Pre-transition effects mediate forces of assembly between transmembrane proteins
We choose the MARTINI model of hydrated dipalmitoyl phosphatidylcholine ( DPPC ) lipid bilayers ( Marrink et al . , 2007 ) to illustrate the orderphobic effect . See Materials and methods . This membrane model exhibits an ordered phase and a disordered phase . Figure 1A contrasts configurations from the two phases , and it shows our estimated phase boundary between the two phases . The ordered phase has regular tail packing compared to the disorganized tail arrangement of the disordered phase . A consequence of the regular tail packing is that hydrophobic thickness of the ordered phase , 𝒟o is larger than that of the disordered phase , 𝒟d . Correspondingly , the area per lipid in the ordered phase is smaller than that in the disordered phase . 10 . 7554/eLife . 13150 . 003Figure 1 . First-order phase transition in a model lipid bilayer . ( A ) Order–disorder phase diagram in the tension–temperature , λ−T , plane . The lateral pressure across the membrane is −λ . Points are estimated from 10 independent heating runs like those illustrated in Appendix 1–figure 1 for a periodic system with 128 lipids . Insets are cross sections showing configurations of a bilayer with 3200 lipids in the ordered and disordered phases . The heads are colored gray while the tails are colored pink . Water particles are omitted for clarity . The hydrophobic thicknesses , 𝒟o and 𝒟d , are the average vertical distances from the first tail particle of the upper monolayer to that of the lower monolayer . A macroscopic membrane buckles for all λ < 0 . Snapshots of the last tail beads in one monolayer of each phase are shown to illustrate the difference in packing . ( B ) Snapshot of a system showing coexistence between the ordered and disordered phases . The gray contour line indicates the location of the interface separating the ordered and disordered regions . The snapshot is a top view of the bilayer showing the tail-end particles of each lipid in one monolayer . h ( x ) is the distance of the instantaneous interface from a reference horizontal axis . ( C ) Fourier spectrum of h ( x ) . The line is the small-k capillarity-theory behavior with γ=11 . 5 pN . DOI: http://dx . doi . org/10 . 7554/eLife . 13150 . 003 Rendering the end particles of all the lipid chains in one of the two monolayers provides a convenient visual representation that distinguishes the two phases . These tail-end particles appear hexagonally-packed in the ordered phase and randomly arranged in the disordered phase . Regions that appear empty in this rendering are in fact typically filled by non tail-end particles or by tail-end particles from the other lipid monolayer . To quantify the distinctions between the two phases , we consider a local rotational-invariant ( Nelson , 2002; Halperin and Nelson , 1978; Frenkel et al . , 1980 ) , ϕl=| ( 1/6 ) ∑j∈nn ( l ) exp⁡ ( 6iθlj ) |2 , where θlj is the angle between an arbitrary axis and a vector connecting tail-end particle l to tail-end particle j , and the summation is over the six nearest neighbors of particle l . The equilibrium average , ⟨ϕl⟩ , is 1 for a perfect hexagonal packing , and it is 1/6 or smaller in the absence of bond-orientation correlations . Small periodically replicated samples of the hydrated DPPC membrane exhibit hysteretic changes in area per lipid and in ⟨ϕl⟩ during heating and cooling . See Appendix , and Marrink et al . ( 2005 ) and Rodgers et al . ( 2012 ) . To establish whether the first-order-like behavior persists to large scales and thus actually manifests a phase transition , we consider larger systems and the behavior of the interface that separates the ordered and disordered phases . Figure 1B shows coexistence for a system size of N = 3900 lipids with an interface between the two phases . To analyze interfacial fluctuations , we first identify the location of the interface at each instant . This location is found with a two-dimensional version of the three-dimensional constructions described in Limmer and Chandler ( 2014 ) and Willard and Chandler ( 2010 ) . Specifically , and as discussed in Materials and methods , the interface is the line in the plane of the bilayer with an intermediate coarse-grained value of the orientational-order density , ( 1 ) ϕ ( r ) =∑lϕlδ ( r−rl ) . where rl is the position of the lth tail-end particle projected onto a plane parallel to that of the bilayer , r is a two-dimensional vector specifying a position in that plane , and δ ( r ) is Dirac’s delta function . We focus on this field rather than the tail-end number density , ρ ( r ) =∑lδ ( r−rl ) , because the difference between the two phases is larger for typical orientational-order than for typical tail-end density . A director density field , u ( r ) =∑lulδ ( r−rl ) , could also be used to distinguish disordered regions from ordered regions . ul would specify the degree to which the hydrophobic chain of lipid l is perpendicular to the average plane of the membrane . A field of this form would be useful for systems where liquid-ordered behavior occurs in the absence of solid-ordered behavior . Multicomponent membranes , for example , can exist in solid-ordered , liquid-ordered , and liquid-disordered states . For constructing the order–disorder interface of the simple one-component membrane considered here , however , u ( r ) offers little more information than ϕ ( r ) . Figure 1C shows the Fourier spectrum of the height fluctuations of this interface , ⟨|δhk|2⟩ . Two different system sizes are studied , with the larger system having approximately double the interface length of the smaller system . The Fourier component δhk is related to the height fluctuation δh ( x ) as δh ( x ) =∑kδhkexp⁡ ( ikx ) where x is a point along the horizontal in Figure 1B . Here , 0⩽x⩽L , and L is the box length . With periodic boundary conditions , k=2πm/L , m=0 , ±1 , ±2 , ⋯ . According to capillarity theory for crystal–liquid interfaces ( Nozières , 1992; Fisher et al . , 1982 ) , ⟨|δhk|2⟩∼kBT/Lγk2 for small k , with kB being Boltzmann’s constant . Given the proportionality with 1/k2 at small k ( i . e . , wavelengths larger than 10 nm ) , comparison of the proportionality constants from simulation and capillarity theory determines the interfacial stiffness ( Camley et al . , 2010 ) , yielding γ=11 . 5±0 . 46 pN . This value is significantly larger than the prior estimate of interfacial stiffness for this model , 3±2 pN ( Marrink et al . , 2005 ) . That prior estimate was obtained from simulations of coarsening of the ordered phase . Because the ordered phase has a hexagonal packing , the interfacial stiffness depends on the angle between the interface and the lattice of the ordered phase . For a hexagonal lattice , there are three symmetric orientations for which the interfacial stiffnesses are equal . We will see that for the model we have simulated there appears to be only little angle dependence . Irrespective of that angle dependence , the stability of the interface and the quantitative consistency with capillary scaling provide our evidence for the order–disorder transition being a first-order transition in the model we have simulated . The system sizes we have considered contain up to 107 particles , allowing for membranes with N ≈ 104 lipids , and requiring 10 μs to equilibrate . As such , our straightforward simulations are unable to determine whether the ordered phase is hexatic or crystal because correlation functions that would distinguish one from the other ( Nelson et al . , 1982 ) require equilibrating systems at least 10 times larger ( Bernard and Krauth , 2011 ) . Similarly , we are unable to determine the range of conditions for which the membranes organize with ripples and with tilted lipids ( Sirota et al . , 1988; Smith et al . , 1990 ) . Presumably , the ordered domain of the phase diagram in Figure 1A partitions into several subdomains coinciding with one or more of these possibilities . With advanced sampling techniques ( Frenkel and Smit , 2001 ) , free energy functions of characteristic order parameters can be computed to estimate the positions of boundaries between these various ordered behaviors . Here , we do not pursue this additional level of detail in the phase diagram because the additional boundaries refer to continuous transitions ( Sirota et al . , 1988 ) . It is only the first-order transition , with its discontinuous change between ordered and disordered phases , that supports coexistence with a finite interfacial stiffness , and it is this stiffness that results in the orderphobic effect , which we turn to now . A disordering ( i . e . , orderphobic ) transmembrane protein is one that solvates more favorably in the disordered phase than in the ordered phase . The disordering effect of the protein could be produced by specific side chain structures . See Appendix . Here , in the main text , we consider a simpler mechanism . In particular , we have chosen to focus on the size of the protein’s hydrophobic thickness and the extent to which that thickness matches the thickness of the membrane’s hydrophobic layer ( Killian , 1998; Sharpe et al . , 2010 ) . See Figure 2 . 10 . 7554/eLife . 13150 . 004Figure 2 . Model proteins in the bilayer . ( A ) Idealized cylindrical protein-like solutes with radius R and hydrophobic thickness ℓ ( magenta ) . The hydrophilic caps of the protein are shown in white . ( B ) Cross section of the lipid bilayer in the ordered phase containing a model protein of radius 2 . 7 nm with a hydrophobic thickness ℓ=2 . 3 nm ≤𝒟d . ( C ) The radial variation of the order parameters ⟨ϕ ( r ) ⟩ ( right axis ) and ⟨ϕ ( r ) ⟩/⟨ρ ( r ) ⟩ ( left axis ) show disorder in the vicinity of the protein of radius 1 . 9 nm . ( D ) Comparison of the radial order parameter variation for three different proteins shows an increase in the extent of the induced disorder region with protein radius . DOI: http://dx . doi . org/10 . 7554/eLife . 13150 . 004 The membrane’s hydrophobic layer is thicker in the ordered state than in the disordered state . For instance , at zero lateral pressure and 294 K in the model DPPC membrane , we find that the average thicknesses of the hydrophobic layers in the ordered and disordered states are 𝒟o=3 . 1 nm and 𝒟d=2 . 6 nm , respectively . A transmembrane protein with hydrophobic thickness of size ℓ≈2 . 6 nm will therefore favor the structure of the disordered phase . If the protein is large enough , it can melt the ordered phase near the protein and result in the formation of an order–disorder interface . To evaluate whether a model protein is nucleating a disordered domain in its vicinity , we calculate the average of the orientational-order density field as a function of r=|r| , ⟨ϕ ( r ) ⟩ ( right axis of Figure 2C ) . It exhibits oscillations manifesting the atomistic granularity of the system . Dividing by the mean density ⟨ρ ( r ) ⟩ largely removes these oscillations . A profile of this ratio in the vicinity of the protein is depicted in Figure 2C ( left axis ) . It changes approximately sigmoidally , connecting its values of 0 . 15 and 0 . 45 in the disordered and ordered phases , respectively . The shape of the profile suggests the formation of an order–disorder interface ( Rowlinson and Widom , 1982 ) . Further , the increase in the spatial extent of the disordered region with the increasing size of the protein , Figure 2D , is indicative of length scale dependent broadening effects brought about by capillary fluctuations . These impressions can be quantified by analyzing fluctuations of the instantaneous interface , which we turn to now . Figure 3A shows a configuration of the instantaneous interface that forms around the orderphobic protein shown in Figure 2B . The interface is identified as described above . A video of its dynamics is provided as Video 1 . As is common in crystal–liquid interfaces , the interface nucleated by an orderphobic protein may exhibit hexagonal faceting ( Nozières , 1992 ) , remnants of which can be observed in Figure 3A . 10 . 7554/eLife . 13150 . 005Figure 3 . Soft order–disorder interface . ( A ) Arrangement of the tail-end particles of the top monolayer corresponding to the protein in Figure 2B . Far away from the protein , the tail-end particles show hexagonal-like packing and are in the ordered state . Proximal to the protein , it can be seen that the tail-end particles are randomly arranged , and resemble the disordered phase . The line connected by the black points denotes the instantaneous order–disorder interface . ( B ) The fluctuations in the radius of the order–disorder interface are consistent with the fluctuations of a free order–disorder interface at coexistence . ℛ0 is the mean radius of the order–disorder interface surrounding a model protein of radius R . DOI: http://dx . doi . org/10 . 7554/eLife . 13150 . 00510 . 7554/eLife . 13150 . 006Video 1 . Instantaneous interface around an orderphobic protein . Also uploaded to https://goo . gl/NBQJP9 . DOI: http://dx . doi . org/10 . 7554/eLife . 13150 . 006 The mean interface is a circle of radius ℛ0 . Fourier analysis of fluctuations about that circle yields a spectrum of components . To the extent that these fluctuations obey statistics of capillary wave theory for a circular interface , the mean-square fluctuation for the kth component is ⟨|δℛk|2⟩=kBT/2πγk2ℛ0 , where k=m/ℛ0 and m=±1 , ±2 , ⋯ , and γ is the order–disorder interfacial stiffness , neglecting the dependence on the angle between the interface and the lattice . The discrete values of k reflect periodic boundary conditions going full circle around the model protein . In Figure 3B , we use the interfacial stiffness from the free interface ( γ=11 . 5 pN ) separating coexisting ordered and disordered phases with the capillary theory expression , and its corresponding spectrum , to compare with the spectrum of the protein-induced interface . The agreement between the theory , the free interface and the protein-induced interface is good , and it improves as the radius of the orderphobic protein increases and the wave vector k decreases . This agreement indicates that the orderphobic protein does indeed nucleate an interface manifesting the order–disorder transition . The deviations of the fluctuations of the free interface from capillary wave theory occur for k≳0 . 8 nm−1 , corresponding to wavelengths 2π/k≲7 nm , and a mean interface radius ℛ0≲1 nm . Indeed , Figure 2 suggests that even a small protein of radius 0 . 5 nm , which supports an interface of radius ℛ0≈1 . 2 nm , is sufficient to induce an order–disorder interface with fluctuations consistent with capillary theory . Figure 4 shows three snapshots from a typical trajectory initiated with two orderphobic proteins of radius 1 . 5 nm separated by a distance of 14 nm . Each induces a disordered region in its vicinity , with soft interfaces separating the ordered and disordered regions . The free energy of the separated state is approximately γ ( P1+P2 ) , where Pi is the perimeter of the order–disorder interface around protein i . On average , ⟨Pi⟩=2πℛ0 . After a few hundred nanoseconds , a fluctuation occurs where the two interfaces combine . While the single large interface remains intact , the finite tension of the interface pulls the two proteins together . Eventually , the tension pulls the two proteins together with a final perimeter , Pf , that is typically much smaller than P1+P2 . A video of its dynamics is provided as Video 2 . 10 . 7554/eLife . 13150 . 007Figure 4 . Demonstration of the orderphobic force: two proteins separated by a center-to-center distance of 14 nm are simulated at 309 K . Snapshots at various times reveal the process of assembly in which the two order–disorder interfaces merge into a single interface . DOI: http://dx . doi . org/10 . 7554/eLife . 13150 . 00710 . 7554/eLife . 13150 . 008Video 2 . Assembly of two orderphobic proteins . Also uploaded to https://goo . gl/HXS0j7 . DOI: http://dx . doi . org/10 . 7554/eLife . 13150 . 008 After the separated interfaces join , the assembly process occurs on the time scale of microseconds . This time is required for the proteins to push away lipids that lie in the path of the assembling proteins . Given this time scale , a reversible work calculation of the binding free energy would best control both the distance and the number of lipids between the proteins . Moreover , the evident role of interfacial fluctuations indicates that the transition state ensemble for assembly must involve an interplay between inter-protein separations and lipid ordering as well as lipid concentration . While we leave the study of reversible work surfaces and transition state ensembles to future work , it seems already clear that the net driving force for assembly is large compared to thermal energies . For example , with a model orderphobic protein radius of 1 . 5 nm , we find γ ( ⟨Pf⟩−2⟨P1⟩ ) ≈−30kBT . The range over which the force acts is given by the average radius of the two interfaces , 2ℛ0 . This range is further amplified by the width of the interface , which is of 𝒪 ( ℛ0 ) for one-dimensional interfaces in two-dimensional systems ( Kardar , 2007 ) . The typical range is ≈10 to 30 nm . In comparison , given the elastic moduli of the membranes we consider , elastic responses will generate attractive forces between transmembrane proteins that are much smaller in strength and range , typically −5kBT and 1 nm , respectively ( Haselwandter and Phillips , 2013; de Meyer et al . , 2008 ) . Moreover , similarly weak and short ranged forces are found from solvation theory that accounts for linear response in microscopic detail while not accounting for the possibility of an underlying phase transition ( Lagüe et al . , 1998 ) . As in the hydrophobic effect ( Chandler , 2005 ) , the strength and range of the orderphobic force leverages the power of a phase transition , depending in this case on the ability of the orderphobic protein to induce a disordered layer in its vicinity . This ability depends upon the proximity to the membrane’s phase transition , and , for the simple protein models considered in this paper , it depends upon the protein’s radius and hydrophobic mismatch with the membrane . The spatial extent of the disordered region increases with proximity to phase coexistence as shown in Figure 5A . 10 . 7554/eLife . 13150 . 009Figure 5 . Strength of the orderphobic force . ( A ) Radial variation of the order parameter showing the extent of the disordered region as a function of temperature , for a protein of radius 1 . 9 nm and hydrophobic thickness 2 . 3 nm . The extent of the disordered region increases as the melting temperature is approached , at zero surface tension . ( B ) Comparison of the radial variation of the order parameter for different hydrophobic mismatches . Proteins with no mismatch do not create any disordered region . ( C ) Arrangement of lipids around a protein with negative mismatch . ( D ) Arrangement of lipids around a protein with zero mismatch . DOI: http://dx . doi . org/10 . 7554/eLife . 13150 . 009 Furthermore , Figure 5B shows that the strength of the effect is maximal for a hydrophobic thickness equal to that of the disordered phase , and it decreases as the hydrophobic thickness approaches that of the ordered phase . In the case of zero mismatch ( i . e . , ℓ=𝒟o ) the value of the order parameter in the vicinity of the protein is consistent with that of a pure bilayer in the ordered state . Therefore , the model proteins with zero mismatch do not induce a disordered region , and the orderphobic effect vanishes . See Figure 5B and D . Figure 4 also shows that the orderphobic effect produces excess mobility , by proteins melting order in a surrounding microscopic layer and by facilitating the motions of neighboring proteins . This finding explains how protein mobility and reorganization can be relatively facile in the so-called 'gel' phases of membranes . Further information on this phenomenon is provided in Appendix . Our prediction of enhanced lipid mobility surrounding orderphobic proteins may be amenable to experimental tests by single molecule tracking techniques ( Eggeling et al . , 2009 ) . Biological membranes and transmembrane proteins are far more complicated than the models considered in this paper . Part of the complexity is associated with multiple components , which allow for more than one order–disorder transition . For example , with a membrane composed of three components , coexistence can be established between liquid-ordered and liquid-disordered phases ( Veatch and Keller , 2005 ) , and both of these phases exist in bio-membranes ( Swamy et al . , 2006; Owen et al . , 2012; Polozov et al . , 2008 ) . The fact that liquid-ordered and liquid-disordered phases can coexist with finite line tension ( Veatch and Keller , 2005 ) implies the existence of a first-order transition between them ( Chandler , 1987 ) and thus the relevance of the orderphobic effect . This effect is much wider in applicability than the Casimir effect ( Machta et al . , 2012 ) , which applies only within the much smaller range of conditions where the first-order transition reaches its limiting case of criticality . A director density for hydrophobic chains serves as the order parameter distinguishing liquid-ordered and liquid-disordered phases . The strength and range of orderphobic effects that will arise from this order–disorder transition merit future investigation . Modeling might build from recent numerical work on the liquid-ordered phase ( Risselada and Marrink , 2008 ) . Bear in mind that the strength and range of the orderphobic effect depends upon the proximity of the order–disorder transition . This proximity can be changed by changing temperature , as illustrated in Figure 5A . With many components in play , the proximity can also be changed by varying membrane composition . One can therefore anticipate that the strength and range of orderphobic effects will depend upon , for example , cholesterol concentrations . It will also depend upon the presence of additional proteins , and the domains formed with those proteins themselves depend upon the orderphobic effect . Another source of complexity is the side-chain structure of transmembrane proteins . These side chains can affect the packing of lipid chains . To the extent that lipid packing is disrupted , even small α-helix proteins can be orderphobic . Evidence for this assessment is provided in Appendix . Thus , the orderphobic effect can lead to clustering of transmembrane α-helices . Moreover , just as the strength and range of the orderphobic effect can be modified by changing the radius and mismatch of our model proteins , the strength and range of the orderphobic effect will also be affected by the structure of protein side chains . Further , an obvious consequence of the orderphobic effect is the existence of a driving force that will move orderphobic proteins from an ordered phase to a disordered phase , and the creation of large disordered domains as a result of clustering orderphobic proteins . Both of these effects have been noted in simulations of disordering α-helix proteins in a membrane exhibiting coexisting liquid-ordered and liquid disordered domains ( Schäfer et al . , 2011; Domański et al . , 1818 ) . Further , there is a dual to the orderphobic effect: a transmembrane protein in the disordered phase that favors the ordered phase can nucleate an ordered region and order–disorder interface . For example , one of our model proteins with a positive mismatch ( ℓ=𝒟o ) would induce order in its vicinity . This effect is illustrated in the Appendix . Interfaces separating the ordered and disordered regions will again provide a force for assembly . This case corresponds to the situation of lipid rafts ( Simons and Ikonen , 1997 ) , which consists of ordered domains floating in otherwise disordered membranes . The stable interface separating domains then serves as a concrete geometrical definition of the raft . This orderphilic effect will depend upon the extent to which the surface of the transmembrane protein is commensurate with the ordered phase structure . Hydrophobic mismatch is but one possibility . β-sheets that align neighboring lipids are others . The fact that the orderphilic effect is a pre-transition effect for the first-order transition between ordered and disordered phases implies it should occur in disordered membranes that are thermodynamically close to coexistence between liquid-ordered and liquid-disordered phases . The orderphobic effect may also be of direct relevance in understanding the behavior of lung-surfactant monolayers . The primary component of these monolayers is the lipid DPPC , with melting temperature higher than physiological temperature ( 41°C ) , and a small proportion of cholesterol , and proteins . These monolayers undergo cyclic surface tension mediated phase transitions between the ordered and disordered phases ( Nag et al . , 1998 ) . The results of this paper are also applicable to lipid monolayers and could govern the diffusion and assembly of proteins embedded within the relatively rigid ordered phases . Finally , we speculate that the orderphobic effect plays important roles in membrane fusion , budding , and cell signaling ( Fratti et al . , 2004; Zick et al . , 2014; Qi et al . , 2001; James and Vale , 2012; Różycki et al . , 2012 ) . In the case of fusion , it would appear that one important role is to promote fluctuations in an otherwise stable membrane . Otherwise , it is difficult to conceive of a mechanism by which thermal agitation would be sufficient to destabilize microscopic sections of membranes . Such destabilization seems necessary for initiating and facilitating membrane fusion . Many proteins are involved in such processes ( Fratti et al . , 2004; Fasshauer et al . , 1998; Wickner and Schekman , 2008 ) , but it may not be a coincidence that the hydrophobic thicknesses of SNARE proteins are 25% smaller than that of the ordered membrane states ( Milovanovic et al . , 2015; Stein et al . , 2015 ) .
We simulate the MARTINI coarse-grained force field using the GROMACS molecular dynamics package ( Marrink et al . , 2007; Pronk et al . , 2013 ) . ‘Antifreeze’ particles are added to the solvent to ensure that the solvent does not freeze over the temperature range considered in the simulations as in Marrink et al . ( 2007 ) . Thermostats and barostats control temperature and pressure , and checks were performed to assure that different thermostats and barostats yielded similar results ( Frenkel and Smit , 2001 ) . The hydrophobic cores of our idealized proteins are constructed using the same coarse-grained beads as the lipid tails ( particle C1 in the MARTINI topology Marrink et al . ( 2007 ) ) . Similarly , the hydrophilic caps are constructed using the first bead of the DPPC head group ( Q0 , in the MARTINI topology ) . The protein beads also have bonded interactions where the bond length is 0 . 45 nm and the bond angle is set to 180° . The associated harmonic force constants for the bond lengths and angles are 1250 kJmol−1nm−2 and 25 kJmol−1rad−2 . Based on the hydrophobic mismatch with the bilayers , the proteins are classified into three categories: ( i ) positive mismatch ( ℓ>𝒟o ) ( ii ) negative mismatch ( ℓ≤𝒟d ) and ( iii ) no mismatch ( ℓ≈𝒟o ) . To create different mismatches , we alter the number of beads in the protein core . These idealized proteins do not contain charges . Proteins are embedded in the equilibrated bilayer at 279 K . The resulting system is then heated to the required temperature and equilibrated for another 1 . 2 µs . All the subsequent averages are performed using 10 independent trajectories each 600 ns long . The assembly of proteins is also performed using the same DPPC bilayer system with 3200 lipids and 50000 water beads . In this case , two proteins are inserted in this bilayer with centers at a distance of 14 nm and the simulation is carried out at 309 K . The flat interface is stabilized by juxtaposing an ordered bilayer equilibrated at 285 K and zero lateral pressure with a disordered bilayer equilibrated at the same conditions corresponding to the cooling and heating curves of the hysteresis loop in Appendix 1—figure 1 , respectively . The system thus constructed is equilibrated in the ensemble with fixed temperature , volume , and numbers of particles . This ensemble allows for maintaining an area per lipid intermediate between the two phases , thus stabilizing the interface . For the purpose of obtaining a smooth and continuous interface , ϕ ( r ) is coarse grained by replacing Dirac’s delta function with a finite-width Gaussian , ( 1/2πξ2 ) exp⁡ ( −|r|2/2ξ2 ) . The replacement changes ϕ ( r ) to ϕ¯ ( r ) . The coarse-graining width , ξ , is chosen to be the average separation between tail-end particles l and j when ⟨ ( ϕl−⟨ϕl⟩ ) ( ϕj−⟨ϕj⟩ ) ⟩/⟨ ( ϕl−⟨ϕl⟩ ) 2⟩ in the ordered phase is 1/10 . This choice yields a value of ξ=1 . 5 nm . The instantaneous order–disorder interface is the set of points s satisfying ϕ¯ ( s , t ) = ( ϕd+ϕo ) /2 . Here , ϕd and ϕo are ⟨ϕ ( r ) ⟩ evaluated in the disordered and ordered phases , respectively . At zero lateral pressure and 294 K , we find ϕd=0 . 4 ± 0 . 02 nm−2 and ϕo=2 . 15±0 . 2 nm−2 . For numerics , a square lattice tiles the average plane of the bilayer , and the coarse-grained field ϕ¯ ( r ) is evaluated at each lattice node . Values between are determined by interpolation . For convenience , the Gaussian function is truncated and shifted to zero at 3ξ . Any value of ξ within the range , 1 nm <ξ< 2 nm gives nearly identical ϕ¯ ( r ) . Outside that range , larger values obscure detail by excessive smoothing , and smaller values obscure detail by capturing a high density of short-lived bubbles of disorder .
We present a mechanism for a generic , powerful force of assembly and mobility for transmembrane proteins in lipid bilayers . This force is a pre-transition ( or pre-melting ) effect for the first-order transition between ordered and disordered phases in the membrane . Using large-scale molecular simulation , we show that a protein with hydrophobic thickness equal to that of the disordered phase embedded in an ordered bilayer stabilizes a microscopic order–disorder interface . The stiffness of that interface is finite . When two such proteins approach each other , they assemble because assembly reduces the net interfacial energy . Analogous to the hydrophobic effect , we refer to this phenomenon as the 'orderphobic effect' . The effect is mediated by proximity to the order–disorder phase transition and the size and hydrophobic mismatch of the protein . The strength and range of forces arising from this effect are significantly larger than those that could arise from membrane elasticity for the membranes considered .
The membrane that surrounds cells provides a selective barrier that allows some molecules through , but blocks the path of others . A cell’s membrane is made up of two layers of molecules with oily tails , and is therefore known as a bilayer . Many proteins are dotted within and on the inner and outer surfaces of the bilayer: some act as channels that control what goes in and out of the cell , while others protrude outside the cell so that they can sense changes in the environment . Membrane proteins can move and interact within the bilayer , and various models have emerged to try to explain this dynamic system . These models are based on the membrane having some fluidity but also having regions where there is more structure , and typically describe the proteins as ordered clusters floating in an otherwise disordered fluid membrane . However , many researchers now think some proteins that pass through both layers of the bilayer ( i . e . , transmembrane proteins ) make membranes more ordered , with a possibly gel-like state . However , it is not clear how transmembrane proteins can move and assemble together within such a relatively rigid membrane . To investigate this , Katira , Mandadapu , Vaikuntanathan et al . carried out computer simulations using a model of a simple bilayer membrane . This membrane can exist in an ordered state , where the oily tails are neatly aligned , or a disordered state , where they are irregularly packed . Virtual ‘heating’ of the membrane caused it to shift from an ordered to a disordered state . When a simple transmembrane protein favoring the disordered state was inserted into the ordered state of the modeled membrane , disordered regions formed locally around the protein and the protein was able to diffuse within the membrane . Modeling what would happen if two transmembrane proteins approached each other revealed that a consequence of the order–disorder transition is a strong attractive force that assembles the proteins together . Katira , Mandadapu , Vaikuntanathan et al . named this new phenomenon the 'orderphobic effect' . The forces arising from this effect were much greater than those currently believed to contribute to the assembly of membrane protein complexes , such as those generated by the elasticity of the membrane . This means that the orderphobic effect may be responsible for generating the protein clusters commonly seen in cell membranes . Future work should next explore the opposite effect , where proteins favoring the ordered state are inserted into the disordered state of a membrane . This is expected to cause clustering of such proteins and thus large ordered regions in an otherwise disordered membrane .