rapid advances in the crystallographic 
 determination of transmembrane 
 ( tm ) protein structures , and especially of g - protein coupled receptors 
 ( gpcrs ) , have established the lipidic cubic phase ( lcp ) as a suitable 
 medium for growing high quality crystals of these proteins ( see ref ( 1,2 ) and citations therein ) . the lcp consists 
 of a highly curved lipid bilayer structure ( see figure 1b  c ) that is continuous in three dimensions ( 3d ) and 
 separates in space two nonintersecting intertwined aqueous channels . because of the continuous nature of both the lipid and the water 
 compartments , lcp has been described as a bicontinuous mesophase with 
 its bilayer midplane tracing a triply periodic minimal surface ( tpms , 
 characterized by zero mean curvature ) ( see refs ( 68 ) and citations therein ) . the particular kind of lcp 
 utilized during in meso crystallization trials is 
 of pn3 m symmetry type , where each aqueous network 
 of the lcp assumes a tetrahedral geometry ( see figure 2d  e ) . ( a ) ball - and - stick representation 
 of the monoolein ( 9.9 mag ) lipid . double bonded 
 carbon atoms ( c9 and c10 ) in the hydrocarbon chain of 9.9 mag are 
 highlighted . the grouping of atoms in martini 
 coarse - grained beads ( eth , gl1 , c1a , c2a , d3a , c4a , and c5a ) is highlighted 
 by ellipsoids . ( b , c ) comparison of the triply periodic minimal surface 
 ( tpms ) of pn3 m symmetry calculated from the improved 
 nodal approximation given in eqs 13 ( b ) , to a snapshot after 6 s - long coarse - grained 
 md simulations of spontaneous self - assembly of 9.9 mag / water complex 
 at 40% ( w / w ) water concentration and at 20 c temperature ( c ) . the figure in c ( see also figures 2 and 3 ) shows the location of the c5a coarse - grained beads 
 from the 9.9 mag lipid ( see a ) ; note the cubic phase structure . the 
 similarity evident in b and c underscores the close relation between 
 the organization of the c5a beads ( representing the density of terminal 
 methylene and methyl groups in 9.9 mag lipids ) and the tpms of the pn3 m cubic phase . the rapid pace of new structural data acquisition using the features of the in meso method 
 underscores the usefulness and success of the technology . thus , structural 
 information obtained in this manner for several rhodopsin - like gpcrs 
 ( reviewed in ref ( 6 ) ) and a gpcr - g protein complex , provided 
 breakthrough insights about the structural basis for signal transduction 
 through these 7-tm helical proteins . however , the present lack of 
 a mechanistic understanding , at the molecular level , of the events 
 that lead to in meso crystallization of membrane 
 proteins , makes the success of the method somewhat enigmatic and results 
 in the need for extensive trials to identify specific conditions , 
 that is , the host and additive lipid , precipitant etc . such in meso trials begin with the target 
 protein 
 being reconstituted into the lcp , and a hypothetical descriptive mechanism 
 based on the membrane curvature and hydrophobic mismatch has been 
 proposed to address 
 the crucial step during in meso crystallization  the 
 preferential partitioning of proteins from the lcp to the lamellar 
 phase . in the original trials , the lipid used to form the lcp was 
 monoolein ( 9.9 mag , see figure 1a ) , however 
 recent experiments have utilized shorter chain analogs of monoolein 
 as well as mixtures of monoolein with cholesterol ( see refs ( 12 , 18,21)and citations 
 therein ) . the reconstituted proteins appear to be well - accommodated , 
 functional , and mobile inside the lcp where they are believed to reside 
 largely in a monomeric state . the addition of a precipitant 
 of suitable composition , which is 
 the next step in the in meso trial , is proposed to 
 result in local structural and compositional changes in the mesophase 
 and likely in protein conformation as well , which alter the interactions between the 
 proteins and the cubic phase lipid bilayer ; ultimately this drives 
 the preferential partitioning of the proteins into the lamellar phase 
 that presumably appears within the lcp . the critical aspect of the nucleation and subsequent crystal growth 
 process that is triggered in this manner , and ultimately determines 
 the quality of the grown crystals , is that once they have diffused 
 from the lcp to the lamellar phase the proteins form tight 3d oligomeric 
 arrays . the oligomerization of 
 tm proteins in lipid bilayers has generally 
 been suggested to involve the hydrophobic mismatch effects originating 
 from the difference in the lengths of the hydrophobic core of the 
 protein and the surrounding lipid hydrocarbon region . such mismatch effects have been shown from experiments to play a 
 role not only in the spatial organization but also in the function 
 of tm proteins , such as the gpcr rhodopsin . the hydrophobic mismatch is considered to drive the membrane to 
 deform in order to alleviate the energetically unfavorable hydrophilic / hydrophobic 
 exposure this mismatch entails . when such membrane deformation can not 
 achieve a complete hydrophobic adaptation , the  residual exposure  
 ( or  residual mismatch  ) to unfavorable interactions 
 gives rise to an unfavorable contribution to the free energy of protein / membrane 
 interaction . the mitigation of this energy 
 term through oligomerization has been suggested as an important mechanistic 
 contribution to membrane - driven oligomerization of multihelical tm 
 proteins , such as gpcrs , in lipid bilayers . therefore , we investigated the potential role of differential hydrophobic 
 exposure of such proteins in lipid bilayers of the cubic and lamellar 
 phases as a possible determinant of the favorable outcome of in meso crystallization trials . here we present the 
 results of a molecular dynamics - based study 
 of a gpcr reconstituted into an lcp environment that reveal differences 
 in the hydrophobic / hydrophilic exposure of the gpcr to lipid the quantitative results 
 reflect the drive for differential oligomerization behavior of these 
 proteins in the two lipid environments that essentially define the 
 end points along the in meso crystallization pathway . 
 with the ability to connect quantitatively the difference in gpcr 
 oligomerization behavior in the two different media , to the curvature 
 and mismatch alleviation capabilities of the two environments , we 
 are able to predict lipid types that would enhance / diminish the difference 
 in the residual exposure for a gpcr in the cubic and lamellar phases . given the recognized importance of protein membrane interaction 
 for the functional properties of gpcrs , molecular dynamics simulation 
 studies using all - atom and coarse - grained representations have been carried out for the prototypical gpcr , 
 rhodopsin , in lamellar membranes . to our knowledge the present work 
 is the first molecular dynamics - based study of a gpcr reconstituted 
 into the lcp . the results enable quantitative inferences about the 
 energetically unfavorable hydrophobic  hydrophilic interactions 
 that differentiate the behaviors of gpcrs in the two different lipid 
 environments , lcp and lamellar membranes . we attain these insights 
 from comparative microsecond time - scale coarse - grained simulations 
 with the martini force - field of rhodopsin in monoolein - based 
 cubic and lamellar lipid bilayers . using our recently reported computational 
 framework for quantifying residual mismatch energies from md trajectories , we evaluated the protein - dependent membrane 
 deformations and their attendant energy cost , and identified specific 
 tm regions and residues of rhodopsin that exhibit differential hydrophobic 
 exposure in the lcp and lamellar bilayers . we show that the bilayer 
 of the lcp more efficiently shields the protein from unfavorable interactions , 
 and that the reduced level of hydrophobic mismatch in the lcp is attributable 
 to the specific , highly curved geometry of the cubic phase . better 
 protection from unfavorable hydrophobic exposure of rhodopsin in the 
 lcp phase is especially evident for residues in tm helix 1 ( tmh1 ) 
 and tmh5 . lcp environment , the planar bilayer of the lamellar phase provides 
 a more favorable setting for gpcr oligomerization as a prelude to 
 nucleation and crystal growth . thus , our findings provide novel energy - based 
 insights into driving elements of in meso crystallization 
 mechanisms and lay the foundation for future quantitative exploration 
 of rational approaches for the generation of structure - quality crystals 
 of membrane proteins . coarse - grained ( cg ) md simulations of the lipid 
 cubic phases ( lcp ) with or without rhodopsin were done with the martini 
 force field , grouping atoms into coarse - grained 
 beads ( see figure 1a ) , and the gromacs 3.3.1 
 package as listed for the various molecular 
 constructs in table 1 . as a host lipid for 
 the lcp , we used monoolein ( 9.9 mag ) which consists of an 18 carbon 
 fatty acid with a cis double bond between carbons 
 9 and 10 in ester linkage to the primary hydroxyl of glycerol ( figure 1a ) . simulation times reported are effective 
 times , taking into consideration typical for martini force - field based 
 coarse - grained simulations factor of 4 . simulations with protein were initiated 
 from self - assembled lipid diamond cubic phase of 500 lipids by replicating 
 the monoolein / water system 27 times ( 3 times in positive and negative x , y , z directions ) and inserting rhodopsin randomly into 
 the expanded lcp phase . in the first phase of the study , we conducted extensive self - assembly 
 simulations of randomly mixed cg 9.9 mag lipids and water molecules 
 into a lipidic diamond cubic phase . as listed in table 1 and detailed in the results , the diamond lipid phase was 
 successfully achieved in multiple simulations conducted at 20 c 
 temperature and 40% ( w / w ) water weight percent . all the self - assembly 
 simulations ( see table 1 ) were initiated from 
 a random placement of 500 cg 9.9 mags ( a number that is within the 
 experimentally determined range of 9.9 mag molecules in the unit cell 
 of the diamond cubic phase ) , and a corresponding number ( determined 
 by the desired percentage ) of cg water beads in a cubic box . each 
 system was first equilibrated for a short period of time with the 
 isotropic pressure coupling scheme and the lennard - jones ( lj ) parameters 
 of all the atoms set to those of the beads representing water molecules 
 in the martini force - field . this setup , implemented as well in earlier 
 cg self - assembly simulations of the lcp , allowed efficient adaptation of the volume of the simulation box 
 to the number of cg molecules in the system , and ensured complete 
 mixing of lipids and waters prior to self - assembly simulation . after this initial equilibration phase , the lj parameters for 9.9 
 mag lipids were reset to their proper values ( see details of force - field 
 parametrization , below ) , and 6 s - long self - assembly md simulations 
 were performed ( the simulation times reported throughout are effective 
 times , taking into consideration the 
 factor of 4 typical for martini force - field - based coarse - grained simulations ) , 
 and using isotropic pressure coupling ( with 3.0 ps and 3e bar time constant and compressibility , respectively ) , 
 and a 30 fs time step . to simulate rhodopsin in the diamond phase , we constructed the system 
 by replicating 27 times ( along positive and negative x , y , 
 z directions ) one of the cubic phase structures obtained 
 from the self - assembly simulations conducted at 20 c and at 
 40% ( w / w ) water . the diamond cubic phase in this enlarged construct 
 remained stable in a microsecond - long cg md simulation ( data not shown ) . a rhodopsin molecule was inserted at a random position in this large 
 cubic phase ( at 122:1 9.9mag / rhodopsin molar ratio , corresponding 
 to 4 mg / ml concentration , see figure s1 in the supporting information ) . after removing overlapping waters 
 and 9.9 mag molecules and adding counterions for electroneutrality , 
 a 4.8 s - long cg md simulation was carried out at 20 c . the total number of cg beads in this large construct was 139  979 , 
 which corresponds to 1  400  000 actual 
 atoms . the simulation utilized the isotropic pressure coupling 
 scheme as above , and a 30 fs time step . rhodopsin 
 was simulated in 9.9 mag lipid lamellar bilayers with the cg martini 
 force - field . the cg model of rhodopsin ( see below ) was inserted into 
 a pre - equilibrated cg 9.9 mag lipid membrane containing 1772 lipids , 
 and after solvation and ionization steps , a 3.2 s - long md simulation 
 was carried out at 20 c temperature using a 40 fs integration 
 time step and the semi - isotropic pressure coupling scheme . the details of the martini cg 
 force fields and the parametrization scheme employed for 9.9 mag lipid 
 and rhodopsin can be found in the supporting methods section of supporting information . the residual exposure energy is the energy 
 cost of the hydrophobic mismatch that persists in the equilibrium 
 state of the membrane - protein interaction , even after membrane remodeling 
 around multi - tm proteins has taken place to reduce the hydrophobic 
 mismatch . this result 
 of the anisotropy of the protein  membrane boundary when multi - tm 
 proteins are inserted in the bilayer has been discussed in detail . for rhodopsin , the residual exposure of residues in the tm segments 
 was quantified with the computational protocol described in ref ( 30 ) in both the lipid cubic 
 and lamellar phases , using solvent accessible surface area ( sasa ) 
 calculations with the naccess software and a probe radius of 1.4  . as described previously , sasa values 
 were obtained either with the solute comprising the protein only ( saprot ) , or with the solute containing the protein and the hydrophobic 
 core of the lipid bilayer ( samem)the latter being 
 defined as the 9.9 mag bilayer region consisting of the gl1-c1a - c2a - d3a - c4a - c5a 
 cg beads ( see above for the cg bead definition , depicted in figure 1a ) . the residual exposure sares was then 
 calculated as sares = saprot  samem for polar residues and as sares = samem for hydrophobic residues ( see ref ( 30 ) for details ) . these accessible surface 
 values were used to obtain the residual mismatch energy penalty gres associated with a particular residue as 
 gres = ressares , where res = 0.028 kcal/(mol ) is a constant of proportionality related to the free energy 
 of transfer of amino acids between aqueous and lipid environments . according to a well - accepted protocol , interfacial trp , arg , and lys residues were not penalized for residual 
 mismatch due to their ability to adapt favorably to hydrophobic / hydrophilic 
 interfaces . for a quantitative assessment of the spatial organization of the 
 bicontinuous lipid / water structures assembled computationally , we 
 utilized an analytical fit of the data obtained from the simulations . 
 to obtain this fit we represented the location of the lipid bilayer 
 midplane of the bicontinuous cubic phase by the triply periodic minimal 
 surface ( tpms , surface with zero mean curvature):1where f(x , y , z ,  ) is the improved nodal approximation given by:2with3 and x = x , y = y , z = z , and p , q , r = c , s ( c and s stand for cosine and sine , respectively ) . the { h , k , l } in the above equations are the crystallographic miller 
 indices , and  is a fitting parameter such that 1/ represents 
 the linear size of the primitive unit cell . this surface was shown 
 to provide an accurate representation of the surface in the pn3 m symmetry group ( see , for example , ref ( 3 ) and citations therein ) . for each 9.9mag / water trajectory listed in table 1 , a fit to the analytical solution was performed on 20 separate 
 frames spaced 350 ns apart . to this end we extracted the coordinates 
 { xi } of n c5a coarse - grained beads representing the location of 
 the terminal methylene and methyl groups in the hydrocarbon tail of 
 9.9 mag lipids for each frame , and the fit was done by optimizing f(xi ,  ) = ci through 
 a search for the translational and rotational transformations that 
 minimized  = ( 1/n)i=1ci . the quality of the fit was assessed from 
 the distribution of { ci } values obtained from the minimization procedure ; this distribution 
 is centered around the theoretical mean , zero ( i.e. , the analytical 
 solution of eq 1 ) . such a distribution was collected 
 for each md trajectory by merging the { ci } coefficients obtained from fitting all 20 separate 
 frames , and the quality was assessed as described above ( see distribution 
 plots and captions in figure 4 and in figure 
 s3 in the supporting information ) . to compare quantitatively the structural arrangement 
 of the lcp bilayer near the protein , to that in the bulk cubic phase , 
 we employed two complementary approaches . in the first , we obtained 
 an analytical solution for the lipid bilayer midplane around the protein 
 by fitting the simulation data ( coordinates of c5a beads ) to f(x , y , z ,  ) = ci , as described above in the tpms representation , 
 except that for this purpose the { ci } coefficients were also considered as variable parameters 
 that must be self - consistently minimized ( see above ) . in this manner , 
 the extent of the deviation of the mean value of the { ci } distribution obtained from the solution 
 for the pn3 m phase ( ci  0 , see above ) , 
 measures the distortion of the diamond cubic phase due to the protein . the alternative approach we developed for quantifying the perturbation 
 of lipid bilayer shape induced by the gpcr departs from the symmetry 
 group definition and focuses on the lipid  water boundary . in 
 this approach , different trajectory frames from the simulation of 
 rhodopsin in lcp are aligned onto snapshots from the simulations of 
 the large protein - depleted 9.9 mag / water system . using a locally defined 
 scoring function ( see below ) , we assess how well the lipid bilayers 
 of the protein - containing and protein - free systems align ( the large 
 protein - free system was constructed by replicating 27 times the equilibrated 
 system composed of 500 9.9 mag lipids/6800 water molecules , and subjecting 
 it to microsecond long cg md simulation during which the pn3 m phase was maintained ; data not shown ) . to align the protein - free 
 and protein - containing structures , we 
 used water densities to first align 10 frames from the last 100 ns 
 of the protein - free simulation , which served as a reference on which 
 we aligned 10 frames from the last 100 ns of the protein - containing 
 trajectory . only water beads at least 50  away from the protein 
 were considered , in order to avoid degradation of the alignment quality 
 due to deformations of the lipid cubic phase near the protein . the alignments utilized the electron density (y ) at any point y of a set of beads with coordinates 
 ri ; the density function was constructed 
 as a superposition of gaussians , so that for a particular trajectory 
 frame4where c and k are the amplitude and width of the gaussians , respectively , chosen 
 to produce a 10  resolution density map . the quality of the superposition of two frames was 
 estimated by 
 measuring the overlap of the water - containing and water - free regions 
 using:5where x = { xj } ( j = 1,2, ... ,m1 , m1 + 1, ... ,n ) are coordinates of beads in 
 the fitted frame which contains m1 water 
 beads and n  m1 lipid beads ; w is the density 
 of water in the reference frame obtained using eq 5 ; sj = 1 for j  m1 , and sj = 1 for j > m1 . the best alignment was calculated by maximizing s with respect to the rotational and translational transformations 
 applied to x. the local quality of the alignment 
 was assessed by a normalized 
 scoring function defined as:6where (xj ) represents the electron density of the 
 beads in the reference frame . the self - assembly of the 9.9 mag / water complex 
 with 40% ( w / w ) water composition was probed at 20 c , which are 
 conditions similar to those used in the original in meso crystallization trials of gpcr proteins ( see ref ( 14 ) and citations therein ) . 
 figures 2 and 3 show several views of 
 9.9 mag / water complexes obtained under these conditions produced by 
 the 6 s - long self - assembly simulation of 500 9.9 mag lipids . 
 in figure 2 , panels a  c offer views 
 of the unit simulation cell , whereas figure 3 depicts various snapshots of the same system replicated 27 times 
 ( 3 times in each orthogonal direction ) . typically , in our simulations 
 the initially randomly mixed 9.9 mag / water system became organized 
 as illustrated in figures 23 within the first 350 ns of simulation , and remained 
 stable for the remainder of the 6 s trajectory . ( a , b ) views of the unit 
 simulation cell containing 9.9 mag / water 
 complex at 40% ( w / w ) water concentration and at 20 c temperature , 
 after 6 s of self - assembly simulation . ( a ) entire system where 
 different components are colored as following : 9.9 mag lipid headgroup 
 beads , blue ; 9.9 mag lipid hydrocarbon tail beads , red ; and water 
 beads , cyan . ( b ) separately 9.9 mag hydrocarbon chain beads from the 
 same perspective as in a. ( c ) view of the tetrahedral arrangement 
 of the water channel from a. the four arms of the tetrahedral geometry 
 are visible . ( d ) snapshot of two nonintersecting continuous intertwined 
 water channels ( in gold and cyan , respectively ) . the representations 
 are for the  expanded  system obtained by replicating 
 the simulated system 9 times ( in positive x , y , and z directions ) . views of the 9.9 mag / water complex from figure 2 replicated 27 times ( 3 times in positive and negative x , y , and z directions ) reveal bicontinuous 
 diamond cubic phase nature of the self - assembled structure . different 
 panels show : the entire system ( a ) , 9.9 mag headgroup and water cg 
 beads ( b ) , 9.9 mag hydrocarbon chain beads ( c ) , only water beads ( d ) . the bicontinuous cubic phase nature of the self - assembled 
 structure 
 is apparent from inspection of the spatial organization of both lipid 
 and water components . indeed , the hydrophobic core of the 9.9 mag 
 lipids ( shown in red in figures 23 ) traces a single highly curved continuous surface 
 in 3d , surrounded by 9.9 mag headgroups ( dark blue shades in figures 23 ) adjacent to aqueous 
 compartments ( cyan colors in figures 2a , c and 3 ) . the aqueous part of the system consists of two 
 nonintersecting three - dimensional continuous networks of water channels 
 ( figure 2d ) , which makes the assembled structure 
 a bicontinuous cubic phase . figure 2c 
 shows that in each of the water 
 compartments four aqueous channels meet in a tetrahedral arrangement . a similar arrangement can be observed as well for the surface traced 
 by the hydrocarbon chains of 9.9 mag lipids ( see figures 2b and 3c ) . thus , the assembled structure 
 resembles closely the doubly bicontinuous lipidic diamond cubic phase . one general feature of such mesophases is that the surface traced 
 by the midplane of the lipid bilayer , which separates the two aqueous 
 compartments , can be approximated by the triply periodic minimal surface 
 ( tpms ) that has a special property : it is characterized at every point 
 by zero mean curvature . consequently , the tpms is saddle - shaped ( figure 
 s2 , supporting information ) . this saddle - shaped 
 geometry of the 9.9 mag / water assembly resulting from our calculations 
 can be appreciated , for example , by tracing the surface formed by 
 the 9.9 mag hydrophobic core ( see figures 23 ) . for a quantitative assessment 
 of the adherence of the bicontinuous 
 structure from our simulations to the organization of the pn3 m diamond cubic phase , we performed a fit of the simulation 
 data to the analytical solution , as described in methods . the quality of the fit was assessed quantitatively 
 from the distribution of the fitting coefficients { ci } ( see methods ) shown in figure 4 . clearly , the distribution is centered around zero mean , and is 
 relatively narrow ( with a standard deviation of 0.4 ) , suggesting 
 that the surface traced by the c5a beads is consistent with the tpms 
 of the pn3 m cubic phase ( see eqs 13 ) . distribution of { ci } coefficients from the fitting 
 of the c5a bead positions from the 
 simulations to the analytical tpms approximation ( see methods ) . the  test established the significance of the fit 
 with probability p < 0.05 . as indicated in table 1 , 
 the self - assembly 
 of 9.9 mag / water mixture was investigated in two additional simulations 
 performed at t = 20 c and 40% ( w / w ) water content . 
 in all cases 9.9 mag and water molecules aggregated in doubly bicontinuous 
 diamond lipid cubic phases , similar to that described in figures 23 ( see figure s3 
 in the supporting information ) . to explore the organization 
 of a gpcr inside the cubic mesophase we simulated the prototypical 
 class - a gpcr , rhodopsin , in 9.9 mag - containing lipidic cubic phase 
 in a large system constructed as described in methods . the simulation results were used as described below to compare 
 quantitatively the interactions of rhodopsin with bilayers of the 
 cubic and lamellar phases , as a basis for interpreting the mechanistic 
 role of the hydrophobic mismatch in triggering the nucleation process 
 during the in - meso crystallization . residues in the rhodopsin tm bundle that participate 
 in unfavorable interfacial hydrophobic / hydrophilic interactions with 
 the lipid bilayers of the cubic and lamellar phase were identified , 
 as described in methods , with a sasa - based 
 approach . the last 1.5 s intervals 
 from the respective cg md trajectories for rhodopsin in the 9.9 mag 
 lcp ( 4.8 s ) , and in the 9.9 mag lipid lamellar phase ( for 3.2 
 s ) were used for the residual exposure analysis ( see methods for definitions ) , and the corresponding 
 residual exposure energies were evaluated as described . table 2 lists rhodopsin 
 tm residues that were found to exhibit different hydrophobic exposures 
 in the lcp and lamellar phase simulations , alongside their respective 
 residual exposure energy penalties ( gres ) . all other residues in the tm bundle experienced a residual 
 penalty of < 1 kbt ( kb denoting the boltzmann constant ) in both lpc 
 and lamellar phases . analysis was carried out on the 
 last 1.5 s time interval of the respective trajectories where 
 sasa values and corresponding residual exposure energies remained , 
 within fluctuations , unchanged . numbering shown in parentheses corresponds 
 to the generic residue numbering scheme for gpcrs defined by ballesteros 
 and weinstein . different residual exposure in the two lipid environments 
 became 
 evident for residues in the extracellular ( ec ) end of tmh1 , and the 
 intracellular side ( ic ) of tmh5 ( table 2 ) . specifically , the pro34/gln36 pair on 
 tmh1 ( superscript numbers identify the residues by the ballesteros 
 and weinstein generic residue numbering scheme for gpcrs ) , and phe228 in tmh5 were found 
 to generate a lower residual exposure energy penalty in the lcp than 
 in the lamellar phase . especially remarkable is the difference in 
 gres for the hydrophobic pro34 residue , that is , 4.7 kbt in the lamellar bilayers , but < 1 kbt in the cubic phase . this difference 
 results from pro34 being largely exposed to the lipid 
 polar headgroups and/or water environment in the planar membrane , 
 but shielded from such unfavorable interactions in the lcp . the large residual exposure predicted for pro34 in 
 the lamellar 9.9 mag bilayers is consistent with findings from our 
 earlier all - atom md simulations of rhodopsin in planar membranes composed 
 of dicn:1pc ( n = 14 , 16 , 18 , 20 ) lipids ; 
 the largest residual exposure for pro34 was found in 
 the bilayers with the thinnest hydrophobic core , that is , dic14:1pc and dic16:1pc membranes . due to the substantial 
 difference between the hydrophobic length of tmh1 in rhodopsin ( 37  , measured along 
 the membrane normal ) 
 and the bulk hydrophobic thickness of the thinner bilayers ( 27.2 
 and 30.5  , respectively for dic14:1pc and dic16:1pc membranes ) , the mismatch was 
 not alleviated by membrane deformations around the protein ( see ref ( 30 ) ) . here , the hydrophobic 
 thickness of the 9.9 mag lamellar bilayers , calculated from the average 
 distance between the lipid gl1 backbone beads on the two leaflets , 
 is 31.5  , so that the hydrophobic mismatch between tmh1 and 
 the bilayer is again substantial , and can be alleviated only partially 
 by membrane deformation , as shown below . the average thickness 
 of the 9.9 mag lamellar bilayer around rhodopsin , 
 calculated from the last 1.5 s of the md trajectory and the 
 membrane thinning around tmh2the helix adjacent to tmh1is 
 evident in figure 5 . interestingly , we find 
 that the bilayer deforms near the polar ser98 residue 
 ( see figure 5d ) , which is apparently shielded 
 from the unfavorable exposure to the hydrophobic lipid core by the 
 local thinning of the lipid membrane . usually , the membrane - facing 
 oh of ser residues in a tm bundle can avoid hydrophobic contact 
 by forming hydrogen bonds with the helix backbone . when this is not 
 feasible energetically , the bilayer thins around this residue . in 
 our simulations we find that in order to shield ser98 from unfavorable hydrophobic exposure , the thinning at tmh2 also 
 constrains the lipids at the adjacent tmh1 ( see ref ( 30 ) for a discussion of this 
 type of constraint ) , so that the membrane can not deform significantly 
 and thus leaves the pro34 residue highly exposed ( see 
 figure 5d and table 2 ) . ( a  c ) views of membrane deformation patterns around the 
 rhodopsin immersed into 9.9 mag lamellar bilayers . the average positioning 
 of membrane leaflets is identified by the two surfaces colored according 
 to the local thickness of the bilayer ( see the bar for color definition ) . the gpcr is shown in van der waals representation with the tm helices 
 colored as follows : tm1 in gray , tm2 in orange , tm3 in white , tm4 
 in pink , tm5 in purple , tm6 in black , and tm7 in lime the rest of 
 the protein ( loops , etc . ) ( d ) magnified view of the 
 region containing residues pro34 and ser98 ( both rendered in green ) showing the membrane deformations in their 
 vicinity . figure 5 shows that the 
 9.9 mag planar bilayer 
 deforms around other tm helices as well . specifically , the membrane 
 thins around tmh4 , alleviating the residual mismatch for this helix , 
 and around tmh7 where the deformation efficiently accommodates the 
 juxtaposed amphipathic helix 8 at the hydrophobic / hydrophilic interface . overall , the heterogeneous pattern of membrane deformations we observe 
 around rhodopsin in the current simulations agrees with earlier findings 
 from all - atom md simulations showing the pattern of membrane remodeling 
 around rhodopsin and the homologous serotonin 5-ht2a gpcr . consistent with those studies , 
 we find here that the deformations of the lamellar 9.9 mag bilayer 
 do not completely alleviate the hydrophobic mismatch with the receptor 
 ( table 2 ) . the different organization of the 
 two lipid phases around the gpcr , as presented below , explains why 
 the residual exposure was larger for rhodopsin in the lamellar 9.9 
 mag bilayer than in the 9.9 mag lcp . figure 6 shows the 
 organization of the lipid cubic phase around rhodopsin in the initial 
 configuration where the protein was randomly inserted in the lcp ( see 
 also figure s1 in the supporting information ) , compared to the organization after 4.8 s of cg md simulations . c with d  f in figure 6 shows the substantial rearrangement of the lipids and solvent 
 around rhodopsin during the long md trajectory . specifically , it appears 
 that the lcp tends to minimize hydrophobic / hydrophilic contacts with 
 the gpcr as the water and lipid headgroup beads organize around polar 
 regions of the protein ( yellow colors in figure 6 ) , whereas the lipid hydrocarbon chain beads surround the hydrophobic 
 core of the protein ( see caption of figure 6 for tm definitions ) . snapshots illustrating initial ( a  c ) , and final 
 ( after 4.8 
 s simulations ) positioning ( d  f ) of rhodopsin ( in yellow / white ) 
 inside the cubic phase . a and d show the organization of water beads 
 ( in cyan ) , b and e illustrate the organization of 9.9 mag lipid headgroup 
 beads ( in pink ) , and c and f show the organization of 9.9 mag lipid 
 hydrophobic core beads ( in purple ) . in all panels rhodopsin transmembrane 
 ( tm ) helices are colored in white and the rest of the protein in yellow . shown are only water and 9.9 mag lipids that are within 20  
 of rhodopsin ; rhodopsin is oriented so that its intracellular and 
 extracellular ends ( as defined in the lipid bilayer ) point toward 
 the bottom and top of the panels , respectively . the snapshots in figure 6 also show that 
 rhodopsin equilibrates , as expected , around the saddle - point inside 
 the lcp so that the tm bundle perpendicularly 
 traverses a narrow but relatively flattened region of the lcp bilayer , 
 much like the gpcr protein does in the lamellar bilayers . however , 
 due to continuous and intrinsically saddle - shaped geometry of the 
 diamond cubic phase , the lcp bilayer is dramatically curved near tmh1 
 and tmh5 , which effectively confines the protein to a small region 
 of the lcp . this special arrangement of the cubic mesophase around 
 rhodopsin shields the gpcr in the lcp from the kind of unfavorable 
 hydrophobic / hydrophilic interactions with the lipid bilayer seen in 
 the lamellar membranes . thus , as illustrated in figure s4 ( in the supporting information ) and reflected in the 
 residual energy penalties ( in table 2 ) , residues 
 pro34 and phe228 are in contact with 
 the hydrophobic core of the lcp bilayer and therefore do not incur 
 a residual mismatch penalty , although gln36 experiences 
 somewhat higher residual exposure in the lcp compared to the lamellar 
 phase ( see figure s4 and table 2 ) . glamellar to be 2.7 kbt for the pro34/gln36 pair , and 1.8 kbt for phe228 ( table 2 ) . notably , the shielding of ser98 that faced the 9.9 mag lipid bilayer in the lamellar phase 
 simulations , occurs in the lcp simulations through interaction with 
 the protein backbone . to relate the calculated residual mismatch 
 energies to the organization 
 of the lipidic cubic mesophase around the gpcr , we quantitatively compared the structural arrangement of the lcp bilayer near the protein , 
 to that in the bulk cubic phase . as described in methods , two different approaches were used : in the first , 
 we obtained an analytical fit of the lipid bilayer midplane shape 
 around the protein by fitting the coordinates of 9.9 mag lipid c5a 
 methyl terminus beads ( figure 1a ) from the 
 simulations to f(x , y , z ,  ) = ci , considering 
 { ci } coefficients as 
 variable parameters that must be minimized during the fitting ( see methods ) . the application of this procedure to spherical 
 lipid shells situated at different distances from rhodopsin ( see figure 7a ) revealed that the organization of the lcp close 
 to the gpcr is different from that expected for the pn3 m lipid bilayer ( ci  0 in figure 7a ) . however , 
 already at distances 50  from the center of rhodopsin 
 ( compare to 3540  linear dimension of a rhodopsin - like 
 gpcr in bilayer x  y plane directions ) , the 
 simulated structure relaxes to the organization of the diamond cubic 
 phase ( ci 
 = 0 , figure 7a ) . the second approach confirms 
 that the distorted lcp bilayer near the protein tends to relax toward 
 the pn3 m phase arrangement at larger distances ( figure 7b ) . in this alternative approach we locally quantify 
 the shape of the lipid bilayer around the gpcr by aligning trajectory 
 frames from the simulation of rhodopsin in lcp onto snapshots from 
 the simulations of the pure lcp system , so as to optimize the overlap 
 of the water - containing and water - free regions ( see methods ) . a scoring function defined locally ( see eq 5 ) is used to assess the quality of the alignment . as seen from figure 7b , this procedure revealed 
 that the fit between the densities of c5a lipid beads in protein - containing 
 and protein - free systems progressively improves with the radial distance 
 from the gpcr ( increasing s in figure 7b ) . indeed , consistent with the result in figure 7a , the scoring function reaches its plateau 50  
 away from the protein , indicating that at these distances the lcp 
 arrangement becomes similar to that expected for the pn3 m phase . distance dependence of the relaxation of membrane deformation near 
 rhodopsin in the cubic phase . ( a ) analytical fit of the density of 
 c5a 9.9 mag lipid beads from cg md simulation of rhodopsin in the 
 lcp phase to the surface defined by the f(x , y , z ,  ) = ci equation ( see methods ) . the mean values 
 of the { ci } fitting coefficient 
 distribution as a function of r , are shown for 10 
  spherical lipid shells located at r = 20 , 30 , 40 , and 50  away the protein center - of - mass . ( b ) reasults from the same measurements performed with an alternative 
 approach based on the alignment of the lipid bilayer midplane in protein - containing 
 and protein - free cubic phases ( see methods ) . the quality of the fit upon alignment was determined for 10  
 spherical lipid shells located at distances r from 
 the protein center - of - mass , and the panel shows the dependence of 
 the scoring function s on r. note 
 that the deformed cubic phase near rhodopsin relaxes to the arrangement 
 that is characteristic of the pn3 m phase 50 
  away from the protein . taken together , our results establish a quantitative 
 link between 
 the residual mismatch penalty and the lipid bilayer deformations around 
 the gpcr in the cubic phase . thus , the areas where the largest perturbations 
 from the diamond cubic phase were observed included residues pro34 and phe228 ( see figure s5 in the supporting material ) , which are also the residues 
 with the largest value of glcp  
 glamellar ( table 2 ) . the main findings from this study offer 
 unprecedented insight into 
 molecular level processes related to in meso crystallization , 
 which are likely leading to protein nucleation and eventually to crystal 
 growth . thus , we found that the difference in the hydrophobic / hydrophilic 
 exposure of the protein to the lipid bilayers of 9.9 mag lcp and lamellar 
 phases is responsible for a lower residual mismatch ( and its corresponding 
 energy penalty ) in the cubic mesophase compared to the lamellar membranes . this indicates why gpcrs reconstituted into the lcp in the initial 
 stages of the in meso trial are well accommodated 
 inside the cubic phase , where they are believed to reside mostly in 
 the monomeric form . indeed , from our findings there is insignificant 
 drive toward rhodopsin oligomerization in the lcp . we showed , however , 
 why this drive is increased in the planar bilayer environment of the 
 lamellar phase where there are several critical mechanistic factors 
 that determine the probability for proteins to oligomerize . one important 
 energy component that has been established from both experiments and computations , involves the hydrophobic mismatch . in particular , a plausible mechanism to relieve the energy penalty 
 due to residual exposure is protein oligomerization , whereby the tm 
 segments incurring the highest energy penalty from the residual exposure 
 come together in the lipid bilayer . for 
 such hydrophobic - mismatch driven association to occur , the 
 residual exposure penalty alleviated by the protein association must 
 counteract other critical factors that could favor proteins in the 
 monomeric state . one such important consideration is the mobility 
 of proteins , because constrained diffusion reduces the chance for 
 protein  protein encounter . as shown in figure 8 , the spatial restriction encountered by the gpcr inside the 
 lcp ( see figure 6 ) significantly affects protein 
 mobility , with the diffusion coefficient of the gpcr in the cubic 
 phase being smaller ( dlcp = 1.1  
 10 m / sec ) than that in the lamellar 
 bilayers ( dlamellar = 5.5  10 m / sec ) . the restricted diffusion mode 
 for rhodopsin in the lcp environment is consistent with the suggested 
 high energy cost in the lcp for gpcr - sized proteins to cross between 
 different regions of the mesophase . this 
 energy penalty was quantified from phenomenological principles to 
 be as high as 1220 kbt in the pn3 m phase with lattice parameter values in the 75110  
 range . this estimated energy cost of protein 
 diffusion in the lcp is significantly higher than the residual mismatch 
 penalty we calculated for rhodopsin in the cubic phase ( table 2 ) . when considered together , the relatively low 
 residual exposure energy for the gpcr in the lcp is not likely to 
 be sufficient to compensate for the high energy barrier for protein 
 mobility in the lcp and drive protein association . the prediction 
 is , therefore , that in the cubic mesophase gpcr proteins will remain 
 largely in the monomeric state . mean - square - displacement ( msd ) as a function of 
 time calculated for rhodopsin in the 9.9 mag lipidic cubic phase ( lcp ) 
 and in the 9.9 mag lamellar membrane ( lamellar ) . corresponding diffusion 
 coefficient values , calculated from the linear fit performed in [ 0 
 s;1 s ] time interval , are dlcp = 1.1  10 m / sec and dlamellar = 5.5  10 m / sec . in contrast to the lcp , the lamellar membrane environment 
 is more 
 favorable for gpcr oligomerization : the residual energy penalty for 
 the rhodopsin tm bundle in the lamellar bilayers is 8 kbt ( table 2 ) and is significantly higher compared to that in the lcp , 
 indicating a stronger drive for gpcr oligomerization in the lamellar 
 membranes . a higher propensity for association in the planar bilayers 
 is also supported by our calculations of the protein mobility in the 
 two lipid environments , whereby we find a 5-fold larger diffusion 
 coefficient for the gpcr in the lamellar bilayers ( figure 8) . taken together , results from our simulations 
 suggest that planar 
 9.9 mag lipid membranes provide a more suitable setting for oligomerization 
 of the gpcr proteins . notably , this prediction is based on results 
 for oligomerization in a plane of the lipid bilayer . this 2d nucleation 
 process together with the formation of protein - enriched lamellar stacks 
 in 3d should ultimately drive the emergence 
 of a bulk crystal . the question remains regarding the mechanism 
 that drives the proteins 
 from the lcp to the lamellar membranes and the potential role of the 
 differential residual interactions between the protein and the bilayers 
 of the two lipidic phases in the nucleation process . we note that 
 the nucleation is triggered by the addition of the precipitant which 
 is known to cause transient water depletion from the lcp interior , resulting in structural changes in the cubic 
 phase bilayer . our current study has not attempted to quantify the 
 effects of the precipitant , but it is reasonable to speculate that 
 dehydration of the cubic phase will only increase the unfavorable 
 residual interactions between the gpcr and the lcp bilayer . this is likely to increase the drive for protein 
 oligomerization inside the lcp upon precipitant addition , which could 
 lead to the formation of locally flattened lamellar bilayers as a 
 prelude to protein crystallization . in order to address quantitatively 
 the structural perturbations due to precipitant , such as changes in 
 curvature of the lcp bilayer , it is critical to calculate the corresponding 
 deformation energies in the presence of the protein . however , the 
 representation of the complex geometry of the lcp around the insertion 
 ( figures 67 ) 
 in the numerical approach developed in the current work for quantifying 
 the lcp shape ( i.e. , fitting the md data to the analytical solution ) 
 is not sufficiently refined to serve in the calculation of reliable 
 energies we note , however , that recent experimental studies showed that , compared to 9.9 
 mag , in meso trials conducted on the gpcr - gs protein 
 complex with a shorter chain mag analog , 7.7 mag ( acyl chain 14 carbon 
 atoms long with the cis double bond between carbon atoms 7 and 8) 
 resulted in better quality crystals . thus , results from our 
 current studies as well as from recently reported all - atom simulations 
 of gpcrs in membranes of different thicknesses , explain why 7.7 mag would provide a better platform in 
 which gpcrs can aggregate more extensively : based on our findings 
 we can predict that , due to the expected 78  
 difference in the hydrophobic thickness of planar 9.9 mag and 7.7 
 mag membranes , the residual exposure of the gpcr will be substantially 
 higher in the thinner 7.7 mag lamellar bilayers . as an example , we 
 recently reported that the rhodopsin tm bundle will exhibit a 12 kbt higher residual energy penalty 
 in dic14:1pc than in dic18:1pc bilayers . a more complete quantitative elucidation of 
 the gpcr nucleation process in relation to the residual mismatch energy 
 component requires further studies that would take into consideration 
 the effect of the different precipitants on various lipids . the structurally specific predictions of the regions where the 
 residual mismatch with lipid bilayers of cubic and lamellar phases 
 differed for the rhodopsin tm bundle made it possible to probe the 
 resulting predictions and general character of the findings by examining 
 crystallographic data . in particular , we showed here that the intracellular 
 ( ic ) end of tmh1 ( pro34 ) , and the extracellular ( ec ) 
 part of the tmh5 including phe228 have higher residual 
 exposure in lamellar membranes , which leads to the prediction ( e.g. , 
 see shan et al . ) that in the planar bilayers 
 rhodopsin will exhibit strong hydrophobic - mismatch driven tendency 
 for oligomerization through tmh1 and tmh5 . to assess this hypothesis 
 in the context of structural information available for gpcrs , we examined 
 crystallographic contact interfaces for 12 different structures of 
 rhodopsin - like gpcrs obtained by means of the in meso technology ( 2 receptor  pbd codes : 2rh1 , 3pds , 3sn6 ; a2a receptor  
 pdb codes : 3eml , 3qak ; chemokine 
 cxcr4 receptor  pdb codes : 3odu , 3oe0 , 3oe6 , 3oe8 , 3oe9 ; dopamine d3 receptor 
  pdb code : 3pbl , and histamine h1 receptor  pdb code : 3rze ) . for all these 
 gpcr structures , we analyzed the content of the unit crystallographic 
 cell and quantified for each tm helix ( 1 ) the frequency of its occurrence 
 at the crystallographic contact interface and ( 2 ) the number of residue 
 interactions each tm helix forms at the contact interface . canonical  interfaces were considered , 
 where the crystallographic contacts were formed exclusively through 
 tm  tm interactions between monomers in parallel orientations . remarkably , we found that tmh1 and tmh5 , the two helices implicated 
 in the largest residual mismatch in our simulations of rhodopsin with 
 9.9 mag lamellar bilayers , contribute to the most common contact interfaces 
 in the crystallographic structures of the homologous gpcrs . more specifically , 
 as illustrated in figure s6a ( see supporting information ) , all but 4 of the 14 distinct intermolecular tm tm interfaces 
 identified in this analysis , involve tmh1 and/or tmh5 . furthermore , 
 examining the residues involved in the interactions at the contact 
 interfaces revealed that the tmh1 stretch of residues in positions 
 1.291.34 ( that would include pro34 in the homologous 
 rhodopsin gpcr ) contribute significantly to the formation of the crystallographic 
 interface . during the final preparation of this manuscript , 
 two new crystal 
 structures of rhodopsin - like gpcrs , the -opioid receptor ( kor ) 
 and the -opioid receptor ( mor ) were reported to be obtained 
 by means of in meso crystallization . consistent with the data presented above , the crystallographic interface 
 of kor consists of tm1 , tm2 and h8 , with the n - terminal end of tm1 
 ( region harboring the residue homologous to pro34 in 
 rhodopsin ) forming extensive dimeric contacts . for mor , two crystallographic 
 interfaces were reported : the more prominent interface consisted of 
 tm5/tm6 helices with the i256 residue ( aligning with 
 the residue next to phe228 in rhodopsin ) residing at 
 the dimer interface , and more limited contacts formed by tm1 , tm2 
 and h8 . taken together , our findings support the mechanistic 
 inferences 
 that ( 1 ) the reduced level of hydrophobic mismatch in the lcp is attributable 
 to the specific highly curved geometry of the cubic phase that provides 
 for more coverage from unfavorable hydrophobic exposure ; ( 2 ) compared 
 to the lcp , lamellar structures provide a more favorable setting in 
 which gpcrs can oligomerize as a prelude to nucleation and crystal 
 growth ; and ( 3 ) the extent of the residual mismatch penalty is likely 
 one of the critical mechanistic factors that determines not only the 
 drive for gpcr proteins to oligomerize during in meso crystallization , but also the structural elements that are likely 
 to participate in contact interfaces . these findings provide 
 novel energy - based insights into in meso crystallization 
 mechanisms and lay a foundation 
 for future computational explorations involving other class - a gpcrs 
 toward designing rational approaches for generation of structure - quality 
 crystals of membrane proteins .
recent successes in the crystallographic determination 
 of structures 
 of transmembrane proteins in the g protein - coupled receptor ( gpcr ) 
 family have established the lipidic cubic phase ( lcp ) environment 
 as the medium of choice for growing structure - grade crystals by the 
 method termed  in meso  . 
 the understanding of in meso crystallogenesis is currently at a descriptive level . 
 to enable an eventual quantitative , energy - based description of the 
 nucleation and crystallization mechanism , we have examined the properties 
 of the lipidic cubic phase system and the dynamics of the gpcr rhodopsin 
 reconstituted into the lcp with coarse - grained molecular dynamics 
 simulations with the martini force - field . quantifying the differences 
 in the hydrophobic / hydrophilic exposure of the gpcr to lipids in the 
 cubic and lamellar phases 
 , we found that the highly curved geometry 
 of the cubic phase provides more efficient shielding of the protein 
 from unfavorable hydrophobic exposure , which leads to a lesser hydrophobic 
 mismatch and less unfavorable hydrophobic  hydrophilic interactions 
 between the protein and lipid 
 water interface in the lcp , compared 
 to the lamellar phase . since hydrophobic mismatch 
 is considered a 
 driving force for oligomerization , the differences in exposure mismatch 
 energies between the lcp and the lamellar structures suggest that 
 the latter provide a more favorable setting in which gpcrs can oligomerize 
 as a prelude to nucleation and crystal growth . 
 these new findings 
 lay the foundation for future investigations of in meso crystallization mechanisms related to the transition from the lcp 
 to the lamellar phase and studies aimed at an improved rational approach 
 for generating structure - quality crystals of membrane proteins .