Theoretical Studies of Structure-Function Relationships in Kv Channels: Electrostatics of the Voltage Sensor
TTheoretical Studiesof Structure-Function Relationshipsin K V Channels:Electrostatics of the Voltage Sensor
Alexander PeyserMarch 27, 2019
Abstract
Voltage-gated ion channels mediate electrical excitability of cellular membranes. Reduced models of the voltage sensor ( VS )of K v channels produce insight into the electrostatic physics underlying the response of the highly positively charged S4 trans-membrane domain to changes in membrane potential and other electrostatic parameters. By calculating the partition functioncomputed from the electrostatic energy over translational and/or rotational degrees of freedom, I compute expectations ofcharge displacement, energetics, probability distributions of translation & rotation and Maxwell stress for arrangements of S4 positively charged residues and S2 & S3 negatively charged counter-charges; these computations can then be comparedwith experimental results to elucidate the role of various putative atomic level features of the VS .A ‘paddle’ model (Jiang et al., 2003) is rejected on electrostatic grounds, owing to unfavorable energetics, insufficient chargedisplacement and excessive Maxwell stress. On the other hand, a ‘sliding helix’ model (Catterall, 1986) with three localcounter-charges, a protein dielectric coefficient of 4 and a 2/3 interval of counter-charge positioning relative to the S4 α -helixperiod of positive residues is electrostatically reasonable, comparing well with Shaker (Seoh et al., 1996). Lack of counter-charges destabilizes the S4 in the membrane; counter-charge interval helps determine the number and shape of energy barriersand troughs over the range of motion of the S4 ; and the local dielectric coefficient of the protein ( S2 , S3 & S4 ) constrains theheight of energy maxima relative to the energy troughs.These ‘sliding helix’ models compare favorably with experimental results for single & double mutant charge experiments on Shaker by Seoh et al. (1996). Single S4 positive charge mutants are predicted quite well by this model; single S2 or S3 negativecounter-charge mutants are predicted less well; and double mutants for both an S4 charge and an S2 or S3 counter-chargeare characterized least well by these electrostatic models (which do not include gating load, unlike their biological analogs).Further computational and experimental investigation of S2 & S3 counter-charge structure for voltage-gated ion channels iswarranted. a r X i v : . [ q - b i o . B M ] D ec ontents ist of Figures Shaker : R Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.14 Shaker : R Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.15 Shaker : R N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.16 Shaker : R Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.17 Shaker : E Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.18 Shaker : D N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.19 Shaker : E Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.20 Shaker : K Q + D N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.21 Shaker : K Q + E Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.22 Shaker : K Q (Imaginary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 ist of Supplementary Material Movie: tile-helix-side.mp4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Movie: tile-helix-top.mp4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Movie: tile-paddle-side.mp4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Movie: tile-paddle-top.mp4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Movie: flat-paddle.mp4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Movie: flat-s4.mp4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 hapter 1
Introduction
Electrical excitability of cells is possible because the move-ment of a few charges can control the flow of many charges.This principle – amplification – led Hodgkin and Huxley(1952b) to their theory of the action potential in terms ofelectrically controlled membrane conductances (for an exam-ple of a computation of their model, see Fig. 1.1). Such con-ductances have been localized to channel proteins conduct-ing Na + , K + , or Ca ions. Besides a conductive port (com-posed of transmembrane domains S5 and S6 , see Fig. 1.2),the channels contain four transmembrane regions (labeled S1 - S4 starting from the amino end). In the S4 region, thereare a total of three to seven positively charged amino acidresidues, each arrayed at every third amino acid position.An extensive electrophysiological data set exists on voltage-controlled ionic conductance and the ‘gating current’ at-tributable to charges controlling the ionic port. A secondextensive data set has emerged from experiments measuringstructure. Both sets provide essential perspectives, but nodirect means to assess physical interactions in the structureand the significance of physical interactions for function. Inthis thesis, I attempt to bridge these perspectives computa-tionally. I focus on electrostatics because the voltage-gatedion channels are controlled by an applied electric field thatacts on intrinsic protein charges.Voltage-gated K + channels ( K v ) are composed of two dis-tinct functional elements, a ‘voltage sensor’ and a ‘poredomain’. As described by Lee et al. (2005), they ap-pear to be “membrane proteins with separate, weakly at-tached membrane-spanning domains”. K v channels mayhave evolved as the concatenation of two separate proteins,one contributing a central tetrameric K + conducting poreand the other contributing a weakly-attached peripheralvoltage sensor which transduces changes in transmembrane potential into a flow-controlling action on the pore domain(Kumanovics et al., 2002). The voltage sensor motif is ho-mologous with voltage-sensitive proton conducting channels( H v , Ramsey et al., 2006). The physics underlying the well-known function of the K + and related Na + and Ca channels(Hodgkin and Huxley, 1952a; Catterall, 1988) in terms of theatomic structures developed over the last 20 years (Doyleet al., 1998; Jiang et al., 2003) has still not been fully elu-cidated. Questions are still open regarding the precise elec-trostatics, thermodynamics and distributions of relative po-sitions & motions of charges at physiological temperatures.Each voltage sensor comprises four largely helical sec-ondary structures. The S4 α -helix bears a positively chargedarginine or lysine residue at every third position while the S2 and S3 helices bear a smaller number of negatively chargedaspartate or glutamate residues (Noda et al., 1986). Theother residues of these membrane-spanning segments aremostly hydrophobic. It has been proposed that the trans-membrane electric field moves the S4 segment through acombination of translation and rotation with respect to theother helices and the lipid membrane (‘sliding helix’ hypoth-esis, Catterall 1986), or alternatively moves the S4 segmentin association with part of the S3 segment across the lipidmuch like a large lipophilic ion (‘paddle’ hypothesis, Leeet al. 2003).In both proposed mechanisms, multiple electric charges ofthe channel protein move in the electric field of the mem-brane and therefore produce a displacement current (‘gatingcurrent’, Hodgkin and Huxley, 1952b; Chandler and Meves,1965; Armstrong and Bezanilla, 1973), as well as electricwork which could drive gating of ionic conductance (Hodgkinand Huxley, 1952b). Translocation of these charges couldbe facilitated by factors such as a local thinning of the lipid5 e m b r ane P o t en t i a l / m V Time / ms
Figure 1.1: Computation of an action potential according to the Hodgkin and Huxley (1952b) model (comparable to Fig. 13, upper curve,same paper). This curve is calculated with I = C M ˙ V + ¯ g K n ( V − V K ) + ¯ g Na m h ( V − V Na ) + ¯ g l ( V − V l ) [Eq. 26, same paper], using thesame constants as the original paper for activation, inactivation, reversal potentials and conductances. The variables n , m and h representthe proportion of ‘activating or inactivating particles’ for a channel in the activating position: n for the activating K + channel particles; m for the activating Na + channel particles; and h for the inactivating Na + channel particles. Since the rate constants for the first derivative ofthese variables were voltage dependent, the physical interpretation was of a charge-carrying voltage sensor driving the opening and closing ofionic conductance. ++++++ - - - S1 S2 S3 S4 S5 S6
VoltageSensor ConductingPoreGatingPore
PoreLoop
N C
Figure 1.2: K v channel schematic topology: a generic K v channel representation highlights the transmembrane domains of a single unitof the K v tetramer. On the left from the amino end of the protein are the voltage-sensor domains, S1 – S4 . On the right are the conductingpore domains, S5 and S6 . The lipid bilayer through which the protein is threaded is represented in pink. The red ‘ + ’ symbols represent theexcess positively charged amino acid residues on S4 that are believed to cross the lipid bilayer at the invagination labeled ‘gating pore’. Theblue ‘ − ’ symbols represent excess negatively charged amino acid residues on the S2 and S3 transmembrane domains. S4 chargesby static residues of S2 and S3 . The existence of a shortgating pore is indicated by the accessibility of modified S4positions to extra- or intracellular cysteine reactants or pro-tons (Yang et al., 1996). The relevance of counter-chargeshas been indicated by neutralization mutants (Seoh et al.,1996) and coordination of S4 charges with residues of op-posite charge in recent crystallographic studies (Long et al.,2007).An understanding of the natural design of the voltage sen-sor needs to be based on a broad exploration of the compo-nents that have been discovered experimentally. The im-portance of design elements that have emerged from theseexperiments should be evaluated. How do such elements de-termine sensor characteristics? How do their specific evolu-tionary layouts bring about voltage sensor behavior as seenin different channel types? And as an important control, canresults of experimental mutations be predicted?It is tempting to base exploration of a molecular device likethe voltage sensor solely on available information of atomicstructures, as represented in Fig. 1.3. The study of atomicstructures, however, is greatly limited in the case of the volt-age sensor. Crystallization destroys the natural (dielectric)environment of the structure and in particular the electricfield that the structure is designed to detect. Atomic-levelcomputations like those based on Molecular Dynamics ( MD )are speculative to the extent that they are based on ques-tionable atomic coordinates. Moreover, they are currentlyrestricted to explorations of sub-microsecond episodes of dy- namics of a few chosen initial configurations (Khalili-Araghiet al., 2010).In this thesis, I analyze reduced electrostatic models (see Concepts, sec. 2.1 ) for the voltage sensor which can be thor-oughly specified: charged rigid bodies moving through piece-wise homogeneous dielectric domains. The sensor model isembedded in a simulation cell mimicking a voltage-clampsetup including electrodes, allowing macroscopically ob-served variables to be predicted from the microscopic model.The electrostatics are solved self-consistently with numeri-cal methods that allow systematic exploration. Specifically,my computations construct an electrostatics-based partitionfunction of charge configuration with two degrees of free-dom: rotation and translation of the S4 segment. Usingthis partition function, I compute the expectation of randomvariables such as charge displacement in response to volt-age for comparison with experiment. These measures havenot been accessible in other computational studies (Bliznyukand Rendell, 2004; Chakrapani et al., 2008; Nishizawa andNishizawa, 2008; Khalili-Araghi et al., 2010).My results show that a voltage sensor ( VS ) model involv-ing a sliding S4 helix is realistic with respect to both gat-ing charge and electrostatic energetics. A crucial compo-nent of this model is negative counter-charges arranged inclose proximity to buried S4 charges. The electrostatic de-sign of this VS tolerates considerable variation resulting inelectrophysiologically interesting variations of sensor charac-teristics. Simulations of S4 , S2 and S3 charge mutants in thiselectrostatic model predict experimentally observed patternsof charge displacement (Seoh et al., 1996).7 E191 D190R189R286R287E183R293R296E226R299 K302R305R163E154E157 K308K247 R240D236D259E274 D220K277R147
Figure 1.3: Crystal structure of a chimeric K v K v K + channel (Long et al., 2007, PDB No. 2r94). This single unit of the channeltetramer is composed of K v S4 and parts of S3 replaced by homologous regions from K v S1 – S4 are represented by ribbons; charged residues (arginines, lysines, glutamates and aspartates) are represented in ball-and-stick format.The approximate S1 region ribbon is in blue; S2 is in red; S3 is in yellow; S4 is in brown; S5 is in cyan; S6 is in purple; and the pore loop is inorange. K + ions near the pore loop are represented by green spheres. Charged residues which may be part of the voltage sensor ( S1 – S4 ) arelabeled by their residue number for this chimera. Graphic produced with VMD (Humphrey et al., 1996) hapter 2 Methods
My approach to voltage sensor electrostatics has three ma-jor elements: (1)
I use reduced physical descriptions of elec-trostatic components; (2)
I compute predictions that corre-spond to experimental conditions and observables ; and (3)
Idevelop a statistical-mechanical description of sensor behav-ior by exploring a wide range of sensor configurations.
Reduced descriptions.
By a ‘reduced model’, I mean amodel where some details are reduced in resolution; for ex-ample, in electrostatic models many charge distributions arereduced to dielectric constants or tensors. The selection ofwhich features are to be reduced and which are to be repre-sented at higher resolution is an iterative problem involvingthe identification of the relevance of those features to themeasures of interest. All models that are not calculated by ab initio quantum mechanical calculations are reduced inthis sense. By explicitly structuring a model with multipletiers of resolution and identifying the relevance of those com-ponents of the system to measures of interest, it is possibleto distinguish the dominant terms in the underlying physics.Atomic matter is made of charged constituents, chargednuclei and charged electrons (Feynman et al., 1963). Thecrucial charges for this study are the uncompensated nuclearcharges in the arginine or lysine side chains of the S4 regionand the excess electronic charges of the glutamate and as-partate side chain in the S2 and S3 regions (Creighton, 1984;Islas and Sigworth, 1999), out of the vast number of nucleiand electrons in the system. Most of the ‘vast number’ ofcharges form neutral atoms or molecules, but at close rangemany molecules reveal spatial asymmetries in their internalcharge distributions. Moreover, the electric field from other components can distort the internal charge arrangements ofmolecules or groups that are overall neutral into asymmet-rical distributions of charge. Both the rotational alignmentof molecules with internal charge asymmetry and the dis-tortion of symmetrical charge distributions in molecules areabstracted as ‘polarization.’ Polarization of matter in anelectric field ‘induces’ charge that is hidden in the absenceof the field (Griffiths, 1999c, see Fig. 22). The most abun-dant polarizable molecule in VS system is the water of thesolutions bathing the membrane.An ab initio (quantum mechanical) description of thecharged nuclei and electrons in a channel protein, membrane,and embedding ionic solutions is not possible — approxima-tions must be made. Approximate physical descriptions canbe made in multiple tiers of resolution: either the results oflower tiers become parameters for higher-tier theory, or inde-pendent experimental results produce parameters for highertiers. For example, an ion is an atom or molecule in whichthe number of nuclear elementary charges is not identicalto the number of electron charges. A classical approxima-tion for an atomic ion is a point charge at the center of asphere where the effective diameter of the sphere that ex-cludes other atoms can be determined from crystallographicmeasurements. I use such a classical description for formalcharges explicitly included in models.Polarization of neutral molecules and groups must also bedescribed by approximations. Voltage sensor charges buriedin the membrane polarize bath water. A reasonable start-ing point for a description of that polarization is a contin-uum description of the water. A space element contain-ing polarizable molecules will exhibit polarization chargeson its surfaces when an electric field is present (Boda et al.,2004). These surface charges represent the polarization of9he molecules in the space element: if these charges are in-cluded in a computation of the electric field, the molecularpolarization is accounted for. The amount of polarizationcharge is proportional to the strength of the applied field(over a range), and depends on the atomic/molecular com-position of the matter in the space element. In the linearrange, the polarization charge at the surfaces of the space el-ement is related to the field strength by a material coefficient(or tensor for anisotropic polarization). Water polarizationcan be described this way by one number, the dielectric coef-ficient. Furthermore since polarization involves a rearrange-ment of charges, polarization takes a finite amount of time todevelop or disappear. However, since voltage sensor relax-ations are slow compared to typical polarization relaxations,polarization can be approximated as instantaneous. For in-stance, the rotational relaxation time of water is on the orderof 10 -11 s (Barthel et al., 1995), while voltage sensor relax-ation times are on the order of > -5 s (Hille, 2001; Sigget al., 2003).The channel and membrane are bathed in electrolyte so-lutions on either side. Electrostatic interactions involvingbath ions are reduced two orders of magnitude by the sol-vent (water) with respect to the vacuum. This reductionis described by the dielectric coefficient of water. Further-more, ions screen one another to an ionic concentration-dependent extent. Screening arrangements in solutions formin nanoseconds (Barthel et al., 1995), which is much fasterthan the time base for VS motion — like polarization, screen-ing can be approximately described as instantaneous.Screening by bath ions is modeled in my simulationcells by placing the electrodes closer to the membranethan would be done in an experiment. A diffuse layerof screening counter-ions is electrostatically equivalent tocounter-charge smeared on a surface a Debye length fromthe screened charge, which is the essential result of theDebye-H¨uckel and Gouy-Chapman descriptions of screen-ing (Debye and H¨uckel, 1923; Gouy, 1909; Chapman, 1913).By varying the distance between the membrane and thescreening electrode surface, variations of bath ionic strengthcan be mimicked (for example, offsets of 3 nm and 0 nmbracket the Debye lengths of dilute [3 mM] and infinitelyconcentrated solutions ). In the simulations reported here Iuse the electrode geometries of Fig. 2.1, roughly equivalentto bath solutions containing 3 mM salt. Experimental vari- See Fig. 3 on pg. 3718 under
Defining electrical coordinates andelectrical travel of Nonner et al. (2004) ation of bath ionic strength has relatively small effects onexperimental gating currents (Islas and Sigworth, 2001). In-deed, control computations (not shown) in which the waterdomain is removed from the simulation and the electrodesare placed directly on the membrane and protein surfaces (tomimic infinite ionic strength) yield results similar to thoseobtained with the simulation cells of Fig. 2.1.
Coupling microscopic VS motion to macroscopic ex-periments. Computer experiments provide insights intomicroscopic systems that are difficult if not impossible toobtain by conventional experiments. To estimate the func-tional competence of hypothetical structures, experimentalobservables of function must be computed, and conditionscomparable to conventional experimental conditions mustbe established. Gating current, the most direct observablereporting VS motion, is recorded experimentally while ap-plying a prescribed voltage across electrodes placed in theelectrolytes bathing the membrane (a voltage clamp). Mycomputational setup is designed to establish a voltage clampand record the charge displaced by VS motion in a mannercomparable to charge displacement recordings with macro-scopic electrodes. From electrostatics to statistical mechanics.
The pri-mary results of my computations are the electrostatic poten-tial energy and gating charge corresponding to a particularlocation of the formal charges of the voltage sensor model ata particular applied voltage. (Another output is the Maxwellstress, see below in
Maxwell stress, subsec. 2.2.6 ). The effi-ciency of my computational methods allows me to computethese variables for a very large number of VS configurationsand applied voltages, thus enabling me to elaborate a sta-tistical (thermodynamic) view of VS configuration.I explore a configuration space that includes the rotationof the S4 helix about its axis and the translation of the S4 charges along that axis at a fixed applied potential. Apartition function in those configurational degrees of free-dom is constructed from the ensemble of Boltzmann fac-tors for each electrostatic potential energy. Using the par-tition function, statistical expectations of equilibrium po-sitions and displaced charges are obtained. In this way, Idetermine both how the model will configure at a particularmembrane voltage, and along which average configurationsthe model will re-configure as voltage is varied in small in-crements – these are equilibrium averages and not trajec-10ory calculations. The relation between gating charge andvoltage is predicted, allowing comparison with experimentalcharge/voltage curves recorded from ensembles of voltagesensors. The typical simulation cell.
A typical geometry for acomputational experiment is shown in 2.1 (a). The simula-tion environment is represented by an axial cross-section ofthe radially symmetric three-dimensional domain swept byrotating that cross-section about its vertical axis. The hemi-spheric boundaries (in green) are electrode surfaces kept atcontrolled electrical potentials. The blue zones representaqueous baths (with a dielectric coefficient (cid:15) = 80). The pinkzone is a region of small dielectric coefficient ( (cid:15) = 2) thatrepresents the lipid membrane. The brown zone representsboth the non- S4 components of the VS protein and the ma-trix of the S4 helix (the central cylinder). The helix crossesthe membrane through a ‘gating pore’ which extends thebaths into the region joining S4 and the rest of the protein.The dielectric coefficient of this protein region is varied inmy simulations to assess its importance for VS motion. Thedielectrics in this reduced model are piecewise uniform andtherefore have sharp boundaries (solid black lines). Pointcharges representing S4 and other charges (not shown) areembedded in the region of protein dielectric. VS motion is simulated in this geometry by moving onlythe S4 charges within the fixed cylinder bounding the S4 helix. Simulating S4 motion this way (by moving chargesand not dielectric boundaries) greatly reduces the computa-tional costs of solving the electrostatics since solving for theeffects of moving dielectric boundaries requires recomputingthe matrix inversion (see Matrix inversion, subsec. 2.2.2 ). Thedielectric geometry as defined is invariant in terms of rota-tion. Only the end caps of the S4 are not invariant in terms ofcentral-axis translation. If those ends of the biological S4 aretranslated, which is not well defined experimentally, then thepositionally fixed end-points of the S4 in these simulationswould not capture the movement of surface charge on those S4 ends. However, the surface charge on those ends wouldmove outside the region of high electrical travel close to thegating pore (Nonner et al., 2004, Fig. 1), therefore addinglittle to either the charge displacement terms or the asso-ciated energy terms. Charge/end-surface interaction termscould only become significant at the extrema of S4 chargemotion along the central axis. My general approach for computing the electric field in thissystem is to superpose the vectorial Coulombic fields ofall charges. Likewise, the electrical potential is computedby scalar superposition of the Coulombic potentials of allcharges (Jackson, 1999b, section 3). The contributions tothe field made by the S4 and other formal charges at knownpositions are readily computed, but the charges on the elec-trodes and the polarization charges at dielectric interfacesare initially unknown. The determination of the contribu-tion of surface charges is a crucial (and computationally ex-pensive) element of solving the electrostatics.Since dielectrics are involved, a precise definition of‘charge’ is needed for the following mathematical treatment.I follow the nomenclature of Jackson (1999a), who distin-guishes between ‘source’ and ‘induced’ charges. The formalcharge on a side-chain of the S4 region (one proton charge)is a source charge, as are charges placed by the externalvoltage clamp circuit on the electrodes. Induced chargescomprise those charges appearing on dielectric boundariesin response to the fields of source charges and other inducedcharges. It is convenient to combine spatially inseparablesource and induced charges into ‘effective’ charges. Specif-ically, I assign to a point source charge ( q s ) embedded in adielectric ( (cid:15) ) an effective charge ( q = q s /(cid:15) ), combining thesource charge and the polarization induced at the dielectricboundary around that source charge ( − q s ( (cid:15) − /(cid:15) ). In otherwords, the effective point charge is the sum of the source andinduced charges, expressing the fact that the field producedby a source charge in a dielectric is (cid:15) times weaker than thefield of the source charge in vacuum. Electrode charges arealso represented in computations by effective charges, whicheliminates the need to specify polarizable matter outside thecell. Finally, induced charge on a dielectric boundary sepa-rated from any source charge is formally treated as an effec-tive charge (lacking a source charge component). Thus, all charges are expressed in computations as effective charges.The field computed from effective charges is identical to thesuperposition of the fields of the source charges and theirpolarization charges. That superposition of fields is the fieldthat must be computed for the computer experiments pre-sented here.11 a) Simulation Environmentfor Sliding Helix ǫ w = ǫ m = ǫ p = S4 . − − . − nm n m Gating PoreGuardElectrode Clamp Electrode
Lipid
S2 & S3 (b)
Simulation Environmentfor Paddle ǫ w = ǫ m = S4 . − − . − nm n m GuardElectrode Clamp Electrode
Lipid S3 Figure 2.1:
Cross-sections of the simulation cells for the sliding helix model (a) and the paddle model (b). The 3D setup is produced byrotating the cross-section about its vertical axis. The setup is bounded by two hemispherical bath electrodes (in green). Their potentials aremaintained at prescribed values as the S4 segment is moved (voltage clamp). The cylindrical electrode (in gold) completing the boundary isa guard. It is divided into rings maintained at potentials that are linearly graded between the potentials of the bath electrodes. The interioris divided into two aqueous baths (in blue) separated by a lipid membrane (in pink) and, in the sliding helix model, a VS protein region (inbrown) forming a gating pore around the sliding S4 helix. In the paddle model, no distinction is made between membrane and protein — the S4 charges (not shown) are embedded in the membrane. The charge configuration of the sliding helix model is shown in Fig. 2.2. To compute the charges on the electrodes and dielectricboundaries, the boundary surfaces are tiled into curvedquadrangular surface elements (Fig. 2.2 for the sliding he-lix model & Fig. 2.3 for the paddle model). The size ofthese surface elements is chosen such that the charge den-sity present on an element can be approximated as uniform on the element. Properly choosing the tile size to conformto that approximation requires numerical controls describedlater (
Gauss box, subsec. 2.2.3 ). There is one unknown to bedetermined for each surface element: its surface charge. Be-low, I will show how one linear equation can be obtained persurface element. Solving the system of these linear equationsyields all unknown charges, which typically involves 4000–10000 surface elements of different sizes (and that number of12nknowns) for numerical accuracy (checked by Gauss’ law torecover the total integral number of charges within volumessurrounded by closed surfaces, see
Gauss box, subsec. 2.2.3 ).The electrodes impose voltage clamp conditions. In thediscrete representation of the electrode surface, the poten-tial at the center of an electrode element ( i ) assumes a pre-scribed value ( V i ). The potential at the center of that ele-ment results from the superposition of the potentials due to all charges ( j ), including that of element i itself: V i = 14 π(cid:15) o (cid:88) j q j | r ij | (2.1)The definition of the distance | r ij | for a surface charge j de-pends on the distance of that element from element i . Thesurface charge of a distant element is combined into a singlepoint charge located at the charge center of that element,and | r ij | is defined as the distance between those points.The surface charge of a proximate element (in particular,element j = i itself) is divided into smaller charges obtainedby sub-tiling element j into a number of subelements (typi-cally 4 × j (cid:54) = i , and 10 ×
10 for j = i ). Each subelementcarries a fraction of element j ’s charge, and 1 / | r ij | is definedas equal to the weighted average by subelement area of theinverse distances for all subelements to the center of i . Thesubelements follow the curvature of the surface element.A correct representation of surface curvature and the cur-vature’s effects on charge distribution is crucial for numericalaccuracy. Inhomogeneities in charge distribution are partic-ularly problematic when source or other induced charges areclose to the surface element. If the element has curvature,the charge induced on one part of the element will inducecharge at close range on other parts of the same element.Inaccuracies due to charge inhomogeneity and surface cur-vature are limited by choosing an adequate initial tiling andsub-tiling as needed.The boundary condition describing the effect on the fielddue to dielectric boundary elements can be expressed in twoequivalent ways. One way of describing the boundary condi-tion is that the normal components of the field strengths oneach side of the boundary ( E ⊥ and E ⊥ ) are inversely relatedto the dielectric coefficients of each region (Griffiths, 1999c,40): (cid:15) E ⊥ = (cid:15) E ⊥ (2.2)The other expression for relating normal field strengths saysthat the field strengths differ by the field of the polarization charge induced on the surface (Jackson, 1999c, 22): E ⊥ + σ(cid:15) n = E ⊥ (2.3)where n is the unit normal vector from the region of (cid:15) = (cid:15) tothe region of (cid:15) = (cid:15) . Furthermore the normal field strength exactly at the surface, without including the field due toinduced charge on that surface ( E ⊥ ), is the average of thenormal field strength at an infinitesimal distance from thesurface including all charges: E ⊥ = E ⊥ + E ⊥ E ⊥ and E ⊥ arises by superpositionwith the field of the polarization charge at the surface whichhas a magnitude of σ/ (cid:15) in both normal directions (Grif-fiths, 1999b, 17).Eqs. 2.2–2.4 can be combined into one expression relatingthe density of induced polarization charge ( σ i ) to the normalfield strength on the surface of dielectric boundary element i : σ i = 2( (cid:15) − (cid:15) ) (cid:15) + (cid:15) (cid:15) o E ⊥ i (2.5)The component of the field strength normal to the tangentplane of the boundary surface element is (Jackson, 1999c,4): E ⊥ i = (cid:88) j q j π(cid:15) o ( r i − r j ) | r i − r j | · n i (2.6)If charge j is the charge of a distant surface element, it istreated as a point charge at r j . Otherwise if element j is asub-tiled surface element, the expression ( r i − r j ) · n i / | r i − r j | in (2.6) is replaced by its weighted average (by subelementarea) taken over all subelements of j . Note that q j is the effective charge of each entity.Eqs. 2.1 and 2.5 suffice to compute all initially unknownsurface (electrode or dielectric) charges. With these chargesknown, all charges in the system are known. The electricfield can then be computed for any location by superpositionof the Coulombic fields of individual charges (using appro-priate sub-tiling for nearby surface charges). My computer code calculates discretely tiled radially sym-metric surfaces or, in other words, boundary surfaces of13hapes that can be produced by turning a piece on a lathe,including hollow shapes (cylindrically symmetrical). Tech-niques for tiling discretely more general surfaces are knownand could be used together with my method for solvingthe electrostatics needed to explore more general geometriesthan those in this study.The electrostatic field is long-range — the field strengthat all locations depends on all charges. Since the coeffi-cient matrix for those relations is dense, the computationalmethod chosen for solving the linear equations to determinesurface charges is LU decomposition (Bowdler et al., 1971,implementation by Whaley and Dongarra, 1997). Invert-ing an N × N matrix by LU decomposition requires order N operations. Fortunately however, the LU decompositionneeds to be executed only once for a given surface geometry.If source charges are moved, the electrostatic equations canbe solved by repeated back-substitutions using the inversematrix, requiring order N operations. The error from approximating the surface charge distribu-tion as piecewise uniform can be assessed using Gauss’ law.(The approximation requires that the field strength at anypoint on a surface element equals the field strength measuredat the center of charge of the element.) Gauss’ law statesthat the integral of the electrical flux normal to a closedsurface over that closed surface is proportional to the totalsource charge contained in the enclosed volume (Jackson,1999d, 39): (cid:73) S (cid:15)(cid:15) E · d a = (cid:88) i ∈V q s,i (2.7)This conservation law holds for any closed surface of any shape, and thus applies to any geometry of interest.Since the density of polarization charge induced by acharge on a dielectric boundary is particularly inhomoge-neous when the charge is close to the surface, the local sur-face region proximate to other charges must be made intosmaller discrete surface elements. On the other hand, anexcessive number of surface elements is computationally ex-pensive. Since S4 charges move relative to dielectric bound-aries in my computer experiments, all S4 positions must betaken into account in designing surface tiling. The adequacyof the surface tiling must be checked for all positions taken by S4 charges, which can be done by verifying Gauss’ lawfor each position of the S4 . For instance, the sum of all S4 and other source charges contained in the region of proteindielectric in Fig. 2.1 (a) must be accurately recovered bysumming the normal field strength multiplied by the per-mittivity and the element area, over the surface boundingthat region ( (cid:80) j ∈S (cid:15) j (cid:15) E ⊥ j a j = (cid:80) i ∈V q s,i , where j variesover all elements of the closed surface S and i varies over allcharges within the volume V enclosed by S ).My approach to solving the electrostatics differs from themore common approach of solving Poisson’s equation ona spatial grid. The approach I use (Boda et al., 2004) isbased on relations describing boundary conditions (makingthe method a ‘boundary element method’, BEM ). The re-sulting boundary integral equations are made discrete on asurface grid. Owing to the relatively small number of sur-face elements compared to the number of volume elements,a full description of the charge distribution and thereforethe electric field can be stored in computer volatile mem-ory. From this information any desired electrostatic outputvariable can be computed. Exhaustive a posteriori tests ofsolution accuracy are possible, such as verification of resultsby checking the consistency of calculated surfaces chargesagainst the integral number of charges enclosed using Gauss’law. S4 charge movements are restricted to a subrange of thedistance between the electrodes. The electrodes record adisplacement current due to the variations of the electricalfield produced at the electrodes by the S4 charges. It isimportant to assess this charge displacement because thedisplacement current (and therefore its integral, the chargedisplacement) can be directly observed in experiments: thedisplacement charge is the ‘gating charge’.The electrode charge displaced by the motion of S4 chargesis assessed in independent ways to check numerical accuracy.One method directly measures the integral of the displace-ment that reaches the internal or external electrode. Con-venient surfaces for measuring this displacement are the in-ternal and external dielectric boundaries of the membraneand protein. Since the electric field strength perpendicularto these surfaces has already been determined in the compu-tation of induced charge, the electrical fluxes through thesesurfaces can be computed via integration of the known nor-14al field strength over the surface area.A method to solve for charge displacement is provided bythe Ramo-Shockley theorem (Shockley, 1938; Ramo, 1939).This method first solves the electric field determining the‘electrical distance’ of a source charge, which is the fieldproduced by the electrodes in the absence of source chargesinside the simulation cell. The gating charge correspondingto an S4 position in a simulation including all charges isgiven by scaling source charges by their respective electricaltravel: Q = −
11 volt (cid:88) j q j [ U ( r (cid:48) j ) − U ( r (cid:48)(cid:48) j )] (2.8)where U is the potential with all source charges removedand the external electrode fixed at 1 volt; and r (cid:48) j to r (cid:48)(cid:48) j arethe endpoints of the trajectory of q j . This method has beendescribed for simulation cells like the ones used in this study(Nonner et al., 2004). An important implication of this the-orem is that the gating charge contributed by an S4 chargeis exclusively determined by the position of the S4 chargein the simulated system. It is entirely independent of othersource charges, fixed or moving, that exist in the system.Another implication of the Ramo-Shockley theorem isused in the computation of potential energy (described belowin Electrostatic potential energy, subsec. 2.2.5 ). The Ramo-Shockley theorem implies that applying a voltage to the elec-trodes modifies the electrical potential at the location of acharge by the fraction of applied voltage corresponding tothe electrical distance of that charge ( U ( r j ) / The charges of the VS move in an electric field that orig-inates from other intrinsic charges, charges induced on di-electric boundaries, and charges delivered from an externalbattery to the electrodes to establish a voltage clamp (orunder natural conditions from the charges that produce theaction potential). There are therefore both internal and ex-ternal sources of electrical work. The biological purpose ofthe voltage sensor is to transduce electrical work from thisenvironment into mechanical or other work that is appliedto other parts of the channel, in particular toward operatingthe gate of the conducting pore.The VS can be viewed as traveling in a force field that isprobed by fixing the positions of the VS while measuring the force on the VS , both translational and rotational. Integrat-ing the force along the path of movement gives the requiredwork for moving the VS between the endpoints of the probedpath: ∆ W = (cid:88) s q s (cid:90) r s, new r s, old E ( r s ) · d r s (2.9)where the summation is over the source charges ( q s ) movingwith the VS , and E is the field produced by all charges of thesystem, excluding q s itself. Note that the electrode charge iscontinually updated by the voltage clamp as the VS movesalong the path.This method of computing work requires that an entirepath be scanned at small intervals for a path integral. Itis computationally expensive and generates cumulative nu-merical errors. A more efficient method based on a directcalculation of work is desirable, one that only needs to probethe endpoints of the path. However, I have used the path in-tegration of force as a useful control for other methods thatcompute work.A highly efficient method to compute work can be con-structed using the Ramo-Shockley theorem. This methodrequires the computation of the configurational energy whenthe electrodes are grounded (hence V m = 0) and the chargedisplacement for a given configuration of source charges.Computation of the charge displacement is described in Com-putation of gating charge, subsec. 2.2.4 , and the method forcomputing configurational energy when V m = 0 follows here.The configurational energy considered in my simulationsis the electrostatic energy (including all implicit polarizationstress). The configurational energy at a given potential is theinteraction energy of every charge in the system with: thesource charges; induced dielectric polarization charges (in-cluding implicit microscopic stress from polarization); andthe charges on the electrodes: W config = W s + W diel + W el (2.10)where the interaction energy (with all charges) of: sourcecharges is W s ; induced charges for dielectrics is W diel ; andelectrode charges is W el .The configurational energy can also be decomposed intothe total electrostatic interaction of all explicit charges inthe system plus the implicit mechanical energy of twisting15nd stretching the polarized molecules: W config = W electrostatic + W diel,stress = 12 (cid:88) j q j V ( r j ) + W diel,stress (2.11)The total electrostatic energy of a discrete charge distribu-tion is (cid:80) j q j V ( r j ), where q j are all the source and inducedcharges in vacuo including dielectric and electrode charges,and V ( r j ) is the potential due to all charges not locatedat r j (Griffiths, 1999b). For dielectric polarization charges,the additional mechanical term W diel,stress corresponds tothe mechanical work of twisting and stretching polarizedmolecules, including any implicit electric fields involved inthese deformations.The configurational energy when the electrodes aregrounded (cid:16) W V m =0config (cid:17) is then: W V m =0config = 12 (cid:88) s q s V ( r s ) (cid:124) (cid:123)(cid:122) (cid:125) W s + W diel,electrical (cid:122) (cid:125)(cid:124) (cid:123) (cid:88) d q d V ( r d ) + W diel,mech (cid:124) (cid:123)(cid:122) (cid:125) W diel ≡ + 12 (cid:88) el q el V ( r el ) (cid:124) (cid:123)(cid:122) (cid:125) W el =0 (2.12)where V ( r ) is the potential due to all charges not locatedat r , q s are the source (not effective) point charges, q d are all induced polarization charges on dielectric boundariesand surrounding source charges, and q el are all charges onelectrodes. The work to polarize dielectric boundaries is W diel ≡ W el = 0) when the electrodes aregrounded: V ( r el ) = 0. Therefore, the total configurational energy when the elec-trodes are grounded is: W V m =0config = 12 (cid:88) s q s V ( r i ) (2.13)In other words, the total configurational energy is equal tothe work to place the source point charges in the potential field produced by all charges, including those produced bydielectric polarization and charges on the electrodes. The Ramo-Shockley based method used for the effi-cient computation of electrostatic potential energy is basedon the fact that the Ramo-Shockley theorem allows the dis-section of the total electrostatic work (including configura-tional and displacement energies) into two components ∆ W and ∆ W , computed separately as follows (He, 2001):1. With zero voltage applied to the electrodes, computethe configurational energies for the given old and newpositions of the VS charges. If the position of VS chargeschange in terms of electrical distance between these twogeometrical positions, displacement charge flows fromone electrode to the other.Since both electrodes are at the same potential, no workis involved in that charge displacement. Hence,∆ W = W V m =0config , new − W V m =0config , old (2.14)2. Apply the desired voltage V m between the electrodes.This will modify the potential energy of the VS charges q i by ∆ W = − QV m (2.15)where Q is the charge displaced at the electrodes whenmoving from the old to the new position (see Eq. 2.8).Therefore, the total potential energy change sensed by theVS is: ∆ W = W V m =0config , new − W V m =0config , old − QV m (2.16)This work defines the ‘electrostatic potential energy land-scape’ in which the VS operates under an applied voltage.Note that the configurational energy only needs to be de-termined for one applied voltage, 0 mV. Landscapes for anyapplied voltage can then be computed once the displace-ment charge for the S4 charge position at that applied volt-age is known. The displacement charge itself is efficientlycomputed using the Ramo-Shockley theorem as describedpreviously (Eq. 2.8). Note that for all graphs, electrostaticpotential energy of the VS will be reported relative to the0 nm translation position (and 0 ◦ rotation for potential en-ergy landscapes).16 .2.6 Maxwell stress Charges buried in the protein polarize water in the bathsand attract induced polarization charges. The polarizationcharges are induced on the water surface, and their attrac-tion produces a pressure (normal component of the Maxwellstress) that tends to bring water toward the charges buriedin the protein. If a lipid bilayer is uniformly charged to400 mV, the electrostrictive pressure across the bilayer isabout 0.3 MPa (3 atm). This condition typically breaksthe bilayer by hypothetically stabilizing the formation andexpansion of transmembrane pores (Melikov et al., 2001;Troiano et al., 1998).The normal component of the Maxwell stress (pressure, P )acting at a dielectric surface is the product of the inducedsurface charge density ( σ ) and the normal component of theelectric field at that surface ( E ⊥ ): P = σ E ⊥ (2.17)Both quantities on the right hand side are computed inmy electrostatic analysis of the system (Eq. 2.5). Maxwellstress may stabilize or destabilize structures composed ofcharges embedded in a weak dielectric. Therefore, I assessthe strength and distribution of the normal component ofMaxwell stress for VS models. A partition function is computed from the electrostatic po-tential energy. The configuration space has two degrees offreedom in VS motion: rotation and translation. The par-tition function (“the key principle of statistical mechanics,”Feynman 1988) in discrete configuration space is: Q = (cid:88) i,j e − E ij /k B T (2.18)where i and j are the indices of the rotational and trans-lational discrete positions; E ij is the electrostatic potentialenergy of the VS in configuration ( i, j ); k B is the Boltzmannconstant; and T is absolute temperature.Movement of S4 charges is restricted in my studies to therotational range of − ◦ to +180 ◦ and a typical translationrange of − .
925 nm to +1 .
925 nm relative to the centralposition. Each degree of freedom is made discrete in 50increments resulting in a total of 2500 energy computationsfor the discrete partition function. With this partition function known, the probability of aconfiguration is: P ij = 1 Q e − E ij /k B T (2.19)and the expectation value of a random variable X is: (cid:104) X (cid:105) = (cid:88) i,j X ij P ij = 1 Q (cid:88) i,j X ij e − E ij /k B T (2.20)The random variables of interest are the rotational andtranslational positions, the associated gating charge and theMaxwell stress. These are computed for 1 mV steps in therange from -100 mV to +100 mV of membrane voltage.I also compute the expectation of electrostatic potentialenergy for the VS at the particular translational positionswith rotational equilibrium established at each translationalposition. The statistical weights are then given by the rota-tional partition function for that translational position. The model comprises a region of inhomogeneous dielectricssurrounded by an eggshell-shaped system of electrodes(Fig. 2.1). The electrode eggshell is composed of three re-gions: two half-spheres (of radius r = 5.0 nm) intercon-nected by an open cylinder (of radius r = 5.0 nm and height d = 3.0 nm). The intracellular hemisphere is at a fixed po-tential ψ/
2, where ψ is the applied transmembrane voltage;the extracellular hemisphere electrode is anti-symmetricallyfixed at − ψ/
2. The cylindrical electrode joining the hemi-spherical electrodes is subdivided into bands, each of whichis held at the potential − ψ ∗ z/d , where z is the height of thecenter of the band relative to the midpoint of the system,and d is the total length of the cylinder. Therefore the poten-tial on the cylinder varies linearly from − ψ/ ψ/ (cid:15) m = 2, representing its lipidcomposition. The protein region is located in the center of17he simulation environment and spans the membrane (itsdielectric is described by a varied dielectric coefficient, (cid:15) p );for the paddle model, (cid:15) p = (cid:15) m . The space between the up-per surface of the membrane/protein and the extracellularelectrode and the space between the lower surface and theintracellular electrode is an aqueous region with a fixed di-electric coefficient of (cid:15) w = 80.For all statistical mechanics calculations, simulation tem-perature is fixed at 30 ◦ C (see eqs. 2.18-2.20).
Embedded at the center of the membrane is the protein re-gion (of (cid:15) p ) representing the S4 and the surrounding trans-membrane domains; (cid:15) p is varied by experiment, but for mostcases is set at 4. The protein region is radially symmetric,of radius 2.15 nm.Moving radially inward from the membrane juncture,there is the ‘gating pore’ from 1.966 nm to 1.266 nm. Thesurface smoothly dips from a depth of ± ± ± S2 and S3 counter-charges are placedequally close to the protein/water interface for all variations.The pore is smoothed by rounded corners of radius 0.15 nm.At the center is the surface of the S4 proper, of radius1.266 nm and length 6.5 nm. The α -helix lies within thisenvelope. The S4 charges are at a radius of 1.0 nm, eachsplit into 3 charges of 1/3 e o on a circle of radius 0.122 nmcentered around the charge position (Fig. 2.2). This reflectsthe structure of arginine residues’ guanido group charge dis-tribution. The S4 charges are separated in the z -direction by0.45 nm and in the xy -plane by 60 ◦ (each sixth residue com-pletes a full turn counter to the direction of the α -helix).There are 6 S4 charges, some of which are eliminated orreplaced with dipoles in specific computations to simulatecharge-neutralization mutants.Counter-charges are located in the protein dielectric ona curve of 1.4 nm radius, concentric to the curve of the S4 charges. Counter-charges are generally spaced at angularintervals different from those of the S4 charges; the interval ratios of counter- and S4 charges are referred to by fractionsbetween 1/2 and 3/2. There are three counter-charges, cen-tered at the midpoint of the membrane. To simulate exper-iments with charge neutralization mutants, some of thesecounter-charges are either eliminated for specific computa-tions or replaced with dipoles.The relationship of S4 charges and counter-charges for agiven translation is given in Fig. 2.4. In figures mappingtranslation to applied voltage (such as Fig. 3.5 a), the trans-lational axis is labeled in terms of that offset — the dis-tance between the center of the S4 charges and the center ofthe model membrane and counter-charges along the trans-lational axis of motion, which is the vertical coordinate inFig. 2.1. Dipole mutants.
To simulate charge neutralization mu-tants from charged residues to glutamine or asparagineresidues, two methods are used: simple elimination of a netcharge as described previously or replacement of the chargeby a dipole. The dipole is centered at the same positionas the center of the original S4 charge or S2 & S3 counter-charge. The orientation of the dipole is radial, with thenegatively charged end pointing towards the central ( z ) axisof the simulation setup and the positive end pointing awayfrom the central axis. The charges of the dipole are sepa-rated by 0.27 nm and the magnitude of each charge is 1/2e (Pauling, 1960; Lozano-Casal et al., 2008). This dipolerepresentation of the mutant produces favorable interactionsnot present in the representation by simple charge deletion. In simulations of the ‘paddle’ model, the S4 region is entirelyburied in the membrane region with the axis of the helixparallel to the membrane plane. The number of S4 charges isfour (as in K v AP ). Since the radius of the α -helix is 1.122 nm,the membrane thickness is extended by 0.5 nm to 3.5 nm,allowing translational motion as well as rotational motion ofthe S4 helix within the membrane. I explore the range ofmotion possible without the emergence of the S4 region intothe baths.18 a) Side (b)
Bottom
Figure 2.2: Source charges and dielectric boundaries in a sliding helix model.
The sliding helix bears six triplets of 1/3 e point charges(shown as red balls for visibility) representing the guanidinium group of arginine residues. These charges are aligned on a superhelix thatis oriented in the direction opposite to that of the S4 helix. The S4 charges are shown at the 0 nm/0 ◦ position of their translational androtational dimensions of motion. Counter-charges are -1 e point charges (blue balls). They are aligned in a superhelix concentric with thesuperhelix of the positive charges (shown for the ‘2/3’ spacing of counter-charges). The dielectric boundaries are shown as a 2D grid; themesh is a dimensional representation of the surface tiling used in solving the electrostatics (tiling is fine near positions that are close to sourcecharges). The change in average position of the S4 charges under a gradual change of voltage from -100 mV to +100 mV can be seen in thesupplementary movies: tile-helix-side.mp4 [suppl.] & tile-helix-top.mp4 [suppl.]. a) Side (b)
Bottom
Figure 2.3: Source charges and dielectric boundaries in a paddle model.
As in the sliding helix of Fig. 2.2, the paddle bears fourtriplets of 1/3 e point charges (shown as red balls for visibility) representing the guanidinium group of arginine residues. These charges arealigned on a superhelix that is oriented in the direction opposite to that of the S4 helix. The S4 charges are shown at the 0 nm/0 ◦ positionof their translational and rotational dimensions of motion. The dielectric boundaries are shown as a 2D grid; the mesh is a dimensionalrepresentation of the surface tiling used in solving the electrostatics (tiling is fine near positions that are close to source charges). The changein average position of the S4 charges under a gradual change of voltage from -100 mV to +100 mV can be seen in the supplementary movies: tile-paddle-side.mp4 [suppl.] & tile-paddle-top.mp4 [suppl.]. /11/22/34/33/2 0 0.2250.45 0.9 1.575 1.8 S4S2 & S3 -1-212
S4 Translation/nm
Figure 2.4: Configurations of S4 charges and counter-charges in sliding helix simulations. Translational positions of S4 charges aremarked in red and of counter-charges in blue . S4 charges are spaced at a uniform and invariant interval. Counter-charge interval is variedbetween simulations and is specified by its ratio with the S4 charge interval (labels below the columns). The S4 helix undergoes translationsuch that its charges line up with counter-charges to varying extents and at varying periods. This is illustrated on the right for a number of S4 positions (translation indicated at the top of columns). The same map applies to the rotational dimension since the rotational intervalsbetween charges are kept in a fixed proportion to the translational intervals (60 ◦ and 0.45 nm respectively for 1/1). hapter 3 Results and Discussion
Can an electrostatically viable voltage sensor model be con-structed on the basis of proposed models? If such a modelis at hand, its sensitivity to model parameter variation canbe explored in order to understand why the model works.Verifiable predictions can be made about the electrostaticconsequences of charge mutants. I will consider three crite-ria in evaluating VS models:1. The VS model produces adequate gating charge. For Shaker K + channels (Jan et al., 1977; Hoshi et al.,1990), a total gating charge movement of > e perchannel subunit is expected (Schoppa et al., 1992; Sigget al., 1994; Aggarwal and MacKinnon, 1996). Whenthe membrane potential changes from a large negativeto a large positive value, this charge determines howlarge the change in electric potential energy is for anindividual VS .2. The model VS moves at the time scale of gating. Sincesuch motion occurs in condensed matter, its rate is lim-ited by friction and possibly energy barriers. Considerthe passage of a single K + ion across the pore of a K + channel: in a large-conductance Ca -activated K + channel, a typical passage time is on the order of 30 ns(corresponding to ≈
20 pA conducted by a queue of3 K + ). If the VS experiences at least as much frictionas the friction in K + motion through a channel pore(even though the VS is considerably larger than K + )and the VS completes its motion within 3 ms (Islas andSigworth, 1999), then it can not encounter electrostaticenergy barriers greater than 11.5 kT ( ≈ VS provides adequate force for operatingparts of the channel molecule constituting the gate ofthe pore. This force is related to the number of VS charges that are in the membrane electric field at agiven moment. That number of charges is also a deter-minant of the slope of the gating-charge/voltage (Q/V)relationship. The VS model therefore must predict theslope of the experimental Q/V curve.My computations simulate an individual, isolated VS . The VS is therefore simulated under ‘zero-load’ conditions: itdoes not drive the gating machinery that a real VS woulddrive when integrated in the channel. Since experimentalgating currents have been recorded only from whole chan-nels, the VS performance parameters derived from these ex-periments likely need to be exceeded by a viable VS modelstudied under zero-load conditions. Specifically, the totalcharge movement, slope of the Q/V curve and rate of mo-tion are expected to be reduced in a VS operating under itsnatural load.In the following simulations, the S4 helix is modeled asa solid body with embedded positive charges which movewith two degrees of freedom, translation and rotation aboutthe helix axis. Counter-charges are kept in fixed positions.These constraints are a first step toward understanding theelectrostatics of the VS . If the VS is deformed in addition tothe translation and rotation of S4 , then total gating charge,internal friction, energy barriers and force developed will beaffected. In the original version of the paddle model (Lee et al., 2003),the S4 helix was embedded in the membrane lipid (like a pad-dle in water) and the proposed motion was like the transfer ofa large lipophilic ion between the two lipid/water interfaces.22his model has been found electrostatically implausible in aprevious computational study because the electrostatic workrequired to translocate the multiply-charged S4 helix acrossthe weak dielectric of the lipid is very large (Grabe et al.,2004). The original paddle model has been modified sinceits inception; a recently proposed version (Tao et al., 2010)has gained features of the sliding helix models that I haveanalyzed and which are described below ( A ‘sliding helix’model, sec. 3.2 ), while losing paddle-like features. This sec-tion presents computations of the original paddle model toassess how unfavorable electrostatic features produce unfa-vorable consequences for the stability and function of a VS design.I have simulated an S4 helix whose movements are com-putationally restricted to not extend the helix beyond theboundaries of the lipid membrane. Electrostatic potentialenergy of the VS (with respect to the central position) iscomputed while varying the position in two degrees of free-dom. The S4 helix is translated between the two membraneboundaries with its axis kept parallel to the boundaries andallowed to rotate fully about its axis. Electrostatic potentialenergy maps for three different applied membrane voltagesare shown in Fig. 3.1 (a-c), with energy represented in falsecolor. Note that these energy maps are similar for all ap-plied voltages, as if applied voltage has a relatively smalleffect relative to the contributions of other simulation pa-rameters.Panel (d) of Fig. 3.1 displays the mean electrostatic po-tential energy of the VS as a function of translation. Thisenergy is computed by averaging over all rotational anglesof the paddle using the statistical weights of the rotationalpartition function (see Statistical mechanics, subsec. 2.2.7 ). Inother words, the energy for each angle is used in a Boltzmannfactor to statistically weigh that energy to derive an overallexpectation energy for that translational position; graphi-cally, one point for a curve at specific potential in panel (d) isthe expectation value calculated from the matching column(by translation) of the respective potential energy graph ofpanels (a-c).The energy in (d) has a large maximum when the S4 axisis in the center of the membrane, more than 0 . VS in essence a bistable structure. When in one extreme posi-tion, the S4 is very unlikely to ever flip to the other position.My energy computations with this paddle model also re-veal that translating the helix produces strong rotationalforces. At the end points of the ± VS is minimalin different rotational positions. This applies for all testedmembrane voltages. The differences between favorable andunfavorable rotational positions approach 1 eV, making ithighly unlikely that the S4 segment could undergo transla-tion without rolling by about half a turn. A proposed modelin which the S4 helix moves by translation alone would pro-duce energetically unstable configurations. If the helix isallowed to follow electrostatic force in the rotational degreeof freedom, the electrostatics of this paddle model antago-nize VS function (Fig. 3.1 d).Lipid bilayers tend to break down when voltages largerthan 400 mV are applied (Melikov et al., 2001; Troiano et al.,1998). The electrostrictive pressure across the bilayer un-der those conditions is ≈ S4 and induced charges attract oneanother; therefore, the lipid/water boundaries are attractedtoward the S4 charges. This is the Maxwell stress. I havecomputed Maxwell stress on the lipid/water interface (see Maxwell stress, subsec. 2.2.6 ) to see how this stress relatesto the electrostrictive pressure known to break lipid bilay-ers. For this paddle model, the Maxwell stress on membraneboundary regions near the S4 charges is very large. Fig. 3.2shows the pressure distributions on the internal and externalbath boundaries with a logarithmic false-color scale (pres-sures range from 10 Pa in blue to 2.5 × Pa in red). Thepeak pressure on the bath boundaries is much larger than asafe electrostrictive stress, even if the S4 axis is centered inthe membrane (the minimal Maxwell stress configuration).The likely consequence of the large Maxwell stresses of thepaddle model is that the lipid retreats and thus exposes thecharged surface of the helix to both baths. Such a configu-ration would not function as voltage sensor.23 a) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − − − − E n e r g y / m e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUUUU (b) +100 mV − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . − . − . − . − . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUUUUU (c) -100 mV − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . − . − . − . − . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUUUUU (d) ∆ Energy − − − . − . . . . R e l a t i v e E l ec t r o s t a t i c E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKH
Ψ = 0 mV C Ψ = +100 mV K Ψ = −
100 mV
Figure 3.1: A paddle configuration is electrostatically bistable.
Panels (a-c): false-color maps of electrostatic potential energy of the VS scanned over two degrees of freedom for three applied voltages. The S4 helix axis is parallel to the membrane plane and translated inthe direction normal to the membrane plane. The S4 segment is buried in the lipid in all scanned translational positions. Rotation is aboutthe S4 axis. Electrostatic potential energy strongly favors positions near the bath interfaces at all applied voltages. Panel (d): Rotation-averaged electrostatic potential energy versus translation for three applied voltages (averaging is based on the rotational partition function).Since electrostatic potential energy depends on rotational position (panel a), allowing the S4 helix to rotate minimizes energy. Nevertheless,the profile of averaged energy is parabolic, and applied membrane voltage does not remove the large barrier to S4 translation. Note thatpanels (a-c) show the electrostatic potential energy of the VS relative to that at translation 0 nm and rotation 0 ◦ of that graph (Eq. 2.16),while panel (d) shows electrostatic potential energy of the VS relative to translation 0 nm (expectation over the rotational degree of freedom). a) Internal at 0 mV (b)
External at 0 mV (c)
Internal at +100 mV (d)
External at +100 mV
Figure 3.2: A paddle configuration is mechanically unstable.
False-color maps of the Maxwell stress acting on the water/lipid interfaces.This stress tends to pull the water boundary toward the S4 charges. Logarithmic color scale goes from 10 Pa ( blue ) through 10 Pa ( green )to 2.5 × Pa ( red ). The surface area shown is 10 nm in diameter. (An electrostrictive pressure of ≈ × Pa is known to break alipid bilayer.) The position in the membrane of the S4 segment is controlled by the applied membrane voltage (0 or +100 mV), and theMaxwell stress shown is the expectation of the Maxwell stress (based on the translational/rotational partition function). A supplementarymovie ( flat-paddle.mp4 [suppl.]) shows how Maxwell stress varies as applied voltage is varied between -100 and +100 mV. .2 A ‘sliding helix’ model The sliding helix models investigated here are VS models inwhich the axis of the S4 helix is oriented perpendicularlyto the plane of the membrane. Two independent kinds ofmotion are allowed: translation along the axis of the S4 he-lix and rotation about the axis. No particular trajectory inthese degrees of freedom is prescribed. Thus both motionsenvisaged for the ‘helical screw’ hypothesis of S4 motion arepossible but are not a priori coupled to one another as theterm ‘screw’ would imply. A concentric invagination of theprotein dielectric around the S4 helix forms a ‘gating pore’.In addition to the S4 positive charges, three negative pointcharges are present in the protein domain. For the spe-cific version of the sliding helix that I consider in this sec-tion, these are aligned in a spiral pattern concentric to thatof the S4 charges, but the angular and translational inter-vals between the counter-charges are chosen to be two-thirds(2/3) that of the S4 charge interval (i.e., 40 ◦ and 0.3 nm).The counter-charge positions are fixed. The dielectric ofthe protein is represented by a dielectric coefficient of 4.These parameters have been chosen via an iterative processto identify the envelope of parameters that are physicallyreasonable and consonant with known biology.The landscape of electrostatic potential energy (Fig. 3.3)is very different from that computed for the paddle model.When a membrane voltage of 0 mV is applied, a trough ofelectrostatic potential energy tends to confine the S4 chargesto a range of positions which can be reached by moving the S4 helix like a screw (Fig. 3.3 a). Thus the S4 segment inthis model tends to be electrostatically stable in its environ-ment (rather than being strongly driven towards the baths,like the paddle considered before). The energy trough isquite shallow however, so additional stabilizing features arerequired to ensure long-term stability.The bottom of the energy trough is nearly flat, allow-ing the helical screw to visit a wide range of positions withnearly uniform probability. When a strong positive or neg-ative voltage is applied to the membrane, the energy troughis shortened to a deep pit at either end of the S4 range ofscrew motion (Figs. 3.3 b & c). Panel (d) shows the expectedelectrostatic potential energy for each translational position(a statistical average over the rotational degree of freedombased on the rotational partition function, see Statisticalmechanics, subsec. 2.2.7 ). There are no significant energy bar-riers to translation. The applied membrane voltage simplytilts the flat bottom of the energy trough. Altogether, the electrostatic energetics of this model are consistent with ascrew motion — the rotation is a physical consequence ofthe electrostatics. This voltage sensor strongly resists eitherexclusively translational or exclusively rotational motion.As I do for the paddle model, I assess the mechanical sta-bility of the dielectric geometry of the sliding helix modelusing the computed Maxwell stress. Fig. 3.4 shows the pres-sure distribution on the protein/water interface (note thatthe invaginated interface has been mapped onto a plane, seefigure legend). Both the intra- and extracellular pressuredistributions are shown for a simulation with applied volt-ages of 0 mV or +100 mV.The Maxwell stress for the sliding helix model is largestwhere charges are close to the bath interface. Large stressesappear at the water interfaces of the S4 helix where chargesface a bath. These stresses do not compromise mechanicalstability since the charged groups are in direct contact withwater. The surface region lining the gating pore receives amoderate Maxwell stress, except for one angular region lo-cated at the bottom of the gating pore close to the innermostand outermost counter-charges (the gating pore is locatedbetween the white and yellow rings in Fig. 3.4). There, localMaxwell stress is on the order of 10 Pa. This magnitude ofstress provides a physical cause for the invagination of thewater boundary into a nano-scale gating pore, which requireswork against the surface tension of the water/protein inter-face. The existence of a gating pore is thus made plausibleby the electrostatics, although the gating pore postulated inthe model is not a computed consequence of the physics atthis level of modeling. The Maxwell stress due to the near-surface counter-charge is narrowly localized so that it, byitself, would not produce a full-circular gating pore like thatassumed in the model. On the other hand, S4 gating chargescould help stabilize a larger gating pore on the side(s) wherethey are exposed.Since this particular sliding helix model has desirableproperties, I compute the partition function over the ro-tational and translational degrees of freedom for stepwisevaried applied voltages. The expectation values of rota-tional and translational positions and of the predicted gatingcharge can thereby be determined as functions of voltage.The relationship between displaced gating charge and volt-age is a prediction of the gating charge per VS displacedwhen a voltage is applied to an ensemble of channels in anexperiment.Figs. 3.5 (a) & (b) show the expectation of position for themodel VS at given voltages. The positions follow the trajec-26 a) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUUUUU (b) +100 mV − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU (c) -100 mV − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU (d) ∆ Energy − − − . . . . R e l a t i v e E l ec t r o s t a t i c E n e r g y / e V Q Q Q Q QR R R R RTTTT UUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKH
Ψ = 0 mV C Ψ = +100 mV K Ψ = −
100 mV
Figure 3.3: A sliding helix configuration has electrostatics suited for a VS . Panels (a-c): false-color maps of electrostatic potential energyof the VS scanned over two degrees of freedom, for three applied voltages. The S4 helix has 6 positive charges, the three counter-chargesare spaced at the 2/3 interval, and the protein dielectric coefficient is 4. The electrostatic potential energy map for 0 mV applied voltageforms a trough favorable to combined translational/rotational (‘screw’) motion of the S4 helix. Applied voltages of -100 or +100 mV convertthe energy trough into a pit at one end of the trough seen with 0 mV. Panel (d): Rotation-averaged electrostatic potential energy versustranslation for three applied voltages (averaging is based on the rotational partition function). Since electrostatic potential energy dependson rotational position (panel a), allowing the S4 helix to rotate minimizes energy. The averaged energy forms a trough that tends to restricttranslation at both ends but is almost flat over intermediate translations (thus allowing diffusive motion of the S4 helix). Applied voltage tiltsthis profile (promoting drift/diffusion of the S4 helix). Note that panels (a-c) show the potential energy of the VS relative to translation 0 nmand rotation 0 ◦ , while panel (d) shows the potential energy of the VS relative to translation 0 nm (expectation over the rotational degree offreedom). a) Internal at 0 mV (b)
External at 0 mV (c)
Internal at +100 mV (d)
External at +100 mV
Figure 3.4: A sliding helix configuration is mechanically stable.
False-color maps of the Maxwell stress acting on the bath interfaces. Thegating pore lies between the yellow and white rings; the top of the S4 segment is delimited by the black ring. The Maxwell stress tends to pullthe water boundary toward the S4 charges. Logarithmic color scale from 10 Pa ( blue ) through 10 Pa ( green ) to 2.5 × Pa ( red ). Thecurved surface of the membrane/protein bath interface (10 nm in diameter) is projected into a plane (preserving path length in walking fromthe center to the periphery). High pressures occur where charges face water and in locations at the bottom of the gating pore (stabilizingthe gating pore near buried counter-charges). The same model parameters are used as in Fig. 3.3. The position in the membrane of the S4 segment is controlled by the applied membrane voltage (0 or +100 mV), and the Maxwell stress shown is the expectation of the Maxwellstress (based on the translational/rotational partition function). A supplementary movie ( flat-s4.mp4 [suppl.]) shows how Maxwell stressvaries as applied voltage is varied between -100 and +100 mV. a) Translation − −
50 0 50 100Potential / mV − − S T r a n s l a t i o n / n m Q Q Q Q QR R R R RTTTTT UUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I I (b)
Rotation − −
50 0 50 100Potential / mV − − − R o t a t i o n / ◦ ( d e g r ee s ) Q Q Q Q QR R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I I (c)
Displacement − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D H Model C Measured
Figure 3.5: The action of voltage on a sliding helix VS : Expectations of translation (a), rotation (b) and displaced gating charge (c) inresponse to varied applied voltage. The expectations of these random variables are computed using the electrostatic partition function for thetwo degrees of freedom. The same parameters for the sliding helix model are used as in Fig. 3.3. Voltage moves the mean position of the VS in a screw-like trajectory and displaces gating charge of the proper magnitude with the proper slope (an experimental charge/voltage curve(Seoh et al., 1996) for Shaker K v channels is shown as the orange line in panel c). VS over the range suggested by the potential energy landscapesin Fig. 3.3. The amount of gating charge displaced overthat range of motion (Fig. 3.5 c, blue line) exceeds 3 e per VS , which is close to the total gating charge measured per VS in Shaker K + channels (orange line). Since a VS drivingcoupled gating machinery might have a smaller range of mo-tion, it is reasonable to expect a model of an uncoupled VS to produce at least as much charge per VS as that observedin channels.Charge is displaced in the model over a voltage range sym-metrical with respect to 0 mV of applied voltage, whereasthe experimental charge displacement is centered about anegative voltage. The slopes of the two charge displace-ment curves are quite similar; the chief difference betweenthe model VS and the real VS is an offset between thecharge/voltage curves. The real VS is integrated into a chan-nel and drives the activation gating of the channel. Mymodel voltage sensor is isolated and drives no load. If thevoltage offset between the two charge/voltage relations isdue to the gating work that the real voltage sensor does onthe rest of the channel, then: (1) the gating work is applied inclosing the channel, and (2) the counter-force exerted by thegate onto the VS is approximately constant over the rangeof VS travel (in contrast to an elastic counter-force).The sliding helix model presented fulfills the criteria for aviable VS model as listed above. Therefore in this study I willuse those model parameters as a basis of further explorationof VS electrostatics. Sliding helix models have many features that can beparametrized and studied: counter-charge position & num-ber, protein dielectric, gating pore size & shape and mem-brane thickness, for example. In this section I present threefeatures of interest that show sensitivity and a significant ef-fect on VS function. In particular, the existence of counter-charges, their spacing and the local dielectric through whichthey interact with S4 charges are strong determinants of theviability of a VS model. The gating pore reduces the length of the S4 segment ex-posed to the weak dielectric separating the two baths. Thesliding helix model studied in the previous section also in-cludes three negative counter-charges in the region of weakdielectric. Either feature is expected to reduce the energeticcost of moving the S4 charges from the baths into the regionof weak dielectric. The role of the counter-charges can beassessed by deleting them from the model. Fig. 3.6 showselectrostatic potential energy of the VS versus translationfor the models with and without three counter-charges. Re-moving the counter-charges produces a large electrostaticbarrier, much like that computed for the paddle model de-scribed above (Fig. 3.1 d, note the distinct shape in themid-range, however).The S4 charges of the sliding helix model induce a sub-stantial charge on the bath interfaces. The induced chargesare negative, attracting the S4 charges toward the baths andthereby destabilizing the buried S4 charges. With the buriedpositive S4 charges neutralized by negative counter-charges,the charges induced on the bath interfaces are greatly re-duced, creating a trough of electrostatic potential energy.These computations indicate that an appropriate number ofcounter-charges are required if the sliding helix model is tofunction as a VS . The gating pore alone does not lower theinduced-charge barrier to the extent needed for the slidinghelix models to function as a VS .Another consequence of the deletion of all counter-chargesfrom the model is that all rotational positions now haveequal electrostatic potential energy. Thus the S4 segmentin the model no longer operates like a screw. If the deliv-ery of torque is important for operating the gate of the realchannel, this would add another consequence to neutralizingmutations of VS counter-charges. The counter-charges of the described sliding helix modelsare arranged following the spiral curve on which the S4 charges are positioned. The intervals of the S4 and counter-charges, however, differed for the previously described model( Counter-charges eliminate the induced-charge barrier, sub-sec. 3.3.1 ): there, the counter-charges were spaced at 2/3the interval between S4 charges. This precludes simultane-ous alignments of more than one S4 charge with a counter-30 − − . − . . . . R e l a t i v e E l ec t r o s t a t i c E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC H C No counter-charges
Figure 3.6: Counter-charges are required for a sliding helix to function as VS . Electrostatic potential energy of the VS averaged overrotation and relative to that at translation zero versus translation. The blue line describes the sliding helix model of Fig. 3.3 which includesthree counter-charges. The orange line describes the same model but with all counter-charges deleted. The energy profile then becomesparabolic like that of a paddle model (compare Fig. 3.1 d). charge (see Fig. 2.4). In order to see how much the close-range electrostatic interactions of charge and counter-chargeaffect electrostatic potential energy of the VS , I have com-puted the consequences of counter-charge intervals of 1/2,1/1, 4/3, and 3/2 times the interval of S4 charges. A latersection ( Charge mutations, sec. 3.4 ) will present computationsof less regular charge spacings obtained by deleting chargesat certain positions of a periodic pattern.Fig. 3.7 shows maps of electrostatic potential energy inthe translational and rotational degrees of freedom, for both0 mV and +100 mV applied voltage. The potential energy ofthe VS expected when rotation is free is plotted versus trans-lation in Fig. 3.8 with 0 mV curves for different spacingssuperimposed. The 1/2, 1/1, and 3/2 intervals yield energylandscapes more hilly than those of the 2/3 or 4/3 inter-vals; the S4 segment of these models with counter-chargesspaced at 1/2, 1/1 or 3/2 intervals tend to dwell in morediscrete positions in energy valleys. When a strong voltageof +100 mV is applied in these models, some of the energyvalleys persist as discrete features.What are the consequences of these different potential en-ergy landscapes for the gating charge displaced in response to applied voltage? How does the expected position of the S4 region respond to voltage? Fig. 3.9 shows the expecta-tion translation (a), rotation (b) and charge/voltage rela-tion (c) based on the partition function over rotational andtranslational degrees of freedom. The charge voltage rela-tions vary in steepness and total displaced gating chargeeven though the numbers of S4 charges and counter-chargesare fixed, as is the dielectric environment. These variationsof the charge/voltage curve must originate from the range oftravel produced by the applied voltage as well as the distri-bution of S4 positions among more or less discrete locations.Predicting these variations requires numerical analysis of theelectrostatics.The expectation values for rotation and translation atdifferent voltages (Figs. 3.9 a & b) reveal a monotonic in-crease in translation as voltage is increased; however, non-monotonic variations of rotation occur in some cases. Thus,for the 3/2 counter-charge interval, the rotation is in theopposite direction to that seen for other intervals. Trans-lational motion (and ability to produce translational force)is robust, while rotational motion (and ability to producetorque) is sensitive to counter-charge alignment. Never-31 a) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUUUUU (b) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU (c) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . − . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU (d) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . − . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU
Figure 3.7: Counter-charge spacing controls electrostatic potential energy landscape (Pt. 1).
The spacing is specified as the ratioof counter-charge spacing to S4 charge spacing (the ratio applies to both the rotational and translational spacing). The protein dielectriccoefficient is 4. Energy of each configuration is represented relative to translation 0 nm, rotation 0 ◦ . Note differences in scale. Figure Continued in Pt. 2. theless, the S4 segment is expected to rotate for all testedcounter-charge intervals.Varying a single parameter of counter-charge configura-tion has strong effects on function in these sliding helix VS models. Counter-charges and their arrangement are crucialfor building a working VS . Simulations were conducted in which the dielectric coeffi-cient of the protein region (including the S4 segment) was varied over the values 2, 4, 8 and 16. Landscapes of elec-trostatic potential energy for dielectric coefficients 2, 8 and16 are shown in Fig. 3.10; the potential energy landscapefor a dielectric coefficient of 4 was presented in Fig. 3.3.The general effect of increasing the dielectric coefficient isto moderate energy variations. Broader ranges of rotationand translation become accessible (note the varying energyscales between these graphs).The expectation values of energy (based on the rotationalpartition function) are shown versus translation in Fig. 3.11.As the dielectric coefficient is decreased, the shallow troughof energy becomes deeper and a pattern of wells and barri-32 e) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUUUUU (f) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU (g) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU (h) − − − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . E n e r g y / e V Q Q Q Q QR R R R RTTTTT UUUUU UUUUU
Figure 3.7: Counter-charge spacing controls electrostatic potential energy landscape (Pt. 2).Figure continued from Pt. 1. ers emerges. In particular, significant wells develop around ± VS as assessed by the steepness at the midpoint of thecharge/voltage curve increases as the protein dielectric coef-ficient is reduced. The reason for this effect of the dielectriccoefficient is evident in Fig. 3.11. With a dielectric coeffi-cient of 2, there are two crisp energy minima at translations ± S4 segment to dwell preferentially nearthese two positions. With a dielectric coefficient of 16, how-ever, there is no significant energy variation (at 0 mV) atany position within the energy trough, so that no S4 posi-tion is preferred. Hence, the distribution of the S4 segmentin the translational degree of freedom varies from a virtual‘two-state Boltzmann distribution’ to a distribution within aspace of uniform potential energy. The uniform-energy dis-tribution of charge approaches hyperbolic rather than expo-nential asymptotic behavior at extreme voltages. The mid-point slopes of the charge/voltage curves can be analyticallydetermined; the midpoint slope of the charge/voltage curveis three times greater for the two-state case than for theuniform-energy case (Neumcke et al., 1978). In this man-33 a) Aligned − − − . . . . R e l a t i v e E l ec t r o s t a t i c E n e r g y / e V Q Q Q Q QR R R R RTTTT UUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKH / C K / (b) Non-aligned − − − . . . . R e l a t i v e E l ec t r o s t a t i c E n e r g y / e V Q Q Q Q QR R R R RTTTT UUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCH / C / Figure 3.8: Counter-charge alignment with S4 charges creates electrostatic barriers and wells. Electrostatic potential energy of the VS versus translation, averaged over rotation and relative to translation 0 nm. The spacing of counter-charges is varied. The protein dielectriccoefficient is 4. The spacings of 1/2, 1/1, and 3/2 allow two counter-charges to align with corresponding S4 charges in certain positions,thereby generating ripples of electrostatic potential energy (panel a). The spacings that allow only one counter-charge to align at a time withan S4 charge produce smoother profiles of energy (panel b). ner, the same structural charges produce up to a three-foldvarying effective gating charge as the protein dielectric coef-ficient (and therefore the potential energy landscape) is var-ied. The force that can be delivered by S4 charge movementis therefore constrained by the local dielectric coefficient. Seoh et al. (1996) reported the results of 9 neutralization mu-tants (over 8 residues) of
Shaker K + channels plus the wild-type in terms of open probability and charge displacementper channel. In single mutants, one of the four outer S4 posi- tive residues or one of three negative charges on the S2 and S3 transmembrane segments were neutralized. In addition, twodouble mutants were investigated. These mutations producea complex pattern of change in the charge/voltage curves,including reductions of total gating charge, shift, and al-teration of slope and shape. Predicting such patterns is achallenge for a physical model.For these comparisons, I use 3 counter-charges set at a2/3 interval to test whether the apparent VS -like behaviorof that model as investigated above responds like a biolog-ical VS to physiological extremes. The positions of thesecounter-charges in the biological VS is ambiguous. Unlike S4 charges which are regularly arrayed on a single transmem-brane domain that is α -helical in nature, counter-charges areirregularly placed on multiple helices connected by amor-phous linking regions and arranged at various orientations34 a) Translation − −
50 0 50 100Potential / mV − − S T r a n s l a t i o n / n m Q Q Q Q QR R R R RTTTTT UUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM M M M M M M M M M M M M M M M M M M M MVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVW W W W W W W W W W W W W W W W W W W W W (b)
Rotation − −
50 0 50 100Potential / mV − − − R o t a t i o n / ◦ ( d e g r ee s ) Q Q Q Q QR R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM M M M M M M M M M M M M M M M M M M M MVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVW W W W W W W W W W W W W W W W W W W W W (c)
Displacement − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM M M M M M M M M M M M M M M M M M M M MVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVW W W W W W W W W W W W W W W W W W W W W H / C / K L / V / Figure 3.9: Counter-charge spacing is important for VS response to voltage: Expectation values of translation (a), rotation (b) anddisplaced gating charge (c) in response to varied applied voltage. The expectations for these random variables are computed using theelectrostatic partition functions for the two degrees of freedom. The same sliding helix models are used as in Figs. 3.7 & 3.8. Counter-chargespacing controls the extent of S4 charge motion, direction of rotation, as well as magnitude of gating charge and shape of the charge/voltagerelation. (Note that 1/2 interval in blue falls closely on top of 2/3 interval in orange ). a ) (cid:15) = t m V − − A x i a l M o v e m e n t / n m − − Rotation/ ◦ (degrees) − . . . . . . . Energy/eV
QQQQQRRRRR TTTTT UUUUU UUUUUUU ( b ) (cid:15) = t m V − − A x i a l M o v e m e n t / n m − − Rotation/ ◦ (degrees) − Energy/meV
QQQQQRRRRR TTTTT UUUUU UUUUUUUU ( c ) (cid:15) = t m V − − A x i a l M o v e m e n t / n m − − Rotation/ ◦ (degrees) − Energy/meV
QQQQQRRRRR TTTTT UUUUU UUUUUU ( d ) (cid:15) = t + m V − − A x i a l M o v e m e n t / n m − − Rotation/ ◦ (degrees) − . . . . . . . Energy/eV
QQQQQRRRRR TTTTT UUUUU UUUUUUU ( e ) (cid:15) = t + m V − − A x i a l M o v e m e n t / n m − − Rotation/ ◦ (degrees) − − Energy/meV
QQQQQRRRRR TTTTT UUUUU UUUUUUUUUUU ( f ) (cid:15) = t + m V − − A x i a l M o v e m e n t / n m − − Rotation/ ◦ (degrees) − − Energy/meV
QQQQQRRRRR TTTTT UUUUU UUUUUUUU F i g u r e . : P r o t e i nd i e l e c t r i cc o e ffi c i e n t c o n s t r a i n s t h e a cc e ss i b l e r a n g e o f m o t i o n . F r o m l e f tt o r i g h t ( p a n e l s a - c ) t h e d i e l ec t r i cc o e ffi c i e n t (cid:15) p o f S a nd t h e s u rr o und i n g p r o t e i n m a t r i xv a r i e s o v e r , & f o r m o d e l s v o l t a g e - c l a m p e d a t m V w i t h a i n t e r v a l ( i s p r e s e n t e d i n F i g . . ) . L i k e w i s e f o r + m V , (cid:15) p i s v a r i e d o v e r , & i np a n e l s ( d - f ) . T h e p o t e n t i a l e n e r g y o f e a c h V S c o n fi g u r a t i o n r e l a t i v e t o t r a n s l a t i o n n m a nd r o t a t i o n ◦ i s r e p r e s e n t e d i n f a l s ec o l o r ; n o t e t h e d i ff e r e n ce i n s c a l e s . A t (cid:15) p = t h e r a n g e b e t w ee n m a x i m aa nd m i n i m aa r e ≈ e V , w h il e a t (cid:15) p = t h i s d i ff e r e n ce i s r e du ce d t o ≈ . e V & . e V . − − . . . . R e l a t i v e E l ec t r o s t a t i c E n e r g y / e V Q Q Q Q QR R R R RTTTT UUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL H ǫ = 2 C ǫ = 4 K ǫ = 8 L ǫ = 16 Figure 3.11: Protein dielectric coefficient constrains the size of energy barriers distinguishing stable configurations.
The electrostaticpotential energy at 0 mV and 2/3 interval relative to the 0 translational position is depicted for each translational position from the rotationalpartition function, as (cid:15) p is varied over 2 ( blue ), 4 ( orange ), 8 ( magenta ) & 16 ( cyan ). The data for (cid:15) p = (cid:15) p is increased, the energetic barrier distinguishing ± S4 to the membrane. − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM M M M M M M M M M M M M M M M M M M M M H ǫ = 2 C ǫ = 4 K ǫ = 8 L ǫ = 16 Figure 3.12: Protein dielectric coefficient constrains the distribution of charge displacement. As (cid:15) p is varied over 2 ( blue ), 4 ( orange ),8 ( magenta ) & 16 ( cyan ), the maximum slope of the charge displacement curve is reduced, while total charge displacement is not. Thecharge displacement for (cid:15) p has been previously presented versus experimental results in Fig. 3.5 (c), blue curve. All curves were calculatedwith a 2/3 interval at 0 mV. VS . Motions ofthe S1 – S3 regions of the VS are experimentally undetermined.The charged residues of the VS have many points of rota-tion and extend relatively far from the α -helical backbonesto which they are attached (particularly arginine and gluta-mate residues, Creighton 1984); for example, in Long et al.(2007, Fig. 4), four charge/counter-charge pairs are shownin direct contact despite the S4 appearing to form an angle(and therefore creating a gap) with the S1 , S2 and S3 do-mains. Additionally, there exists ambiguity regarding whichcharges are permanently situated in the intra- and extra-cellular solutions and which ones are within the gating poreregions where the ratio of electrical travel to geometricaltravel is larger (Nonner et al., 2004, Fig. 2).However, it is known that three or four counter-chargeson the S2 and S3 transmembrane domains are highly con-served depending on which K v channels are included (Islasand Sigworth, 1999; Jiang et al., 2003; Tao et al., 2010),and that neutralization of three counter-charges have pro-found effects on VS behavior (Planells-Cases et al., 1995;Papazian et al., 1995; Seoh et al., 1996). From the resultsshown in Counter-charges eliminate the induced-charge bar-rier, subsec. 3.3.1 , three counter-charges are sufficient to pro-duce VS -like behavior, and fewer charges fail to stabilize thesliding helix in the membrane (data not shown for single anddouble counter-charge simulations).The biologically possible configuration space is of high di-mensionality, so as a test of the variety of conditions underwhich the previously elucidated model can function I be-gin with a reduced model of counter-charge positions andmutations. As described in Sliding helix model, subsec. 2.3.1 ,charge and counter-charge mutants are modeled both as asimple elimination of the point-charges associated with theresidues and by the point-charges’ replacement by a dipoleto represent the polarizability of glutamine and asparagineresidues. The envelope contained by the curves representingthese two extreme models of each mutation can be used to in-vestigate the robustness of components of this model (chargeelimination is a maximally energetically unfavorable model,whereas the chosen dipole representation is minimally ener-getically unfavorable).I have calculated the behavior of models with correspond-ing charge deletions, testing to what extent the experimen-tal charge/voltage curves are predicted. This requires as- signing the S2 and S3 negative charges neutralized by Seohet al. (1996) to counter-charge positions in the sliding helixmodels. The residues E D E
293 are assignedto the outermost, central, and innermost counter-charge re-spectively. The counter-charges are spaced at the 2/3 in-terval, and the protein dielectric coefficient is set to 4. Inthese models when one of the counter-charges is removed,the trough of electrostatic potential energy that confines S4 charge motion (see Fig. 3.6) becomes inverted; therefore the S4 segment is no longer stable in the weak dielectric. Onthe other hand, mutants with one S2 or S3 negative residueneutralized are functional, indicating that their S4 segmentsare not dislocated. Therefore, I confine the S4 segment ofthe models presented in this section within the translationalrange of ± VS sampled over the rotational andtranslational degrees of freedom.Figs. 3.13 through 3.21 summarize the results. Panel (b)of each figure shows the experimental charge/voltage curvesfor the wildtype channel ( orange squares ) and the mutant( blue circles ), as reported in Fig. 2 (A-J) of Seoh et al.(1996). Panel (a) of each figure shows the correspond-ing curves computed for the model. The experimentalcharge/voltage curves are vertically aligned so that theirmidpoints correspond to zero displaced charge. There are nohorizontal alignments or normalizations except in Fig. 3.13(Seoh et al., 1996 report in their Fig. 2 E a normalizedcharge/voltage curve for mutant R Q ). All other curvesrepresent charge per VS .In comparing the computed and experimental results, it isuseful to consider differences in the size of wildtype and mu-tant charges, shifts between wildtype and mutant voltagedependencies, and slope and shape changes between wild-type and mutant. My computations apply to an isolated VS , while the experiments were done on channels; therefore,the ‘idle’ VS is being compared to the naturally gate-coupled VS . To the extent that the coupling to the gate restricts S4 charge motion and requires work to be done by the VS , it isexpected that the gating charge of the model VS is greaterthan that of the channel VS , and that the charge/voltagecurves are shifted with respect to one another. Because VS model and channel charge/voltage curves are expected tobe shifted with respect to one another, it is useful to focuson the differences between mutant and wildtype curves. Towhat extent does the model account for the mutation-versus-wildtype changes?38 .4.1 Mutation of a positive charge Model and experimental results of successively mutating one S4 charge are shown in Figs. 3.13 through 3.16, startingwith the outermost arginine residue. In real Shaker chan-nels, these mutations produce substantial changes to thecharge/voltage relation including reduction of total charge,changes of slope, deformations of the charge/voltage rela-tion and shifts along the voltage axis. The experimentalpattern of change varies from mutation to mutation. Thepredictions from the model reflect the varying experimentalpatterns very well. Only one qualitative difference is seen:the model does not predict the shift toward negative voltagesseen in the R N mutant (Fig. 3.15). The results of neutralizing one of a group of putativecounter-charges, two glutamate residues of S2 and an as-partate residue of S3 , are shown in Figs. 3.17 through 3.19.In the model, these mutations are mimicked by deleting theoutermost ( E D E E
283 & E
293 andbecause of the symmetry in the model’s charge configura-tion, the simulations predict that the Q/V curves for E Q and E Q should reflect such symmetry, but in fact the ex-perimental curves are not symmetric. In particular E Q has a substantially smaller total gating charge than E Q .Thus the VS of the channel has an asymmetry in its struc-ture or operates under asymmetrical constraints that arenot included in these VS models. E Q and D N re-veal activation curves (not shown) of the ionic current thatare strongly shifted to positive voltages. The charge/voltagecurves reported by Seoh et al. (1996) do not extend over thisvoltage range, so they can not reveal the full gating charge. The two double mutants investigated by Seoh et al. (1996)are represented in Figs. 3.20 and 3.21. The charge/voltage curves of the mutants are shifted to negative voltages; theactivation curves are also strongly altered (with a left shift inthe case of K Q + D N ). The VS of these mutant chan-nels appears to have difficulty in moving into fully-closed po-sitions. The predicted curves reveal limitations of the model.A double mutant lacking two VS charges is expected to beparticularly sensitive to the geometry assigned to chargesdue to the positions and orientations of potential gaps be-tween charges and counter-charges, apart from the alreadymentioned distinction regarding gating load. The double mutations of K Q + E Q and K Q + D N by Seoh et al. (1996) were partially motivated by thelack of functional expression by K Q mutants. Since ionicconductance was blocked at all potentials, charge displace-ment per channel could not be estimated. One could spec-ulate as to the cause of this — whether the VS proper wasnon-functioning or whether some folding pathology blockedproper expression. Experimentally, there is limited accessi-bility for non-expressive behavior; however, computationalexploration is still possible. In Fig. 3.22, a computationalanalog of K Q is tested: (1) charge displacement is greatlyreduced in panel (a), with some increase extending towardslarge positive potentials; (2) S4 position is constrained to theintracellular, closed positions in panel (b); and (3) no pathfor fully moving to the open position is apparent from thepotential energy landscapes in panels (c) & (d). The fail-ure of K Q to function biologically is consistent with thepredicted electrostatic limitations of K Q as a VS . These model studies of charge mutants show that the qual-ity of the model predictions varies in a pattern. S4 chargemutants are described well, S2 and S3 (counter-charge) mu-tants less well, and double mutants least well. S4 chargeslikely form a regular array of charges because of the S4 ’shelical structure. Thus the model assumption of uniformspacing of the S4 charges is probably sound. The arrange-ment of putative counter-charges provided by the S2 and S3 segments has a much larger range of uncertainty, of which Ihave explored only a small subrange — more exploration isneeded.39 a) Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH st ‘+’ removed C ‘Wildtype’ K st ‘+’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured R Q C Measured ‘Wildtype’
Figure 3.13: Effective gating charge is reduced by mutating the outermost arginine.
The slope of the QVrelation, the effective gating charge, is reduced relative to the wildtype for both the computed and experimentalresults. Note that Seoh et al. (1996, Fig. 2 E) report for this case normalized charge displacement, so totalgating charge is not comparable with wildtype results. Dipole and deletion representations are similar. (a)
Model − − −
100 0 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q QR R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGH nd ‘+’ removed C ‘Wildtype’ K nd ‘+’ to dipole (b) Seoh et al. (1996) − − −
100 0 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q QR R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured R Q C Measured ‘Wildtype’
Figure 3.14: Mutation of the 2 nd arginine significantly reduces effective charge. Both computed andexperimental results show a left shift. Note the inflection point for the model mutations below -100 mV and thesensitivity to charge representation at extreme negative voltages. a) Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH rd ‘+’ removed C ‘Wildtype’ K rd ‘+’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured R N C Measured ‘Wildtype’
Figure 3.15: Mutation of the 3 rd arginine reduces total gating charge. In this case, experimental resultsshow a left shift that is not apparent in the computational results, combined with an even more significant totalreduction in charge. R N also shows a left shift in open probability (data not shown). Dipole and deletionrepresentations are similar. (a) Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH th ‘+’ removed C ‘Wildtype’ K th ‘+’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUH H H H H H H H H HHH H H H H H H HI I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured R Q C Measured ‘Wildtype’
Figure 3.16: Mutation of the 4 th arginine reduces total gating current. Neither results shows a shift in thecharge/voltage relationship. Dipole and deletion representations are similar. a) Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH st ‘-’ removed C ‘Wildtype’ K st ‘-’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHH HHHHHHHHHHHHHHH H HHH H HI I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured E Q C Measured ‘Wildtype’
Figure 3.17: Mutation of the outermost counter-charge produces a right shift with no gating chargereduction.
Experimental results also show a right shift for the open probability (data not shown). Bothrepresentations of the mutation are right-shifted and have similar slopes, but the magnitude of the right-shift isdistinct. (a)
Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH nd ‘-’ removed C ‘Wildtype’ K nd ‘-’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHH HHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHI I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured D N C Measured ‘Wildtype’
Figure 3.18: Mutation of the middle counter-charge has a mild effect.
Experimental results show a smallreduction in gating charge & slope, while computational results predict a slight increase for both due to theexclusion of S4 from central positions. Dipole and deletion representations are similar. a) Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH rd ‘-’ removed C ‘Wildtype’ K rd ‘-’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHI I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured E Q C Measured ‘Wildtype’
Figure 3.19: Mutation of the innermost counter charge produces a left shift.
Computational results aresymmetrical with 3.17 (a), while experimental results show a large reduction in gating charge which is notapparent in E Q . Experimental results also show a left shift for open probability, as opposed to to the rightshift for E Q (data not shown). Both representations of the mutation are left-shifted and have similar slopes,but the magnitude of the left-shift is distinct. (a) Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH th ‘+’ & 2 nd ‘-’ removed C ‘Wildtype’ K th ‘+’ & 2 nd ‘-’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured K Q + D N C Measured ‘Wildtype’
Figure 3.20: Mutating the outermost lysine & central counter-charge produces a small reduction intotal charge displacement.
Experimental results show a left shift that computational results do not reproduce. K Q + D N also shows a left shift in open probability (data not shown). The dipole representationdisplays a much larger right-shift and a slightly larger reduction in total charge displacement relative to thedeletion representation. a) Model − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH th ‘+’ & 3 rd ‘-’ removed C ‘Wildtype’ K th ‘+’ & 3 rd ‘-’ to dipole (b) Seoh et al. (1996) − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHI I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D DH
Measured K Q + E Q C Measured ‘Wildtype’
Figure 3.21: Mutating the outermost lysine & the innermost counter-charge produces a reduction in totalgating current.
The computational results for deletion, however, predict a reduction in slope that is replaced bya larger reduction in total gating charge in the experimental results. K Q + E Q also shows a left shift inopen probability (data not shown). The dipole representation fails to predict the left shift of the experimentalresults reproduced by the deletion representation; however, the dipole representation reproduces the higher slopeof the experimental mutant. a) Displacement − − −
50 0 50 100Potential / mV − − − C h a r g e D i s p l a ce m e n t / e Q Q Q Q Q QR R R R R RTTTTTTT UUUUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH th ‘+’ removed C ‘Wildtype’ K th ‘+’ to dipole (b) Translation − −
50 0 50 100Potential / mV − − S T r a n s l a t i o n / n m Q Q Q Q QR R R R RTTTTT UUUUUHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHI I I I I I I I I I I I I I I I I I I I ICCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCD D D D D D D D D D D D D D D D D D D D DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKG G G G G G G G G G G G G G G G G G G G GH th ‘+’ removed C ‘Wildtype’ K th ‘+’ to dipole (c) − . − . − . . . . . − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . . . . E n e r g y / e V Q Q Q Q Q Q QR R R R R R RTTTTT UUUUU UUUUUUUU (d) +100 mV − . − . − . . . . . − − R o t a t i o n / ◦ ( d e g r ee s ) − . . . . . E n e r g y / e V Q Q Q Q Q Q QR R R R R R RTTTTT UUUUU UUUUU
Figure 3.22: The non-functional K Q mutant is electrostatically incompetent. Charge displacement inpanel (a) is marginal, only reaching 0 e at highly positive potentials. The associated translation in panel (b)tracks the gating charge, never moving past the central position until the internal relative electrode potentialsapproaches +100 mV. In panels (c & d), a large energy barrier at 1 nm is apparent even at +100 mV. Panels (c)and (d) were produced with the deletion representation. Both mutant representations show similar results. hapter 4 Perspectives
I have developed a computational approach for studying theelectrostatics of the voltage sensor ( VS ) controlling conduc-tion in voltage-dependent ion channels. The VS is describedin a microscopic physical model that is reduced in detail tothose features whose relevance for VS function is to be inves-tigated. The microscopic model is complemented by a sim-ulation system that establishes voltage-clamp conditions andrecords gating charge movements in a manner analogous toand comparable with a macroscopic experimental setup. Us-ing efficient computational methods that allow a statistical-mechanical analysis of VS behavior, characteristics that areexperimentally accessible (such as the charge/voltage rela-tion) are computed for different models of the VS . My ap-proach thus substantially extends the computational meansfor studying structure-function relationships of the VS systemwhile making comparisons with experimental results. Thepresented results provide the following perspectives on the VS . How do charged residues ‘contribute’ to gatingcharge?
Published experiments have sought answers tothis question by neutralizing a formally charged residue ofthe VS and measuring the slope of voltage dependence of ac-tivation (St¨uhmer et al., 1989) or recording ensemble gatingcharge while counting channels with an independent methodsuch as noise analysis of ionic current (Aggarwal and MacK-innon, 1996; Seoh et al., 1996; Baker et al., 1998; Ledwelland Aldrich, 1999). The results of the latter are expressed asthe change (typically reduction) in gating charge per channel.My computational studies show how charge neutralization canmodify gating charge in a manner amenable to several modesof analysis: (1) the S4 charge motion carries less charge asa direct consequence of removing one of its charges — the simplest mode; (2) the range of S4 charge motion becomeselectrostatically restricted, leading all S4 charges to travel ashorter distance, as they are part of a solid body; and (3) theelectrostatic potential energy landscape for S4 travel is alteredin such a manner that the probability distribution of positionsis altered, with consequences for the shape of Q/V relation-ships. Modes (2) and (3) may apply regardless of whether theneutralized residue is mobile or fixed in position. Under thelatter conditions, the total gating charge can be reduced by anamount greater than the neutralized charge since the motionof all charges is modified by altering one charge. Counter-charges to the positively charged S4 residuesare essential. Much discussion of how the S4 helix withits array of positively charged residues might be stabilizedin the membrane has focused on the polarizability of the S4 matrix and its environment. The primary requirements forallowing the S4 charges to act as a VS are the proper numberand positions of counter-charges. The importance of nega-tively charged residues for VS function is well established; therespective residues of the S2 and S3 segments are highly con-served (Islas and Sigworth, 1999; Tao et al., 2010), and neu-tralization mutants show strong alterations of function (Seohet al., 1996).Computationally, I find that a viable VS model includescounter-charges. Removing counter-charges from the modelcreates a landscape of electrostatic potential energy that tendsto exclude the S4 charged region from the membrane region— a large barrier develops from charge induced on the bathinterface by buried, unbalanced S4 charges. This happensin the presence of deep gating pores that reduce the num-ber of buried S4 charges to no more than three. My com-putations show that much of the observed consequences of46eutralization mutants on S2 , S3 , and S4 can be understoodon electrostatic grounds as resulting from the charge/counter-charge interactions among the buried residues and the chargesthat they induce on dielectric boundaries. The computa-tional results for a reduced electrostatic model suggest thatthe charge/counter-charge interactions in the VS warrant de-tailed investigation. Electrostatics dominate.
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