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Copyright © 2008 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol. 5, 1–39, 2008

Concepts in Bionanomachines: Translocators

Artur BaumgaertnerInstitut für Festkörperforschung, Forschungszentrum Jülich, 52425 Jülich, Germany

This review summarizes, without pretension on completeness, a perspective on some biological rel-evant biomolecular machines. The attempt to give a kind of general overview would be necessarilyincomplete, since this scientific field is one of the most rapidly expanding and changing in compu-tational sciences. Any attempt to be as complete as possible would be obsolete anyway very soon.However, the kind of synopsis may still be valid in the future. The review is restricted to membraneproteins which translocate ions, small molecules and macromolecules across a cellular membrane(“translocators”). The report is focused on computational studies and theories, but references aremade to experimental work as well.

Keywords: Nanomachines, Ratchets, Collective Excitations, Entropic Barriers, Cooperativity.

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. Disruptive Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Helix Shutters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Voltage-Driven Gates . . . . . . . . . . . . . . . . . . . . . . . . 102.4. Light-Driven Gates . . . . . . . . . . . . . . . . . . . . . . . . . 10

3. Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1. Cooperativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2. Entropic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4. Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1. Collective Motions . . . . . . . . . . . . . . . . . . . . . . . . . 164.2. Brownian Ratchets . . . . . . . . . . . . . . . . . . . . . . . . . 294.3. Entropic Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . 35References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1. INTRODUCTION

Nanotechnology is perfectly realized in biological sys-tems. Cells are essentially biological assemblers that buildthousands of custom-designed molecules and constructnew assemblers. This view1 was pioneered by RichardFeynman2 and popularized by Eric Drexler’s3�4 evocativeidea of a self-replicating assembler5 building nanoscaledevices atom by atom. Bionanotechnology, being a syn-onym for nanobiotechnology, is a rapidly growing fieldthat encompasses contributions from various disciplines,ranging from engineering and computational sciences tophysics, chemistry and biology. Hence, there is an increas-ing need for reviews and textbooks that provide an intro-duction to biomolecular sciences and their impact onnanotechnology.

It is now clear that most functions in the cell arenot carried out by single protein enzymes, colliding ran-domly within the cellular jungle, but by macromolecular

complexes containing multiple subunits with specificfunctions.6 Living cells are made up of these complexes,which carry out many of the functions essential for theirexistence, differentiation, and reproduction. In many casesthe malfunction of these proteins can be a source of dis-ease; for example, myosin mutations, particularly in thehead and neck region of the molecule, can result in inher-ited diseases such as familial hypertrophic cardiomyopa-thy. An understanding of the mechanisms of these proteinsmay provide a guide for therapy.

Many of these complexes can be described as “molec-ular machines” or “molecular motors” or “moleculardevices,” depending on their sizes, complexity and tasks.Indeed, this designation captures many of the aspectscharacterizing these biological complexes: modularity,complexity, cyclic function, and, in most cases, the con-sumption of energy. In molecular machines or motors, arotary or linear movement is used for motility, nucleicacid processing, folding or unfolding, or as transducersof light or chemical energy. Examples of such molecularmachines are the replisome, the transcriptional machinery,the spliceosome, the ribosome, but also smaller machines,residing in the plasma and organellar membranes, areknown as ion and protein transporters, or as the bacterialflagellar motor.

The size of these macromolecular complexes, alone,often makes them inaccessible to X-ray crystallography.The structure determination of large, complex proteinensembles will pose a particularly difficult problem forstructural biologists. As a result, many efforts have concen-trated on determining the structures of individual subunitsand domains within the machines. Consequently, informa-tion on the organization of the assembly subunits, their

J. Comput. Theor. Nanosci. 2008, Vol. 5, No. 9 1546-1955/2008/5/001/039 doi:10.1166/jctn.2008.003 1

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Concepts in Bionanomachines: Translocators Baumgaertner

interactions, and sometimes their precise function withinthe context of the fully functional complex is often lost.

This may lead to a bare accumulation of many factswhich may not support our ‘understanding.’ According toa saying of the mathematician Henri Poncaré: Sience isbuilt up of facts, as a house is with stones. But a collectionof facts is no more a science than a heap of stones is ahouse., this review tries to order some biological nanoma-chines by their common physical principles, as far as theyare known. It is the hope that following this line, one dayin the future it will be possible to summarize the principlesof bionanomachines similar as in solid state physics, for-mulated, e.g., in the brilliant book by Anderson,7 Conceptsin Solids.The Context. In the post-genomic era, the recognition

that computational methodologies will play a critical rolein biology is widespread. Cell and molecular biologistshave realized the impracticality of trying to successfullypredict complex molecular mechanisms using intuition.Accordingly, the molecular and cell biology communitiesare now seeking suitable avenues for enabling them to addcomputational tools to their research kits.

The essential question in understanding biomolecularmachines is concerned with the explanation of the macro-scopic phenomenology in terms of the atomic structuresand forces involved. Although a complete description isnot yet available even for the best-characterized system,considerable progress has been made recently, not onlyfrom an experimental point of view, but also with respectto computational and theoretical achievements. Some ofthese machines, which have been studied by simulationsand mathematical methods are listed in Table I. An essen-tial element of the progress has been improvements inthe experimental methodology, of which single-moleculeexperiments are arguably the most important. However,a full understanding, in particular, of the kinetics of thedynamical mechanism requires theoretical and computa-tional techniques that make use of atomic-resolution struc-tures and detailed calculations of the molecular energeticsand dynamics. For example, how are the conformational

Artur Baumgaertner received his doctoral degree in physics from the University ofSaarbrücken (Germany) under the supervision of K. Binder. He spent most of his profes-sional career at the Research Centre Jülich (FZJ). In 1982 he received an IBM Fellowshipfor two years and worked at the IBM Research Laboratory Almaden with Do Yoon and withPaul J. Flory (Stanford University) on dense polymer systems. After two years he returned tothe FZJ and worked with K. Binder on modeling and simulation of polymers. His researchinterests at that time were in soft matter including topics as polymers in disordered media,liquid crystal polymers, polymer dynamics, and membranes. In 1992 he received a JosephDrown Fellowship from The Scripps Research Institute (La Jolla, USA) where he started towork in the area of molecular and cell biology. In 1996 he established at the FZJ the firstgroup on theoretical biophysics (Forum Modellierung). The main focus of his work is on

function of membrane proteins (bionanomachines) and on the self-organisation of cells (motility, migration, intracellularsignaling). He is a member of the faculty of physics at the university of Duisburg-Essen.

changes that engender the motion of rotary motors such asATPases produced by ATP as the energy source?

Considerable progress has been made in the past fewyears by a combination of biophysical techniques and the-oretical analysis. Single-molecule studies have played animportant role for a variety of motors including kinesin,myosin, and polymerases. The understanding of vari-ous ATPase complexes, among the smallest biomolecularrotary motors, has made particular progress by the inter-play of experimental and theoretical studies.

The term ‘motor’ or ‘machine’ is used to describe somebiomolecular complexes because they transduce one formof energy to another, e.g., chemical binding to mechanicalwork. These machines make use of chemical energy froma variety of sources, of which the most common is thebinding energy of ATP, H2O and its hydrolysis productsADP, H2PO4: ATP4− +H2O → ADP3− +H2PO4− Protonand ion gradients, as well as redox potential differences,also serve as the energy source in some cases.

Biomolecular machines range from single subunits (e.g.,DNA polymerases) through the smallest rotary motor, F1-ATPase, composed of nine subunits in mitochondria, tothe flagella motor of bacteria, which can be composed ofseveral hundred subunits of a number of different proteins.

Each protein machine possesses its specific function andoften it forms an element of the chemical network ofwhich the cell is composed. The motors have a wide rangeof functions, including chemical (e.g., ATP) synthesis,organelle transport, muscle contraction, protein folding,and translocation along DNA/RNA and protein filaments.They play an important role in cellular signaling, cell divi-sion, and cellular motion.Classification. In the present report some of the recent

advances in modeling and simulating active transportprocesses across bilayer membranes are outlined. Theexchange of substances across cellular membranes may beaccomplished by diffusion, facilitated diffusion, or activetransport. The result of unaided diffusion is an equilibra-tion of concentrations on both sides of the membrane.In facilitated diffusion the establishment of equilibrium

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Baumgaertner Concepts in Bionanomachines: Translocators

Table I. List of some biomolecular machines which have been stud-ied theoretically and by simulations. Abbreviations: ATP = hydrolysisof adenosine triphosphate, n-phos = nucleotide phosphorylation. �� =membrane potential and ion gradient.

Biomolecular devices Function Energy source

Membrane transducersGPCR Signal relay Ligands

RhodopsinCCR5, CXCR4

Catalytic receptors Signal relay Photons, ligandsInsulin, EGF

Chemotactic sensors Signal relay Photons, ligandssR

Receptor-ligand Signal relay Photons, ligandsTition, GroEL, etc.

Membrane channelsPorins Valve for ion, water Gradients

OmpF, OmpT,OmpA, FhuA

AQPMaltoporin

Ion channels Selective ion valve Gating by� � �AChR ligandClCgA ��

GluR LigandKcsA pHKvAP VoltageMscL, MscS Strain

Membrane transportersIon translocons Ion pumps Photons

bR H+

hR Cl−, anionIon translocases Ion pumps

V-ATPase H+ ATPCa2+-ATPase Ca2+ ATPNa+-K+-ATPase Na+, K+ ATP

Solute translocases Solute transport ATPHisP, MsbA, GLUT3 ATP

Protein translocases Protein transport ATPMitochondrial pore ATP, ��ER pore ATP, GTPNuclear pore

Membrane motorsATP synthase ATP synthesis ��

Flagellar motor Bacterial motility ATP, ��

is sped up by the function of a protein. Active trans-port is necessary to accumulate a substance against aconcentration gradient. Thermodynamics require somekind of energy to perform this, so there has to be anotherdownhill gradient that may be dissipated or some otherform of chemical energy. Depending on the source ofenergy primary transport is differentiated from secondarytransport. Primary transport uses energy directly: light orchemical energy is converted to electrochemical energyas electrochemical potential of the substances to be trans-ported. This category comprises photosynthetic electrontransport, light driven ion pumps, redoxenergy depen-dent respiratory chains, transport ATPases and sodiumpumps utilizing decarboxylation energy. In secondarytransport the electrochemical energy originates from the

electrochemical potential of another substance that is usedup in symport or antiport. Normally a molecule passesthe membrane unchanged. However, in group transloca-tion there is a chemical modification. The structures of fewtransport systems are known at atomic resolution. Amongthese are rhodopsins, some components of the PTS, theproton transporters bacteriorhodopsin and V-ATPase, theion transporter Ca-ATPase, and the protein transporters asmitochondrial and endoplasmatic pores, and the nuclearpore complex. Table I lists some of the known biomolec-ular machines, which are localized in cellular membranes,with their functions and energy sources. There exists oth-ers and that all the functions of the known motors are notyet recognized. Table I includes the class of membranechannels. In a strict sense, these proteins are not machines,but rather ‘valves’ whose gating are controlled by someform of external free energy.

2. GATING

In contrast to transporters, channels are passive deviceswhose opening and closing (‘gating’) is controlled exter-nally by ligands, voltage, pH value or mechanical stress.Most ion channels have moving parts that ‘gate’ the chan-nel and directly control ion permeation, often in responseto a specific external stimulus. Understanding the molecu-lar mechanisms underlying ion channel gating requires thecharacterization of two interrelated processes: first, that theenergy transduction machinery converts multiple types ofphysical stimuli (voltage, ligand binding, force and others)into protein motion, and second, the structural rearrange-ments that define these gating motions. A channel cangennerally assume two stable conformations, the open andthe closed one. The structural part of the channel respon-sible for the gating controls the accessability of ions to acentrally located water-filled pore. The opening and clos-ing of the gate is accompanied by conformational changesin the protein during gating. The structural and dynami-cal details of the gating mechanism are the least knownproperties of ion channels, mostly because of the fact thatan opened state of the channel is a transient one, thus noteasily fixed to be isolated by cristallization. Consequently,for the majority of known structures no direct comparisonof X-ray structures is available and one has to use otherexperimental techniques that reveal the structural determi-nants of the gating mechanism.

2.1. Disruptive Gates

Gramicidin. The dissociation of gramicidin A (gA)channels into monomers is the simplest example of achannel gating process. Gating presumably occurs via dis-sociation and association of the monomers with the clos-ing transition triggered by breaking the dimer’s stabilizinghydrogen bonds (Fig. 1). The deletion of a single hydrogen

J. Comput. Theor. Nanosci. 5, 1–39, 2008 3

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Fig. 1. Disruptive gate of gramicidin.

bond at the junction between the monomers destabilizesthe gA dimer and reduces its conductance. The initial stepsin this process have been studied8 via a computationalmodel that simulates the reaction coordinate for dimer-monomer dissociation. The nonbonded interaction energybetween the monomers is determined, allowing for theirfree relative translational and rotational motion. Lowestenergy pathways and reaction coordinates of the gatingprocess are determined. Partial rupture of the six hydro-gen bonds at the dimer junction took place by couplingmonomer rotation and lateral displacement. Coupling rota-tion with axial separation was far more expensive energet-ically. The transition state for channel dissociation occurswhen monomers are displaced laterally by 4–6 Å, sep-arated by 1.61–2 Å, and rotated by 120 degrees, break-ing two hydrogen bonds. In membranes with significanthydrophobic mismatch there is a much greater likelihoodof forming 4 hydrogen bonds (HB) and possibly even2 hydrogen bond states. In the 4HB state the pore remainsfully open and conductive. However, transitions from the6HB to 4HB and 4HB to 2HB states take place via inter-mediates in which the gA pore is closed and nonconduc-tive. These lateral monomer displacements give rise totransitory pore occlusion at the dimer junction, which pro-vides a rationale for fast closure events (flickers). Localdynamics of gA monomers also leads to lateral and rota-tional diffusion of the whole gA dimer, giving rise to dif-fusional rotation of the dimer about the channel axis.

Aquaporin-1 (AQP1). Aquaporin-1 (AQP1) was thefirst member of the family to be functionally characterizedas a water channel. While substrate permeation throughmonomeric pores of aquaporins is well characterized, lit-tle is known about the possible tetrameric pore. AQP1 hasbeen suggested to function as an ion channel upon cGMPactivation, although this idea has been controversial. Tak-ing a theoretical and experimental approach, it has beendemonstrate by MD simulations9 that the current mightarise through the tetrameric pore and a plausible mecha-nism for conduction and gating was proposed. In responseto simulated ion permeation, immediate hydration of theputative central pore was facilitated by moderate confor-mational changes of pore-lining residues. cGMP is foundto interact with an unusually arginine-rich, cytoplasmicloop facilitating its outward motion, which is hypothesizedto trigger the opening of a cytoplasmic gate.

2.2. Helix Shutters

2.2.1. Gates of Mechanosensitive Channels

Mechanical forces act on living organisms from all direc-tions throughout the biosphere, making mechanosensorytransduction one of the fundamental sensory transductionprocesses in the biological world. As molecular switches,mechanosensitive (MS) ion channels convert mechanicalforces exerted on cellular membranes into electrical or bio-chemical signals in physiological processes ranging fromcellular turgor control in bacteria to touch and hearing inmammals. Mechanosensitive channels are a class of ubiq-uitous membrane proteins gated by mechanical strain inthe cellular membrane. To adapt to osmotic stress, sev-eral types of MS channels in the bacterial cell membrane,including the channel of small conductance MscS, canopen when osmotic forces stretch the membrane. Withoutthis response, the bacteria lyse. In recent years, the molec-ular details of how MscS and other bacterial MS chan-nels are gated by mechanical forces have been a subjectof intensive research. Some toxic peptides are selectiveinhibitor of MS channels. The mechanism of inhibitionremains unknown, but it is known10 that they modify thegating, thus violating a trademark of the traditional lock-and-key model of ligand–protein interactions. Suspectinga bilayer-dependent mechanism, the effect of toxins ongramicidin A (gA) channel gating have been examinedexperimentally.10 It was shown that the inhibition increaseswith the degree of hydrophobic mismatch between bilayerthickness and channel length, meaning that the toxic pep-tide decreases the energy required to deform the bound-ary lipids adjacent to the channel. These results suggestthat modulation of membrane proteins by amphipathicpeptides (mechanopharmacology) involves not only theprotein itself but also the surrounding lipids. This hasimportant therapeutic implications.


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(a)

(b)

Fig. 2. Ribbon representations of the structures of MscL. The top part(a) depicts the view from the plane of the membrane (gray area), whereasthe transmembrane region viewed down the membrane normal is illus-trated at the bottom (b). Individual subunits are represented in differentcolors (PDB code 1MSL).

Mechanosensitive Channel of Large Conductance(MscL). MscL, a bacterial mechanosensitive channel oflarge conductance, is the first structurally characterizedmechanosensor protein.11–15 The protein is a pentamer(Fig. 2), approximately 50 Å wide in the plane of the mem-brane and 85 Å tall. Each 151-residue subunit consists oftwo transmembrane helices, labeled TM1 and TM2, anda cytoplasmic helix that extends some 35 Å below themembrane. The TM1 helices are arranged so as to blockdiffusion through the channel at their N-terminal ends.Excision of the cytoplasmic domains has been found tohave little effect on the gating properties of the channel.In prokaryotes the channel plays a crucial role in exocy-tosis and in response to large osmotic pressure changes.In general, it is believed that the gating of meachanosen-sitive channels is induced by changes in the intra-bilayerpressure profiles which originate from bilayer deforma-tion. In order to change the membrane tension it has beensuggested13 that different hydrophobic mismatches at theprotein–lipid interface induced by different types of lipidsmay cause an asymmetry of tension across the bilayermembrane and hence lead to a spontaneous curvaturewhich controls the open and the closed state. At least, it

is clear from experiments with different types of mixturesof lipids13�14 that the protein–lipid interaction must play afundamental role in defining the physical principles thatunderly MscL gating. Recently, MD simulations16�17 havebeen performed in order to contribute to the experimentaldata on gating mechanism. Standard MD simulation in thenanosecond time regime and the analysis of the structuralfluctuations have indicated that the least mobile part of theprotein could be identified as the gate, on the same locationsuggested by experimental findings. This part comprisesthe first 5 residues of the TM1 helices, which are shownto be pinched together to form a non-leaky occlusion. Inaddition, steered MD simulation16 of the bare protein with-out membrane and without water have been carried out.The effect of the membrane on the protein has been takeninto account by applying a constant surface tension on theprotein. Under a range of conditions, it has been shownthat the transmembrane helices tilted considerably as thepore opened. The protein refolded into an open confor-mation, where the transmembrane helices flattened as thepore widened, with a minimal loss of secondary structure.The rate at which the protein refolded has been nearlyinversely proportional to the applied surface tension. Theresults indicate that membrane thinnning and hydropho-bic mismatch within the transmembrane helices my indeeddrive gating.

More recently18 steered molecular dynamics simulationshave been used to investigate how forces arising frommembrane tension induce gating of the channel. A homol-ogy model of the closed form of MscL from was sub-jected to external forces of 35–70 pN applied to residuesnear the membrane-water interface. The magnitude andlocation of these forces corresponded to those determinedfrom the lateral pressure profile computed from a lipidbilayer simulation. A fully expanded state was obtainedon the 10-ns timescale that revealed the mechanism fortransducing membrane forces into channel opening. Theexpanded state agrees well with proposed models of MscLgating,13–15 in that it entails an irislike expansion of thepore accompanied by tilting of the transmembrane helices.The channel was most easily opened when force wasapplied predominantly on the cytoplasmic side of MscL.Comparison of simulations in which gating progressed tovarying degrees identified residues that pose steric hin-drance to channel opening.

The crystal structure of the Mycobacterium tuberculo-sis homolog of the bacterial mechanosensitive channel oflarge conductance (Tb-MscL) provides a unique opportu-nity to consider mechanosensitive signal transduction atthe atomic level. Molecular dynamics simulations of theTb-MscL channel embedded in an explicit lipid bilayerand of its C-terminal helical bundle alone in aqueous sol-vent have been performed by Elmore and Dougherty.17

C-terminal calculations imply that although the helix bun-dle structure is relatively unstable at physiological pH, it


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may have been stabilized under low pH conditions suchas those used in the crystallization of the channel. Specificmutations to the C-terminal region, which cause a simi-lar conservation of the crystal structure conformation, havealso been identified. Full channel simulations were per-formed for the wild-type channel and two experimentallycharacterized mutants. The wild-type Tb-MscL trajectorygives insight into regions of relative structural stability andinstability in the channel structure. Channel mutations ledto observable changes in the trajectories, such as an alter-ation of intersubunit interactions in one of the mutants. Inaddition, interesting patterns of protein–lipid interactions,such as hydrogen bonding, arose in the simulations. Theseand other observations from the simulations are relevantto previous and ongoing experimental studies focusing oncharacterization of the channel.Mechanosensitive Channel of Small Conductance

(MscS). MscS, the mechanosensitive channel of small con-ductance, is found in the inner membrane of Escherichiacoli and its crystallographic structure in an open formhas been recently solved.19�20 Much of what we knowabout the molecular mechanisms of gating in MscS chan-nel is derived from its crystal structure. Only recently havemolecular dynamics and experimental studies shed addi-tional light on the structural changes that occur upon MscSgating. MscS was the first bacterial ion channel shownto respond to membrane stretch. Since its cloning, MscShas emerged as a prototype of a diverse family of MSchannels encompassing several representatives from bac-teria, archaea, fungi and plants. The crystal structure ofEscherichia coli MscS solved at a resolution of 3.9 Åreveals that the channel folds as a homoheptamer and hasa large cytoplasmic region. Each subunit contains threeTM domains (Fig. 3). Of these, the TM3 helices line thechannel pore, whereas the TM1 and TM2 helices consti-tute the sensors for membrane tension and voltage. Theprecise conformation of MscS in the crystal form is con-troversial at present, as a recent study using moleculardynamics simulations21 implied that water and ions can-not pass through the channel pore, suggesting that thecrystal structure may reflect an inactive or desensitizedstate rather than the open state. Nevertheless, the crys-tal structure strongly suggests that the channel was cap-tured in an open state, because the transmembrane porereveals an opening of 11 Å, which could account forthe 1 nS conductance observed for the MscS activity.22

Booth and colleagues20 proposed a new structural modelof MscS gating that involves rotation and tilt of pore-lining transmembrane helices. They also show how dis-ruption of a conserved pattern of glycine and alanineresidues and their interactions along the pore-lining trans-membrane helix TM3 affects the gating of this channel.They proposed a model for the closure of MscS based onmutagenesis experiments affecting structural properties ofthe pore-lining transmembrane TM3 helix (Fig. 4). TM3

(a)

(b)

Fig. 3. Ribbon representations of the structures of MscS. The top part(a) depicts the view from the plane of the membrane (gray area), whereasthe transmembrane region viewed down the membrane normal is illus-trated at the bottom (b). Individual subunits are represented in differentcolors (PDB code 1MXM).

is rich in glycine and alanine residues that the authorspropose form a structural motif that plays a crucial rolefor gating of this channel. The position of the glycine-alanine pattern on the helix faces is conserved in the MscSfamily of proteins, further supporting the notion that thisstructural motif is important for proper function in thesechannels. Gating transitions in MscS thus must involvecooperative action of glycines and alanines along the TM3helix. The current model of MscS closure indicates thatall TM helices decrease in tilt, becoming more verticalin the transition from the open to the closed states. Dur-ing channel closure TM3 domains must slide against eachother and rotate, resulting in a change in the packing ofglycines and alanines along the helix. This leads to a muchmore compact channel with a smaller pore in which cer-tain leucines come into close proximity, thus forming a


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Fig. 4. The pore structure of the MscS channel in the open and closedstates. The illustration shows how the tilting and rotation of the pore-lining TM3 helices could open and close the channel.

hydrophobic barrier preventing ion flow. The experimen-tal evidence is consistent with the gating model in whichopening of MscS is facilitated by slight iris-like rotationsand tilt of the pore-lining helices (Fig. 4). A smooth sur-face formed by four glycines should facilitate the sliding ofalanine ‘knobs,’ which underlies the conformational tran-sition required for channel gating. Rotation of the trans-membrane domains in MscS closely resembles a currentmodel for gating of MscL,15 the bacterial MS channel oflarge conductance. However, consistent with its smallerconductance, the structural changes in MscS are of smallermagnitude compared with that in MscL, whose conduc-tance is three times larger than that of MscS. A search forstructurally conserved motifs in other ion channels sim-ilar to the conserved pattern of alanines and glycines inMscS channels and examination of the extent of structuralplasticity tolerated in other channels may provide insightsinto how evolution has solved steric hindrance problemsto achieve a required protein function. With this modeland the structures at hand we can expect the research onbacterial MS channels to continue to contribute to noveland significant insights into the structure and function ofion channels. By means of molecular dynamics simula-tions Schulten and Sotomayor21 have studied the stabilityof the channel conformation of MscS suggested by a pre-vious X-ray crystallographic structure at 3.9 Å resolution19

in a fully solvated lipid (POPC) bilayer. The combinedsystem encompassing 224, 340 atoms. When restrainingthe backbone of the protein, the channel remained in theopen form and the simulation revealed intermittent per-meation of water molecules through the channel. Abol-ishing the restraints under constant pressure conditionsled to spontaneous closure of the transmembrane chan-nel, whereas abolishing the restraints when surface ten-sion (20 dyn/cm) was applied led to channel widening.The large balloon-shaped cytoplasmic domain of MscS

exhibited spontaneous diffusion of ions through its sideopenings. Interaction between the transmembrane domainand the cytoplasmic domain of MscS was observed andinvolved formation of salt bridges between residues Asp62and Arg128; this interaction may be essential for the gat-ing of MscS. K+ and Cl− ions showed distinctively dif-ferent distributions in and around the channel. To gaininsight into the effect of the lipid membrane compositionand geometry on MscL structure, a fully solvated, all-atommodel of MscL in a stress-free curved bilayer composedof double- and single-tailed lipids was studied using MDsimulation.9 The bilayer was modeled as a domed structureaccommodating the asymmetric composition of the mono-layers. During the course of the simulation a spontaneousrestructuring of the periplasmic loops occurred, leading tointeractions between one of the loops and phospholipidheadgroups.

2.2.2. pH-Gated Channels

Potassium Channel (KcsA). The Streptomyces lividanspotassium channel (KcsA) is pH regulated.23 A gatingmechanism was proposed by Perozo and coworkers24–26

by using site-directed spin-labeling methods and elec-tron paramagnetic resonance spectroscopy. Results fromthese experiments indicate that the channel undergoes a“twisted” motion where each of the four TM2 helices tiltsaway from the permeation pathway, towards the mem-brane plane, and rotates about its helical axis, supportinga scissoring-type motion, similar as depicted in Figure 4,with a pivot point near residues 107–108. These move-ments result in a large increase of the diameter of theintracellular mouth up to the central water-filled cavity.Although the possible collective motion of the helices canbe constructed, mainly based on steric consicderations, theorigin of the pH-mediated driving force is still unclear.It seems likely that the four long cytoplasmic C-terminiof residues 123–160 may play the crucial role, becausesome of the charged residues may change their protonationstate during pH variation and hence provide the necessaryvariation in their inter-chain Coulomb interaction. Anothercandidate for pH-mediated conformational transitions arethe four N-termini 1–23 carrying also titrable groups. It isunclear how these chains interact with each other, eventu-ally indirectly via the lipid head groups of the membranewhich are in general pH sensitive. Unfortunately the gatingmechanism cannot be studied by standard MD simulationbecause the time scale of gating is in the order of at leastmicroseconds. The simulation of channel gating is there-fore still one of the great challenges in biophysics.

The structure of the bacterial potassium channel, KcsA,corresponds to the channel in a closed state. Severallines of experimental evidence suggest that the channelmust widen its intracellular mouth when in an open state(Fig. 5). Thus it is known that the protein moves in this


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Openconductive

Inactivated

H+ H+H+H+

Fig. 5. Gating in the KcsA channel.

region, but it is unclear by how much and the mecha-nisms that are involved. To address this issue Sansom andcoworkers27 have applied a novel approach to generateplausible open-state models of KcsA. The approach canbe thought of as placing a balloon inside the channel andgradually inflating it. Only the protein sees the balloon,and so water is free to move in and out of the channel.The balloon is a van der Waals sphere whose parameterschange by a small amount at each time step, an approachsimilar to methods used in free energy perturbation calcu-lations. They have shown that positioning of this balloonat various positions along the pore axis generates similaropen-state models, thus indicating that there may be a pre-ferred pathway to an open state. They also showed thatthe resulting structures from this process are conforma-tionally unstable and need to undergo a relaxation processfor up to 4 ns. They showed that the channel can relaxinto a new state that has a larger pore radius at the regionof the intracellular mouth. The resulting models may beuseful in exploring models of the channel in the contextof ion permeation and blocking agents. Other studies onblocking agents of K+ channels have been performed forthe docking of the scorpion toxin Lq2 on binding sitesof potassium channels28 using Brownian dynamics sium-lations. More recently, it was shown29 that there exist aninner gate which is located at the selectivity filter. Theinner gate shows large structural excursions of the selec-tivity filter during ion conduction and opens maximally atlow pH regardless of the magnitude of the single-channel-open probablility. Thisa establish a mechanistic basis forthe role of the selectivity filter during channel activationand inactivation.M2 Channel. The transmembrane fragment of the

influenza virus M2 protein forms a homotetrameric chan-nel that transports protons. Molecular dynmaics simu-lations have been performed by Sansom et al.30 andSchweighofer and Pohorille.31 In the more recent paper31

simulations of M2 were reported which helped to eluci-date the mechanism of channel gating by four histidinesthat occlude the channel lumen in the closed state. Theauthors tested two competing hypotheses. In the “shut-tle” mechanism, the nitrogen atom on the extracellularside of one histidine is protonated by the incoming pro-ton, and, subsequently, the proton on the nitrogen atom is

released on the opposite side. In the “water-wire” mech-anism, the gate opens because of electrostatic repulsionbetween four simultaneously biprotonated histidines. Thisallows for proton transport along the water wire that pene-trates the gate. For each system, composed of the channelembedded in a hydrated phospholipid bilayer, a 1.3 ns tra-jectory was obtained. It was found that the states involvedin the shuttle mechanism, which contain either single-protonated histidines or a mixture of single-protonated his-tidines plus one biprotonated residue, were stable duringthe simulations. Furthermore, the orientations and dynam-ics of water molecules near the gate are conducive to pro-ton transfer. In contrast, the fully biprotonated state is notstable. Additional simulations showed that if only two his-tidines are biprotonated, the channel deformed but the gateremained closed. These results support the shuttle mecha-nism but not the gate-opening mechanism of proton gatingin M2.Protein Translocon. The heterotrimeric SecY/Sec61

complex is a protein-conducting channel that provides apassage for proteins across the membrane as well as ameans to integrate nascent proteins into the membrane.32�33

While the first function is common among membrane pro-tein channels and transporters, the latter is unique. Thistranslocon performs its function in concert with a varietyof other macromolecular assemblies: the ribosome, SecA(in bacteria) or BiP (in eukaryotes), and others. Thesemacromolecules provide the driving force necessary tofeed the polypeptide through the channel and prevent itfrom sliding back. Insertion of nascent membrane pro-teins, one transmembrane segment at a time, by SecYlikely occurs through a lateral gate in the channel. Molec-ular dynamics simulations have been used to investigatethe mechanism of gate opening.34 Opening and closingthe gate under different conditions allowed us to identifystructural elements that resist opening as well as those thataid closure. SecE, considered to act as a clamp keepingthe lateral gate closed, was found to play no such role.Loosening of the plug by lateral gate opening, a potentialstep in channel gating, was also observed. The simulationsrevealed that lipids on time scales of up to 1 s do not floodchannels with an open lateral gate.

2.2.3. Ligand Sensors

Acetylcholine Receptor (nAChR). The nicotinicacetylcholine receptor (nAChR) is one of most intensivelystudied ligand gated ion channels, and thus provides aparadigm for the molecular mechanism of fast synap-tic neurotransmission. Depolarization of the presynapticmembrane causes release of acetylcholine into the synapticcleft. Acetylcholine diffuses to the postsynaptic membranewhere it binds to the extracellular domains of the nAChR.Details of the structure of the nAChR have been unveiledrecently.35


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The nAChR is pentameric, the five subunits arrangedaround a central pore (Figs. 6(b, c)). The transmembranepore is lined by the second transmembrane helix (M2) ofeach subunit. Binding of acetylcholine to the extracellular

(a)

(b)

(c)

Fig. 6. (a) Proposed model for the gating mechanism. The ACh-inducedrotations in the �-subunits are transmitted to the gate—a hydrophobicbarrier to ion permeation—through the M2 helices. The rotations desta-bilize the gate, causing the helices to adopt an alternative configurationwhich is permeable to the ions. The helices move freely during gatingbecause they are mainly separated from the outer protein wall and con-nected to it by flexible loops, containing glycine residues (G). S–S is thedisulphide-bridge pivot in the ligand-binding domain, which is anchoredto the fixed outer shell of the pore. The relevant moving parts are shaded.The height of the membrane-spanning pore is about 40 Å, the �-sheetstructure about 60 Å. (b) Side view and (c) top view of the pore, as seenfrom the synaptic cleft, with subunits shown in different colors (PDBcode 1OED).

domains is thought to initiate a wave of conformationalchange that is propagated through the protein into thetransmembrane domain and the inner (M2) helices. Thisresults in a ‘twist to open’35 mechanism of channel gat-ing whereby the central pore is opened to the perme-ation of cations. A leucine side chain is highly conservedthroughout the nAChRs and other members of the super-family and plays a central role in the gating mechanismof these receptors. Unwin and coworkers35 have proposedthe following gating mechanism (Fig. 6(a)). Binding ofACh opens the channel by initiating rotational movements(arrows) of the inner �-sheets of the � subunits in theligand-binding domain. The rotations destabilize the gate,causing the helices to adopt an alternative configurationwhich is permeable to the ions. These movements are com-municated to the inner (M2) helices lining the pore andbreak apart the gate—a hydrophobic girdle in the middleof the membrane—so that ions can flow through. A tryp-tophan side chain in the ligand-binding domain identifiesthe ACh-binding region; a valine side chain links the innersheet to the inner helix; leucine and valine side chains onthe inner helices make the gate; the locations of the mem-brane surfaces are indicated by broken lines. The helicesmove freely during gating because they are mainly sep-arated from the outer protein wall and connected to itby flexible loops, containing glycine residues (G). S–S isthe disulphide-bridge pivot in the ligand-binding domain,which is anchored to the fixed outer shell of the pore.

The self-assembly of M2 peptides to form a pentamericbundle of transmembrane helices is a model of the pore-lining region of the nicotinic acetylcholine receptor. Thishas been studied36 by molecular dynamics simulations ofa model of the M2 bundle in a POPC bilayer in orderto explore the conformational dynamics of the channelassembly. On the timescale of the simulation, the bundleremains relatively stable, with the polar pore-lining sidechains remaining exposed to the lumen of the channel.Fluctuations at the helix termini, and in the helix curva-ture, result in closing/opening transitions at both mouthsof the channel, on a timescale of 10 ns. On average, waterwithin the pore lumen diffuses 4× more slowly than wateroutside the channel. Examination of pore water trajectoriesreveals both single-file and path-crossing regimes to occurat different times within the simulation.

The analysis of short MD simulations by theMcCammon group37 shows a pattern of motions whichindicate how ligand binding may correlate with larger-scalesubunit motions that would connect with the transmem-brane region that controls the passage of ions.

MD simulations were performed by Sansom andcoworkers38 of the nAChR embedded in a bilayer mimeticoctane slab. The M2 helices and M2–M3 loop regionswere free to move, whereas the outer (M1, M3, M4)helix bundle was backbone restrained. The M2 heliceslargely retain their hydrogen-bonding pattern throughout


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the simulation, with some distortions in the helical endand loop regions. All of the M2 helices exhibit bendingmotions, with the hinge point in the vicinity of the centralhydrophobic gate region. The bending motions of the M2helices lead to a degree of dynamic narrowing of the porein the region of the proposed hydrophobic gate. Calcula-tions of Born energy profiles for various structures alongthe simulation trajectory suggest that the conformationsof the M2 bundle sampled correspond to a closed confor-mation of the channel. Principal components analyses ofeach of the M2 helices, and of the five-helix M2 bundle,reveal concerted motions that may be relevant to channelfunction.

2.3. Voltage-Driven Gates

KvAP. The functional unit of a voltage-gated channelis an assembly of four proteins, or subunits; in each, thepolypeptide chain snakes back and forth across the mem-brane six times39�40 (Fig. 7). This ‘six-transmembrane’structure is seen in the voltage-gated potassium, sodiumand calcium channel families, and also in other chan-nel types. The resulting X-ray structure39 of KvAP showsthe expected potassium channel (Fig. 7) core, consistingof transmembrane segments S5 to S6, surrounded by S1through to part of S3. What was unexpected is that the S4helix, along with the second part of S3, forms an �-helicalhairpin—a ‘paddle’ that extends out from the channel coreinto the membrane’s fluid interior. The paddle has a flex-ible connection to the rest of the channel, as MacKinnonand coworkers showed by comparison with another crys-tal structure, of segments S1 to S4 alone. This flexibilityexplains the difficulty that the authors encountered in crys-tallizing the protein; it also suggests a mechanism for volt-age sensing. The paddle is a hydrophobic, charged particlethat can move in the membrane interior, transporting itsfour positive charges from one membrane surface to theother. It is the location of S4—not embedded in the proteincore, but loose in the membrane—that is the big surprisehere. It explains an old puzzle, that small lipid-soluble

Fig. 7. Structure of the voltage-gated ion channel KvAP (PDB code1ORQ). The front subunit is omitted to unveil the three ions (green balls).

molecules somehow have ready access to ion-channel volt-age sensors. Such molecules include local anaesthetics,the alkaloid nerve toxins and the well-known insecticidesallethrin and DDT. It is now easy to imagine them diffus-ing up to the voltage-sensor paddle from within the lipidmembrane interior. An X-ray crystal structure is like aposed photograph; in the KvAP crystal, for instance, thevoltage-sensor paddle is held firmly in place by an anti-body scaffold.

Still a few questions have been left to be answered. Theactual conformation of the channel in the membrane willneed to be clarified, because in the crystal the membraneis replaced by a blanket of detergent molecules. Questionsalso remain about the disposition of the amino-terminalend of the protein (thought to be intracellular) and of theloop between the S3 and S4 segments in related chan-nels (in the well-studied Shaker potassium channel, thisloop is always accessible from the outside surface). More-over, details of the motions of the voltage sensor—in somechannels the charge movement occurs in several discretesteps—remain to be worked out, as does the energeticissue of moving the quadruply charged paddle through themembrane interior. Still MD simulations have to be per-formed on this voltage-gated channel in order to elucidatesome of the unsolved problems.

2.4. Light-Driven Gates

Bacteriorhodopsin (bR). The membrane protein bacte-riorhodopsin (bR) serves as a light driven proton pump inbacterial cells. Bacteriorhodopsin is a heptahelical trans-membrane protein (Fig. 8) that is found in extremelyhalophilic archeae like Halobacterium salinarum, and ineubacteria. Photoisomerization of the all-trans retinal chro-mophore, covalently attached to Lys216 through a pro-tonated Schiff base, to the 13-cis, 15-anti configurationinitiates ion translocation across the cell membrane, andestablishes an electrochemical gradient for ATP synthesisand other energy-requiring membrane processes. The cyclicreaction, the “photocycle,” that follows the photoisomer-ization of the retinal produces distinct, spectroscopicallyidentifiable photointermediates, which have been defined inkinetic and spectroscopic terms as BR570, K590, L550, M412,N560, O640. The photoreaction induces a vectorial transferof a proton across the membrane, leading to the release of aproton at the extracellular side and an uptake from the cyto-plasmic side. Our current knowledge of the structure andthe photocycle of bR has been reviewed in detail by severalauthors (see e.g., Ref. [41]). Three decades of site-directedmutagenesis, static and time-resolved spectroscopy, andlow-resolution projection and 3D maps have revealed manydetails of the photocycle. However, understanding howthe energy absorbed by the chromophore leads to unidi-rectional (vectorial) ion transport requires detailed struc-tural descriptions of each photointermediate at the atomic


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(a)

(b)

Fig. 8. Trimeric structure of bacteriorhodopsin, (a) side view and(b) from the top (PDB code 1CWQ).

level. Bacteriorhodopsin was successfully crystallised atvery high resolution using lipidic cubic phase methods.42

But even at highest resolution, X-ray crystallography can-not capture all details, in particular dynamical processesduring the coupling of conformational transitions and pro-ton translocation. Therefore molecular dynamics simula-tions have proven to provide valuable insights. Amongothers, molecular dynamics simulations have been usedto elucidate conformational fluctuations43–46 and bR-watermobility.47–50

Early intermediates of bacteriorhodopsin’s photocy-cle were modeled by means of ab initio quantummechanical/molecular mechanical and molecular dynam-ics simulations.51 The photoisomerization of the retinalchromophore and the formation of photoproducts corre-sponding to the early intermediates were simulated bymolecular dynamics simulations. By means of the quan-tum mechanical/molecular mechanical method, the result-ing structures were refined and the respective excitationenergies were calculated. Two sequential intermediateswere found with absorption maxima that exhibit red shiftsfrom the resting state. The intermediates were thereforeassigned to the K and L states. In K, the conformation ofthe retinal chromophore is strongly deformed, and the NHbond of the Schiff base points almost perpendicular to themembrane normal toward Asp-212. The strongly deformed

conformation of the chromophore and weakened interac-tion of the Schiff base with the surrounding polar groupsare the means by which the absorbed energy is stored. Dur-ing the K-to-L transition, the chromophore undergoes fur-ther conformational changes that result in the formationof a hydrogen bond between the NH group of the Schiffbase and Thr-89 as well as other rearrangements of thehydrogen-bond network in the vicinity of the Schiff base,which are suggested to play a key role in the proton transferprocess in the later phase of the photocycle.

The planarity of the polyene chain of the retinal chro-mophore in bacteriorhodopsin was studied using molecu-lar dynamics simulation techniques and applying differentforce-field parameters and starting crystal structures.52 Thelargest deviations from a planar structure are observed forthe C13 = C14 and C15 = N16 double bonds in the retinalSchiff base structure. The other dihedral angles along thepolyene chain of the chromophore, although having lowertorsional barriers in some cases, do not significantly devi-ate from the planar structure. The results of the simulationsof different mutants of the pigment show that, among thestudied amino acids of the binding pocket, the side chainof Trp-86 has the largest impact on the planarity of reti-nal, and the mutation of this amino acid to alanine leadsto chromophore planarity. Deletion of the methyl C20,removal of a water molecule hydrogen-bonded to H15, ormutation of other amino acids to alanine did not show anysignificant influence on the distortion of the chromophore.The results from this study suggest the importance of thebulky residue of Trp-86 in the isomerization process, inboth ground and excited states of the chromophore, and infine-tuning of the pKa of the retinal protonated Schiff basein bacteriorhodopsin. The dark adaptation of the pigmentand the last step of the bacteriorhodopsin photocycle implylow barriers against the rotation of the double bonds in theSchiff base region. The twisted double bonds found in thepresent study are consistent with the proposed mechanismof these ground state isomerization events.

Although the molecular structure of bR in its groundstate is now well determined, details of intermediate pho-tocycle states are still controversial.42�53 In addition, theamount of buried (“internal”) water molecules in bR whichare assumed to play a decisive role in providing protonpathways and to be involved in the molecular mecha-nism leading to proton translocation, is still unclear. Veryrecently49�50 new details of the amount and the distributionof internal water molecules, and of the related hydrogen-bonded networks in bR that constitute proton pathwayshas been reported. The simulations reveal a much higheraverage number of internal water molecules per monomerthan observed in crystal structures. The spatial distribu-tion of water molecules (Fig. 9) and their correspondinghydrogen-bonded networks inside bacteriorhodpsin in itsground state (G) and late M intermediate conformationswere determined. In addition, these studies49�50 provides


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Fig. 9. Accessible volumes for internal water molecules of G-state bR(left) and M-state bR (right). The surfaces of the volumes for trapped anddiffusive water molecules are represented by yellow and blue triangulatednets, respectively. The red balls represent the positions of trapped watermolecules as identified by crystallographic studies.

important information on hydrogen-bonded networks inbR fluctuating on the ps to ns time scale50 which was notseen in crystallographic studies.

The proton transfer occurring immediately after reti-nal trans-cis photoisomerization is still highly controver-sial. The gradual release of stored energy is inherentlynonequilibrium: which photocycle intermediates are pop-ulated depends not only on their energy but also ontheir interconversion rates. To understand why the pho-tocycle follows a productive (i.e., pumping), rather thansome unproductive, relaxation pathway, it is necessaryto know the relative energy barriers of individual steps.To discriminate between the many proposed scenariosof this process, all the possible minimum-energy pathshave been computed.54 The results show that not one, butthree very different pathways have energy barriers con-sistent with experiment. This result reconciles the con-flicting views held on the mechanism and suggests astrategy by which the protein renders this essential stepresilient. Using quantum mechanical/molecular mechan-ical minimum-energy reaction-path calculations the firstproton transport step following photon absorption in bacte-riorhodopsin, from the 13-cis retinal Schiff base to Asp85,have been investigated.55 The results suggest that the bar-rier may arise from breaking and forming of hydrogenbonds involving the Schiff base, Asp85, Asp212, and waterw402, and from nonbonded interactions involving proteingroups that respond to the charge rearrangements in theSchiff base region.

Data for bR indicate that its permanent dipol is not ofsome interfacial character but is due to a real asymmetryof the charge distribution. Electron-crystallographic refine-ment of bR53 provide a dipole moment of ∼570 D. Itis suggested56 that the permanent dipole of bR supports

proton transport by attraction of protons inside and repul-sion of protons outside of the cell.Halorhodopsin (HR). Anion transport through

biological membranes is less well understood than protontransport. Whereas several anion exchangers have beendiscovered and functionally characterized in eukaryoticand bacterial membranes, only a few anion pumps havebeen reported. To date, halorhodopsin (HR) is the onlyknown chloride pump that is energized by light. HRoccurs in halophilic archaea and translocate into thecell against the electrochemical gradients. Halorhodopsinshows sequence homology to bacteriorhodopsin (BR) andsensory rhodopsins. The structure of HR halorhodopsinwas determined at 1.8 Å,57 a very high resolution for amembrane protein. This study took advantage of the factthat a chloride ion is much easier to determine in thecrystal structure than a proton. A single chloride ion wasfound in the transport site between Lys242 and the retinalchromophore. The protein structure, in combination withenergetic calculations, explained why chloride and protontranslocation modes are mechanistically equivalent inarchaeal rhodopsins.

The mechanism of chloride transfer was determined bycomputing multiple pathways for this process.58 It wasargued that in addition to driving the ionic transport, theprotein must possess ‘valve’ mechanisms that prevent dis-sipation of the ion gradients they create. The calcula-tions reveal two conditions of a ‘valve’ mechanism. In theground state, the structure is transiently opened by chloridepassage, then this activated opening, which is achieved byflexible deformation of the surrounding protein, was shownto significantly raise the chloride translocation barrier dur-ing the photocycles, thus preventing chloride backflow.Unlike macroscopic ‘valve’ designs, the protein allowsdifferential ion flows in the pumping and resting statesthat are tuned to match the physiological timescales ofthe cell, thus creating a ‘kinetic’ valve. In engineering,pump valves often possess a spring, which keeps the valveclosed and whose force must be overcome during pump-ing. In the case of halorhodopsin, the force required toopen the lumen for chloride passage plays the role of thespring. This creates a kinetic valve that slows down chlo-ride backflow in the resting state of the protein, whileallowing chloride transfer during pumping without becom-ing the rate-limiting step of the photocycle. This suggeststhat the protein architecture is designed so as to recon-cile two conflicting objectives: optimizing the productivepumping and preventing unwanted ion leakage.58 Proteinflexibility may play a similar role in other pumps transport-ing sterically demanding ions. A possible indication forthis may be when the crystal structure shows no obviouspassage for the ion. Furthermore, there might be additionalvalves (not necessarily kinetic ones) along the chloridepathway in halorhodopsin.58 The pathway calculations,minimum-energy paths, were computed using the Conju-gated Peak-Refinement (CPR) algorithm59 as implemented


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in the module of the simulation module of CHARMM. Theadvantage of using CPR is that it does not need the defini-tion of an a priori reaction coordinate. With this method,the mechanisms of several complex reactions in proteinshave been determined.

3. SELECTIVITY

3.1. Cooperativity

For every experimentally identified ion channel, there isa well defined ion selection sequence, according to whichchannels are usually named as potassium, sodium, chlorineor calcium channels. It appears that for the monovalent-selective channels the main selection criterion is the sizeof the ion, whereas for the calcium channel it is the mag-nitude of ion charge (calcium channel highly discrimi-nates Na+ over Ca2+, although the radii of Na+ and Ca2+

do not differ much—0.95 vs. 0.99 Å, respectively). Thus,there are dominantely two different selection mechanismsat play; the one operates on the basis of ion size, and theother on the basis of ion charge. The deciding factor onselectivity in channels is the free energy of permeation,namely the variation in free energy of the system as dif-ferent ion species pass through the channel. In the simplecase one can observe the differences in potential energiesat various points along the pore, but for a more quantitativedescription, a free-energy calculations are needed.Aquaporin (AQP1). The extraordinary permeation rate

of 3 billion water molecules per second per singleaquaporin-1 (AQP1) molecule, combined with the strictselectivity for water, have challenged several MD simu-lations in order to elucidate the relation between struc-tural determinants and selectivity in porins.60–64 Bacterialporins, e.g., from the outer membrane of E. Coli allowdiffusion of hydrophilic molecules with molecular weightup to 600 Da and exhibit modest ionic selectivity. Theexistence of membrane water channels was predictedin the 1950s.65 Today the detailed structures of severalhuman66�67 and bacterial68 porins are known.

The nature of the control of water/proton selectivity inbiological channels is a problem of fundamental impor-tance. Most studies have proposed that an interference withthe orientational requirements of the so-called proton wireis the source of selectivity. The elucidation of the struc-tures of aquaporins, which have evolved to prevent protontransfer (PT), provided a clear benchmark for exploringthe selectivity problem. Previous simulations of this sys-tem have not examined, however, the actual issue of PT,but only considered the much simpler task of the transferof water molecules. In a paper of Burykin and Warshel69

aquaporin is taken as a benchmark and the origin of thewater/proton selectivity in this and related systems wasquantified. This was done by evaluating in a consistentway the free energy profile for transferring a proton alongthe channel and relating this profile to the relevant PT rate

constants. It was found that the water/proton selectivityis controlled by the change in solvation free energy uponmoving the charged proton from water to the channel. Thereason for the focus on the elegant concept of the pro-ton wire and the related Grotthuss-type mechanism wasalso considered. It was concluded that these mechanismsare clearly important in cases with flat free energy sur-faces (e.g., in bulk water, in gas phase water chains, and ininfinitely long channels). However, in cases of biologicalchannels, the actual PT mechanism is much less importantthan the energetics of transferring the proton charge fromwater to different regions in the channels.Potassium Channel (KcsA). Another focus of MD sim-

ulations of the KcsA potassium channel has been the selec-tivity filter and the permeation process from the cavity tothe filter. The question of selectivity against Na+ ions hasbeen addressed in several studies through free-energy per-turbation calculations, where a K+ ion in one of the bind-ing sites is alchemically transformed into a Na+ ion. thecalculated free-energy barrier range from 11 kT to 8 kT,70

and 5 kT71 which are in rough agreement with the experi-mental value of 9 kT extracted from the K+/Na+ selectivityratio of about 104. It has been suggested70 that hydropho-bic residues lining the intrapore and cavity are responsiblefor the relative high diffusion of ions in those segments.Despite the large suppression of the diffusion coefficient inthe filter region, permeation through this segment has beenfound by BD simulations to be the fastest step in a fullconduction cycle thanks to the Colomb repulsion. RecentMD simulation of Sansom and coworkers72 indicate thatK+ ions and water molecules within the filter undergo con-certed single-file motion in which they translocate betweenadjacent sites within the filter on a nanosecond time scale.In contrast, Na+ ions remain bound to sites within the fil-ter and do not exhibit translocation. Furthermore, entry ofa K+ ion into the filter from the extracellular mouth isobserved, whereas this does not occur for a Na+ ion. It isargued72 that these differences in interactions in the selec-tivity filter may contribute to the selectivity of KcsA forK+ ions in addition to the differences in the dehydrationenergy between K+ and Na+ and the block of KcsA byinternal Na+ ions.

The selective interactions of various cations, Na+, K+,Rb+, and Cs+, within the selectivity filter of the KcsApotassium channel have been investigated73 via multiplemolecular dynamics simulations (total simulation time,48 ns) based on the high resolution structure of KcsA,embedded in a phospholipid bilayer. As in simulationsbased on a lower resolution structure of KcsA, concertedmotions of ions and water within the filter are seen.Despite the use of a higher resolution structure and theinclusion of four buried water molecules thought to sta-bilize the filter, this region exhibits a significant degreeof flexibility. In particular, pronounced distortion of fil-ter occurs if no ions are present within it. The two most


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readily permeant ions, K+ and Rb+, are similar in theirinteractions with the selectivity filter. In contrast, Na+

ions tend to distort the filter by binding to a ring of fourcarbonyl oxygens. The larger Cs+ ions result in a smalldegree of expansion of the filter relative to the X-ray struc-ture. Cs+ ions also appear to interact differently with thegate region of the channel, showing some tendency tobind within a predominantly hydrophobic pocket. The fourwater molecules buried between the back of the selectiv-ity filter and the reminder of the protein show comparablemobility to the surrounding protein and do not exchangewith water molecules within the filter or the central cav-ity. A preliminary comparison of the use of particle meshEwald versus cutoff protocols for the treatment of long-range electrostatics suggests some difference in the kinet-ics of ion translocation within the filter. More recently, itwas shown by MD simulations74 that the classical view ofa snug structural fit of K+ inside the narrow and rigid poreis not the origin of the ion selectivity seen in potassiumchannels. But rather, the carbonyl groups coordinating theion in the narrow pore (Fig. 10) are very dynamic and thatthier intrinsic electrostatic properties control selectivity.Selectivity is concluded to emerge as a robust feature ofa flexible fluctuating pore lined by carbonyls. The resultsof MD simulations show75�76 that the KcsA channel doesnot select for K+ ions by providing a binding site of anappropriate (fixed) cavity size. Rather, selectivity for K+

arises directly from the intrinsic local physical propertiesof the ligands coordinating the cation in the binding site,and is a robust feature of a pore symmetrically lined bybackbone carbonyl groups. Further analysis reveals that itis the interplay between the attractive ion-ligand (favoring

S0

S1

S2

S3

S4

Sext

Fig. 10. The structure of the KcsA channel in ribbon representation,and the definition of the binding sites Sj in the selectivity filter.

smaller cation) and repulsive ligand–ligand interactions(favoring larger cations) that is the basic element gov-erning Na/K selectivity in flexible protein binding sites.Because the number and the type of ligands coordinatingan ion directly modulate such local interactions, this pro-vides a potent molecular mechanism to achieve and main-tain a high selectivity in protein binding sites despite asignificant conformational flexibility.

3.2. Entropic Traps

The Nuclear Pore. The eukaryotic nucleus is sur-rounded by a protective nuclear envelope, which isperforated by trafficking machines termed nuclear porecomplexes (NPCs). The NPCs are the sole mediators ofexchange between the nucleus and the cytoplasm. Smallmolecules pass through the NPCs unchallenged. How-ever, large macromolecules are excluded unless chaper-oned across by transport factors. Specific carrier molecules,many of which belong to the karyopherin superfamily,mediate transport of cargoes through NPCs and also havean emerging role as regulators of other cellular processes,including mitosis and gene expression.Structure and Function. Very recently the crystal struc-

ture of one carrier molecule, exportin Cse1p complexedwith its cargo (Kap60p) and RanGTP, has been resolved.77

The NPC78 has a 8-fold rotational symmetry perpendic-ular to the membrane, and are asymmetric with respectto the plane of the nuclear envelope. The NPC is com-prised of a cylindrical core from which numerous periph-eral filaments project toward the nucleus and cytoplasm.The complex can be minimally characterized as havingthree substructures: the cytoplasmic fibrils, a central core,and the nuclear basket. The central core in the NPC con-tains eight spokes sandwiched between the cytoplasmicand nuclear rings. The spoke structures collectively encir-cle the central region through which all active transportoccurs. One can picture the NPC as a tubular hole in thenuclear envelope, bristling at each entrance with numerousfilaments carrying a multitude of binding sites for trans-port factors. One of the basic questions is how such asimple structure can mediate all the complexities of gatedtransport. Operationally, NPCs are composed of proteinscalled nucleoporins (or Nups) forming the stationary phasefor nucleocytoplasmic exchange, whereas the mobile phaseconsists of soluble transport factors and their cargoes. Asnucleocytoplasmic transport is driven by a series of spe-cific interactions between components of both phases, it isfrequently difficult to determine which proteins are perma-nent constituents of the NPC. Nevertheless, to understandhow transport occurs, it necessary to characterize the play-ers in both phases and understand how their interplay leadsto the coordinated vectorial exchange of macromoleculesacross the nuclear envelope. Transport involves the recog-nition of nuclear localization signals on cargos destined


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for the nucleus and nuclear export signals on cargos des-tined for the cytoplasm. Most import and export signals arerecognized by the �-karyopherin (kap) family of solubletransport receptor proteins (also known as importins andexportins). Once bound to a cargo, a kap negotiates theNPC, releases its cargo in the destination compartment andreturns for another round. Because this cycle is directionaland can accumulate cargos against a concentration gradi-ent, an energy source and a directional cue are needed.Both are provided by the small GTPase Ran, which iscontrolled by two regulators—Ran guanosine nucleotideexchange factor (RanGEF), and Ran-specific GTPase acti-vating protein (RanGAP). But, how does the NPC selectonly the karyopherin-cargo complexes and exclude othermacromolecules? In other words, how does the NPC actas a sorting machine?Models and Theory. Several functional models have

been proposed. The oily-spaghettimodel,79 the selectivephase model,80–82 the molecular-latch model,83 and the“virtual gating” model.84 The virtual gating model hasbeen suggest84 to explain the mechanism of the rapid andselective macromolecular trafficking through the NPCs.One of the basic assumptions is that the entropic barrier fortransporting macromolcules across the NPC plays a piv-otal role. But how is the entropic barrier set up? Beyondsimple occlusion, other factors could add to the barrierproperties of the NPC. The intrinsically disordered nucle-oporins (or nups) could act as entropic bristles. Moleculesthat are large enough to occupy a significant portion ofthis volume and move on the same timescale as the bris-tles tend to be excluded from this volume. The disorderedfilamentous sidearms of neurofilaments and microtubule-associated proteins act as entropic bristles, whose ‘push’might help to keep the parallel arrays of their associatedfilaments regularly spaced. Similarly, the ‘push’ from thenups could keep macromolecules away from the centralchannel, and the larger the macromolecule, the more itwould feel this push. The appeal of this model is that ituses a well-studied polymer phenomenon consistent withthe reported structure of nups.

Based on the observation that the nuclear pore complexcontains a selective permeability barrier and since exper-iments on the physical properties of this barrier appearto be in conflict with current physical understanding ofthe rheology of reversible gels, the paper by Bickel andBruinsma82 contains the proposal that the NPC gel isanomalous and characterized by connectivity fluctuations.A simplified model is developed to demonstrate the possi-bility of enhanced diffusion constants of macromoleculestrapped in such a gel.

To reconcile the observed selectivity and the high rateof translocation through nuclear pores, Kustanovich andRabin proposed85 that the core of the nuclear pore com-plex is blocked by a metastable network of phenylalanineand glycine nucleoporins. Although the network arrests the

Fig. 11. Schematic diagram of polymer threading to a hole.

unfacilitated passage of objects larger than its mesh size,the cargo-importin complexes act as catalysts that reducethe free energy barrier between the cross-linked and thedissociated states of the nups, and open the network. UsingBrownian dynamics simulations they have calculated thedistribution of passage times through the network for inertparticles and cargo-importin complexes of different sizesand discuss the implications of their results for experi-ments on translocation of proteins through the nuclear porecomplex.Biomimetic Systems. The absorption and insertion of a

polymer into a small pore has been the subject of theoreti-cal considerations.86�87 From a principal point of view thissituation can be modeled by the dynamics of absorbtioninto a narrow potential well (Fig. 11). Although similarto the physics of a driven polymer translocation through amembrane channel,88−90 it has distinct features and ressem-bles more the selectivity process encountered by a proteinentering a membrane pore. The scaling estimates of thecharacteristic time � for the absorbtion process in immo-bile solvents have been proposed.86 The absorption timefor a Rouse chain on the grounds that Rouse time �R∼N 2,where is the length of the chain, is the solitary relevanttime scale, and all other relevant parameters appear in thedimensionless combination

� = �R��fR/kBT � (1)

where the factor � is a function of the pulling force f andthe average size of the unperturbed polymer coil, R∼ N� ,and � is the excluded volume exponent. It was shown86

that the absorbtion process develops as sequential straight-ening of folds of the intial polymer conformation. By esti-mating the typical size of the fold involved into the motion,it was predicted that

� ∼ N 1+�/f (2)


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4. TRANSPORT

4.1. Collective Motions

Membrane channels are proteins, which reside in mem-branes acting as “valves” thereby permitting selectiveor non-slective flow of molecules along their concentra-tion gradients across the membrane. One may distinguishbetween pores and ion channels. In contrast to poresexhibiting weak ion selectivity, ion channels are highlyspecific filters regulating the ion balance of living cells.As far as the molecular structure determination of ionchannels is concerned, obtaining a crystallized functionalform of the membrane protein is still a generally non-solved problem; up to now, only few structures have beenresolved up to the atomic details. Consequently, modellingand simulations of membrane processes has been long fac-ing the problem of lack in structural data, not to mentionthe problem of computational limitations for such complexsystems. In the last few years, the most rapid develop-ment in the field of membrane proteins simulations hasbeen seen specifically in the area of ion channels.91�92 Thedetailed insights into the molecular architecture providedby breakthroughs in determination of crystal structures atthe atomic resolution, as well as advances in computersimulations (thanks to the increased speed and power ofcomputational facilities), have been gained. This is, by andlarge seen in a recent succesfull structural determinationof the bacterial potassium channel KcsA93–96 accompaniedby the computational simulation of the structure,97–102 thatalmost completed the detailed picture of structure-functionrelationship for the potassium ion transport process.

MD simulations provide a valuable tool for studyingmembrane proteins, enabling us to probe their conforma-tional dynamics in both membrane and detergent micelleenvironments.91�92 They are of particular value in enablingus to extrapolate from the essentially static (time- andspace-averaged) structure revealed by X-ray diffraction toa more dynamic picture of the behavior of a single proteinsin an environment mimicking a small patch of the bacte-rial membrane. MD simulations have been employed in anumber of studies, most notably to probe protein and sol-vent dynamics in relationship to permeation mechanismsin porins.

4.1.1. Porins

Some types of bacteria are characterized by the construc-tion of their cell wall: the cytoplasmic or inner mem-brane engulfes the cytoplasm, a second, outer membranesurrounds the cell with few contacts to the inner mem-brane. In between is the periplasm, where some metabolicactivities take place. The exchange of substances betweencytoplasm and periplasm is regulated by highly spe-cific transport systems, usually ion channels, whereas theexchange between periplasm and environment occurs via

outer membrane proteins (OMPs), which may be non orweekly selective for groups of substances. Perhaps thebest-characterized family of OMPs would be the porins.These include both relatively nonspecific general diffu-sion pores and also more specific passive pores (e.g.,for oligosaccharides) across the outer membrane. Thus,they are an important component of the transport prop-erties of the bacterial membrane. Other transport proteinsin outer membranes include those for ferric ions and forvitamin B12, and export pathways for polypeptide toxinsand hydrophobic drugs. In addition to transport proteins,outer membranes include a number of membrane-boundenzymes. Several structures of such enzymes have beendetermined, including those of a protease OmpT and twoOMPs acting on lipid substrates, OMPLA and PagP.Structure. The construction principle of porins is the

same irrespective of their type: a chain of 300–420 aminoacids folds to an antiparallel �-barrel of 16 or 18 strands(Fig. 12). The wall of the pore has a thickness of one aminoacid only. On the side of the barrel facing the periplasm thebeta strands are connected by short loops or turns. On theother side the loops directed to the environment are largerand variable. The loop connecting beta strands 5 and 6 is ofspecial importance: it is folded into the barrel and constrictsthe cross section. At the narrowest point there are someionizable amino acids. The filter properties of the pore aredefined at this point. Porins are inserted in the outer mem-brane as trimers. Amino and carboxy termini of the singlemolecules face the threefold symmetry axis of the complex.As found in other transmembrane proteins there are twobelts of aromatic amino acids pointed to the surfaces of themembrane.

To date, the structures of 20 such OMPs have beensolved by X-ray diffraction and by nuclear magnetic reso-nance. Thus, OMPs provide an opportunity for moleculardynamics simulation studies to explore the conformationaldynamics of a whole family of structurally related mem-brane proteins, to define both common dynamic properties

Fig. 12. Ribbon view of OmpF porin from Rhodobacter capsulatus(PDB entry 2omf).


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and functionally important differences between individualspecies of OMPs.103�104

Function. Porins are tubes with a diameter of about1 nm which are filled with water. Nonspecific porins allowthe diffusion of ions and molecules up to a molecularweight of 600. The diffusion speed depends on both thedifference of concentration in the periplasm and outsideand the molecular weight of the solute. The passing ofions may be regulated electrically. If (in vitro) a volt-age of + or −100 mV is applied, the channel is closedfor ions (voltage gating). This phenomenon is also foundin another class of beta-barrel pores, the toxins. Frommutation experiments it was concluded that there is nomovement of the loop constricting the inside of the chan-nel. Probably the applied voltage changes the electrostaticproperties of the interior wall. A physiological functionof voltage gating is not obvious. Because they are wellcharacterized, both structurally and functionally, the porinsrepresent ideal systems for addressing questions about thefundamental principles underlying ion flow in molecu-lar pores at the molecular level using theoretical models.There have been a number of theoretical studies of porins,focusing on different aspects of porin activity.OmpF. Realistic simulations of porins have been per-

formed with explicit ions, solvent molecules105 and alsophospholipid bilayer membrane.106 To explore the mech-anism of ion conduction, Suenaga et al. simulated OmpFin the presence of an applied transmembrane potential.105

The translocation of a single Na+ through the channelwas observed in 1.3 ns under the influence of a poten-tial of 500 mV. Tieleman and Berendsen106 generated a1 ns MD simulation of an atomic model of OmpF trimerembedded into an explicit phospholipid bilayer membranewith a few explicite counterions (those needed to balancethe total charge of OmpF). Since the MD simulationsare computationally intensive, and yet, the time-scale ofion permeation is significantly longer than can be cur-rently simulated, a number of studies have been usedBrownian dynamics (BD) to explore the ion conductionmechanism.107–109 In this approach, the channel and theions were represented explicitly while the influence of thesurrounding solvent molecules was incorporated implic-itly via a stochastic random force, a friction coefficientdamping the velocity, and some effective potential (gen-erally calculated on the basis of a continuum electrostaticapproximation). A good correlation was achieved betweencalculated single-ion transmission probabilities and exper-imental ion selectivity. Although approximations such asBD and PB calculations are less computationally intensivethan all-atom MD and hence very attractive, their valid-ity, however, is still unknown in the context of molecularpores. For example, it is unclear whether a structurelessdielectric continuum is a valid representation. Further-more, the channel structure is generally kept rigidly fixedduring BD simulations and the coupling to protein fluc-tuations is ignored. To answer some of these questions,

Im and Roux110 have performed molecular dynamics sim-ulations of OmpF porin embedded in an explicit (DMPC)bilayer bathed by a 1 M [KCl] aqueous salt solution110

and have explored the microscopic details of the mecha-nism of ion permeation. The atomic model includes theOmpF trimer, 124 lipids, 13470 water molecules as wellas 231 K+ and 201 Cl−, for a total of 70,693 atoms. Thestructural and dynamical results are in excellent agreementwith the X-ray data. In the narrowest part of the aque-ous pore it was observed that the water molecules aremarkedly oriented perpendicular to the channel axis due tothe strong transversal electrostatic field arising from pore-lining residues. On average the size of the pore was smallerduring the simulation than in the X-ray structure, under-going small fluctuations. Remarkably, it was observed thatK+ and Cl− follow two well-separated average pathwaysspanning over nearly 40 Å along the axis of the pore. Inthe center of the monomer, the two screw-like pathwayshave a left-handed twist, undergoing a counter-clockwiserotation of 180 degree from the extracellular vestibule tothe pore periplasmic side. In the pore, the dynamical dif-fusion constants of the ions are reduced by about 50 %relative to their value in bulk solvent. Analysis of ion sol-vation across the channel reveals that the contributionsfrom the water and the protein are complementary, keep-ing the total solvation number of both ions nearly constant.Analysis suggested that ion–ion pairs play an importantrole. In particular, it is observed that the passage of Cl−

occurs only in the presence of K+ counterions, and iso-lated K+ can move through the channel and permeate ontheir own. The presence of K+ in the pore screens thenegative electrostatic potential arising from OmpF to helpthe translocation of Cl+ by formation of ion pairs. RecentMD simulations111 have been analyzed employing a firstpassage time approach which allows characterizing the dif-fusive properties of a well-defined region of this channel.It was found, in addition to the expected regular behavior,a gradient of the diffusion coefficient at the channel ends,witness of the transition from confinement in the channelto bulk behavior in the connected reservoirs. The overalldiffusive behavior of water in OmpF shows remarkablesimilarity with that in a homogeneous channel. However, asmall fraction of the water molecules appears to be trappedby the protein wall for considerable lengths of time. Thedistribution of trapping times exhibits a broad power lawdistribution ��∼ �−2�4, up to � = 10 ns, a bound set bythe length of the simulation run.OmpT. Molecular dynamics simulations112 have been

performed on the outer membrane protease OmpT embed-ded in a lipid bilayer. It was observed that the �-barrelwas tilted relative to the bilayer plane. The greatest degreeof conformational flexibility was seen in the extracel-lular loops. A complex network of fluctuating H-bondswas detected between the active site residues, which sup-ports a catalytic mechanism. A configuration yielded by


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docking calculations of OmpT simulation snapshots and amodel substrate peptide Ala-Arg-Arg-Ala was used as thestarting point for an extended Hückel calculation on thedocked peptide. The trajectories of water molecules revealexchange of waters between the intracellular face of themembrane and the interior of the barrel but no exchangeat the extracellular mouth. This suggests that the pore-likeregion in the center of OmpT may enable access of waterto the active site from below. The simulations appearedto reveal the presence of specific lipid interaction sites onthe surface of the OmpT barrel, revealing the ability ofextended MD simulations to provide meaningful informa-tion on protein–lipid interactions.OmpA. OmpA is a relatively simple bacterial outer

membrane protein, for which two X-ray structures areavailable. The bacterial outer membrane protein OmpA iscomposed of an N-terminal 171-residue �-barrel domain(OmpA171) that spans the bilayer and a periplasmic,C-terminal domain of unknown structure. OmpA has beensuggested to primarily serve a structural role, as no con-tinuous pore through the center of the barrel can bediscerned in the crystal structure of OmpA171. How-ever, several groups have recorded ionic conductances forbilayer-reconstituted OmpA171. To resolve this apparentparadox Sansom et al.113 have used molecular dynamicssimulations on OmpA171 to explore the conformationaldynamics of the protein, in particular the possibility oftransient formation of a central pore. A total of 19 nsof MD simulations of OmpA171 have been run, and theresults were analyzed in terms of (1) comparative behaviorof OmpA171 in different bilayer and bilayer-mimetic envi-ronments, (2) solvation states of OmpA171, and (3) porecharacteristics in different MD simulations. Significantmobility was observed for residues and water moleculeswithin the �-barrel. A simulation in which putative gateregion side chains of the barrel interior were held in anon-native conformation led to an open pore, with a pre-dicted conductance similar to experimental measurements.The OmpA171 pore has been shown to be somewhat moredynamic than suggested by the crystal structure. A gat-ing mechanism was proposed to explain its documentedchannel properties.FhuA. One of the more complex members of the OMPs

is FhuA. Its primary function is to provide a bindingsite on the outer membrane surface for siderophores,such as ferrichrome, and subsequently to facilitate theirenergy-dependent transport across the membrane, presum-ably powered by the TonB-ExbBD protein complex thatresides in the cytoplasmic membrane. Crystal structuresof FhuA with and without a bound ferrichrome moleculehave provided some clues as to the initial stages of thesiderophore transport mechanism. Sansom et al.114 haveperformed molecular dynamics simulations of FhuA andof the FhuA-ferrichrome complex, both embedded in a

phospholipid bilayer, to probe the short timescale dynam-ics of this integral membrane protein, and to explore pos-sible mechanistic implications of this dynamic behavior.Analysis of the dynamics of the protein suggests that theextracellular loops move as relatively rigid entities rela-tive to the transmembrane �-barrel. Comparison of the twosimulations (with and without bound ferrichrome) revealedsome ligand-induced changes in loop mobility. These sim-ulations support the proposal that binding of ferrichromeinitiates a signaling mechanism that ultimately leads tothe TonB-mediated partial or total removal of the coredomain from the �-barrel, thus opening up a permeablepore. These simulations are among the longest that havebeen performed on a complex membrane protein. How-ever, a simple analysis of sampling reveals that the descrip-tion of protein motions is far from complete.Aquaporin �AQP1�. The extraordinary permeation rate

of 3 billion water molecules per second per singleaquaporin-1 (AQP1) molecule, combined with the strictselectivity for water, have challenged several MD simula-tions in order to elucidate the relation between structuraldeterminants and selectivity in porins.60–64 Bacterial porins,e.g., from the outer membrane of E. coli allow diffusion ofhydrophilic molecules with molecular weight up to 600 Daand exhibit modest ionic selectivity. The existence of mem-brane water channels was predicted in the 1950s.65 Todaythe detailed structures of several human66�67 and bacterial68

porins are known.The aqueous pore and the mechanism of ion conduc-

tion have been studied by MD simulations with explicitions and solvent molecules115 and also with a lipid bilayermembrane.106 The complete translocation of a single Na+

ion in the OmpF under a potential of 500 mV was observedin 1.3 ns115 The simulation106 revealed that a strong electricfield oriented transverse to the pore axis influences the iontransport. The flow of one ion through various bacterialporins (OmpF, PhoE, OmpK36 and mutants thereof) hasbeen studied using BD simulations107–109 which showed agood agreement between calculated transmission probabil-ities and experimental ion selectivity.

The MD simulations of the protein tetramer for severalnanoseconds62 reveal that the water molecules are stronglyoriented in the channel interior with their dipoles rotat-ing by about 180� during permeation. The dipoles arealigned with the electric field which originates from thedipoles of two specific helices, HB and HE, of the pro-tein. Hydrogen competition between water molecules anda few polar groups in the pore was found to dominate thepermeation process. During the course of a 10 ns simu-lation, a total of 16 such permeation events are observed,a result that is in good agreement with the very high rateof permeation known from experimental work. Hydrogenbond statistics for water molecules show that there are twomajor interaction sites inside the channel: the Asn-Pro-Ala


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(NPA) region and the aromatic-arginine (Ar/R) constric-tion region. The two highest enthalpic barriers to the pas-sage of water molecules are located directly adjacent toNPA region. this together with the water rotation that alsooccurs here, suggest that the NPA region is a major selec-tivity filter. Porins are often designed such that protonconduction is strictly prevented in order to maintain theproton gradient across the cell membrane that serves as amajor energy storage mechanism. In the simulations, fre-quent simultaneous hydrogen bonding of water moleculesto the two NPA asparagines has been observed, therebyweakening interactions among adjacent water molecule inthe pore. As contiguous hydrogen bonded water chains areknown to be efficient proton conductors, it was suggestedthat this region is the main proton filter.63

Water permeation and electrostatic interactions betweenwater and channel have been investigated64 in theEscherichia coli glycerol uptake facilitator GlpF, a mem-ber of the aquaporin water channel family, by moleculardynamics simulations. A tetrameric model of the chan-nel embedded in a lipid bilayer membrane was used forthe simulations. During the simulations, water moleculespass through the channel in single file. The movementof the single file water molecules through the channelis concerted, and it was shown that it can be describedby a continuous-time random-walk model. The integrityof the single file remains intact during the permeation,indicating that a disrupted water chain is unlikely to bethe mechanism of proton exclusion in aquaporins. Spe-cific hydrogen bonds between permeating water and pro-tein at the channel center, together with the protein elec-trostatic fields enforce a bipolar water configuration insidethe channel with dipole inversion at the NPA motifs. At theNPA motifs water–protein electrostatic interactions facili-tate this inversion. Furthermore, water–water electrostaticinteractions are in all regions inside the channel strongerthan water–protein interactions, except near a conserved,positively charged Arg residue. It was found that varia-tions of the protein electrostatic field through the channel,owing to preserved structural features, completely explainthe bipolar orientation of water. This orientation persistsdespite water translocation in single file and blocks pro-ton transport. Furthermore, It was found that for perme-ation of a cation, ion–protein electrostatic interactions aremore unfavorable at the conserved NPA motifs than at theconserved Arg, suggesting that the major barrier againstproton transport in aquaporins is faced at the NPA motifs.

The difference between osmotic permeability (pf) anddiffusion permeability (pd) of single-file water channelshas been studied by Schulten et al.116 They have demon-strated that the pf/pd ratio corresponds to the number ofeffective steps a water molecule needs to take to permeatea channel. While pd can be directly obtained from equi-librium molecular dynamics simulations, pf can be bestdetermined from simulations in which a chemical potential

difference of water has been established on the two sidesof the channel. In light of this, they suggested a methodto induce in molecular dynamics simulations a hydrostaticpressure difference across the membrane, from which pfcan be measured. Simulations using this method were per-formed on aquaporin-1 channels in a lipid bilayer. Thepf/pd ratio is explained in terms of channel architectureand conduction mechanism.Glycerolporin Facilitator (GlpF). Aquaporins consti-

tute a class of membrane proteins with more than150 members, including a glycerol-conducting subclass(aquaglyceroporins). The E. coli glycerol facilitator, GlpF,selectively conducts glycerol and water, excluding ionsand charged solutes. The detailed mechanism of the glyc-erol conduction and its relationship to the characteristicsecondary structure of aquaporins and to the Asn-Pro-Ala(NPA) motifs in the center of the channel are unknown.The X-ray structure of the Escherichia coli glycerol facil-itator, GlpF, a homotetramer highly similar to AQP1, wasreported at a resolution of 2.2 Å.117 A complete con-duction of glycerol through the GlpF was deduced frommolecular dynamics simulations,60 and key residues facili-tating the conduction were identified. The nonhelical partsof the two half-membrane-spanning segments expose car-bonyl groups towards the channel interior, establishing acurve-linear pathway. The conformational stability of theNPA motifs is important in the conduction and criticalfor selectivity. Water and glycerol compete in a randommanner for hydrogen bonding sites in the protein, andtheir translocations in single file are correlated. The sug-gested conduction mechanism should apply to the wholefamily.Maltoporin. Gram-negative bacteria are enclosed by an

inner and an outer membrane. Whereas transport throughthe inner membrane is accomplished by active transporters,the passage across the outer membrane occurs by pas-sive diffusion through porin channels. Nonspecific porins(e.g., the matrix porin OmpF) allow the uptake of ionsand small molecules. Certain larger solutes are translo-cated by solute-specific porins such as maltoporin which ispermeable to maltose and longer maltodextrin sugars. Thebinding affinity of maltodextrins to maltoporin increaseswith increasing chain length, while the permeation ratedecreases for longer maltodextrins. There is a high energycost associated with desolvating a polar sugar and cross-ing the nonpolar interior of a membrane. Maltoporin pro-vides the glucosyl units with tight binding sites within thepore. This lowers the energy of the sugar halfway acrossthe membrane, thus enhancing the local maltodextrin con-centration, and allows for substrate specificity by exclud-ing larger or less elongated compounds. In facilitating thetranslocation, maltoporin has to reconcile two conflictingobjectives: (1) tightly binding the sugar in the pore and(2) facilitating the motion of the sugar chain through thepore. In a recent study,118 it was shown how maltoporin


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achieves this by the creation of a delocalized binding site,which allows relatively free diffusion of the maltodextrinchain along the pore while maintaining strong interactions.The minimum-energy path of maltohexaose translocationwas obtained by the conjugate peak refinement method,which optimizes a continuous string of conformers withoutapplying constraints. This reveals that the protein is pas-sive while the sugar glides screw-like along the aromaticlane. Near instant switching of sugar hydroxyl H bondpartners results in two small energy barriers (of 4 kcal/moleach) during register shift by one glucosyl unit, in agree-ment with a kinetic analysis of experimental dissociationrates for varying sugar chain lengths. Thus, maltoporinfunctions like an efficient translocation enzyme and theslow rate of the register shift is due to high collisionalfriction.

4.1.2. Ion Channels

The membrane lipid bilayer is highly impermeable to ions,since the energy barrier for transferring a hydrated ionto the low dielectric environment of acyl chain region isprohibitively large. Nevertheless, every cell has to have apathway for ionic flow through the membrane, in orderto maintain proper physiological functioning. Ion channelsare transmembrane protein structures119�120 which ensurethe continuous transport pathway for diffusive flow of ionsacross the membrane. Since the net flow of electric chargesgives rise to a rapid change in transmembrane potential,ion channels play a key role in generating and propa-gating action potentials in nervous system. Moreover, itis well established that improper ion channel function-ing leads to a number of pathological conditions, therebypresenting those structures as the primary targets for phar-macological use. Ion channels are functionally and struc-turally simplest among the membrane transport proteins,since the transport itself doesn’t call for a (major) con-formational changes of protein structure. Conformationalchanges occur in the process of gating, i.e., are neededfor discrimination between opened (permeable to selectedions) and closed (impermeable to all ions present) channelstates. The tranformation is energy dependent. In contrastto pores, which exhibit weak ion selectivity, ion channelshave highly specific filters, which regulate the ion balanceof living cells.

From a general point of view with respect to nanotech-nology, voltage-gated ion channels are of particular inter-est. They control electrical activity in nerve, muscle andmany other cell types. The crystal structure of a bacte-rial voltage-gated channel reveals the astonishingly sim-ple design of its voltage sensor. The voltage-gated ionchannels, residing in membranes of living cells, from bac-teria to humans, behave like field-effect transistors. Intransistors, the flow of electrons through a semiconductor‘channel’ is governed by the voltage applied to a ‘gate’

0

0.2

0.4

0.6

0.8

1.0

−150 −100 −50 0 50

Membrane voltage (mV)

Con

duct

ance

Transistor

K+ channel

Fig. 13. Voltage sensing in a potassium ion channel. The control ofion flow through voltage-gated channels is very sensitive to the voltageacross the cell membrane. By comparison, an electronic device such asa transistor is much less sensitive to applied voltage.

electrode. As voltage-sensing devices, these channels canperform much better than their electronic counterparts(Fig. 13). Their high sensitivity to voltage is important,because cellular voltage changes are small.

With the protein equivalents, voltage-gated ion channels,an appropriate voltage, imposed across the cell membrane,causes the channels to open and allows a current of ions tocross the membrane. The molecular structures within ionchannels that sense the membrane voltage have remainedobscure for the 50 years since Hodgkin and Huxleyfirst described121 their function. The voltage sensors haveat last been made visible, in the X-ray structure of apotassium ion channel by MacKinnon and cowrkers39�40

and a hypothesis for voltage-sensor motion has beensuggested.

One of the most famous examples of an ion chan-nel is the potassium specific channel from Streptomyceslividans KcsA, which is regulated by pH. Because ofsequence homologies and similar physiologic propertiesit may be regarded as a prototype for an ion channel.Another class of ion channels, which reside mostly in thecytoplasmic membranes of bacteria, exhibit a significantresponse to mechanical stress. Upon changes of pressurethese mechanosensitive channels alter the diameter of thepores without discrimination of the kind of ions passingthrough. One outstanding example is McsL which is dis-cussed below.

In order to assure the precisely controlled transmem-brane ionic flow, ion channels have to fulfill three specificpoints;(i) high selectivity towards specific ion type (or have tohighly discriminate one ion type over all the other ionspresent),(ii) permeation at the diffusion rate (or to provide an ener-getically barrierless pathway which ensures the rapid trans-port of selected ions), and


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(iii) well defined gating control mechanism, in which theconformation of the open state has to be acomplished uponspecific stimulus; the known mechanisms include changein transmembrane potential (voltage-gated ion channels),binding of another molecule (ligand-gated or receptor-gated channels) and mechanical stress (mechanosensitivechannels or stretch-gated channels).

At present, no single computational technique candescribe all the functional properties of an ion chan-nel structure. The choice of the level for the theoreti-cal approach and the use of the computational techniqueto describe processes of selectivity, permeation and gat-ing of an ion channel, depend essentially on the timescaleof the process itself. While the timescale of the perme-ation process for the typical channel is in the order of fewtenths of nanoseconds, the gating takes the time in theorder of milliseconds. The present state of computationalspeed does not permit the completion of neither of theprocesses using MD techniques, calling for more coarse-grained modelling. On the other hand, BD and contin-uum theories do not distinguish between monovalent ions,so the modelling of channel selectivity appeals for theMD approach. Although Brownian dynamics (BD) sim-ulations have provided usefull data on the total perme-ation process of ions through channels,122 their reliabilityconcerning details is limited due to the coarse-grainingof the channel structure and applications of mean-fieldapproximations of water and lipids. More details, includ-ing fluctuations of all constituents of the channel and itsenvironment (all atom model of channel, water and lipids)have been elucidated by use of molecular dynamics (MD)simulations91�123 albeit quite limited in time to the fewnanoseconds time regime.Structure. In order to support the reliability to the

results of MD simulations of a full all-atom model of anion channel (including the channel, the water and the lipidbilayer), it is important to carefully prepare the startingconfiguration. In particular, the embedding of the crystalstructure into a pre-equilibrated membrane-water system isa non-trivial task, since the insertion of the structure intoa prepared cavity in the membrane-water system leavesmany local vacancies between the two systems. A toorapid subsequent equilibration renders the channel struc-ture to unfold. The typical r.m.s. deviation between theoriginal and this type of equilibrated structure can be ofthe order of few Å. Obviously, this type of modified struc-ture may not provide correct insights into permeation andselectivity mechanism of the channel. Therefore a com-parison between the crystal structure and the simulation-mediated equilibrated structure is important.Permeation. The goal of modelling the permeation pro-

cess is to understand the physical processes underlyingthe permeation of ions through the channel, with the abil-ity to reproduce present and predict future experimen-tal observations. The time scale of permeation is too

long to enable reproducing the experimental data on theprocess using MD calculations, so the modelling whichattempts to reproduce experimental data usually employsmore coarse-grained methods like continuum electrostaticsand/or BD simulations. Nevertheless, among the compu-tational methodologies used on describing the permeationprocess, only molecular dynamics has the advantage ofproviding the data on the molecular kinetic details at theatomic resolution. However, none of the simulation stud-ies with explicit lipid environment was able to completethe permeation process for the K+ ion within KcsA chan-nel, due to the current computational limitations for sim-ulations of such a large systems. The permeation processdeals with an open state conformation of the protein; thechallenge for the theoretical approach is to explain thechannel’s ability to conduct the ion movement at very highrate and relate it to the detailed molecular structure of thesystem. In developing the realistic models of ion perme-ation, one is faced with remarkable complexities that needto be considered: (1) structure determinants of channel-mediated ion transport, i.e., the geometry of the protein–water interface along the permeation pathway, and (2) thedistribution of charges in the protein wall, which determineion–protein interactions along the pathway. Although theprecise geometry of ion-transport pathway varies amongdifferent channels, there are two characteristic regions thatappear to be a general feature of all biological channels: awider cavity that accomodates hydrated ion and short andnarrow selectivity filter. Those two regions seem to be evo-lutionary conserved, at least in specific channel type (e.g.,potassium channels). The mechanism of ion permeationthrough the channel is generally determined by interac-tions of permeating ion with the channel wall and ion–ion interaction within the channel pore. Additionally, theinteractions are modified with the physical determinants ofwater molecules within the narrow channel structure, thatcan be very much different from the bulk water outsidethe protein structure.

The next paragraphs deal with some results obtainedon specific ion channel structures and the simulation tech-niques applied to resolve their specific features.Alamethicin. Alamethicin is an antimicrobial peptide

that forms stable channels with well-defined conduc-tance levels. Sansom and coworkers124 have used extendedmolecular dynamics simulations of alamethicin bundlesconsisting of 4, 5, 6, 7, and 8 helices in a POPC bilayerto evaluate and analyse channel models and to link themodels to the experimentally measured conductance lev-els. Their results suggest that four helices do not forma stable water-filled channel and might not even form astable intermediate. The lowest measurable conductancelevel is likely to correspond to the pentamer. At higheraggregation numbers the bundles become less symmetri-cal. Water properties inside the different-sized bundles are


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similar. The hexamer is the most stable model with a sta-bility comparable with simulations based on crystal struc-tures. The simulation was extended from 4 to 20 ns orseveral times the mean passage time of an ion. Essentialdynamics analyses were used to test the hypothesis thatcorrelated motions of the helical bundles account for high-frequency noise observed in open channel measurements.In a 20-ns simulation of a hexameric alamethicin bundle,the main motions are those of individual helices, not ofthe bundle as a whole. A detailed comparison of simu-lations using different methods to treat long-range elec-trostatic interactions (a twin range cutoff, Particle MeshEwald, and a twin range cutoff combined with a reactionfield correction) shows that water orientation inside thealamethicin channels is sensitive to the algorithms used.In all cases, water ordering due to the protein structureis strong, although the exact profile changes somewhat.Adding an extra 4-nm layer of water only changes thewater ordering slightly in the case of particle mesh Ewald,suggesting that periodicity artifacts for this system are notserious. Subsequent studies of the same group of scien-tists have performed MD simulatrions on alamethicin K18,which is a covalently linked alamethicin dimer in whichthe glutamine residue at position 18 in each helix has beenreplaced by a lysine residue. As described in their pre-vious work, channels formed by this peptide show pH-dependent selectivity. The maximum anion selectivity ofthe putative octameric conducting state is obtained at pH 7or lower. Inasmuch as no change in selectivity is seenbetween pH 7 and pH 3, and because protons are expectedto be in equilibrium with the open state of the channelduring a selectivity measurement, the channel is believedto be fully charged (i.e., all eight lysines protonated) atpH 7. In an effort to understand how such a highly chargedchannel structure is stable in membranes and why it is notmore selective for anions, Sansom and coworkers125 haveperformed a number of computer simulations of the sys-tem. Molecular dynamics simulations of 10 ns each of theoctameric bundle in a lipid bilayer environment were pre-sented, with either zero, four, or eight lysines charged inthe absence of salt, and with eight lysines charged in thepresence of 0.5 M and 1 M KCl. When no salt is presentand all lysines are charged, on average 1.9 Cl− ions areinside the channel and the channel significantly deforms.With 0.5 M KCl present, 2.9 Cl− ions are inside the chan-nel. With 1 M KCl present, four Cl− ions are presentand the channel maintains a regular structure. Poisson-Boltzmann calculations on models of the octameric chan-nel also predict an average of 2–4 Cl− ions near the lysineresidues as a function of ionic strength. These counteri-ons lower the apparent charge of the channel, which mayunderlie the decrease in selectivity observed experimen-tally with increasing salt concentrations. They suggestedthat to increase the selectivity of Alm K18 channels, posi-tive charges should be engineered in a narrower part of thechannel.

Chloride Channels. The ClC family of chloride chan-nels is present in virtually all tissues and organisms,and is widely expressed in most mammalian cells.126

It is now recognized that chloride channels regulate avariety of important physiological and cellular functions.Chloride-conducting proteins are vital for regulating pH,cell volume, electrical impulses, transport of salts acrosscells, and voltage stabilization of excitable muscle cells.126

Mutations in ClC channels cause diseases. The originalstructures127�128 permit theoretical study of chloride perme-ation, and determination of the bacterial systems’ curvi-linear ionic pathways. Even though the bacterial systemsare transporters, not channels, the prokaryotes share sig-nature sequence identities with their eukaryotic relatives.Consequently such atomic level analyses might help inunderstanding ion transport in eukaryotes and ClC chan-nel function generally. Despite a wealth of experimentaldata, major issues need to be resolved at the atomic level.What is the exact chloride pathway? Which amino acidscoordinate the translocating ion and what are their roles?What is the effect of charged amino acids located withinthe pore mouths on the anion translocation? Identificationof the role of pore-lining amino acids is important forunderstanding both conduction and gating, as mutationsof charged residues believed to be chloride-coordinatingaffect both these properties simultaneously.

Jordan and coworkers129 have determined the lowestenergy curvilinear pathway, identified anion-coordinatingamino acids, and calculated the electrostatic potentialenergy profiles. They have studied the effect of mutatingthe charge of some strongly conserved pore-lining aminoacids on the electrostatic potential energy profiles. Whena certain residue (E148) is neutralized, it creates an elec-trostatic trap, binding the ion near midmembrane. Thissuggests a possible electrostatic mechanism for control-ling anion flow: neutralize E148, displace the side chainof E148 from the pore pathway to relieve the steric bar-rier, then trap the anion at midmembrane, and finally eitherdeprotonate E148 and block the pore (pore closure) orbring a second Cl ion into the pore to promote anion flow(pore conductance). The authors have developed a newcomputational approach, the Monte Carlo ion channel pro-teins (MCICP) method, to simulating water–protein, ion–protein, and protein–protein interactions, for applicationto the study of permeation and gating in ion and waterchannels. It exploits the Metropolis Monte Carlo methodand kinetic Monte Carlo techniques. The implementationof the computational model for ClC Cl− crystal structureis based on various assumptions. Simulations are based onthe four crystallographic X-ray structures of bacterial ClCCl− assemblies: StClC (Protein Data Bank (PDB) entry1KPL), EcClC (PDB entry 1OTS), and two EcClC mutants(PDB entries 1OTT).

Schulten and coworkers130 have investigate Cl− conduc-tion by means of an all-atom molecular dynamics sim-ulation of the ClC channel in a membrane environment.


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Based on their simulation results, they proposed a king-of-the-hill mechanism for permeation, in which a loneion bound to the center of the ClC pore is pushed outby a second ion that enters the pore and takes its place.Although the energy required to extract the single centralion from the pore is enormous, by resorting to this two-ionprocess, the largest free energy barrier for conduction isreduced. At the narrowest part of the pore, residues Tyr-445 and Ser-107 stabilize the central ion. There, the boundion blocks the pore, disrupting the formation of a continu-ous water file that could leak protons, possibly preventingthe passage of uncharged solutes.Gramicidin. Gramicidin A (gA), an antibiotic from

Bacillus Brevis, has remained for almost twenty years theonly channel with a known structure, therefore presentingthe favourite object for theoretical modelling. From physi-ological, as well as structural and functional point of view,it is not considered a biological channel. It is a 15-residuepeptide (Fig. 14), which forms a head-to-head dimer in themembrane, assuming a form of a 25 Å long cylinder witha central pore of 2 Å radius. The structure is permeant tomonovalent cations (H+, Li+, Na+, K+, Rb+, Cs+), bindsdivalent cations and rejects all anions. The featureless poreof gA is lined only by peptide backbone atoms, with sym-metrical binding sites of moderate affinity near the entryand the exit. Alkali metal cation transport is considered,in simplest terms, to be carried out in three steps: two pri-marily extrachannel steps (diffusion through bulk aqueoussolution up to the channel combined with entry into thebinding site and exit from the exit site followed by aqueous

Fig. 14. Structure of gramicidin A dimer. The cylindrical pore containsa file of water molecules (H and O atoms in white and red) surroundedby peptide bonds (yellow) (PDB code 1MAG). The lipids are depictedas gray bonds, the K+ and Cl− ions as green and gray spheres.

diffusion away from the channel) and one intrachannelstep (usually referred to as a cation translocation). The rateof the intrachannel step strongly depends on membranepotential, causing current–voltage relationships (I–V s) tobe superlinear when the intrachannel step is rate limiting.Thus I–V s are sublinear at low permeant ion concentra-tions, where the extrachannel entry process is rate limiting,and become more superlinear with higher concentrationsof permeating ion.131 Detailed studies on different gram-icidin structures have been performed using MD simula-tions recently132 indicating the great utility of MD studiesin the analysis and interpretation of solid-state NMR data.Earlier simulations of a gramicidin channel inserted intoa fluid phase DMPC bilayer with 100 lipid moleculeswith constant surface tension boundary was reported byJakobsson and coworkers.133�134 Among other results, itwas found that hydrocarbon chains of lipids adjacent tothe channel had higher-order parameters than those fartheraway. The thickness of the lipid bilayer immediately adja-cent to the channel was greater than it was farther away.The thickness of the hydrocarbon region adjacent to thegramicidin was much thicker than what other studies haveidentified as the ‘hydrophobic length’ of the gramicidinchannel.Permeation. The permeation properties of gA chan-

nel seem to be also very dependent on the nature ofthe lipid environment (which is less likely to be thecase for much larger, biological channels). Some inter-esting results appeared on effects of membrane electro-statics on gA conductance properties. The data providedthrough experiments135�136 were, to a large extent, con-firmed by continuum theory studies (3D Poisson-Nernst-Planck calculations).137 The calculated large changes inconductance of the channel (e.g., ionic current throughgA is six fold larger in charged phosphatidylserine mem-brane than in a phosphatidyilcholine membrane) causedby changed properties of the membrane (including lipidcharges and dipoles, as well as membrane thickness) wereconsistent with experimental results.

By use of the formal correspondence between diffu-sion and random walk, the permeability for water throughthe gramicidin channel was calculated,134 and was foundto agree closely with the experimental results for per-meation of water through gramicidin in a phospholipidmembrane. By using fluctuation analysis, components ofresistance to permeation were computed for movementthrough the channel interior, for the transition step at thechannel mouth where the water molecule solvation envi-ronment changes, and for the process of diffusion up tothe channel mouth. The majority of the resistance to per-meation appears to occur in the transition step at thechannel mouth. A significant amount is also due to struc-turally based free energy barriers within the channel. Onlysmall amounts are due to local friction within the chan-nel or to diffusive resistance for approaching the channelmouth.


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Among other results obtained from simulations of gA,also some important aspects of H+ transport were revealed.Generally, proton fluxes across membranes constitute oneof the most fundamental aspects of cell physiology, sincethe passive flow of H+ through H+ conducting channelsdrives ATP-synthesis. The non-equilibrium electrochemi-cal potential for protons is achieved by H+ pumps (e.g., inbacteria, via bacteriorhodopsin, later discussed in the text)which utilize different forms of energy (from photochem-ical or redox reactions) to perform the uphill transport.Proton transport is postulated to occur via mechanismsimilar to one for salt-based conductivity suggested in1805 by von Grotthuss138 and therefore termed Grotthussconductance. Within aqueous channels, proton transportoccurs as covalent bonds exchange with hydrogen bondsbetween hydronium and a neighboring water to produce acharge transport, which is followed by a slower reorienta-tion of the electroneutral water molecule. Ion-free gram-icidin channels contain eight or nine water molecules ina single file131 which are free to interact and rotate. Thewater molecules thus form an ideal “proton wire,” withfully aligned dipole moments as the lowest energy confor-mation. Channel permeability to protons is 40- to 60-foldhigher than permeability to Na+, and single-channel protonconductance at infinite dilution is 9.1-fold higher than Na+

conductance. Water flux is not induced by proton-mediatedcurrents, as it is with alkali metal currents thus suggest-ing that H+ transport through the gA channel occurs bymeans of Grotthuss conductance.131 A theoretical anal-ysis of proton transport in a nine-water wire in vacuoemphasizes the difference between the time scales for pro-ton passage (<ps) and the subsequent water reorienta-tion (>ns).139 Positive charge transport first requires that a(hydrated) proton approach a channel containing a watercolumn aligned with water oxygens toward the channelentry. After rapid exchange of hydrogen bonds and cova-lent bonds, which results in the release of a hydrogenat the channel exit, the water column becomes alignedin the opposite direction and must completely reorientbefore the next proton transport can occur. Reorientationis expected to be rate-limiting and occurs by propagationof a hydrogen-bonding defect through the channel. Fur-thermore, it was found139 that the mobility of H+ in thegramicidin channel is essentially determined by the finestructure and the dynamic fluctuations of the hydrogen-bonded network. The process of H+ permeation is medi-ated by thermal fluctuations in the relative positions ofoxygen atoms in the wire. When permeating proton is notpresent, the water chain adopts one of the two polarizedconfigurations, each corresponding to an oriented donor-acceptor H-bond pattern along the channel.140 As the watermolecules are ubiquitous in biological systems and formmodulable H-bonded networks, it is expected that similarfeatures in the coordination of these networks apply also toall the membrane proteins that provide efficient pathwaysfor H+ transport.

Roux and coworkers141 have developed a model for pro-ton conduction through gramicidin. based on moleculardynamics simulations. The transport of a single protonthrough the gramicidin pore is described by a potential ofmean force and diffusion coefficient obtained from molec-ular dynamics. In addition, the model incorporates thedynamics of a defect in the hydrogen bonding structure ofpore waters without an excess proton. Proton entrance andexit were not simulated by molecular dynamics. The singleproton conduction model includes a simple representationof these processes that involves three free parameters.

Multinanosecond molecular dynamics simulations ofgramicidin A embedded in a dimyristoylphosphatidyl-choline bilayer have been performed by Grubmülleret al.142 Water permeability was found to be much higherin the double helical conformation, which was explainedby lower hydrogen bond-mediated enthalpic barriers atthe channel entrance and its larger pore size. Free-energyperturbation calculations indicated that the double helicalstructure is stabilized by the positive charges at the N ter-mini, whereas the helical dimer is destabilized. Togetherwith the experimental observation that desformyl grami-cidin conducts water hundredfold better than gramicidin,this suggests that desformyl gramicidin A predominantlyoccurs in the double helical conformation.

In a recent study Chung, Kuyucak and coworkers143

used the well-known structural and functional propertiesof the gramicidin A channel to test the appropriateness offorce fields commonly used in molecular dynamics (MD)simulations of ion channels. For this purpose, the high-resolution structure of the gramicidin A dimer was embed-ded in a POPC bilayer, and the potential of mean force ofa K+ ion was calculated along the channel axis using theumbrella sampling method. Calculations were performedusing two of the most common force fields in MD sim-ulations: CHARMM and GROMACS. Both force fieldslead to large central barriers for ion permeation, that aresubstantially higher than those deduced from the physio-logical data by inverse methods. In long MD simulationslasting over 60 ns, several ions were observed to enter thebinding site but none of them crossed the channel despitethe presence of a large driving field. The present results,taken together with many earlier studies, highlighted theshortcomings of the standard force fields used in MD sim-ulations of ion channels and calls for construction of moreappropriate force fields for this purpose.Glutamate Channels GluR. Fast synaptic transmission

between nerve cells in mammals is carried out predom-inantly by ionotropic glutamate receptors (iGluR). Thesereceptors are a family of ligand-gated ion channels thatopen in response to the binding of glutamate. Glutamateis released presynaptically and binds to a post-synapticreceptor gating a cation-selective channel, thus depolar-izing the post-synaptic cell. Although glutamate is thenatural ligand, the various iGluRs identified by sequence


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comparisons may also be classified in terms of their ago-nist pharmacology. For example, those receptors that showgreatest sensitivity to the agonist AMPA are termed AMPAreceptors (GluR2), those that show greatest sensitivity tokainate are referred to as kainate receptors (GluR5-7).In all of these GluRs the agonist/antagonist binding siteis located within the extracellular region of the protein.Recently Sansom and coworkers144 have used multiplemolecular dynamics simulations of 2–5 ns duration toexplore the structural dynamics of GluR2 in the presenceand absence of glutamate and in a complex with kainate.Their studies indicate that not only is the degree of domainclosure dependent upon interactions with the ligand, butalso that protein/ligand interactions influence the motion ofthe S2 domain with respect to S1. Differences in domainmobility between the three states (apo-S1S2, glutamate-bound, and kainate-bound) are surprisingly clear-cut. Theydiscuss how these changes in dynamics may provide anexplanation relating the mechanism of transmission of theagonist-binding event to channel opening. They also showhow the glutamate may adopt an alternative mode of bind-ing not seen in the X-ray structure, which involves a keythreonine (T480) side chain flipping into a new conforma-tion. This new conformation results in an altered patternof hydrogen bonding at the agonist-binding site.

Excitatory synaptic transmission is mediated by iono-tropic glutamate receptors (iGluRs) through the inducedtransient opening of transmembrane ion channels. Thethree-dimensional structure of the extracellular ligand-binding core of iGluRs shares the overall features of bacte-rial periplasmic binding proteins (PBPs). In both familiesof proteins, the ligand-binding site is arranged in twodomains separated by a cleft and connected by two pep-tide stretches. PBPs undergo a typical hinge motion of thetwo domains associated with ligand binding that leads toa conformational change from an open to a closed form.The common architecture suggests a similar closing mech-anism in the ligand-binding core of iGluRs induced by thebinding of specific agonists. Starting from the experimen-tally determined kainate-bound closed form of the S1S2GluR2 construct, Mendieta et al.145 have studied by meansof molecular dynamics simulations the opening motion ofthe ligand-binding core in the presence and in the absenceof both glutamate and kainate. Their results suggest thatthe opening/closing interdomain hinge motions are cou-pled to conformational changes in the insertion region ofthe transmembrane segments. These changes are triggeredby the interaction of the agonists with the essential Glu209 residue. A plausible mechanism for the coupling ofagonist binding to channel gating has been discussed.Calcium Channels. Our understanding of ion perme-

ation and selectivity in calcium channels is still poor. Inthe absence of a molecular sieve mechanism which selectsbetween ions on the basis of ionic radii, reconciling theirhigh selectivity and high conductance has been a difficult

problem. Calcium channels are very selective against Na+

ions and exhibit a multi-ion Coulomb repulsion mecha-nism to achieve a high conduction of Ca2+ ions. The factthat the two ions have similar raddii but different chargesindicates that selectivity must be based on charge. Sincethe detailed structure of a calcium channel is not avail-able, Chung and coworker146 have studied a coarse-grainedmodel using BD simulations. They have used the availableinformation on the structure and conductance properties toconstruct a model channel consisting of inner and outervestibules and a selectivity filter. The filter was designedsuch that two critical elements, its size and charges on itswall, were determined from experimental data. The radiuswas set to 2.8 Å according to the size of tetramethylam-monium, the largest permeable ion. Four negatively glu-tamate resiedues were assigned to the filter region. TheBD simulations have quite successfully reproduced exper-imental current–voltage curves, saturation of conductancewith concentration, selectivity against Na+, the anomalousmole fraction effect, attenuation of the calcium current byexternal sodium ions, and the effect of mutating glutamateresidues on blocking of sodium current.Potassium Channel KcsA. A major breakthrough in

our understanding of ion channels at the atomic level hadtaken place after the recent successful structural determina-tion of the bacterial potassium channel KcsA from Strep-tomyces lividans by MacKinnon and coworkers.93–96 Thestructure is depicted in Figure 15.

First MD simulations99�101 have shown that the simu-lated atomic structure remained stable and very close tothe original structure. The r.m.s. deviation of the C� atomsbetween the crystal structure and the simulated struc-ture was about 3.7 Å, which is comparable to the crys-tal structure of 3.2 Å resolution93 and 2.0 Å resolution.95

Fig. 15. Structure of the KcsA potassium channel embedded in a fullyhydrated lipid bilayer membrane. The narrow selectivity filter is the encir-cled region.


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Deviations between crystal structure and simulated struc-ture had been found at the extracellular mouth where asignificant widening had been observed. It was speculatedthat the absence of polarization force fields in the simu-lation protocol caused this effect. All simulations indicatethat the KcsA structure remains closed at the intracellu-lar mouth. It should be noted that the fluctuation profileof the radius of gyration of each residue along the chan-nel shows that the residues in the hydrophobic core of thebilayer are relatively more stable than those located closeto the head group region and the aqueous phase. This mayexplain the unstable position of the ion near the exit ofthe selectivity filter.147�153 Since the potassium ions, whichare localized in the selectivity filter, represent an integralpart of the crystal structure, it was of interest to comparetheir possible locations as predicted by X-ray analysis93�95

to simulation findings. Based on the most recently pub-lished structure,95 it was conjectured that the K+ ions mayoccupy seven different locations: 4 inside the selectivityfilter, one in the cavity and 2 ions weakly bound to theextracellular mouth. MD simulations, however,99�101�147�153

indicate that only one or at most two ions may occupy thefilter simultaneously. Some MD results report on only oneion in the filter, where the second ion exits the filter eitherto the intracellular side99 or to the extracellular side.148

However, in the latter both cases, it cannot be excludedthat the findings have to be attributed to incorrectly equili-brated structures. In particular, the results of the simulationshowed that the potassiun ion K1, which originally bindsto the oxygen atom of the Tyr78 carbonyl group, finallydissociates into the aqueous phase. The K2, which bindsto the oxygen atom of the Thr75 carbonyl group and K3,which is located in the cavity, both remain bounded withinthe channel for the whole 3 ns of simulation. At equilib-rium the K2 ion remains located between Gly77 and Val.78

Indeed, more elaborate simulations are necessary in orderto clarify the details on ion localization in the selectiv-ity filter. The MD simulations predict an average numberof water molecules within the cavity of about 21, whichis enough to solvate the cavity ion K3, but which is incontrast to the conjecture of 50 water molecules basedon solving the finite difference Poisson equation.149 How-ever, the latter result has to be viewed in the light of therecently reported shortcomings of the Poisson-Boltzmanntheory applied to inhomogeneous systems.150

The mechanisms underlying transport of ions acrossthe potassium channel have been examined151 usingelectrostatic calculations and three-dimensional Browniandynamics simulations. In order to build the open-state con-figurations of the channel with molecular dynamics sim-ulations, the transmembrane helices were pulled outwarduntil the channel attains the desired interior radius. Togain insights into ion permeation, the potential energy pro-files experienced by an ion traversing the channel in thepresence of other resident ions were constructed. These

profiles reveal that in the absence of an applied fieldthe channel accommodates three potassium ions in a sta-ble equilibrium, two in the selectivity filter and one inthe central cavity. In the presence of a driving potential,this three-ion state becomes unstable, and ion permeationacross the channel was observed. These qualitative expla-nations are confirmed by the results of three-dimensionalBrownian dynamics simulations. It was found that thechannel conducts when the ionizable residues near theextracellular entrance are fully charged and those nearthe intracellular side are partially charged. The conduc-tance increased steeply as the radius of the intracellu-lar mouth of the channel was increased from 2 Å to5 Å. The simulation results reproduced several experi-mental observations, including the current–voltage curves,conductance-concentration relationships, and outward rec-tification of currents.

Our current understanding of ion transport through bio-logical potassium ion channels is based on the concept ofthe multi-ion transport mechanism: permeating ions line ina queue in the narrow channel pore and move in a single filethrough the filter. The multi-ion concept had been acceptedfor many decades, its molecular mechanism, however,remained still elusive. More insights have been obtainedrecently using molecular dynamics simulations,92�102�152

and in particular steered MD simulations.148�153 It wasshown that without the repulsion from the incoming ions,the outermost ion would be tightly bound to its site and itsexit toward the extracellular side would require a signifi-cant actication of free energy, and therfore, repulsive forcesare absolutly essential for rapid conduction.102�92 However,the role of the pore lining carbonyl atoms and the watermolecules was not clear, but had been elucidated by meansof steered MD simulations. Steered molecular dynamicssimulation permit to monitor the collective motion of ionsand water molecules through the narrow selectivity filter,which is shown in Figure 10. The whole atomic structureof the KcsA channels including lipids and water moleculesis shown in Figure 15. The simulations reveal that the highconductivity is based on the cooperative diffusion of ionsand water molecules mediated by the charged flexible car-bonyl groups lining the selectivity filter. A detailed anal-ysis has shown the following microscopic mechanism ofion permeation. At rest, the queue of ions and intercalatedwater molecules, e.g., K1-W1-K2-W2, are residing at theminima of the periodic pore potential made by the porelining carbonyl groups, C O. The pore exiting ion K1reorients and attracts the neighboring water molecule W1,which thereby induces a local transformation of the neigh-boring pore potential where the ion K2 is located. Thisneighboring pore potential is an asymmetric double wellpotential, bistable, and shelters the K2 in the lower one ofits two minima. The W1-induced transformation switchesthe lower minimum from one to the other well, towardsthe position of W1. Hence, K2 moves towards this new


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minimum, thereby following the movements of K1 andW1.

The very remarkable property of this narrow filter isits high conductivity: once the outer ion is pulled outof the filter, all other ions and water molecules fol-low the same direction, apparently at almost no cost ofenergy. This becomes more obvious during the transportthrough very long artificial nanotubes constructed in anal-ogy to the selectivity filter of this ion channel, which isshown in Figure 16. From top to the bottom this figureshows at subsequent times, in steps of 50 picoseconds,six successive configurations of ions and water molecules

Fig. 16. Sequence of snapshots of water molecules and K+ ions movingin a defect-like motion through a nanotube constructed as a periodicallycontinuated KcsA selectivity filter.

(a) ν = 1.0

(b) ν = 4.0

(c) ν = 1.33

Fig. 17. Schematic distributions of ions with varying density in a staticperiodic potential. In (a) and (b) the equilibrium distance of the ionscoincides with the periodicity of the potential, so all ions rest insidethese minima. In (c) these distances differ, forcing some ions out of theminima.

passing through the nanotube. It is observed that the trans-port occurs by the hopping of vacancies from the right tothe left, as indicated by the arrows. One aspect of the hop-ping of ions can be attributed to the incommensurabilitybetween ion repulsion and the periodicity of the periodicpore potential. Figure 17 shows different ion configura-tions, three with a commensurable occupation and two thatare incommensurable. The ion occupation number � is adensity-like measure for the commensurability,

� = number of relevant binding sites−1

number of ions−1(3)

For commensurable distributions the number is an integer.This effect is known from the Frenkel-Kontorova

model,154 which is here a one-dimensional chain con-nected by harmonic springs (mimicking a layer of atoms)in a spatially periodic potential (mimicking a substrate).This chain is driven by a constant force and dampedby a velocity-proportional damping, emitting waves intothe substrate. It is originally used as a model for atomicfriction but can also be applied to describe superionicconductors.155 It is outlined in Figure 18(a).

The equations of ion motion in a harmonic potential are:

xj +�xj = xj−1 +xj+1 −2xj −b sin xj +F � j = 1 � � �N (4)

with certain boundary conditions (periodic or open).There is a modification of this model called the Frenkel-Kontorova-Tomlinson model.156 It adds springs that con-nect the atom to a rigid plate that moves along one axiswith a constant velocity (see Fig. 18(b) for a visualiza-tion). A variation of this is used for example to under-stand earthquakes. This model is not very suitable for ionic


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Fig. 18. A sketch of the Frenkel-Kontorova model154 (a) and it’s exten-sion, the Frenkel-Kontorova-Tomlinson model156 (b). In both cases thedistance of the ions does not commensurate with the periodicity of thepotential.

conductors and is only mentioned here for the sake ofcompleteness.

The computer simulations of the ion conduction in sucha nanotube153�157 reveal the same effects as observed forthe ion channel. Therefore, since the molecular mechanismof the ion-water transport in ion channels and correspond-ing nanotubes is based on general physical principles, theefficiency of this type of linear organic ionic conduc-tor has to be compared to known solid state superionicconductors.155�158–160

Potassium Channel KirBac1.1. Inward rectifier (Kir)channels have two main physiological roles: they regu-late cell excitability by stabilizing the membrane potentialclose to the potassium equilibrium potential, and they areinvolved in potassium transport across membranes.119 Forexample, Kir3.1/Kir3.4 channels modulate cardiac electri-cal activity, and Kir6.2 is involved in insulin release frompancreatic cells. Two recent structures, of the intracellulardomain of mammalian Kir161 and of a complete bacterialKir homolog,162 open up the prospect of a detailed under-standing of structure/function relationships in this impor-tant family of K channels. KirBac is the first bacterial Kirwhose structure has been solved at 3.65-Å resolution162

(Fig. 19). The overall transmembrane topology is similarto that of the simple bacterial channel KcsA, with the addi-tion of a “slide” helix N-terminal to the M1 helix. How-ever, unlike KcsA, the crystal structure of KirBac revealsan intracellular domain consisting mostly of �-sheets.The channel is assumed to be in a closed nonconduct-ing conformation because the intracellular pore mouth ishydrophobic and very narrow (only 0.05 nm in radius,

Fig. 19. Structure of the potassium ion channel KirBac1.1 (PDB code1P7B).

compared with the 0.133 nm radius of a K+ ion). Theselectivity filter of KirBac is quite similar to that of KcsAand is supposed to contain a succession of five ion bind-ing sites formed by cages of eight oxygen atoms. Thereis considerable interest in the conformational dynamicsof the filter of Kir channels. In particular it has beensuggested that, in addition to the intracellular gate, theremay also some gating at the selectivity filter. This is notrestricted to Kir channels: this phenomenon of inactiva-tion of voltage-gated (Kv) channels is also thought toinvolve changes in conformation of the selectivity filter. Itis therefore of some interest to characterize the conforma-tional dynamics of the filter region of KirBac and to com-pare with the experimental and computational behavior ofother K channels. Molecular dynamics simulations of Kir-Bac have been performed by Sansom and coworkers163�164

on a 10-ns timescale, which is comparable to the timescale of ion permeation. The results of five simulations(total simulation time 50 ns) based on three different ini-tial ion configurations and two different model membraneswere reported. These simulation data provide evidencefor limited (<0�1 nm) filter flexibility during the con-certed motion of ions and water molecules within the filter,such local changes in conformation occurring on an 1-nstimescale. In the absence of K+ ions, the KirBac selec-tivity filter undergoes more substantial distortions. Theseresemble those seen in comparable simulations of otherchannels (e.g., KcsA and KcsA-based homology models)and are likely to lead to functional closure of the chan-nel. The suggested filter distortions may provide a mech-anism of K-channel gating in addition to changes in thehydrophobic gate formed at the intracellular crossing pointof the M2 helices. The simulation data also provide evi-dence for interactions of the “slide” helix of KirBac withphospholipid headgroups.Concluding Remark. There are several computational

problems in simulationg membrane channels. One prob-lem is related to the common three-dimensional periodicboundary conditions, which seems not to be appropri-ate for the two-dimensional symmetry of the channel-membrane system. A systematic study on this problem hasbeen made recently.165 Slab geometric boundary conditionswere applied in the molecular dynamics simulation of asimple membrane-channel system. The results of the sim-ulation were compared to those of an analogous systemusing normal three-dimensional periodic boundary condi-tions. Analysis of the dynamics and electrostatics of thesystem showed that slab geometric periodicity eliminatesthe artificial bulk water orientational polarization that ispresent while using normal three-dimensional periodicity.Even though the water occupancy and volume of a simplechannel is the same when using either method, the elec-trostatic properties are considerably different when usingslab geometry. In particular, the orientational polarizationof water was seen to be different in the interior of the


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channel. This gave rise to a markedly different electricfield within the channel. The implications of slab geom-etry for the future simulation of this type of system andfor the study of channel transport properties have beenstressed.165

4.2. Brownian Ratchets

All cells transport proteins into (“insertion”) and across(“translocation”) membranes. In doing so, the cells mustmaintain the membrane’s permeability barrier betweenthe cytoplasm and either the extracellular medium orother intracellular compartments. Since proteins are macro-molecules, their passage through the membrane creates anontrivial problem: how can a macromolecule pass throughthe membrane without creating a hole that destroys the per-meability barrier? This is one of the fundamental problemsin biology.166 Relatively small proteins such as toxins andantimicrobial peptides are known to insert spontaneouslythemselves by physicochemical processes, where the pep-tides first get adsorbed to the membrane surface wherethey change their conformation into an �-helix, and thenstart to insert driven by the hydrophobic nature of the pep-tide segments.167–170 Large proteins, however, accomplishtheir insertion into or translocation accross membraneseither assisted by a complex proteinaceous machinery171

or through fusion of membranes.172 Protein translocationassisted actively by special membrane proteins (“proteintranslocases” or “translocons”) is the basic mechanismfor protein import of various cellular organelles and bac-teria, for example, into mitochondria,173�174 peroxisomes,chloroplasts, from the rough endoplasmic reticulum,175 andacross the nuclear envelope.84

All protein translocases possess several essential fea-tures. First, they contain intrinsic signal recognition sitesthat act as the docking sites for the targeting signals oftranslocation substrates. The docking sites may act asthe primary receptors for polypeptides, or they may actdownstream of primary signal receptor systems that targetpolypeptides from their site of synthesis (cis compartment)to the translocases. Second, translocases form selectivelypermeable protein-conducting channels that mediate trans-port from the cis compartment to their destination (transcompartment). Finally, translocons must be coupled to atranslocation driving force. In most cases, the associa-tion of molecular chaperones with the polypeptide in thetrans compartment provides the energy for translocation.Viewed from the perspective of classical membrane trans-porters, translocases represent remarkably flexible trans-port complexes. The translocation channels of translocasescan accommodate hundreds or thousands of distinct pro-tein substrates while maintaining the permeability barrierof the membrane. In addition to providing a conduit forcomplete translocation of polypeptides across membranes,

many translocases sense stop-transfer signals within inte-gral membrane proteins and gate laterally to allow the dif-fusion of transmembrane segments into the bilayer. Thus,the channels are not passive players in the translocationprocess.

Two classes of translocases have evolved in responseto these extraordinary demands. The most common classof translocons is envisioned to function as derivatives ofgated ion channels.176�177 In this class, the nascent or newlysynthesized polypeptide is threaded vectorially through agated protein-conducting channel in a largely unfoldedconformation with the aid of molecular chaperones. Bymaintaining the polypeptide in an unfolded conformation,a single translocase of defined dimensions can accommo-date a vast array of substrates. As such, the transloca-tion reactions can be modeled as the transport of polyionsthrough modified channels in a manner analogous to ion ormetabolite transport. We refer to these systems as ‘signal-gated translocases.’ The second class of translocases is dis-tinguished by their ability to transport fully folded and/oroligomeric proteins of large dimensions while maintain-ing the membrane permeability barrier. Stable translocasechannels have not been detected in these systems, lead-ing to the proposal that translocases of variable aperturesare assembled in response to the size of the translocationsubstrate. We refer to this class as the ‘signal-assembledtranslocons.’Theory. In pioneering works by Oster and

coworkers178�179 it was proposed that protein translocasesmove proteins across membranes using biased thermalmotion, the “Brownian ratchet” mechanism, which is basedon the chemical asymmetry between the cis and trans sidesof the membrane such as the binding and dissociation ofchaperonins to the translocating chain, or chain coilinginduced by pH or/and ionic gradients, or glycosylation,or disulfide bond formation. This concept dates back toFeynman180 who showed that a “Brownian ratchet” can usethermal fluctuations to perform directed work given a tem-perature gradient. A quantitative analysis of experimentaldata for posttranslational translocation into the endoplas-mic reticulum was performed by Elston.181 This analysisreveals that translocation involves a single rate-limitingstep, which is postulated to be the release of the signalsequence from the translocation channel. The Brownianratchet and power stroke models of translocation are com-pared against the data. The data sets are simultaneouslyfit using a least-squares criterion, and both models werefound to accurately reproduce the experimental results.A likelihood-ratio test reveals that the optimal fit of theBrownian ratchet model, which contains one fewer freeparameter, does not differ significantly from that of thepower stroke model. Therefore, the data considered therecannot be used to reject this import mechanism. The mod-els were further analyzed using the estimated parametersto make experimentally testable predictions.


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Mitochondrial Pore. The mitochondrial inner mem-brane is the central energyconverting membrane of eukary-otic cells. The electrochemical proton gradient generatedby the respiratory chain drives the ATP synthase. Tomaintain this proton-motive force, the inner membraneforms a tight barrier and strictly controls the transloca-tion of ions1. However, the major preprotein transportmachinery of the inner membrane, termed the presequencetranslocase, translocates polypeptide chains into or acrossthe membrane. Different views exist of the molecularmechanism of the translocase, in particular of the cou-pling with the import motor of the matrix8, 10, 11. Wehave reconstituted preprotein transport into the mitochon-drial inner membrane by incorporating the purified prese-quence translocase into cardiolipin-containing liposomes.We show that the motor-free form of the presequencetranslocase integrates preproteins into the membrane. Thereconstituted presequence translocase responds to targetingpeptides and mediates voltage-driven preprotein translo-cation, lateral release and insertion into the lipid phase.Thus, the minimal system for preprotein integration intothe mitochondrial inner membrane is the presequencetranslocase, a cardiolipin-rich membrane and a mem-brane potential. Mitochondria fulfill a large variety ofmetabolic tasks in eukaryotic cells, such as the genera-tion of ATP, the biosynthesis of haem, and amino-acidmetabolism. Similar to proteins that are destined for otherorganelles, such as chloroplasts or peroxisomes, mito-chondrial proteins are synthesized on cytosolic ribosomesand are subsequently transported to mitochondria post-translationally. Pore-forming membrane-embedded mul-tiprotein complexes (“translocases”) that are found inthe outer and inner mitochondrial membrane mediatethe transport of proteins across and into the outer orinner mitochondrial membranes. The TOM (translocase ofthe outer mitochondrial membrane) complex consists ofcytosol-exposed receptors and a pore-forming core, and itmediates the transport of proteins from the cytosol acrossand into the outer mitochondrial membrane. The SAMprotein complex (sorting and assembly machinery), isinvolved in the biogenesis of �-barrel proteins of the outermembrane. Two translocases of the inner mitochondrialmembrane (TIM complexes) mediate protein transportat the inner membrane. The TIM23 complex accom-plishes the transport of presequence-containing proteinsacross and into the inner membrane. The TIM22 complex(a twin-pore translocase) catalyses the insertion of multi-spanning proteins that have internal targeting signals intothe inner membrane. The TIM23 complex requires thePAM complex (presequence-translocase-associated motorcomplex) and a membrane potential, ��, for the com-plete transport of proteins into the mitochondrial matrix.The TIM22 complex mediates the membrane insertion ofmultispanning inner-membrane proteins that have internaltargeting signals, and it uses a �� as an external driv-ing force. Membrane insertion by the TIM22 complex

is a multistep process, in which the preprotein initiallytethers to the translocase in a step that is independentof ��. Subsequently, in steps that require a ��, posi-tive charges in the matrix-exposed loops of the precur-sor protein allow docking in the twin-pore translocaseand, eventually, the precursor inserts into the inner mem-brane. The protein complex TIM23 is of particular interest.The transport of proteins across the inner membrane intothe matrix requires two energy sources—the membranepotential of the inner membrane and the ATP-dependentactivity of the presequence-translocase-associated import-motor (PAM) complex. The so called “mitochondrial heat-shock protein-70” (mtHsp70) is the central component ofthe PAM complex and it binds directly to an incoming,unfolded polypeptide chain. Translocase of the inner mito-chondrial membrane, Tim44, serves as a membrane anchorfor mtHsp70. The function of mtHsp70 has to be tightlyregulated to allow it to cycle between an ADP-bound stateand an ATP-bound state, which correspond to states ofhigh and low substrate affinity, respectively. This regu-lation requires the cooperation of mtHsp70 with cofac-tors. The regulated cycling of mtHsp70, however, is stillunknown.

Although the functional role of mHsp70 could not pre-cisely definded, it was clear that this protein complexis required for post-translational translocation into mito-chondria (and the endoplasmic reticulum). Therefore itwas assumed that mHsp70 should act as translocationATPase. Two operating principles have been suggestedand examined.178�179�182–184 The “power stroke model” pro-poses that mHsp70 undergoes a conformational change,which pulls the precursor protein through the translocationpore, whereas, in the “Brownian ratchet model,” the role ofmHsp70 is simply to block backsliding through the pore.Theories for Mitochondrial Pore. Soon after the iden-

tification of mHsp70 as the translocation ATPase, it waspostulated that protein import is driven by a “Brownianratchet” mechanism.178�179�182�183 According to this model,a precursor chain diffuses within the translocation pore;soluble mHsp70 molecules bind to the chain on the matrixside of the pore, thereby hindering reverse diffusion andcausing a net unidirectional movement of the chain into thematrix. A quantitative study suggested that this mechanismcould account for the observed rates of translocation.178�179

However, three experimental facts had to be taken intoaccount in the theory: (1) mHsp70 drives translocationin conjunction with Tim44, a membrane-associated sub-unit of the inner membrane import machinery; (2) someprecursors fold before import, and their unfolding canbe accelerated by the ATP-dependent action of mHsp70;(3) the diffusion of precursor proteins within the transloca-tion pore is far slower than originally assumed. Attemptshave been made to bring the Brownian ratchet model andthe power stroke model in accord with the experimentalobservations. Of course, a theoretical approach will not


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by itself reveal how proteins are imported into mitochon-dria. However, this kind of analysis serves two purposes.First, it helps to define possible translocation mechanismsthat are consistent both with the experimental data andwith the laws of mechanics. Second, it highlights ques-tions that will generate further insights into the import pro-cess. However, the major conclusion of the study by Osterand Glick183 was that the interactions of precursor pro-teins with the translocation pore are very significant. Basedon existing experimental systems, they developed a simplemathematical procedure to calculate diffusion coefficientsfor precursor chains in the translocation pore. Remarkably,the calculated diffusion coefficients are 106-fold lowerthan for a freely diffusing chain. This phenomenon hasreceived little attention, but it should be incorporated intoany mechanistic description of translocation. Strong dragforces in the pore greatly reduce the predicted efficiencyof a Brownian ratchet, and under these conditions a powerstroke motor could outperform a Brownian ratchet. Theirsimulated power stroke motor did not match the transloca-tion rates observed in actual mitochondria, indicating thatthe current translocation model was too simplistic. Thereare two key a assumptions underlying this model. Thefirst assumption is that precursor-pore interactions exert adrag force of constant magnitude. In reality, different seg-ments of the precursor chain will interact differently withthe pore walls. They performed simulations in which theprecursor chain experienced occasional strong resistanceinstead of a constant, moderate drag force. This modi-fication had virtually no effect on translocation rates inthe ratchet model, but it significantly accelerated translo-cation in the power stroke model. The second importantassumption is that the chain segment in the pore is fullyextended, capable neither of compression nor of furtherextension. In fact, the mitochondrial translocation poreis quite flexible, and translocating precursor chains maybe able to form helices or other elements of secondarystructure. Hence it would be more accurate to model theprecursor as a flexible chain than as a rigid rod. Thismodification would probably make little difference in theBrownian ratchet model, but in the power stroke model itwould be expected to accelerate translocation. The strengthof the simulated Brownian ratchet is inversely proportionalto the size of the ratcheting interval, which is the dis-tance the chain diffuses between the binding of successivemHsp70 molecules. With a very small ratcheting intervalit would be possible to overcome the drag forces causedby strong pore interactions. However, based on the knownproperties of the mitochondrial translocation machinery, itseems unlikely that such a small ratcheting interval couldbe achieved in this system. Whether the power strokemodel is a plausible representation of mHsp70 functionis still a matter of debate. No information is yet avail-able about the putative mHsp70 power stroke. But evenif this power stroke turns out to be weaker than we have

assumed, it is conceivable that two mHsp70 molecules canpull simultaneously on the precursor chain. The strength ofthe mHsp70 motor could be examined directly by adapt-ing optical laser trap technology to study protein translo-cation into mitochondria. A separate issue concerns theATPase cycle time of mHsp70. The power stroke motormust turn over between strokes. Therefore, to be compati-ble with the observed translocation rates, the ATPase cycletime for the complete mHsp70 chaperone system must besmaller than 5 seconds. This number can be measured onceall of the components of the mHsp70 chaperone systembecome available in purified form. Although the calcula-tions favor a power stroke mechanism, they do not allowto distinguish conclusively between the Brownian ratchetand power stroke models. A recent mathematical analy-sis of both mechanisms has been performed by Elston,184

which revealed that qualitative differences between thetwo models, Brownian ratchet and power stroke, occur inthe behavior of the mean velocity and effective diffusioncoefficient as a function of mHsp70 concentration. How-ever, a final distinction of applicability between ratchetand power stroke will come from other lines of investi-gation. For example, most studies have focused on mea-surements of average translocation times, but additionalmechanistic information could be obtained by examiningthe variances of translocation times. A second approachshould focus on the details of the mHsp70 reaction cycle.In the Brownian ratchet model, mHsp70 must dissociatefrom Tim44 while remaining bound to the precursor chain.In the power stroke model, mHsp70 must remain boundto both Tim44 and the precursor chain. One of the mostpromising ways to examine the mechanism of transloca-tion is to characterize the mHsp70-dependent unfolding ofprecursor proteins. Indeed, the power stroke notion wasinspired by the finding that mHsp70 could unfold a stablyfolded precursor protein. Although the models investigatedso far assume that the precursor chain is already unfolded,they could be extended to incorporate an unfolding event.It seems likely that mHsp70 causes unfolding by pullingon the N-terminus of a folded domain This novel reactioncould eventually be examined by a combined theoreticaland experimental study.Endoplasmic Pore. A decisive step in the biosynthesis

of many secretory and plasma-membrane proteins is theirtransport across the endoplasmic reticulum (ER) mem-brane in eukaryotes or across the cytoplasmic membranein prokaryotes.185 These polypeptides are first targetedto the membrane by hydrophobic amino-acid sequences,which are either cleavable signal sequences or trans-membrane segments (TM) of membrane proteins. Sol-uble proteins, such as those destined for secretion, aresubsequently transported across the membrane through aprotein-conducting channel with a hydrophilic interior. Inthe case of membrane proteins, when a hydrophobic TMarrives in the channel, it is released through an opening


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in the channel wall into the surrounding lipid phase. Thecapacity of the channel to open laterally towards the lipidand the wide variety of substrates that it must transportdistinguish it from many other channels. An evolutionarilyconserved heterotrimeric complex of membrane proteins,called the Sec61 complex in eukaryotes and the SecY com-plex in eubacteria and archaea, forms the channel.32 The�-subunit forms the channel pore, and it is the crosslinkingpartner of polypeptide chains passing through the mem-brane. Reconstitution experiments have shown that theSec61/SecY complex is the essential membrane compo-nent for protein translocation. The channel itself is a pas-sive conduit for polypeptides and must therefore associatewith other components that provide a driving force.186 Inco-translational translocation, the major partner is the ribo-some. The elongating polypeptide chain moves directlyfrom the ribosome into the associated membrane channel.The energy for translocation comes from GTP hydrolysisduring translation. Many (or perhaps all) cells also havepost-translational translocation, in which polypeptides arecompleted in the cytosol and then transported across themembrane. In yeast (and probably in all eukaryotes), thepost-translational translocation partners are another mem-brane protein complex (the tetrameric Sec62/63p complex)and the lumenal protein BiP, a member of the Hsp70 fam-ily of ATPases. BiP may promote translocation by actingas a molecular ratchet or power stroke machine, preventingthe polypeptide chain from sliding back into the cytosol.In the eubacterial post-translational pathway, the cytosolicATPase SecA pushes polypeptides through the channel. Inaddition, an electrochemical gradient across the membranestimulates translocation in vitro and is essential in vivo.

An understanding of the mechanisms that underlie pro-tein translocation requires detailed structural information.Low-resolution structures have been obtained by single-particle electron microscopy (EM) of either the isolatedSec61/SecY complex or the Sec61 complex bound to theribosome. Rapoport and coworkers32 have reported thestructure of the SecY complex from the archae M. jan-naschii, determined by X-ray diffraction at 3.2 Å resolu-tion, and have proposed based on the X-ray structure thefollowing refined model for the translocation of secretoryproteins. Initially, the channel is closed because the plugblocks the pore (Fig. 20, stage 1). Next, a channel part-ner binds; depending on the mode of translocation and theorganism, this can be a ribosome, the Sec62/63p complexor SecA (stage 2). Although part of an oligomer, only onecopy of the SecY/Sec61p complex forms the active pore.The closed state of the channel may be destabilized by aconformational change, but binding of the partner aloneis insufficient to completely open the channel. In the nextstep, the substrate inserts as a loop into the channel, withits signal sequence intercalated between TM2b and TM7,and with its mature region in the pore (stage 3). Insertionrequires a hinge motion to separate TM2b and TM7, and

Fig. 20. Different stages of translocation of a secretory protein.

displacement of the plug to its open-state position close toone od the subunits. The mature region of the polypeptidechain is then transported through the pore, and the sig-nal sequence is cleaved at some point during translocation(stage 4). While the polypeptide chain is moving from anaqueous cytoplasmic cavity to an external one, the porering forms a seal around the chain, hindering the perme-ation of other molecules. Finally, when the polypeptide haspassed through, the plug returns to its closed-state position(stage 5).Theories for Endoplasmic Pore. In order to understand

the physical basis of the translocation, two mechanisticmodels for post-translational translocation been proposedand have been referred to as the “Brownian ratchet” andthe “translocation motor” 178�182 or “power stroke motor.”A detailed modeling and mathematical analysis of bothmodels181 have revealed that the two models make quali-tatively different predictions. The differences are observedin the behavior of the mean velocity and effective diffu-sion coefficient as a function of organellar Hsp-70 concen-tration. Since direct measurement of these two quantitiesis not possible, a method has been proposed for deter-mining the mean velocity and effective diffusion coeffi-cient that only requires monitoring the fraction of proteinreleased from the translocation channel as a function oftime. This method has applied to one of the most com-monly studied post-translational translocation systems inthe ER. Similar as for the mitochondrial pore, a signalsequence near the amino terminus targets the precursorprotein for import. In the ER, the central channel-formingprotein is believed to be Sec-61. On the lumenal side of themembrane Sec-61 associates with another protein complex(Sec-62/63p). A certain domain of Sec-63p interacts withorganellar BiP, which is a member of the Hsp-70 family


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of ATPases. BiP also binds to the precursor protein andis responsible for providing directionality to the process.The mechanism that drives post-translational translocationis not known, and two different roles for organellar Hsp-70have been suggested. In the first scenario, Hsp-70 asso-ciates with both the membrane bound complex (Sec-63p orTim-44) and the precursor protein. Hsp-70 then undergoesan ATP-dependent conformational change that pulls theprecursor through the pore.182 In the second scenario, therole of Hsp-70 is simply to prevent backward diffusion ofthe precursor protein,178 and import relies on biased ther-mal diffusion. Both models represent ‘molecular motors’in that chemical free energy is used to produce directedmotion. Elston has studied181 two types of models for ERtranslocation pore developed on the basis of experimen-tal information. A quantitative analysis of experimentaldata for post-translational translocation into the endoplas-mic reticulum reveals that translocation involves a singlerate-limiting step, which is postulated to be the release ofthe signal sequence from the translocation channel. TheBrownian ratchet and power stroke models of translocationare compared against the data. The data sets are simulta-neously fit using a least-squares criterion, and both modelsare found to accurately reproduce the experimental results.A likelihood-ratio test reveals that the optimal fit of theBrownian ratchet model, which contains one fewer freeparameter, does not differ significantly from that of thepower stroke model.V-ATPase. There are several membrane proteins that are

thought to be rotary motors. These “rotarons” accomplishtheir primary functions by rotating one group of subunitswith respect to the rest. Examples include the V-ATPase,the F1F0-ATPase and the bacterial flagellar motor whichare described in the following chapter.Structure and Function. Intracellular compartments

exhibit a significant difference between their lumenal pHand that of the bulk cytoplasm. This pH difference ismaintained mostly by the vacular ATPases. The vacuo-lar H+-ATPases (or V-ATPases) are a family of ATP-dependent proton pumps responsible for acidification ofintracellular compartments and, in certain cases, protontransport across the plasma membrane of eukaryotic cells.Vacuolar acidification is essential for a variety of cel-lular processes187�188 including ligand-receptor dissocia-tion and receptor recycling following receptor-mediatedendocytosis, intracellular targeting of newly synthesizedlysosomal enzymes, protein processing and degradation,and coupled transport of small molecules, such as neu-rotransmitters. Acidification of endosomal compartmentsis also necessary for budding of transport vesicles andinfection by certain envelope viruses, such as influenzavirus. V-ATPases have also been identified in the plasmamembrane of certain cells where they function in such pro-cesses as renal acidification, pH homeostasis, bone resorp-tion and coupled potassium transport. They have also been

Fig. 21. Structural model of the V-ATPase.

implicated in a variety of disease processes, includingrenal tubular acidosis, osteopetrosis and tumor metastasis.V-ATPases are multisubunit complexes (Fig. 21) composedof a peripheral domain V1 (yellow and orange) responsi-ble for ATP hydrolysis and an integral domain V0 (blueand grey) responsible for proton translocation. Based upontheir structural similarity to the F1F0-ATP synthases,189 theV-ATPases are thought to operate by a rotary mechanismin which ATP hydrolysis in V1 drives rotation of a ringof proteolipid subunits in V0. The peripheral V1 domaincontains eight different subunits (subunits A-H). The inte-gral V0 domain, responsible for proton translocation, iscomposed of five subunits (a, b, c, c′, c′′). The V1 and V0

domains are connected by both a central stalk (composedof the D and F subunits) and a peripheral stalk (composedof the C, E, G and H subunits). These stalks play a cru-cial role in the assumed rotary mechanism of ATP-drivenproton transport.

Vacuolar (V-)ATPases operate by a rotary mechanism.Protons are first engaged at the cytoplasmic side ofthe hemi-channel in subunit a and protonate buried Gluresidues in the proteolipid subunits (E137 in subunit c,E145 in subunit c′ and E108 in subunit c′′). These residuesremain protonated as a result of their forced rotation intothe hydrophobic phase of the bilayer. Rotation is facili-tated by ATP hydrolysis at catalytic sites that are locatedat the interface of subunits A and B. This drives rota-tion of a central rotor that includes subunits D and F(orange) of V1 connected to subunits d and the ring ofproteolipid subunits of V0 (blue). The orientation of sub-units a and e of V0 (grey) are fixed relative to the A3B3head by the peripheral stalks. Following rotation, protons


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are displaced from the proteolipid subunits into a lume-nal hemi-channel of subunit a as a result of the interac-tion between the Glu side chains and a crucial buried Argresidue (R735) in transmembrane helix-7 (TM7) of sub-unit a. Other buried charged residues in the C-terminaldomain of subunit a (H743, E789, R799) line the lumenalhemi-channel through which protons exit the membrane.

The V-ATPases are structurally related to the F-ATPasesthat normally function in ATP synthesis in mitochondria,chloroplasts and bacteria. This relationship between theV- and F-ATPases is evident both in the overall struc-ture of the complexes and in the sequence of certain sub-units. Electron microscopy has revealed a similar overallshape for the two proteins, with the peripheral domainof each complex connected to the corresponding integraldomain by both central and peripheral stalks. These stalksare believed to play an important role in the mechanismby which these enzymes catalyze ATP-dependent protontransport. On the other hand, there are also significantstructural differences189 between the F- and V-ATPasesrevealed by electron microscopy. For example, images ofthe V-ATPase reveal a complex array of projections ema-nating from the V1 domain, and a cuff-like cytoplasmicregion on the V0 domain. Some studies suggest that V1 andV0 may be connected by as many as three stalks. Func-tionally, although the V-ATPases have been shown to bereversible, they normally operate, unlike the F-ATPases,in the direction of ATP-driven proton transport. In termsof primary sequence, three subunits show clear sequencehomology indicating a common evolutionary origin for theV- and F-ATPases. These include the two nucleotide bind-ing subunits (A and B), which are approximately 20–25%identical to the and subunits of the F-ATPases, and theproteolipid subunits (c, c′ and c′′), which are homologousto the c subunit of F0.Theory and Simulation for V-ATPase. A mechanochem-

ical model of V-ATPase proton pump has been proposedby Oster, Grabe and Wang.190 They proposed two mod-els: the two-channel model and the one-channel model forthe a-subunit. In the two-channel model two half-channelspenetrate the stator to the level of the rotor sites; all otherparts of the rotor-stator interface are hydrophobic. A hor-izontal polar strip connects the channels to allow the pas-sage of an unprotonated site, but protons are blocked fromleaking by the stator charge. Protons enter the input (basic)channel and bind to a rotor site, largely neutralizing it.Rotation carries the protonated site one complete revolu-tion (to the left) where the site enters the output (acidic)channel (from the right). The stator charge forces the siteto relinquish its proton into the lumen. Note that the sizesof the rotor and stator are such that two rotor sites cannotfit in the rotor-stator interface at once. In the one-channelmodel the rotor sites now lie above the level of the mem-brane and are in equilibrium with the cytoplasm when notin the stator. A single channel penetrates the stator to the

level of the rotor sites; all other parts of the rotor-statorinterface are hydrophobic. A horizontal polar strip con-nects the channel to the cytoplasm to allow passage of anunloaded site. The stator charge blocks passage of protonsthrough the strip. Rotation brings a protonated site into theoutput channel where the stator charge forces it to releaseits proton to the lumen. The unloaded site then rotatesthrough the polar strip, past the stator charge, exiting therotor-stator interface to the left.

In their paper, Oster et al.190 used a mathematical formu-lation of the model where the motion of the rotor, in termsof its rotation angle ��t�, was computed from a force bal-ance equating the viscous drag on the rotor to the torquesthat act on the rotor and the Brownian force modeling therotor’s thermal fluctuations by a Langevin-type equation

�d�

dt= �Q�� s�+ �� s�+ �"�� s�− �D��+ �B�t�

The various terms correspond to the rotator-stator chargeinteraction, the membrane potential, the dielectric barrier,the driving torque from V1, and the Brownian torque,respectively. The chemical states of the rotor are describedby the binary variable si for full or empty, and determinedby a Markov equation ds/dt = K��s, where K��) is thetransition matrix. Using appropriate parameters, the modelcalculations reproduced a variety of experimental measure-ments of performance of the V-ATPase proton pump.

Alternative approaches could follow similar lines asproposed recently for the rotary motor F0F1-ATPase.F0F1-ATPase also known as ATP synthase, which is anadenosine triphosphate (ATP) producing protein commonto most living organisms.6 In vivo, the two parts of ATPsynthase, F0 and F1, are attached to each other andmechanically coupled by the central shaft. ATP synthesisoccurs in natural systems when an ion gradient (commonlyH+, or Na+ in some cases depending on the ATPase type)passes through the membrane embedded F0 part of theprotein. The chemical ion gradient induces the rotationof F0 and the shaft. The rotating shaft creates a torqueon the static F1 part, which leads to a sequential confor-mational change of some sub-parts. F1-ATPase is com-posed of six subunits spatially alternated and arrangedaround a central shaft giving it a three-fold symmetry.This complex protein mechanism leads to ATP catalysiswhen substrates adenosine diphosphate (ADP) and inor-ganic phosphate (Pi) are bound to one of the three cat-alytic sites. This phenomenon is crucial in the energeticmetabolism of the cell since it provides the energy nec-essary for the endothermic ATP synthesis reaction. TheF1 motor has been modeled by several groups. Recently,Oster and coworkers191 proposed a quantitative model thataccounts for these substeps. This very complete modelwith 64 chemical states describes the mechanochemicalbehavior of the F1 motor as a crankshaft escapement thatconverts the elastic bending of some subunits into the rota-tion of the shaft. Another model including the different


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conformations of some subunits was developed by Karplusand coworkers.192 More recently The F1 motor was mod-eled as a stochastic process for the angle of its shaft andthe chemical state of its catalytic sites.193 The stochasticprocess is ruled by six coupled Fokker-Planck equationsfor the biased diffusion of the angle and the random jumpsbetween the chemical states. The model reproduces theexperimental observations that the motor proceeds by sub-steps and the rotation rate saturates at high concentrationsof adenosine triphosphate or at low values of the frictioncoefficient. In each chemical state & , the shaft of the motoris submitted to an internal torque caused by some free-energy potential U&�� which depends on the angle �. Themotion of this angle is thus described by an overdampedLangevin-type equation

�d�

dt=−(U&��

(�+ �ext + �fluct�t� (5)

where �ext is some external torque and the fluctuatingtorque is taken as a Gaussian white noise related tothe friction coefficient � according to the fluctuation-dissipation theorem. The chemical state & and its associ-ated potential U&�� change accordingly at each one ofthese random events. The mechanical motion of the shaftis thus coupled to the chemical reaction in such chemo-mechanical processes. The state of the system is describedby the probability density p&�� t� to find the motor ineach chemical state with its shaft forming the angle � attime t. The time evolution of the probability densities isruled by a set of coupled Fokker-Planck equations includ-ing terms for the description of the random jumps betweenthe discrete chemical states & .

4.3. Entropic Barriers

Translocation of polymeric macromolecules through nar-row biological pores is very complex involving manymachineries. Although several essential features of translo-cation are richly documented in systems like nuclear porecomplex, the mitochondrial and endoplasmatic pore, a sim-ple system has only recently been identified for followingthe single-file passage of one isolated polymer throughone channel (see e.g., Ref. [194] for review). The idea isthat the ionic current through the channel, which accom-panies the passage of the polymer, is blocked to a certainextent during the event of translocation of the polymer. Itis thought that the extent and duration of the current block-ade are unique signatures of the identity of the polymer,both in terms of the polymer’s chemical characteristics andphysical length. Even in the simplest setup, where identi-cal molecules undergo translocation, has generated severalpuzzling results. The distribution, P��), of the duration �of blockade of ionic current Ib is very broad and appearsto exhibit at least two peaks. In addition, there are severallevels of ionic current blockade for the same molecule. It is

standard practice in experimental investigations to com-bine the histograms of � and Ib.

194 The resultant scatterplots always yield two groups of data even for monodis-perse homopolymers.

In several theoretical papers the the distribution, P��),has been derived88�89�195–200 theoretical activity, based onentropic barrier dynamics201–204 all of which lead to ageneric P��) unlike in experiments. Nonetheless it is veryintructive to follow these theoretical considerations, whichis summarized below.

Motivated by experiments in which a polynucleotide isdriven through a proteinaceous pore by an electric field,Lubensky and Nelson196 have studied theoretically the dif-fusive motion of a polymer threaded through a narrowchannel with which it may have strong interactions. Theyshowed that there is a range of polymer lengths in whichthe system is approximately translationally invariant, andthey developed a coarse-grained description of this regime.From this description, general features of the distribu-tion of times for the polymer to pass through the porehad been deduced. They also introduced a more micro-scopic model, which provides a physically reasonable sce-nario in which, as in experiments, the polymer’s speeddepends sensitively on its chemical composition, and evenon its orientation in the channel. They pointed out that theexperimental distribution of times for the polymer to passthrough the pore is much broader than expected from sim-ple estimates, and speculated on why this might be. Moregeneral aspects of the dynamics of polynucleotide trans-port through nanometre-scale pores have been reviewedrecently.194

A theory of channel-facilitated transport of long rod-like macromolecules through thin membranes under theinfluence of a driving force of arbitrary strength hasbeen developed by Berezhkovskii and Gopich.205 Analyticexpressions were derived for the translocation probabil-ity and the Laplace transform of the probability densityof time that a macromolecule spends in the channel bytranslocating and nontranslocating (returning back) macro-molecules. These results were used to study how the dis-tribution of the macromolecule lifetime in the channeldepends on a polymer chain length and the driving force.It was shown that depending on the values of the parame-ters, the lifetime probability density may have one or twopeaks. This theory is a generalization of the theory devel-oped by Lubensky and Nelson.196 Using Brownian molec-ular dynamics simulations the dynamics of the passage ofa stiff chain through a pore into a cell containing particlesthat bind reversibly to it have been studied by Gelbart andcoworkers.206 The mean first-passage time as a function ofthe length of the chain inside for different concentrationsof binding particles has been estimated. As a consequenceof the interactions with these particles, the chain experi-ences a net force along its length whose calculated valuefrom the simulations accounts for the velocity at which it


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enters the cell. This force can in turn be obtained from thesolution of a generalized diffusion equation incorporatingan effective Langmuir adsorption free energy for the chainplus binding particles. These results suggest a role of bind-ing particles in the translocation process that is in generalquite different from that of a Brownian ratchet. Further-more, nonequilibrium effects contribute significantly to thedynamics; e.g., the chain often enters the cell faster thanparticle binding can be saturated, resulting in a force sev-eral times smaller than the equilibrium value. Monte Carlosimulation studies of the translocation of homopolymersof length N driven through a channel have been performedby Loebl et al.200 They found that the translocation time� depends on temperature in a nontrivial way. For tem-peratures below some critical temperature Tc, � ∼ T −1�4,whereas for T > Tc, � increases with temperature. The lowtemperature results are in good agreement with experimen-tal findings as is the dependence of on the driving fieldstrength. The velocity of translocation displays the samecharacteristics as found in experiment but the N depen-dence of � shows the linear relationship observed in exper-iment only for large values of N . A possible reason for thisdiscrepancy may be attributed to the simple intermonomerpotential used and the assumption that the drag is the sameat all temperatures. Kantor and Kardar89 considered thepassage of long polymers of length N through a hole in anartifial thin membrane (Fig. 22). If the process is slow, it isin principle possible to focus on the dynamics of the num-ber of monomers s on one side of the membrane, assumingthat the two segments of the chain are in equilibrium. Thedynamics of s�t� in such a limit would be diffusive, witha mean translocation time scaling as the Rouse time N 2

in the absence of a force, and proportional to N when aforce is applied. It was shown that the assumption of equi-librium breaks down for sufficiently long polymers, andprovide lower bounds for the translocation time by com-parison to unimpeded motion of the polymer. These lowerbounds exceed the time scales calculated on the basis ofequilibrium, and point to anomalous ‘subdiffusive’ charac-ter of translocation dynamics. This was explicitly verified

Fig. 22. Schematic depiction of a polymer translocation from one sideto the other.

by numerical simulations of the unforced translocation ofa self-avoiding polymer. Forced translocation times areshown to strongly depend on the method by which theforce is applied. In particular, pulling the polymer by theend leads to much longer times than when a chemicalpotential difference is applied across the membrane. Thebounds in these cases grow as N 1+� , respectively, where� is the exponent that relates the scaling of the radius ofgyration to N . This confirms their previous suggestion,199

and in particular the anomalous diffusion of the two chainsegments. For time scales shorter than the equilibrationtime ∼ N 1+� , the fluctuation of s�t� is

�+s�t′ + t�− s�t′�,2� ∼ t2� (6)

where the anomalous exponent is

� = 1/�1+2�� (7)

Since all theories lead to a generic P��) unlike inexperiments, in order to gain insight into this puzzle,Muthukumar and Kong90 have conducted simulations of aminimal model of polymer translocation through the �HLchannel. Their minimal model incorporates enough chemi-cal details of the polymer and channel to evaluate the poten-tial roles played by secondary structures of the polymer,orientation of the chain, binding sites inside the lumen ofthe channel, etc. In addition to resolving the above puz-zle, the details emerging from their simulations are so vividthat additional experimental protocols can be formulated tosubstantially narrow down P��, and thus enable a muchfaster sequencing scheme. The key result of their simula-tions is that conformational entropy of the polymer playsa dominant role in dictating � and P��. The breadth ofP�� can be directly attributed to the entropic trap arisingfrom the vestibule of the �HL pore. Another observationis that there is no direct correlation between translocationtime and blocked current. For each value of � , there are twodominant values of blocked current. Finally, an experimen-tal setup is proposed that would enable precise sequencingof polymers by several orders of magnitude faster than cur-rent techniques. P�� can be made dramatically sharper,sufficient to facilitate sequencing of polymers, by deletingthe vestibule or dragging the polymer in a direction normalto the pore axis.

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Received: 27 October 2007. Accepted: 10 November 2007.


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