a continuum method for determining membrane protein insertion energies and the problem of charged residues

They calculated the free energy profile for tratisladng the charged residtte from the upper leaf-let to the lower leafleaf-let revealing that an insertion energy of 17.8 kcal/mol is requ

A Continuum Method for Determining Membrane Protein Insertion Energies and the Problem of Charged Residues

Seungho Choe,'' Karen Ạ Hecht/ and Michael Grabế^

^ Department of Bioiogical Sciences and ^Department of Computational Biology in the School of Medicine, University of

Pittsburgh Pittsburgh, PA 15260

Continuum electrostatic approaches have been extremely successful at describing the charged nature of soluble proteins and how thfy iiitrraci with bindinj^ parlners However, il is unclear whether ((nilinnuni mctliods can be used lo quaniitativt'ly undt'istand tht- t-ncrgctics of mcinbrant' protein insertion and siabiliiỵ Rcct-ni iranslaiion expt rinuMU-s su^gt-st that thf eiierg)- required to insert charged pcptides into membranes is much smaller tliau predicted b\ present c<»ntinuum theories Atomistic simulations have poinied to bilayer inhomogeneity and mem-brane deformation around buried charged groups as two critical features that are neglected in simpler models Here, we deveiop a fully couiiuuum method that circumvents both of ihese shortcomings by using elasticit) theoi7

to determine the shape ofthe deforiTied membrane and tlien subsequently uses this shape to (arrvout coiUiruium electrostatics calculations Our method does au excellent job of quantitatively matching results from detailed nio-leculur d\n;unics simulations al a tiny fraction ofthe computational cost We expect that this method will be ideal for studying large membrane piotein complexes

INTRODUCTION

The central role of the cell membrane is to act as a

selec-tive barrier sepatating the ceil iVoin its environtnent

(Ịipowslcy atid Siickinann, 1995) The architecture of tlie

lipid bilayer is such that hydrophobic alkyl chains are

sand-uithed between lipid head grou|}s (Taiiford I99i) This

ari-angenieiu siiit-Uis the lueuiijrauc's hydrophobic core

from exposure to water and other polar or charged species

in tlie surrounding einiroiinieiii (Tanfortị 1991; Ịipowsky

and Sackuiauu, 1995) In ađition to lipid uioleculcs, tlie

ceil membrane is host to membrane proteins that must be

euibedcied witiiiti tlie iiiiayer without disnipting iLs

sttitc-turai iiitegt itỵ The iiydiopiiobic uauueot u-ausmembrane

(TM) segments aliows membrane proteins to be

main-tained wiihiu the lipid biiayer wiihoiit compromising

cei-tulat iionietistasis (Etigeiniau et aị, 1986)

Membrane proteins account for a third of all proteins in

a ceil and are im'olved in ntuiierous imporUint biological

iuucUotts, sucii as ion conciuctiou, ceii receptor signaiing,

and nutrient transport (Lipowsky and Sackmaun, i995;

vou Heijut' 2007) Aithougii predoniiuautiy hydrophobit

in nalutc, many TM scgtuetits cotmiin poiar and charged

residues Notabiy, voitage-gated K* channeis, cystic fibrosis

iiaiisnieniliraue couductatice regulator, and tlie giycitie

rfcept(M, GLR-\l, are aii known to contiiiu cliarged

resi-dues within their TM domains (Jan an(i jau, i990; Hessa

c-t aị 2()05b; Bakker et aị, 2006: ịinsdeii, 200(3) Aceutrai

(|ueslion iu ihe study of niettii)tane proteitis is how

ciiar god residues can iie stabiy accommodated witliiti tlie

iipid biiayer The success of continuum electrostatics at describing tiie ijasic biophysicai pro]XTties of soluble pro-teitis leads us to ask if tliese approaches can also be used

to understand the energetics of membrane proteins Biochemical partitioning experiments performed on amino acids in two-phase buik soiuliotis liave produced amino acid hydi <^phobicity scaies that predict a high en-ergetic ixirrier ibr inserting charged residues into low-dielectric etnironnients sitnilar to the hydrocarhon core

of the membranẹ For instance, solvatioti enetgies for liie positiveiy ciiarged residue atgiuine ratige from 44 to

60 kcai/moi (Wiice et aị, 1995), and continuttm eiec-trostatics calcuiations match these vaiues weii (Silkoff

et aị, i996) hi tiie iate 60s Adrian Parsegian used con-tinuum metiiods to arrive at simiiatiy iargc euerg\' har-riers when considering the movement of ciiarged ions across the niembrane {Parsegian, 1909) However, a re-cent study introduced a ijiological iiydropliobicity scale tiiat chaiienges the iong-heid notion tiiat charged resi-dues are not easiiy accomtnodated iu the iow-dieiectric core of the biiayer Hessa et aị (2005a) measured the ability ofthe Sec61 translocon to insert a wide range of designed poi\peptide sequences (H-segments) into the niembrane of rough tnicrosomes Surprisitigiy, tiiese ex-periments reveaied that there is a very iow apparent free energy for inserting charged residues into t h e membranẹ By these methods, the apparent free energy for arginine was determined to be -^2.5 kcai/mol

Corrcspnndcncc to Michai-l C.nibe: mdgiabc@piiịcdu

VUv oiiliiie veísion ol ihis article coiiLaiiis supptcniculal material.

Abbreviations used in this paper: MD iTKik-tiilar dynamics; SASA, sol-vent accessible surface area; TM, Lrdiismcmbninẹ

O 2008 Choe el al.

The Rockefeller University Press (30.00

563

Reference peptide

solution membrane

Figure 1 Cartoon diagram depicting the states used to calculate

amino acid insertion energies (A) The total energ\'ofa reference

peptide haibonng between zero and seven TM leticine residues

in a background of TM alanine residues is calculated in solution

(tell) and then in the presence ofthe membrane (riglu) In both

stales, the three terms in Eq I or four temis in Eq 3 are

cal-culated Helices were constructed using MOLDA (Yoshida and

Matsuura, 1997) (B) The central residue {green} was

systemati-cally replaced by all other residues, except proline, and the

en-ergy calculations between solutioii anfl membrane were repeated.

Carefully subtracting energv' values comptited from B with those

from the reference peptide in A removes contiibtitions to the

in-sertion energ)' from the backgrotind residues as disctissed in the

online supplemental material.

More recently, Dorairaj and Allen (2007) performed

detailed molecular dynamics (MD) simulations of a

poly-leucinc TM a-helix harboring a single charged

ar-ginitie to probe the energetics of charged residues in

lipid bilayers They calculated the free energy profile

for tratisladng the charged residtte from the upper

leaf-let to the lower leafleaf-let revealing that an insertion energy

of 17.8 kcal/mol is required to move the arginine to the

center of the bilayer This energ)' is remarkahly lower

than esdmates based on die partitioning of side-chain

analogues between bulk phases, and it makes

impres-sive strides in understanding lhe iranslocon-derived

bi-ological energ)' scale In accord widi observations from

the Tobias and Tieleman laboratories, one reason that

this enetgy is mtich lower than previously thotight is

that tlie membrane utidergoes sigtiificant benditig to

allow water access to the charged residue as it enters the

hydrophohic core ofthe membrane (Freiies et al., 2005; MacCallum et al., 2007) This distortion aiso permits the polar and charged lipid head groups to remain in favorable contact with the ciiarged residtte However, these favorable interactions occur at the expense of in-troducing strain into the lipid bilayer

The physical insight gained from cateftti MD simula-tions is indispensable, and the eticrgetic analysis serves

as a benchmark for other simulation metliods How-ever, calctilations performed with MD are coinputation-ally expensive, and we wanted to determine if much faster continiumi methods could be used to qitandta-tively reproduce the energetics of memhrane protein insertion In the 1990s, Honig's lahoratoiy developed the PARSE parameter set of partial charges and van der Waals radii specifically to reproduce the energetics of partidoning small molecules and analogue side chains between bulk phases (Sitkoffel al., 1994) Uliile consid-ering only the electiostatic and nonpolar components of the free energy of transfer, this model reproduces the ex-perimentally detennined energies quite well (Wolfenden

et al., 1981) Unforttinately, applying this technique di-rectly to membrane proteins is not straightforward The small size and fast difftisional motion of most solvent molecules facilitate die continuum approximation of replacing the atomic details of the s(^)ltiti()ti by a single dielectric valtie This approximation cannot be made for lipid membranes due to their inhomogeneity The head grotip region is characteiized hy a zone of high-dielectric vahte, the hydrocarbon core by a low-dieleclric value, and as discussed above, the shape of the mem-brane-water interface can be far from planar The largest difficulty in applying continuum elecuostatics toward understanding memhrane proteins is describing the geometr)' of the lipid bilayer around the protein Here

we use elasticity theory to determine the shape and strain energy associated with bilayer deformations Elas-ticity theory has been employed to understand the ge-ometry of membrane systems (Helfrich, 1973), and at smaller length scales it has successfully described tnem-brane protein-bilayer interactions (Nielsen et al., 1998; Harroun et al., 1999; Grabe et al 2003) Itnportantly, once the membrane shape has been determined, we feed this shape into an electrostatics calcttlation to de-termine the solvation energy of the TM helix We show that by mat rying these two theories, we are able to qtian-dtadvely match recent MD simulations at a fracdon of the comptttational cost, demonstrating that continuum tnethods can be used tC) understand ihc- energetics of charged membrane proteins

MATERIALS AND METHODS

Determination of Amino Acid Insertion Energies

We designed similar peptides to those used in Hessa et al (2005a) and calculated their electrostatic, membrane dipole, membrane

564 Modeling Membrane Insertion Energies

TABLE I

Parameter Valites Used in AU Calculations

WaiiT diclcctrir

Protein dit-lfctrit

Mi'mbrane core dielfctric

t.ipid head group dit-lectric

K({iii]il>riiiiii [iiL-inhiane w i d t h

Head group width

liin sciccninR coiircntralion

Hulk inicrfiicial sin face- tension

Area compression-expansion modulus

Bending or splay-<fistonion nindulus

a/K,

U

SASA prefactor for nonpolar t-ncrgy

Consuint term for nonpolar energy

80 2 2 80

42 A

8 A

100 inM 1.425 X 1 0 " N/A

2 8 3 X 1 0 " ' " N A

1.05 X 10"^ A^

5.66 X ] 0 ' ' A ^ 0.028 kcal/mol A 1.7 kcal/mol

Rons etal., 1991 Clallicchio etal., 2000 Ung etal., 2007 Long etal., 2007 Nielsen and Andersen, 2000 Upowsky and Sarkman, 199!> Shrake and Rupley 1973 Jacobs and White 1989 Jacobs and Wiiitf, I9«9 Jacobs and White, 1989 Jacobs and Wliiie 1989 Jacobs and White, 1989 GalHcchio etal 2000 Gallicchioei il , Junn

dfformation, and nonpolar energies in solution and in the

mem-braiu- as described below The dittercnce beuvecn these terms

computed in solution and in the membrane is the total helix

in-sertion energ)' (see Fig 1) The single amino acid energy scales

shown iti Figs 2 and 7 were obtained by comparitig the insertion

ctKTgifs of two sitnilar segments harboring different cenlial

;nnini) iicitis The etiergy difTerence bet\veen the helices is altril>

ntcd tn the eiuTg)' diflerence between tlie central residnes Exact

details for ptodticitig the amino acid energy scales are given in

the online supplemental material (av"dilable at http://www.jgp

.org/cgi/content/full/Jgp.2()08()9959/DCl).

Electrostatic Calculations

All <H1( iil.iiiniis weic performed using tbe program APBS O.f).!

(haker ft al., 2001) Three levels of focusing were employed

start-ing with ati initial sysiem si/e of 300 A on each side The spatial

discieti/;ition at the finest level was 0.3 A per grid point We

in-rltidfd sytnmetiic counter-ion concentrations of 100 inM in all

calculations In the presence of the membrane, APBS was first

used to generate a set of dielectric, ion-concentration, and charge

maps describing the system in solution We developed code to

then edit these 3D maps to add the effect of the membrane with

the geometry detennined tVoin the soltition of a separate

elastos-uitiis cakuhitioii Once edited, tlie maps were read back into

APBS U) finisb the electrostatic cakuhuions Dipole charges used

to create tbe membrane dipole potential were added at the two

adjacent grid points closest to the membrane core-lipid head

group interface These charges were added in plus/minus pairs

Sl) that the net charge of tbe system was not affected, and the

value of each charge was scaled to reproduce the desired internal

potential of+3()0mV Our calculations only incltide the influence

of the membrane dipole potential on the menibiane protein

charges; we do not consider the secondary effect of bending a

sheet of dipole charges or the reversible work associated with

moving charges apart for inserting the membrane protein For

more details on editing these maps please see Grabe et al (2004)

and the online suppletnental material.

Elastostatic Calculations

The sliape and strain enei-g)' of the membrane was calculated

us-ing ekisticit)' theory with typical tnembrane parametei^s taken

from Nielsen et al (19*18) (Table I), hi ibe supplemental

mate-rial, we derive tbe finite difference scheme used to ruimerically

solve for the membrane shape after distortion, and we discuss the

solution's cotivorgence propetties We also discuss the integration

tnethod tised to determine the strain energ\' frotn the mem-brane shape Finally, the hfigbt of the memmem-brane was fed back into the electrostatic calculations to determine the dielectric envi-ronment around the inset ted helix There are many ways tbat this fma! step cati be performed, hut we determined that the mem-brane-water ititerface coutd be well fit over the entire domain by

a high-Older polynomial The polynomial lit is used in the code

to edit tbe dielectric ion-coiKentration and cliaige maps The exact equation for the membrane shape is given in the online

supplemental material

Nonpolar Energy Calculations The solvent accessible surface area (SASA) of each transmem-brane helix was call tilated by treating each atom in the molecule asasolvatedvaiiderWaals' sphere (Lee and Richards 1971) The surface of each sphere was reptesentecl by a set of 100 evenly dis-tributed test points assigned as described in Rakliin;mo\ et al (1994) Tbe fraction of test points tiot occhided by neighboring atoms or tnembrane was used to calctilate the SASA of each atom, and these values were summed to give tbe touil SASA of the pro-tein (Shrake and Rupley, 1973) Tbe nonpolar contribution to the solvation energy was assumed to be linearly pioportionat to the SASA as described in the Results.

Online Supplemental Material Tbe online supplemental material is availahle ai hitp://www.jgp org/cgi/content/full/jg|x200809959/DCl We provide a detailed discussion ofthe calculations used to produce the computational insertion etietgy scales, and we show tbe relationship between the luimher of leucine residues in a TM helix and tbe total insertion energy" in Fig SI Table SI provides the energy valties used to make the bar graphs in Fig 2 B and Fig 7 A while Table S2 gives these valttes using lielices optimized with the SCWRI roiamer li-braiy rather tliaii SCCOMP Fig S2 shows the calculated pK., nf an arginine side chain as it penetrates the membrane, and Fig S3

reproduces Fig 5 D and Fig 7 A of the main text usitig a head

group dielectric value of 40 rather than 80.

RESULTS

Electrostatics of a Flat, Low-dielectric Barrier Both the Honig atid White grotips have carefttlly con-sidered the essential etiergetic temis relevant to inserting

= 80

B 40

35

I 25

8 20

^ 15

<

5

0

-5

^jj-lli

I LFVCMAWTYGSNHQREKD

Amino Acid Figure 2 f "omputed biological hydrophobicity scale tising a

clas-sical view of the membrane (A) A model peptide with an arginine

at Lhe central position (green) is shown spanning a low-dielectric

region reptesenting the membrane Helical segments with this

geometry were used to produce the bar graph in B (B)

Inser-tion energies for 19 ofthe 20 naturally occurring amino acids.

.Ajiiino acids are ordered according to the translocon scale (Hessa

et a!., 2005a) All in<ilecular drawings were rendered using VMD

(Humphrey etal 1996).

a helices into membranes (Jacobs and White, 1989:

Ben-Tal et al 1996) hi our present study, we statt with

the following components:

(I)

where the terms oti the right hand side cortespond to

the electrostatic, nonpolar, and membrane defonnation

energies, respectively Each tctm is calculated with

re-spect to the state in which the helix is in bulk aqueous

solution far from the unstrained membrane

The electrostatic contribution to helix insertion is

de-termitied by solving the Poissoti-Boltztiianti equation:

(2)

where 4> = c^/kuT is the reduced electrostatic potential

and <J> is the electrostatic potential; K is the

Debye-Huckel screening paratneter, which accounts for ionic

shielding; e is the dielectric constant for each ofthe

dis-tinct microscopic regimes in lhe system; atid p is the density of charge witliin the protein moiety For a given 3D structure, we tised the PARSE parameter set to as-sign the atomic partial chatges, p and the van der Waals radii to each atom Following the PARSE protocol, we

assigned e - 2.0 to protein and e = 80.0 to water As can

be seen in the Fig 2 A for a flat bilayer, we started by treating the entiie length of the bilayer as a

low-dielec-tric environment with s = 2.0 For a specific protein

configitration and dielecttic environment 0 can be cal-culated and the- unal electrosuitic c-nerg) determined as

GHIT = Z 't(ri)p(ri), where the sum rtttis over all charges

in the system

There is a large driving force for nonpolar solutes to sequester themselves from water This phenomenon

is associated with the hydrophobic collapse of protein cores during protein folding, atid it is a tnajor consider-ation for membrane protein insertion into the hydro-phobic bilayei Traditionally, the hydrohydro-phobic ity of an apolar solute is described in terms of the entropy loss and the enthalpy gaiti associated with the formation of rigid networks of water molecules around the solute Theoretical treatments of the nonpolar energy- gener-ally assume that tlie transfer free energy from vacuum

to water scales linearly with the SASA (Ben-Tal et al., 1996) However, detailed computational studies of small alkane molecules liave shown that for cyclic alkanes the SASA is only weakly con elated with the total free energy

of solvation This is due to cyclic alkanes having a more favorable solute-solvent interaction enetg)', per unit sur-face area, as compared with linear alkanes (GalHcchio

et al 2000) Recently, it was shown that continuum methods can correctly describe these effects if they in-corporate solvent accessible volume terms and disper-sive sokile-solvent intetactions in addition to the SASA tettiis (Wagoner and Baker, 2006) Fortunately, helices and chain-like molecules have volumes and surface ar-eas that are nearly proportional, and a pioper parame-terization of the prefactor multiplying the SASA term can effectively account for the voltime dependence Therefore, following the work of Sitkofl and coworkers,

we calculated AG,,,, as the difference between the SASA ofthe TM helix embedded in the niembrane, Am^,,,, and tbe value in solution A^,,,: AG,,p = a-(An,em ~ A^oi) + b, where the values a = 0.028 kcal/moI-A^ and b = -1.7 kcal/mol were obtained by using the equation for AG,,,,

to Ht the experinientally determined transfer ft ee ener-gies of several alkanes between water and liquid alkane phases (Sitkoff et al., 1996)

Next, we wanted to use Eq 1 together with calcula-tions like those shown in Fig 2 A to address the appat ent free energy of insertion for amino acids as determined

by the translocon experiments (Hessa et al., 2005a) We used MOLDA to create ideal poly-alanine/leucine a he-lices with tbe central position chosen to be one ofthe

20 natural amino acids, excluding proline (Yoshida and

566 Modeling Membrane Insertion Energies

Matsitura 1997) Subsequently, we optimized tlie

side-chain rotatner confonnations with SCCOMP (Eyal et a!.,

2004) The ftee energ\' difference of the helical

seg-ment in solutioti versus the membrane, AG,,,, was

calcu-lated, and the insertion propensity of the central amino

acid was determined by correcting for the backgrottnd

alanine-leucine helix (see Materials and methods)

Next, we ordered the amino acid insertion energies

along the x-axis according to the findings of Hessa et al

(2005a) in wliich the far left residue inserts the most

fa-votably and the insertion energetics increase

monoion-ically going to the right (Fig 2 B) It can be seen that

the flanking residues are in qualitative agreement; our

theoretical calculations show that isoleucine is one of

the most favorably inserted amino acids and aspartic

acid is the least favorable Our scale is not strictly

mono-tonic, but it does exhibit a similar trend of increasing

insertion energy from left to right The most obvious

deviation from this trend occurs for tlie polar residues,

which decrease in energ)' from N to Q Also, while our

calculations predict that glycine is among the least

fa-vorahlv inserted nonpolar residues, Hessa et al found

that glycine is clustered with the polar residues, which

produces a noticeable dip in the bar graph between Y

and S (Fig 2 B)

Interestingly, the PARSE parameter set was developed

U) qitantitatively reproduce the solitbilities of side-chain

analogues (SitkofTet al., 1994), yet when we extend this

work to describe tbe results ftoni the translocon scale, the

computational and experimental values lack

quantita-tive agieenient Most not;tbly the ttiuislocon scale predicts

a very small apparent ftee energ)' difTerence between

incorporating charged lesidues and polar residues,

while our continuum electrostatic calculations predict

that charged residues require 30-35 kcal/mol more

en-i-igy to itisert into the membrane (Fig 2 B) While it is

possible that the energetics of the translocon

experi-ment.s are skewed by tbe close proximity ofthe extruded

seguienl to the membrane, atnong olher possibilities,

part ofthe discrepancy surely lies in our simplified

treat-ment ol the membrane

Elasticity Theory Determines the Membrane Shape and

Strain Energy

Moleculai- simulations highlight two features that are

missing in our continuutn model First, the membrane

is not a slab of pure hydrocarbon, but rather the lipid

head gtoups are quite polar and interact intimately

with charged moieties on the protein; and second,

lip-ids adjacent to the helix bend to allow significant water

penetration into the plane of the membrane (Freites

et al., 2005; Dorairaj and Allen, 2007) Both of these

features can potentially reduce the energy required to

in-set l charged tesidues into the membrane Phospholipid

head groups are polar and mobile, making their

electro-static natttre much tnoie like water than livdiocarbon

A

n

|u,/2 th(r,e)

increasing r

B

maximum

upper leaflet 0 lower leaflet

upper contact curve

— minimum

lower contact curve

TM helix

Figure 3 System geometiy corresponding to electrostatic and elasticity calculations (A) Cross section of the deformed mem-brane showing the flat lower leaflet and cur^'ed upper leaflet (solid red) Liiis theeqtiilibritnn membrane width aud h is the hciKhlof the tipper leaflei The inidplane is assigned / = tt (dashed black) The radius ol tlie "I'M lielix is i,, and only half Of" tlu' helix is pic-tured (B) The idealized helix is shown and ilie contact cur\'es of the tipper and lower membrane leaflets are shown in red The lower curve is Hat; however, the upper curve dips down wilh a minimum value at lhe position where ihe central residue resides

on the full molecular helix when present.

For dipahnitoylphosphatidylcholine (DFPC) bilayers the width of the head grotLp region is between 6 and

9 A based on synchrotron studies (Hehn et al., 1987), and MD simulations estimate thai the corresponding dielectric value for this i egion is higlier than bulk water (Stern and Fellei; 2003) This additional level of com-plexity is easily accounted for by adding an 8-A-wide iniermediate dielectric region, wbich we assigned lo GhR = 80, on each side of the inetnbtane core It is also known that membrane siiTJiems exliibil an interior elec-trical potential ranging from +300 to +600 iiiV that is thought to arise from dipole charges at the itpper and lower leaflets (Jordan, 1983) While the exact nature of this potential is not known, it is thought that interfa-cial watei, ester linkages, and head gtotip charges may play a role Following the work of Jordan and Coalson,

we have adopted a physical tnodel of the membrane di-pole potential in which a thin layer of didi-pole charge

is added to the upper and the lower leaflet between the head group region and hydrocarbon core (Jordan, 1983; Cardenas et al., 2000) The strength of the dipoles was acljusted so that the v-aliie of the polenlial was +300 mV

at lhe center of lhe bilayer far from the TM helix While this additional energy could be considered part

of AG,.],., we chose to call it ACi,ij[«,i,., and we amended

Eq 1 as follows:

site of helix

B

•t-l

D)

CD

X

20

0

-20

minimum compression

maximum compression

-20 0 20 lower leaflet X-axis [A]

0 50

Radial Position [A]

E

0)

E

(3 '^

8 6 4 2 0 Leaflet

10 Thickness

20 [A]

Figure4 Sliapi'(i[ tlifnicmbraiif from claslitity tlieory (A) The solution to Eq 5 was computed

to detennine the height of the upper leaflet of the membrane given a deformiilion near the ori^n The metiihrane-water interfaces of the upper and lower leaflets are represented as two surfaces We only show the surfaces within a 20 A radius of the origin The centers of the surfaces are missing since we assumed that the membrane terminates on a cylindrical helix witli a radius of 7.5 A; full molecular delail is noi incorporated

at this point C:a!culations were perfbiincd with parameters found in Table I (B) The membrane profile for the solution in A is shown along the axis of largest deformation The cylinder repre-senting tlie TM segment is shown at the origin (solid hlack lines), and the surface of the mem-brane is shown in red The equilibrium height of Uie upper leaflet is also picliited (dashetl black Hue) to show that a significant amouul of water peuetration (blue shatle) accompanies membrane bending (C) The membrane deibruiation en-ergy increases as the leaflet bends with the defor-mation reaching a maximmn of~.5 kcal/mol,

Tlie much more difficult lask is lo detciTnine how lo

model membrane defects at the continuum level We chose

to use elasticity tlieoiy which treats ihe membrane surface

as an elastic sheet characterized by material properties

that describe its resistance to distortions such as benditig

(Hclfrich, 1973) This method has been successfully

ap-plied to tnemhnine mediated protein-protein interactions

(Kim et al., 1998; Grabe et al., 2003) and membrane

sort-ing based on height mismatcli between tiie hydrophobic

length of tlie protein and the membrane (Nielsen et al.,

1998; Nielsen and \ndersen, 2000) These later models are

often termed "mattress models" since diey allow the

mem-brane to compress vertically much like a mattiess hed does

when sat upon Simtilations indicate that the membrane

undergoes significant compression around huried charged

groups, so we closely followed the formulation of the

mat-tress models in oui" present work The membrane

deft^nna-tion etiergy is written in terms of die compression, bending,

and stretch of the membrane as foUows:

compression bending slretch

(4)

dii,

where K^ is the bilayer compression modulus, K^- is the

bilayer bending nKJdnlus, a is the surface tension, and

u is the deviation of the leaflet height, h, from its

equi-librium value, u ^ h — Lo/2 The total energy is given by

the double integral of the deformation energ\' densit)^

over the plane of the membrane A cartoon cross

sec-tion of the membrane bending around an embedded

TM helix illustrates the geometiy in Fig 3

The energy tninitna of Eq 4 cotrespond to stable membrane configurations The equation for these equi-librium shapes is determined via standard functional differentiation of Eq 4 with respect to u:

(5)

where 7 - a/K and P = 2K«/(L,Mi,) We solved Eq 5 to

determine the shape of the membrane, but before do-ing this we had to specify ihe values of ihe membrane height and slope both far from the helix and where it meets the helix We assumed that the membrane asymp-totically approaches its ttnstressed eqtulibi ium height,

u = 0, far from the TM helix Unfonunately, we do not know the height of the membrane as it contacts the he-lix We started by positing the shape of the contact curve based on visual inspection of MD simulations, but later

in the manuscript we will describe a systematic method for determining the true membrane-protein contact curve, which requites no a priori knowledge

In the unstressed case, as in Fig 2 A, we assumed that AG,,n.,,, is zero and that the metnbrane meets ihe helix without bending From the MD work of Dorairaj and Allen, we observed that the ttpper leaflet bends to con-tact charged residues embedded in llie outer half of the metnbrane {Dorairaj and Allen, 2007) The membrane appears to contact the helix at the height of the chaiged residue, while remaining unpertiubed on the backside

of the helix and along ihe entirety of the lower leaflet Therefore, for the present set of calculations we only con-cerned otirselves with deformations in the tipper leallet and the e n e i ^ in Eq 4 was ^vritten to reflect tliis assump-tion (see online supplemental material, available at litlp:// www.jgp.org/cgi/content/full/jgp.200809959/DCl)

568 Modeling Membrane Insertion Energies

dipole layer

dipole layer

dipole layer

dipole layer

0 10 20

Arginine Positron [A]

Figure 5 Helix insertion energy' of a model polyieu-ciTif helix with a cenUiii arginiiK", As ihc iirginiiie en-Itrs the nicinbrane, ihe upper leallet Ix-iids to allow water penetration At tlie upper leaflet, the arginine height is 21 A, and it is 0 A at the center (A) The total elecnosliilic energy remains nearly constant upon in-sertion (B) The nonpolar energy increases linearly to

18 kcal/mol as tlie membiane bending exposes buried

TM residues lo water (tl) The membrane delniinalion

energy icdrawii from Fig 4 C (D) The total helix

in-sertion energy is the sum of A-f^ plus ACiipoi.- which

is not shown (solid red line) Correcting tor the opti-mal membrane deformation at a given arginine depth,

as shown in Fig 6 B, produces a noticeably smaller inserlion eneig)' (dashed red line) Onr continuum computatiiinal model matches well wilh results from fully ;itomistic MD simulations on ihe same syslem (di-anioiuls taken Ironi Doraiiaj and Allen 1^0(17) The re-sult trom a classical continuum calculation is shown for reference (solid blue line) (E) System geometry when the arginitie (green) is positioned at the upper leaflet This configiiraiion represents the tar right posilion ()n A-D Gray surfaces repiesem ihe lipid head grou|>-wa-ter ingrou|>-wa-terface, purple suifaces lepreseni the lipifl head grt)up-liydi()earboii core intei face The membrane is not deformed in this instance (F) System geometry when the arginine is ai the center of ihe membrane This configuration represents the far left position on A-D The shape of the upper membrane-water iriier-face (gray) was determined by solving Eq 5.

We constrained the contact ciirvt- of the upper leaflet

to form a sinusoidal cttrve against the helix with the

mini-nuiin \A\UV being the height of the C,, atom of ihe

(barged residue and the tiiaxintum\aluc' being ihe heigiit

of the unstressed bilayer as shown in Fig 3 B As the

dt'|)lh ()( the charged residue changed, we adjttsted

the niiiiiniuni value of the c i m e to reflect this chatige

We solved Eq 5 numerically as detailed in the Materials

aud methods ttsitig a Finite difTerence scheme in polar

coordinates witli tlie standard bilayer parameter values

provided in Tahle I and taken frotn Huang (1986) and

Nielsen etal (1998)

The shape of the membrane when the charged

resi-due is located at its center can be seen in Fig 4 A As

discussed above, only the upper leaflet is deformed, and

the helix is not explicitly represented at tliis slage, ratlier

the membrane originates from a cylinder with a

ra-dius approximately the size of the TM segtnent (7.5 A)

The metnbiaue-water interface along the radial line

corresponding lo the point of largest deformation is

de-pi* ted in Fig 4 B By comparing the flashed hlack lines

to llie ted line, it is clear thai a sigttificant atiiotitu of

water penetration from the extracellular space

accom-patiies tliis deformation, solvating residtte.s at the helix

center However, this deformation comes at a cost; tlie

compression and curvature introduced to the

tnem-biane tesiilts in a strain energy The magnittide of the

sttain energ)', AGn,f,,,, is plotted in Fig 4 C; where the far

left corresponds to a compression of half the bilayer width, as shown in Fig 4 A, and the far right corre-sponds to no compression at all hnportantly the strain energy for the most drastic tU foriiiatitm is only ^ 5 0 kcal/mol, and the energy falls off faster than linear as the upper leaflet height at the helix is increased to its eqttilibrittm valtie Localizing the penetialion to one side of the membrane and one side of the heiix dramat-ically reduces the strain energy, which is essential for this to be a viable mechanism for reducing the insertion energedcs of charged residues while minimizing mem-hiane defoitnation

The Electrostatics of Charge Insertion Is Minimal Next, we used the metiibtane shape predicted ftom our elastostatic calctilations to revisit the electrostatic calcu-lations with an arginine at the central position We as-stitiied that the deformation of the tipper leaflet does not alter the width of the high-dielectric region corre-sponding to the polar head groups The helix was posi-tioned such that the C,, atom of arginine was at the tnetiibi^atie-waler interface (21 A, Fig 5 E), and then it was ti~anslated down until the Q, atom reached a height of

0 A at the center of the membrane For each ( onfigu-ralion, the membrane leaflet was deformed iis in Fig 4 B

so that it contacted the Cu, atom, resulting in an ever-iticieiLsitig tiienihriuie deformation as the arginine leached die bilayer core The final geometiy is pictured in Fig 5 F

0 5 10 15 20 Leaflet Thickness [A]

Figure 6 Delennining lhe optimal nicmbranf shape for a fixed

charge in the core of ihe inemhrane (A) Starling with the

argi-nine at the center of the membrane as in Fig 5 F, the helix was

held fixed, and the point of contact of the upper

membrane-water interface with the helix was varied from a height of 0 A to

its equilibrium uidth of 21 A The electrostatic ctierg)' decreases

by 35 kcal/mol as the arginine residue gains access to the polar

head groups and extracellular water The majority of lhe decrease

in electrostatic energy occurs from 21 to 5 A, and there is

pro-nounced flattening in the curve between 0 and 5 A (B) The

to-tal insertion energy exhibits a well-defined energy minimum at a

contact membrane height of 5 A.

Dtiring peptide insertion, we calculated each lemi of tlie

free energy in Eq 3 Remarkably, the electrostatic

com-ponent ofthe energy is essentially flat when the

mem-brane is allowed to bend (Fig 5 A) This result is not

obvious, and it highlights the importance of the local

geometry with regard to the solvation energ\' of"charged

proteins This restth will have important consequences

for the mechanistic workings of membrane proteins that

move charged residues in the membrane electric field

as part of their normal ftinction Nonetheless,

defbrm-ing the membrane exposes the surface ofthe protein to

water; and therefore, it incurs a significant nonpolar

energy, AGn,,, that is proportional to the change in the

exposed surface area (Fig 5 B) Additionally, AG,,,,,,,,

increases as discussed above (Fig 5 C) Adding together

panels A-(' and AGrfjp,,!^, we obtain the contintitini

ap-proximation to the potential of mean force (PMF) for

inserting an arginine-containing helix into a membrane

(Fig 5 D, solid red curve)

Continuum Calculations Match MD Simulations

Our peptide sysiem is the one exploied by Dorairaj and

Allen, which allowed us to directly compare our

contin-uum PMF with their PMF (diamonds in Fig 5D adapted

from Dorairaj and Allen, 2007) The shapes of both

curves and their dependence on the arginine depth in the membrane are incredibly similar, despite our results (solid red line) being 3-5 kcal/moI higher than the MD simulations Redoing the continuum cakulatit)n with-out allowing the membrane to bend, and neglecting the polarity of the head gnmps, results in the classical calcti-lation shown in solid blue For this model, the insertion energetics quickly rise once the charged residue pene-trates the memhrane-water interface, and then the energy plateaus to a valtie ^15 kcal/mol larger than vai-ues predicted by the membrane-bending model or MD simttlations Allowing for niembrane bending and the proper treatment ofthe polar head grtnip region, firings the qualitative shape of the continiuim calctiiations much more in-line with the MD simulations, suggesting that otu" calctilations are capturing the correct physics

of charged residue insertion into the bilayer Addition-ally, we calculated the pK,, of the arginine side chain as described in the online sttpplemental matei ial and found

il to be close to 7 at the center ofthe membritne, wliicb

is in excellent agreement with the MD simulations of

Li etal (2008)

The most tnisatisfying aspect of otir ctu rent approach

is that we have to first posit the contact curve of the membrane height against the protein, hi realitv; the mem-brane will adopt a shape that minimizes the system's to-tal energy A priori we have no way of knowing what that shape is, nor do we know if the shapes shown in Fig 5 (E and F) are stahle for the given arginine positions

We circumvented this shortcoming by canning out a set

of calculations in which the helix remained fixed with the arginine at the membrane center, hut we varied the degree of leaflet compression As the leaflet compresses from its eqtnlibritnn value, the electrostatic component

of the helix transfer energ\^ chops by 35 kcal/mol (see Fig 6 A), again highlighting lhe importance of water penetration and lipid head grotip bending Over 90%

of tbis energ)' is gained by compressing the metnbrane down to a height of 5 A, and vei7 little further electro-static stabilization is gained by compressing farther The reason for this can be seen from F'ig 5 F in which the arginine "snorkels" up toward the extracellular space Therefore, once the membrane height comes down

to ^ 5 A, the charged gtianidinium group is completely engulfed in a high-dielectric environment When the nonpolar, lipid deformation, and membrane dipole en-ergies are added, lhe energy profile produces a notice-able free energy minimum at 5 A (Fig 0 B) Thus, by extending our simulation protocol we have shown that these strtictures with deformed membranes are me-chanically stable, and we have removed the uncertainty regarding the proper placement of the membrane at the protein interface

Initially, we expected the curve in Fig 6 B to exhibit a sharp jump in energy as the positively charged arginine side chain moved across the dipole charge layer interface

570 Modeling Membrane Insertion Energies

between the membrane core and the lipid head group.

I lowever, the depth of the residue and the shape of the

l)fiiding ineiiihraiif snuxilhes out the transition and

re-sulls in a 5-6 kcal/mol stabilization as the charge exits

the interior of the core rather than the full 7 kcal/mol

corn'spondiiif^- to a charge in a +300 mV potential

Inipoi latitly, when the arginine is at the center of the

membrane, the free energy is 5 kcal/mol lower if the

mtMTihranc only conipicsscs down to 5 A rather than

com-pressing the lull halt-width ol the membratie (Fig 6 B)

Ihis realization shows that the original helix insertion

cnci^rv' (Fijif 5 D, red r u n e ) is wrong—it is too high For

each vertical posiLion of Uie helix, we then performed a

set of calculations in which we swept throtigh all values

of the leaflet thickness from 0 to 21 A to determine the

optimal compression of the membrane These coiTected

tiKig)' values were tised to detennine the PMF of

argi-nine helix insertion into tlie memhrane (Fig 5 D, red

dashed curve) The results of the MD simulations and

our contintiuni approach are now in excellent

agree-ment, and even at the center of the bilayer the two

approaches give values that differ by <1 kcal/mol A

no-talilf (liffcrencf between these tnethods is that we can

compute the dashed red cui"ve in Fig 5 D in several

Iiours on a single desktop computer, while detailed MD

simulations require c()ni]>utfr clusters and several

or-ders of magnitude more time

Revisiting the Calculations of Amino Acid

Insertion Energies

Our initial tlccision to incorporate membrane bending

into our computational model was motivated by the

model's inability to reproduce the low apparent free

en-crg\' of insertion for charged residties as determined

by the ttanslocon scale (Hessa et al., "iOOira) Thus, we

wanted to readdress the translocon data with our

mem-brane-bending model As hefore, we started with the

lii'lical segments inserted in tlie membrane witli the amino

acid of interest positioned at the center of the

mem-brane as in Fig 5 F The upper leaflet of the tncmmem-brane

was then deibrmed until it rcacbcd the midplane of

the bilayer as in Fig 5 F, and the total insertion

en-ergy of the helix was calculated even' 1 A as the leaflet

liciglit was varied from 21 to 0 The energy minimum

was identified as in Fig 6 B and used to calculate the

in-scition eiietgy of rhe central amino acid The infhience

of the backgromid helix was accounted for as discussed

previously Only when the centra! residue is charged

do we obst'r\'e stable minima with nonflat geometries

Therefore, our new calculations only marginally affect

the energetics of the polar and hydrophohic amino

ac-ids Tin- model prcdic ts tbat the insertion energetics of

tbe charged residues are 7-10 kcal/mol, corresponding

to a 25-30 kcal/mol reduction over the flat memhrane

getJinetry calctilatiotis Interestingly, as observed by Hessa

et al (2005a), our model shows that a minimal amount

20 15 10 5 0 -5

B 40

O

MacCallum et al (2007) Hessa et al (2005) Tfiis work

T ^

LFVCMAWTYSNQREKD

Amino Ac\6

30

20

10

0

-10

^m elec+np+dipole

1=1 dipole

^m elec

^ np

N R(flat) R(betit)

Figure 7 Tlu* infliifiict' of iiK'iiihraiic bcntiingon coiiipiitiiig the bioliiji;i(iil liydrophobiriiy sciilf and itic iiiicrphiy ol clccirosiatics and nonpolar forces (A) Aminoacid insnlion ( nergicsfor 17 resi-dues calrnialed usiiiff onr iiiembninc-bt-nding model (green bars) and compared witli die translocon scale (Hessa et al., 2005a) (red bars) and a scale developed from MD simulations of lone amino acick (MacCallum cl af., 2007) (blue bai^s) All three scales were shilied by a constant factor to sei the inserlion energ>' of alanine

lo zero (1.97 kcal/mol green, -O.I I kcal/mol red 2.02 kcal/mol blue) Tlie insertion energy' of cliarged residues is ledured by

25-30 ktal/niol \>\ penniiting membiane bending Calcnlalions

simi-lar [n rhose in Fig (i li indicale llial only (haiged residues result ill distorted mcmbi-.mes (li) 1 he energy' diffeieiue between veiy polar amino acids and charged amhio acids is quite small; however, our m<ide! predicts thai the physical scenarios are quite dUftrcnt The Insertion penalty for asparagine is primaiily eleetrostatic while the nonpolar component stabilizes the amino acid (left bars), (.on-versely, ihere is veiy little electrosiatic penaltv for inserting argi-nine and mosi of the <ost is associaled wilh the nonpolai' enei^gy required to expose llie TM domain to waier (right liars) Foi't om-parison, we show thai llie elassifal Hat membrane gives rise to a huge elecirostaiic penalty and a significant (i kcal/mol membrane dipole potential penalty for arginine (middle bars).

of additional t-tiergy is reqttired to insert charged resi-dues compared with some polar resiresi-dues For instimce, our calculations predict that inserting lysine requires only a little more than 1 kcal/mol more t'tirrg>' tban in-set titig asparagine This result is impressive, but it is im-portant to remember that the physical scenarios are quite different for these two cases The pcnalt)' for in-serting asparagine is primarily the electrosuitic cost of movitig a polar residue into a flat low-dielectric environ-ment; however, since the tnembrane bends in tbe pres-ence of a charged rcsidtie, the penalty for arginitie is

not electrostatic, but rather it is the nonpolar cost of

ex-posing the helix to water and bending tbe membrane

(see Fig 7 B)

While our membrane-bending model reconciles many

of the ob.sei"vations from the translocon experiments,

the magnitude and the spread of the insertion energy

values are still larger for our computational scale There

are many reasons why this may be the case, and the most

obvious reason i.s that our hypothetical situation witb a

single TM belix and a reference state in pure aqueous

solution is an oversimplification of tbe lull complexity of

tbe translocon system Tbus, we wanted to compare our

continuum calculations to MD simulations, wbich bave

equally weil-deliued states The insertion energetics of

amino acids in TM peptides have not been

systemati-cally computed via MD so we compared our values to

energies calctilated for single side chains in DOPC

bila-yers (MacCallum et al., 2007) It is immediately evident

that the insertion energetics from the continuum

cal-culations (green bars) and the MD simulations (blue

bars) bave a similar spread and are well correlated with

each other (Fig 7 A) Since all three scales in Fig 7 A

represent different physical situations (a single TM

helix [green], a lone side chain [blue], and a 3 TM

pro-tein [red]), we sbifted eacb scale by a constant factor

that sets the alanine insertion energy to zero

Compar-ing the 17 valties ofthe corrected MD scale to the

con-tintium scale, we see that 10 valties agree to within 1.3

kcal/mol Notably, the values for tjTosine and arginine

are the largest outliers, exhibiting a difference of 5-7.5

kcal/mol between the two metbods The agreement

be-tween tbe continunm metbod developed bere and tbe

fully molecular approacb is encouraging, and with tbe

proper parameterization, it is likely tbat a purely

con-tinuum approach will accurately describe the energetics

oi membrane protein-lipid interactions Finally, in tbe

online supplemental text we explored the robustness of

our conclusions with respect to changes in key

parame-ters sucb as the dielectric value of the lipid head group

region, tbe partial cbarges and atomic radii, and the

choice of rotamer libraiy: in all cases, our main

conclti-sions are insensitive to tbese parameters

D I S C U S S I O N

We have developed a method for calculating tbe

ener-getics of membrane proteins that merges two

contin-uum theories that bave on their own been very successful

at describing biophysical pbenomena: elasticity tbeory

and continiuim electrostatics Electrostatic calculations

with flat membranes have been considered in the past

(Roux, 1997; Grabe et al., 2004), but recent

experi-ments and simulations suggest a more dynamic role for

the membrane and tlie charge properties of die lipid

head groups (Freites et al., 2005; Hessa et al., 2005a;

Ramu et al., 2006; Schmidt et al., 2006; Dorairaj and

Allen, 2007; Long el al., 2007) Shape changes tbat coire-.spond to deforming tbe membrane can be determined

by solving tbe elasticity equation, Eq 5 Tbis determines the energ)' of bending, but it also predicts tbe sbape of the membrane This shape is then fed into electrostatics calculations to define the distinct dielectric regions ofthe membrane and tbe extent of water penetration around tbe protein Tbis added degree of freedom dras-tically reduces the electrostatic penalty for inserting cliarged residues into the center of the membrane at a marginal cost associated with membrane deibrmation

We sbowed in Fig 6 B tbat tbe contact curve of the membrane along tbe protein surface can be determined

by considering multiple botindaries and picking tbe one that minimizes the free energy Tbus, the proce-dure is self-consistent, and it results in phvsically stable strtictures Importantly, our continuum approacb can

be scaled up to treat larger protein complexes tbat have charged residues in or uear the TM domain such as voltage-gated ion channels However, we feel that there are several major considerations tbat must be addressed before extending otir present methodology' First, wben multiple charged residues are present on large protein complexes the contact curve will be complicated A very efficient searcb algorithm must be implemeuted to de-termine the membrane shape ol'lowest energy Second, for large deformations the linear approximation to mem-brane bending used in Eq 5 may not be adequate; therefore, we need to consider more sophisticated mod-els tbat allow for bigbly deformed geometries sucb as the finite element model of Tang et al (2006) Third, it has been sbown tbat tbe dispersion terms are needed to properly describe tbe nonpolar contribution to the free energy of solvation (Eloris et al., 1991; Gallicchio etal., 2000; Wagoner and Baker 2006), and tbe existence of fast continuum methods to do tbese calculations should

be incorporated (see (Wagoner and Baker, 2006)) Lastly, we believe tbat our metbod will be a valuable as-set to researchers considering tbe quantitative energet-ics of membrane proteins and the stability of membrane protein complexes; aud fui ibermore, a systematic study

of these systems and their interaction with the mem-brane as they undergo conformational changes will re-quire fast computational methods currently not offered

by molecular dynamics simulations

We would like to tlKuik Chiirlcs Wolgt-nnilh (University of Con-neciictiu Siorrs, ( T ) aiici Hi»ng)iiii Wiiiig {Uiiivfisity ofCliilifornia, Santa Crtiz, CA) for lielpltil rli.scussions regartling etiisiirity theory and The Center ior Moleciilai and Materials Simtilalions forconi-piiter related resources.

This work was supported by National Science Foundation gram MCB-0722724 (to M Grabe).

Olaf S Andersen served as editor.

Submilled: 9January 2008 Accepted: 24 April 2008

572 Modeling Membrane Insertion Energies