashers and just all the good times!

Jag vill också passa på att tacka alla som har sett till att jag emellanåt harägnat mig åt aktiveter långt bort från min dator. Marcus som alltid har ettnytt äventyr på gång och som sett till att utrustningsförrådet fyllts på medfrisbees, skridskor, mountainbike, klätterprylar och vandringsattiraljer - alltegentligen bara lysande ursäkter för att få komma ut och fika! Alla andraklätterkompisar som Hanna, Jenny, Malin och Ola som har gjort Karbintill en fast punkt i tillvaron. Min cykelkompis Fredrik som inte kan släppatanken på att det visst finns fisk i Ekoln och Carolina som ser till att det finnsfika under väntan. Åsa som nappade på förslaget att prova på rugby en soligsommarpubskväll. Mitt rugby lag, Uppsala RFC, ingen nämnd ingen glömd,vad tråkigt mitt liv måste varit innan jag träffade er - och så mycket tid jagmåste haft! Emma och Anna som sett till att vara mammalediga i Uppsalaoch gett mig bra anledningar till lunchutflykter under jobba-hemma-dagar.

Min familj, mamma och pappa för att ni alltid trott på mig vad jag än germig in på, mina syskon Bengt, Ida och Malin som alltid finns där och för attvi alltid har så roligt tillsammans när vi väl träffas och Malin för att du harden goda smaken att följa mina fotspår, i alla fall när det gäller viktiga sakersom rugby.

Mattias för att du har tillhandahållit köksbordet där stora delar av dennabok är skriven och för din korrekturläsning, men mest för att du är du och föratt vi är tillsammans.

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