hydrophobic match or mismatch in transmembrane 
 ( tm ) helices ( or 
 proteins ) refers to the match or mismatch between the length of the 
 hydrophobic core of the helix and the native thickness of the hydrocarbon 
 region of the membrane ( figure 1 ) .
positive mismatch refers to a situation in which the helix is longer 
 than the membrane thickness ( figure 1a ) , and 
 negative mismatch refers to a situation in which it is shorter ( figure 1c ) .
hydrophobic mismatch is a fascinating example 
 of mutual protein  membrane interaction . in cell membranes , 

hydrophobic mismatch is one of the mechanisms driving the formation 
 of microdomains ( lipid rafts ) , in which membrane lipids and proteins 
 of compatible length diffuse laterally and cluster together .
microdomains usually have important functional 
 implications , for example in cell division and signal transduction .
moreover , hydrophobic mismatch is thought to 
 be important in cellular processes such as the sorting of lipids and 
 tm proteins into cellular compartments .
this notion is supported by 
 a recent survey that found differences in the lengths of tm helices 
 from various cellular organelles , which were compatible with the differences 
 in the thicknesses of the respective lipid bilayers .
helix - membrane configurations with ( a ) positive hydrophobic mismatch , 
 ( b ) perfect match , and ( c ) negative hydrophobic mismatch .
the helix 
 is represented as a cylinder , with the hydrophobic core in purple 
 and the hydrophilic termini in white .
( a ) at positive mismatch , the 
 tm helix tilts and the membrane expands to match the helix hydrophobic 
 core .
( b ) at perfect match , the helix tilts because of the favorable 
 increase in precession entropy , and the membrane thins so that the 
 polar helix termini can remain in the lipid headgroup region rather 
 than partition into the hydrocarbon region of the membrane .
( c ) at 
 slight negative mismatch ( lower left panel ) , the tm helix tilts and 
 the membrane thins locally as in perfect match . in cases of excessive 
 mismatch ,
the helix adopts a surface orientation rather than forcing 
 the membrane to thin beyond its elastic limit ( lower right panel ) .
both tm helices and lipids adapt to mismatch by 
 minimizing the 
 exposure of the polar side chains and backbone of the helix to the 
 hydrophobic membrane environment while maximizing the favorable interaction 
 between the hydrophobic amino acids and the lipid .
several means of 
 system adaptation to both positive and negative mismatch have been 
 observed in experiments . in positive mismatch , 

the helix tilts from the membrane normal and decreases its effective 
 hydrophobic length ( figure 1a ) .
another adaptation to positive mismatch is kinking or flexing of 
 the tm helix . alternatively
( or jointly ) , 
 the acyl chains of the phospholipids surrounding the helix can stretch
. the helix may also migrate to membrane regions 
 with a better match to its hydrophobic length and/or interact with 
 other tm helices / proteins . in cases of 
 negative mismatch
, the acyl chains have been shown to contract and 
 reduce the bilayer thickness ( figure 1c ) . in extreme cases of negative mismatch , where 
 tm orientation involves severe membrane deformation
, the helix can 
 be oriented in parallel to the membrane surface and reside at the 
 water  membrane interface .
recent molecular 
 dynamics ( md ) simulations have demonstrated that 
 tilting of tm helices occurs also at perfect match ( figure 1b ) and negative mismatch ( figure 1c ) . using potential of mean force calculations , 
 i m and co - workers have attributed tilting under such conditions to 
 the gain in precession entropy associated with the rigid - body translational - rotational 
 motion of the tilted helix in the membrane .
the authors demonstrated that , due to the 
 precession entropy gain , tm helices tilt at least 10 from the 
 membrane normal even under negative mismatch conditions .
for instance , 
 under negative mismatch conditions , helix  lipid interactions 
 are energetically unfavorable and oppose the tilt .
this might be related to the fact that under such conditions 
 tilting involves a desolvation free energy penalty due to the transfer 
 of the polar helix termini from the aqueous phase into the hydrocarbon 
 region of the membrane . to avoid the associated free energy penalty , 
 the membrane thins .
the reduction in the entropy of the lipid chains 
 due to membrane thinning can be balanced by the increase in precession 
 entropy . incorporating this free energy balance
, we present below 
 a simple theoretical model , an equation of state , for estimating the 
 tilt angle according to the helix length and membrane thickness .
the 
 theoretical model integrates insights gained from monte carlo ( mc ) 
 simulations using the method presented in refs ( 1317 ) and below .
we use the equation of state to estimate the tilt 
 angles of 17 
 synthetic peptides of the walp / kalp / gwalp series of different lengths 
 ( table 1 ) in six membrane types with various 
 native thicknesses ( table 2 ) .
the peptides 
 feature hydrophobic cores , composed of alanine and leucine amino acids , 
 flanked by lysine ( in kalp peptides ) or tryptophan residues ( in walp 
 and gwalp peptides ) .
the results agree with experimental data , previous 
 calculations , and our mc simulations .
the lengths of their hydrophobic 
 cores were estimated assuming a translation of 1.5  per residue 
 along the helix axis , as in a perfect -helix .
the width of the hydrophobic 
 core is calculated as described in ref ( 22 ) .
the peptide was described in 
 a reduced way ; each amino acid was represented by two interaction 
 sites , corresponding to the -carbon and side chain .
the membrane hydrophobicity was represented 
 as a smooth profile , corresponding to the native thickness of the 
 hydrocarbon region .
the total free 
 energy difference between a peptide in the aqueous phase and in the 
 membrane ( gtot ) was calculated as1where gcon is the free energy change due to membrane - induced 
 conformational changes in the peptide . at constant ( absolute )
temperature t , it can be calculated as gcon = e  ts , where e is the internal 
 energy difference between the peptide in water and in the membrane . 

the internal energy is a statistical potential derived from available 
 three - dimensional ( 3d ) protein structures .
the energy function assigns a score ( energy ) to each peptide conformation 
 according to the conformation s abundance in the protein data 
 bank .
common conformations are assigned high scores ( low energy ) , 
 while rare conformations are assigned lower scores ( higher energy ) . 

s refers to the entropy difference between 
 the water and membrane - bound states , while the entropy ( s ) in each state is determined by the distribution of the virtual 
 bond rotations in the reduced peptide representation .
gdef is the 
 free energy penalty associated with fluctuations of the membrane thickness 
 around its native ( resting ) value , calculated following the estimation 
 of fattal and ben - shaul .
their calculations 
 were based on a statistical - thermodynamic molecular model of the lipid 
 chains .
their model fits a harmonic potential of the form gdef = l , where l is the difference between the native 
 and actual thickness of the membrane and  is a harmonic - force 
 constant related to the membrane elasticity and is equal to  
 = 0.22 kt /  , where k is the boltzmann constant . in 
 the model ,
the membrane is allowed to deform within its elastic range , 
 that is , 20% of its native thickness .
gcoul stands 
 for the coulombic interactions between titratable residues of the 
 peptide and the ( negative ) surface charge of the membrane .
chapman theory that describes how the electrostatic 
 potential depends on the distance from the membrane surface in an 
 electrolyte solution .
we used electrostatically 
 neutral membranes , corresponding to the zwitterionic lipid phosphatidylcholine , 
 so gcoul = 0 .
gsol is the 
 free energy of transfer of the peptide from the aqueous phase to the 
 membrane .
it takes into account electrostatic contributions resulting 
 from changes in solvent polarity , as well as nonpolar effects , both 
 resulting from differences in the van der waals interactions of the 
 peptide with the membrane and aqueous phases , and from solvent structure 
 effects .
gimm is 
 the free energy penalty resulting from the confinement of the external 
 translational and rotational motion of the peptide inside the membrane . 

glip is the free 
 energy penalty resulting from the interference of the peptide with 
 the conformational freedom of the aliphatic chains of the lipids in 
 the bilayer while the membrane retains its native thickness .
the latter 
 three terms , i.e. , gsol , gimm , and glip , are 
 calculated using the kessel and ben - tal hydrophobicity scale .
the scale accounts for the free energy of transfer 
 of the amino acids , located in the center of a polyalanine -helix , 
 from the aqueous phase into the membrane midplane .
in order to avoid 
 the excessive penalty associated with the transfer of charged residues 
 into the bilayer , in the model the titratable residues are neutralized 
 gradually upon insertion into the membrane , so that a nearly neutral 
 form is desolvated into the hydrophobic core . to calculate the 
 free energy change in eq 1 , we conducted mc 
 simulations of the peptide in water and in membrane 
 environments . in water ,
the peptide is subjected solely to internal 
 conformational modifications . in one mc cycle ,
the number of internal 
 modifications attempted is equal to the number of residues in the 
 peptide . in the membrane ,
each mc cycle includes additional external 
 rigid body rotational and translational motions to allow the peptide 
 to change its location in the membrane and its orientation with respect 
 to the membrane normal .
a helical peptide in a membrane typically 
 adheres to one of the two following configurations : tm orientation 
 with the helices principal axis roughly along the membrane 
 normal or surface orientation with the axis approximately in parallel 
 to the membrane surface .
the transition between the two configurations 
 is associated with a high free energy barrier .
thus , for simulations 
 in the membrane environment , each of the two configurations is used 
 as the initial orientation for three independent simulations of 500  000 
 mc cycles .
simulations in water ( i.e. , without the membrane ) are also 
 carried out in three independent runs of 500  000 mc cycles 
 each .
we conducted mc simulations with 16 
 peptides of the walp and kalp series interacting with six membrane 
 types of varying thicknesses .
the results for walps , which are more 
 commonly used in experiments and md simulations , are presented below 
 ( figure 2 ) , and the results for kalps are presented 
 in the supporting information ( figure s1 ) . 
 throughout the simulations , the peptides were , in essence , helical 
 in both the aqueous and membrane environments ( figure s2 ) , which is anticipated for these peptides , composed 
 mostly of ala and leu , two amino acids with high helix propensity .
mc simulations and comparison to the theoretical model . 
 the results 
 were obtained from mc simulations of eight walp peptides ( table 1 ) interacting with membranes of six different types 
 ( table 2 ) .
the standard errors are marked ; 
 in many cases , the error bars are smaller than the symbols .
( a ) the 
 dependence of the tilt angle on the hydrophobic mismatch . a tilt angle 
 of 0 corresponds to a helix with its principal axis perpendicular 
 to the membrane plane ; a tilt angle of 90 corresponds to a helix 
 with its principal axis parallel to the membrane plane .
the inset 
 shows the theoretical dependence of the tilt angle of walp21 on the 
 hydrophobic mismatch ( solid curve ) in comparison to the values obtained 
 from the mc simulations ( triangles ) .
( c ) location 
 of the flanking residues in the membrane vs hydrophobic mismatch . 
 for clarity , the data for only three peptides are shown .
z is the shortest distance between the average position 
 of the -carbon of the flanking residue and the boundary of 
 the hydrocarbon region of the membrane .
( d ) correlation between the theoretically predicted 
 tilt angles of walps and the values estimated from the mc simulations ; theoretical_tilt = 0.97  mc_tilt + 
 0.6 , r = 0.99 .
the values based on the mc simulations were reproduced from 
 a. figure 2a shows the 
 dependence of the helix tilt angle on the hydrophobic mismatch .
as anticipated , helices whose hydrophobic cores were longer than 
 the membrane thickness were in tm orientation with their principal 
 axes tilted with respect to the membrane normal . in cases of negative 
 mismatch , the helices assumed a tilted tm orientation as well , provided 
 that the difference between the length of the helical peptide and 
 the thickness of the bilayer did not exceed approximately 10  . 
 to accommodate helical peptides of greater negative mismatch in tm 
 orientation , the membrane would be forced to deform beyond its elastic 
 limit , which was not allowed in the simulations .
water interface 
 as helices with their principal axes approximately parallel to the 
 membrane surface ( e.g. , figure s3 ) , in 
 line with previous experimental studies . within the boundaries of the elastic region of the 
 membrane ,
the tilt angle decreased as the mismatch decreased , with 
 a minimal value of 10 at a mismatch of 10  .
we compared the tilt angles calculated in the mc simulations to 
 the available data , obtained using various experimental techniques 
 and md simulations .
results for walps are shown in table 3 ; table s1 shows results for kalps . 

good agreement was observed in all cases , but it is noteworthy that 
 in some cases the range of tilt values obtained in previous studies 
 is rather large . in particular , the range of tilt values obtained 
 in previous studies of walp23 in dmpc and dlpc membranes exceeds 20 
 ( table 3 ) .
the method used is listed in 
 parentheses . where possible , the values are shown as average  
 standard deviation .
besides helix tilting , an additional 
 possible mechanism of adaptation to hydrophobic mismatch is stretching 
 and contraction of the acyl chains of the lipids surrounding the peptides . 

we estimated the 
 membrane adaptation as the average deviation of the thickness of the 
 hydrocarbon core during the simulation from its initial value , deduced 
 from x - ray studies of pure ( peptide - free ) lipid bilayers .
for helical peptides 
 that were too short to span the membrane and resided on the surface , 
 the membrane thickness fluctuated around the initial native value , 
 as it should .
interestingly , membrane thinning of up to 1  was 
 detected also when the helix s hydrophobic core was up to 5 
  longer than the thickness of the hydrocarbon region of the 
 membrane ( figure 2b ) .
finally , helices with 
 hydrophobic tm cores that were more than 5  longer than the 
 membrane thickness caused the membrane to expand slightly to improve 
 the fit to their long tm cores ( figure 2b ) . 

clearly , the changes in the membrane thickness upon incorporation 
 of a tm helix are small and might appear to be negligible in view 
 of the implicit and crude representation of the membrane in the model . 

however , the same pattern was repeatedly observed in simulations of 
 various peptide - membrane systems , consolidating the observation .
we inspected the membrane location of the trp ( lys ) residues at the 
 edges of the hydrophobic core of each walp ( kalp ) peptide , namely , 
 the residues in the third positions from the n- and from the c - termini 
 of the peptides . figures 2c and s1c show the average deviation of the -carbons 
 of these residues from the nearest membrane boundary as a function 
 of the hydrophobic mismatch .
when the hydrophobic mismatch was strongly 
 negative , the c of these residues remained at the membrane 
 boundary .
however , as the mismatch became less negative , the c 
 position extended farther away from the boundary into the polar headgroup 
 region . at a positive mismatch of around 5  , the c positions 
 of trp saturated at their maximal values of 3  ( figure 2c ) from the membrane boundary , and lys saturated 
 at a value of 4  ( figure s1c ) . using figures 2c and s1c ,
we estimated peff , i.e. , 
 the length of the helix interacting with lipid chains , to be used 
 in the theoretical model ( table s3 ) . here
, we develop a simple theoretical 
 model as a closed - form expression to estimate the dependence of  , 
 the angle at which the principal axis of the tm helix tilts from the 
 membrane normal , on the length of the hydrophobic core of the helix 
 ( p ) , on the effective length of the helix , i.e. , 
 the length of the helix portion that interacts with the lipid chains 
 ( peff , peff < p ) , and on the native ( peptide - free ) thickness 
 of the hydrocarbon region of the membrane ( l ) .
first , we deal with the two limiting cases : helices that are substantially 
 longer , or shorter , than the native thickness of the lipid bilayer . 
 for ( p  l )
 5  , 
 i.e. , positive mismatch of 5  or more , the hydrophobic effect 
 dominates , and  is determined mostly by the tendency of the 
 hydrophobic core of the helix to be buried in the hydrocarbon region 
 of the membrane ( figure 1a ) . because of the 
 low free energy penalty of membrane expansion , the membrane may slightly 
 expand ; indeed , our mc simulations showed 
 that it expands by approximately 1  ( figure 2b ) .
in addition , the ends of the hydrophobic core of the helix 
 extend out of the hydrocarbon region of the membrane such that only peff interacts with the lipids ( figure 2c ) .
the other limit addresses 
 the case of large negative mismatch , 
 i.e. , when the hydrophobic core of the helix is substantially shorter 
 than the hydrocarbon region of the membrane ( figure 1c , lower right panel ) . in this case
, the membrane would need 
 to deform beyond its elastic range ( 20% of its native thickness ) to accommodate the helix in tm orientation , 
 which is energetically unfavorable .
thus , when peff is less than 80% of the membrane s native thickness , 
 the helix resides in surface orientation , and  = 90 , 
 i.e. , when peff < 0.8l , cos  = 0 .
l < 5  
 and peff  0.8l . according to the mc 
 simulations and previous calculations , tm helices 
 tilt even at negative mismatch ( figure 1c , 
 lower left panel ) , in spite of the free energy penalty due to membrane 
 deformation ( gdef ) . the driving 
 force for this is the free energy gain from the increase in precession 
 entropy ( ts ) . here , we exploit 
 the balance between these opposing contributions to derive an expression 
 for . we make several assumptions .
first , in accordance 
 with the mc simulations , 
 we assume that the peptide adopts , in essence , the same ( helical ) 
 conformation regardless of the tilt angle , and that its internal energy 
 ( e ) is independent of the tilt
. additionally , 
 we assume that all residues are preserved in the same local environment , 
 i.e. , the hydrocarbon or headgroup region of the lipid bilayer or 
 the aqueous phase .
we also limit the possible changes in the thickness 
 of the hydrophobic region of the membrane to up to 20% of its native 
 value .
finally , we assume that helix 
 librations in the membrane ( maximum amplitude denoted as  ) 
 are independent of the tilt  ( figure 3 ) . under these assumptions
, eq 1 reduces to 
 the free energy balance:2 the precession entropy gain associated with 
 tm helix tilting in 
 the membrane .
( a ) schematic illustration of the spherical surface 
 area corresponding to the precession entropy of a hypothetical helix 
 that spans the membrane vertically .
helix librations around the membrane 
 normal , with a maximum amplitude of  , generate rotational entropy 
 that is proportional to the dark cap - like surface area .
( b ) the precession 
 entropy of a helix , which is tilted at an angle  from the normal , 
 is larger ( larger area ) . assuming that  is independent of  , 
 the rotational entropy of the tilted helix corresponds to the dark 
 belt - like area .
the helix is represented as a cylinder with the hydrophobic 
 core in purple and hydrophilic termini in white .
l is 
 the native ( peptide - free ) width of the hydrocarbon region of the membrane .
peff is the length of the portion of the helix s 
 hydrophobic core that spans the hydrocarbon region of the membrane .
figure 3 illustrates the 
 precession entropy 
 ( s ) of the helix in vertical vs tilted configurations ; 
 in each case , the entropy is proportional to the logarithm of the 
 shaded area in the corresponding panel of the figure .
this estimation 
 of the precession entropy is somewhat different from the derivation 
 of i m and lee .
in particular , it includes 
 the contribution of helix librations also in the vertical orientation . 
 in the vertical configuration ,
the precession entropy is proportional 
 to the surface area of a small sector of a sphere calculated as 2rh , where r = 1/2peff is the sphere s radius and h is 
 the height of the small sector ( figure 3a ) . 

the surface area of the sphere sector is 1/2peff(1  cos  ) . for a helix tilted 
 by 
, the entropy corresponds to a larger belt - like section 
 of the same sphere ( figure 3b ) .
the surface 
 area of the belt - like section is the difference between the areas 
 of two sphere sectors , denoted , respectively , by 2r(1 
using r = 1/2peff and 
 one of the trigonometric identities , the area of the belt - like section 
 can be written as peff  sin   sin . for negative mismatch , 
 one has to compare between two alternative 
 states : in the first state , the ( short ) tm helix is in vertical orientation , 
 and the membrane thins to match its hydrophobic core ( figure 4a , i ) .
the precession entropy of this state corresponds 
 to the area of the small sphere sector of figure 3a . in the alternative state ,
the helix tilts away from the 
 normal , and the membrane thins further ( figure 4a , ii ) .
here , the precession entropy is higher and corresponds to 
 the area of the larger belt - like region in figure 3b .
substitution in eq 2 gives3where the left - hand side is associated with 
 the precession entropy and the right - hand side with the membrane deformation . 
 in eq 3 ,  is a harmonic - force constant 
 reflecting the membrane elasticity .
for a cylinder 
 of radius 5  , approximating the helix ,  = 0.075 kt /  . the derivation 
 of eq 3 with respect to  gives a simpler 
 expression:4 derivation of the theoretical model .
( a ) two 
 hypothetical configurations 
 of the system at ( small ) negative mismatch : ( i ) the helix is in vertical 
 orientation , and the membrane thins to match the helix s hydrophobic 
 length .
( ii ) driven by the precession entropy gain , the helix tilts 
 by  from the normal , and the membrane thins further .
( b ) two 
 alternative configurations of the system for ( small ) positive mismatch 
 in the range of 05  : ( i ) the helix is tilted by an angle 
  from the membrane normal to match the native width of the 
 hydrocarbon region of the membrane .
( ii ) driven by the precession 
 entropy contribution , the tilt angle increases to  (  
 >  ) , and the membrane slightly thins . the helix is represented 
 by a cylinder , with the hydrophobic core in purple and the polar termini 
 in white .
the membrane normal is marked by the vertical dashed line ; 
 the helix s principal axis is marked by the solid line .
the mc simulations indicated that the tilt is driven 
 by the precession 
 entropy also in cases of positive mismatch of up to about 5  . 
 in this respect ,
this notion is based on the trends of membrane adaptation 
 ( figure 2b ) and the location of the flanking 
 hydrophilic residues relative to the membrane s hydrophobic 
 core ( figure 2c ) . in both cases ,
different 
 patterns were observed for different degrees of mismatch , and the 
 border between them was at a positive hydrophobic mismatch of approximately 
 5  rather than at a perfect match .
figure 4b shows two hypothetical helix - membrane configurations for a small 
 positive hydrophobic mismatch in the range between 0 and 5  . 
 in the first
, the membrane retains its native thickness , and the helix 
 tilts by an angle  from the membrane normal to match the width 
 of the hydrocarbon region of the membrane ( figure 4b , i ) . in the alternative configuration , the membrane thins , 
 and the tilt angle increases to its final value of  (  
 >  ) to facilitate the favorable increase in helix precession 
 entropy ( figure 4b , ii ) .
substitution in eq 2 gives5conveniently , the derivation of eq 5 with respect to  leads to eq 4 . to summarize:6
equation 6 can 
 be viewed as an equation of 
 state of the helix in the lipid bilayer . to understand it better
, 
 we plotted  as a function of  , l , and peff in the physiologically relevant region of 
 parameter space ( figure 5 ) .
the tilt angle 
  decreases with an increase in  ; i.e. , the membrane 
 rigidity limits the tilt , as anticipated .
additionally , 
 the tilt angle  increases with increases in peff and with decreases in l , as it should 
 ( figure 5a ) . to explore these relations further
, 
 we plotted  as a function of peff / l at constant  ( figure 5b ) .
this revealed a linear relation between  and the peff - to - l ratio at constant peff 
l , with an increase 
 in  when the helix length decreases ( and the membrane thins ) . 

the increased tilt for shorter helices ( in thinner membranes ) is due 
 to the decrease in the perturbation to the lipid ; the lipid chains 
 are shorter .
the theoretical tilt angles agree well with previous 
 calculations and measurements ( tables 3 and s1 ) . to examine the equation of state further , 
 we compared the results to mc simulations of various peptides within 
 lipid bilayers over a broad range of hydrophobic mismatch scenarios .
( a ) the dependence 
 of  on peff and l , for three different  values .
the tilt angle (  ) was 
 calculated using the second line of eq 6 in 
 the range 15  < peff < 50 
  and 20  < l < 40  , using 
 the appropriate limitations on l and peff ( i.e. , ( p  l ) < 5  and peff  0.8l ) . clearly , the dependence of  on  is marginally 
 weak .
the dependence 
 of  on peff / l for 
  = 0.075 kt /  .
the tilt angle 
  increases with an increase in peff at the same l ( symbols with the same color ) .
interestingly , 
 there is a linear relation between  and the peff - to - l ratio at constant ( peff  l ) .
figure 2a ( inset ) shows the good correlation between the 
 theoretical model and mc simulations for the walp21 peptide in bilayers 
 of various thicknesses , and figure 2d shows 
 that the agreement extends throughout the walp series ( correlation 
 coefficient of 0.99 , slope of about 1 , and small intercept ) .
similar 
 agreement was obtained also for the kalp series ( figures s1a and s1d ) .
interestingly , in both cases , the minimal 
 tilt angle was approximately 10 , in agreement with previous 
 calculations .
in addition , we also studied 
 gwalp23 , a representative peptide from the newly introduced series 
 of gwalp peptides , which feature a single trp residue at their termini ( table 1 ) .
the tilt values obtained via the 
 theoretical model correlated well with the mc simulations and available 
 data ( figure s4 , table s2 ) .
we presented 
 a theoretical derivation of an equation of state relating the tilt 
 of a tm helix to the helix length and bilayer thickness .
the equation 
 of state ( eq 6 ) was utilized to investigate 
 17 peptides of various lengths interacting with membranes of six different 
 thicknesses , covering hydrophobic mismatch in the range of approximately 
 25  , much broader than ranges used in previous studies .
the tilt angles calculated using the theoretical model correlated 
 well with our mc simulations and with data from previous experiments 
 and calculations ( tables 3 and s1 ) . in this respect , it is important to note that the first 
 nuclear magnetic resonance ( nmr ) studies in walp and kalp peptides 
 reported very small tilt angles .
however , more recent 
 publications have shown that this is because the nmr experiments were 
 interpreted using an overly simplified model of helix motion ; interpretation 
 of the same data using several dynamic models revealed larger tilt 
 angles that are closer to those obtained in computational 
 studies , including our theoretical model and mc simulations . the proper 
 model for interpretation of the nmr data is debatable , but clearly 
 the external helix motion in the membrane should be considered .
the equation of state captures the thermodynamic determinants 
 of tilting of all hydrophobic -helical peptides , regardless 
 of their sequences .
it provides a back - of - the - envelope estimate of 
 the tilt angle of any arbitrary peptide , given the peptide s 
 ( effective hydrophobic ) length and the membrane thickness .
it could 
 be useful for the design and interpretation of experiments , as well 
 as for preparation of initial peptide - membrane conformations to be 
 used , for instance , in mc or md simulations .
this way , the initial 
 configuration of the system should be close to its energetic minimum , 
 which should facilitate rapid convergence .
although the theoretical 
 tilt estimation agreed with the available 
 data , one should keep in mind that the crude estimation has inherent 
 limitations .
the equation of state is based on the balance between 
 the precession entropy and gdef , but other terms in eq 1 may 
 also contribute .
for instance , the helix may tilt even further , inserting 
 the polar termini deeper into the membrane core , and the precession 
 entropy may compensate for the associated desolvation penalty . in 
 addition , the assumptions made to derive the theory clearly simplified 
 the mechanisms affecting helix tilt .
for instance , the model assumes 
 that the internal energy ( e ) is independent of the tilt .
in fact , at larger tilt angles , the 
 side chains of the helix become more restrained , thus causing entropy 
 reduction .
similarly , the assumption of the same helix librations in the vertical 
 and tilted configurations is questionable .
moreover , the equation 
 of state does not include possible specific peptide  lipid interaction . 

finally , the theoretical model assumes that the helix can be approximated 
 by a perfectly symmetrical cylinder with no preference to any rotational 
 angle around the principal axis .
the results of the theoretical 
 model and mc simulations are compatible with previous systematic studies 
 of hydrophobic mismatch , performed using explicit md simulations and 
 potential of mean force calculations .
in particular , our simulations 
 fully agree with previous calculations showing that the helix tilt 
 from the membrane normal is at least 10 , regardless of the extent 
 of the hydrophobic mismatch . regarding membrane 
 adaptation , there are some discrepancies between the studies .
i m and 
 kim reported membrane thinning up to 7  and membrane thickening 
 up to 5  .
kandasamy and larson , 
 simulating kalp peptides , demonstrated membrane thinning of up to 
 6  and thickening of up to 3  , values closer to our estimations ( figures 2b and s1b ) , as well as to previous experimental 
 assessments .
regardless of the exact 
 values , the three studies agree that the magnitude of membrane - thinning 
 is larger than that of thickening , as anticipated .
additionally , 
 kandasamy and larson demonstrated that in kalps the lysine side chains 
 extend further into the lipid polar headgroup region as the hydrophobic 
 mismatch increases .
we observed a similar effect in the flanking trp 
 residues in walps ( figure 2c ) and in the flanking 
 lys residues in kalps ( figure s1c ) .
kandasamy 
 and larson attributed this phenomenon to specific interactions between 
 the lipids phosphate oxygen atoms and ammonium groups of the 
 flanking lys side chains .
peptide interactions in the balance between 
 precession entropy gain and free energy penalty driving the tm helix 
 tilt ; they did not demonstrate the exact nature of these interactions .
in contrast , our theoretical model is based on the balance between 
 the precession entropy and nonspecific helix  lipid interactions . 

peptide interactions 
 are also not taken into account in our mc simulations , and yet the 
 results of both the theoretical model and simulations agree well with 
 experimental data .
peptide 
 interactions play only a secondary role , at least for the walp / kalp / gwalp 
 peptides . the theoretical model , 
 derived from the mc simulations , showed that if the effective length 
 of the helix is shorter than the minimal thickness of the bilayer , 
 the helix resides at the water  membrane interface . clearly , the situation is slightly more complicated 
 than this .
a surface configuration of 
 a hydrophobic helix such as a walp or kalp is energetically favorable 
 irrespective of the hydrophobic mismatch .
in contrast , the free energy 
 of membrane association of a helix in tm configuration depends on 
 the hydrophobic mismatch ( figure s3 ) .
therefore , 
 regardless of the hydrophobic mismatch , the peptide partitions between 
 the two configurations , and the partition ratio depends on the free 
 energy of the two states .
a similar trend was demonstrated for polyleucine 
 peptides , either ln or ggpg - ln - gpgg , where n is the number of 
 leu residues , as well as for peptides with the sequence of the 
 form kk-(la)n - kk . in both walps and kalps ,
the helices surface and tm 
 configurations differ from each other in their helical content .
in contrast , the surface configuration has lower helical 
 content with decreased helicity in the helix core ( figure s2 ) .
this is because formation of a perfect -helix 
 in the surface configuration involves insertion of the polar trp or 
 lys side chains and helix termini into the membrane core , which is 
 energetically unfavorable .
this is in agreement with the all - atom 
 molecular dynamics simulations of ulmschneider and colleagues .
it has been suggested that walps tilt 
 to a larger degree compared with kalps of similar length , since trp 
 residues partition deeper into the headgroup region than do lys residues .
this proposition is guided by 
 the difference in the free energy penalty of transfer of trp and lys 
 from the aqueous phase into the membrane ( 1.3 versus 7.4 kcal / mol 
 in the hydrophobicity scale used here ) .
the transfer free energy difference is not taken into account explicitly 
 in the theoretical model , but it is considered implicitly since it 
 affects the effective hydrophobic length of the helix peff ( table s3 ) .
application 
 of a paired t - test to the mc simulations showed that 
 the tilt angles of the walps were larger than those of the kalps at 
 a confidence level of 0.95 .
however , the average difference was only 
 2.8 , which is probably below the resolution of the mc model 
 because of the use of a reduced representation for the peptide .
following previous works on tilting under 
 hydrophobic mismatch 
 conditions , we demonstrated 
 that precession entropy can contribute to the tilting of tm helices 
 under conditions of perfect match and negative mismatch , despite the 
 unavoidable membrane deformation .
we utilized the energy balance between 
 the precession entropy s and 
 free energy of membrane deformation gdef to derive an equation of state describing 
 the dependence of the tilt on the helix length and membrane thickness . 

the theoretical tilt values are similar to measurements , previous 
 md simulations , and our mc simulations .
thus , the equation of state 
 can be used for a quick estimation of the helix tilt .
notably , 
 our simple theoretical model managed to reproduce the 
 tilt angles observed for 17 different peptides in membranes of various 
 thicknesses .
this supports the model s underlying assumption , 
 namely , that the tilt is determined by the free energy balance between 
 the helix precession entropy and lipid perturbation .
however , one 
 should take into account that the 17 peptides are synthetic and very 
 similar to each other .
it may well be that in reality more free energy 
 contributions should be taken into account .
 hydrophobic helical peptides interact with lipid bilayers 
 in various 
 modes , determined by the match between the length of the helix s 
 hydrophobic core and the thickness of the hydrocarbon region of the 
 bilayer . 
 for example , long helices may tilt with respect to the membrane 
 normal to bury their hydrophobic cores in the membrane , and the lipid 
 bilayer may stretch to match the helix length . 
 recent molecular dynamics 
 simulations and potential of mean force calculations have shown that 
 some tm helices whose lengths are equal to , or even shorter than , 
 the bilayer thickness may also tilt . 
 the tilt is driven by a gain 
 in the helix precession entropy , which compensates for the free energy 
 penalty resulting from membrane deformation . using this free energy 
 balance 
 , we derived theoretically an equation of state , describing 
 the dependence of the tilt on the helix length and membrane thickness . 
 to this end , we conducted coarse - grained monte carlo simulations of 
 the interaction of helices of various lengths with lipid bilayers 
 of various thicknesses , reproducing and expanding the previous molecular 
 dynamics simulations . 
 insight from the simulations facilitated the 
 derivation of the theoretical model . 
 the tilt angles calculated using 
 the theoretical model agree well with our simulations and with previous 
 calculations and measurements . 