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.

This paper presents implications of first-order order–disorder phase transitions in lipid bilayers. The fluid mosaic model (Singer and Nicolson, 1972) and the lipid raft hypothesis (Simons and Ikonen, 1997; Munro, 2003) have guided intuition on how proteins diffuse and assemble in biological membranes—ordered clusters floating in an otherwise disordered fluid membrane (Simons and Toomre, 2000; Lingwood and Simons, 2010). However, recent advances show that a significant proportion of the membrane is liquid-ordered (Swamy et al., 2006; Owen et al., 2012; Polozov et al., 2008), with coexistence between the liquid-ordered and disordered phases. This coexistence suggests that effects of an order–disorder transition might be at play in the assembly of proteins. This possibility is studied here by examining the effects mediated by the simplest related order–disorder transition, that between solid-ordered and liquid-disordered phases.

Specifically, with molecular simulation, we study a coarse-grained model of a hydrated one-component bilayer and proteins that are added to the membrane. The model membrane exhibits two distinct phases—a solid-ordered phase and a liquid-disordered phase—and a first-order transition between them. We find that a transmembrane protein in the ordered bilayer can induce effects that resemble pre-melting (Lipowsky, 1982; 1984; Limmer and Chandler, 2014). In particular, within the otherwise ordered membrane phase, mesoscopic disordered domains surround proteins that favor disordered states. Importantly, the boundary of the domains resembles a stable, fluctuating order–disorder interface. The dynamic equilibrium established at the boundary allows the protein and its surrounding domain to diffuse. Moreover, because the interface has a finite stiffness, neighboring proteins can experience a membrane-induced force of adhesion, an attractive force that is distinctly stronger and can act over significantly larger lengths than those that can arise from simple elastic deformations of the membrane (Dan et al., 1993; Goulian et al., 1993; Phillips et al., 2009; Kim et al., 1998; Haselwandter and Phillips, 2013).

This force between transmembrane proteins is analogous to forces of interaction between hydrated hydrophobic objects. In particular, extended hydrophobic surfaces in water can nucleate vapor–liquid-like interfaces. In the presence of such interfaces, hydrophobic objects cluster to reduce the net interfacial free energy. This microscopic pre-transition effect manifesting the liquid–vapor phase transition can occur at ambient conditions (Chandler, 2005; Lum et al., 1999; Willard and Chandler, 2008; Stillinger, 1973; ten Wolde and Chandler, 2002; Mittal and Hummer, 2008; Patel et al., 2011; 2012). In the transmembrane case, we show here that a protein favoring the disordered phase creates a similar pre-transition effect. In this case it manifests the order–disorder transition of a lipid bilayer. Like the raft hypothesis, therefore, clusters do indeed form, but the mechanism for their assembly and mobility emerge as consequences of order–disorder interfaces in an otherwise ordered phase. We refer to this phenomenon as the 'orderphobic effect'.

While considering the effect with one specific order–disorder transition, one should bear in mind its generic nature. The orderphobic effect should be a general consequence of a first-order transition, whether the transition is between solid-ordered and liquid-disordered phases as considered explicitly herein, or between liquid-ordered and liquid-disordered phases as in multicomponent membrane systems. More is said on this point in the Implications section of this paper.

The order–disorder transition is a first-order phase transition

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, ${\displaystyle {\mathcal{𝒟}}_{\mathrm{o}}}$ is larger than that of the disordered phase, ${\displaystyle {\mathcal{𝒟}}_{\mathrm{d}}}$. Correspondingly, the area per lipid in the ordered phase is smaller than that in the disordered phase.

Figure 1

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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), ${\displaystyle {\varphi }_l=|(1/6){\sum }_{j\in \mathrm{n}\mathrm{n}(l)}\exp (6i{\theta }_{lj}){|}^2}$, where ${\displaystyle {\theta }_{lj}}$ is the angle between an arbitrary axis and a vector connecting tail-end particle ${\displaystyle l}$ to tail-end particle ${\displaystyle j}$, and the summation is over the six nearest neighbors of particle ${\displaystyle l}$. The equilibrium average, ${\displaystyle ⟨{\varphi }_l⟩}$, 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 ${\displaystyle ⟨{\varphi }_l⟩}$ 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) ${\displaystyle \varphi (\mathbf{r})=\sum _l{\varphi }_l\delta (\mathbf{r}-{\mathbf{r}}_l).}$

where ${\displaystyle {\mathbf{r}}_l}$ is the position of the ${\displaystyle l}$th tail-end particle projected onto a plane parallel to that of the bilayer, ${\displaystyle \mathbf{r}}$ is a two-dimensional vector specifying a position in that plane, and ${\displaystyle \delta (\mathbf{r})}$ is Dirac’s delta function. We focus on this field rather than the tail-end number density, ${\displaystyle \rho (\mathbf{r})={\sum }_l\delta (\mathbf{r}-{\mathbf{r}}_l)}$, because the difference between the two phases is larger for typical orientational-order than for typical tail-end density.

A director density field, ${\displaystyle u(\mathbf{r})={\sum }_l{u}_l\delta (\mathbf{r}-{\mathbf{r}}_l)}$, could also be used to distinguish disordered regions from ordered regions. ${\displaystyle u_l}$ would specify the degree to which the hydrophobic chain of lipid ${\displaystyle l}$ 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, ${\displaystyle u(\mathbf{r})}$ offers little more information than ${\displaystyle \varphi (\mathbf{r})}$.

Figure 1C shows the Fourier spectrum of the height fluctuations of this interface, ${\displaystyle ⟨|\delta h_k{|}^2⟩}$. Two different system sizes are studied, with the larger system having approximately double the interface length of the smaller system. The Fourier component ${\displaystyle \delta h_k}$ is related to the height fluctuation ${\displaystyle \delta h(x)}$ as ${\displaystyle \delta h(x)={\sum }_k\delta h_k\exp (ikx)}$ where ${\displaystyle x}$ is a point along the horizontal in Figure 1B. Here, ${\displaystyle 0⩽x⩽L}$, and ${\displaystyle L}$ is the box length. With periodic boundary conditions, ${\displaystyle k=2\pi m/L}$, ${\displaystyle m=0,\pm 1,\pm 2,\cdots }$. According to capillarity theory for crystal–liquid interfaces (Nozières, 1992; Fisher et al., 1982), ${\displaystyle ⟨|\delta h_k{|}^2⟩\sim k_{\mathrm{B}}T/L\gamma k^2}$ for small ${\displaystyle k}$, with ${\displaystyle k_{\mathrm{B}}}$ being Boltzmann’s constant.

Given the proportionality with ${\displaystyle 1/k^2}$ at small ${\displaystyle k}$ (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 ${\displaystyle \gamma =11.5\pm 0.46}$ pN. This value is significantly larger than the prior estimate of interfacial stiffness for this model, ${\displaystyle 3\pm 2}$ 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.

Transmembrane proteins can disfavor the ordered membrane

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.

Figure 2

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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 ${\displaystyle {\mathcal{𝒟}}_{\mathrm{o}}=3.1}$ nm and ${\displaystyle {\mathcal{𝒟}}_{\mathrm{d}}=2.6}$ nm, respectively. A transmembrane protein with hydrophobic thickness of size ${\displaystyle \ell \approx 2.6}$ 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.

An orderphobic protein nucleates a fluctuating order–disorder interface

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.

Figure 3

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Video 1

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The mean interface is a circle of radius ${\displaystyle {\mathcal{ℛ}}_0}$. 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 ${\displaystyle k}$th component is ${\displaystyle ⟨|\delta {\mathcal{ℛ}}_k{|}^2⟩=k_{\mathrm{B}}T/2\pi \gamma k^2{\mathcal{ℛ}}_0}$, where ${\displaystyle k=m/{\mathcal{ℛ}}_0}$ and ${\displaystyle m=\pm 1}$, ${\displaystyle \pm 2,\cdots }$, and ${\displaystyle \gamma }$ is the order–disorder interfacial stiffness, neglecting the dependence on the angle between the interface and the lattice. The discrete values of ${\displaystyle k}$ reflect periodic boundary conditions going full circle around the model protein.

In Figure 3B, we use the interfacial stiffness from the free interface (${\displaystyle \gamma =11.5}$ 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 ${\displaystyle k}$ 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 ${\displaystyle k\gtrsim 0.8}$ nm⁻¹, corresponding to wavelengths ${\displaystyle 2\pi /k\lesssim 7}$ nm, and a mean interface radius ${\displaystyle {\mathcal{ℛ}}_0\lesssim 1}$ nm. Indeed, Figure 2 suggests that even a small protein of radius 0.5 nm, which supports an interface of radius ${\displaystyle {\mathcal{ℛ}}_0\approx 1.2}$ nm, is sufficient to induce an order–disorder interface with fluctuations consistent with capillary theory.

The orderphobic effect generates forces of assembly and facilitates protein mobility

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 ${\displaystyle \gamma (P_1+P_2)}$, where ${\displaystyle P_i}$ is the perimeter of the order–disorder interface around protein ${\displaystyle i}$. On average, ${\displaystyle ⟨P_i⟩=2\pi {\mathcal{ℛ}}_0}$. 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, ${\displaystyle P_{\mathrm{f}}}$, that is typically much smaller than ${\displaystyle P_1+P_2}$. A video of its dynamics is provided as Video 2.

Figure 4

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Video 2

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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 ${\displaystyle \gamma (⟨P_{\mathrm{f}}⟩-2⟨P_1⟩)\approx -30k_{\mathrm{B}}T}$. The range over which the force acts is given by the average radius of the two interfaces, ${\displaystyle 2{\mathcal{ℛ}}_0}$. This range is further amplified by the width of the interface, which is of ${\displaystyle \mathcal{𝒪}({\mathcal{ℛ}}_0)}$ 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 ${\displaystyle -5k_{\mathrm{B}}T}$ 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.

Figure 5

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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., ${\displaystyle \ell ={\mathcal{𝒟}}_{\mathrm{o}}}$) 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).

In this Appendix, we examine hysteresis in the bilayer system, show results for mean-square displacements as functions of time, and provide evidence that a small transmembrane ${\displaystyle \alpha }$-helix protein can be orderphobic if the side chains of the protein disrupt the packing of lipid tails.

The lipid bilayer system exhibits hysteresis

Appendix 1—figure 1 shows the change in area per lipid with temperature while heating and cooling a bilayer. There are finite jumps in area per lipid as the system transitions between the two phases, suggesting a first-order phase transition. Hysteresis occurs because ordering from the metastable disordered phase is much slower than disordering from the metastable ordered phase. Due to the difference in time scales, when contrasting melting and freezing from heating and cooling runs, the melting points from heating runs as shown in Appendix 1—figure 1A provide the more accurate estimates of the actual phase boundaries. Systematic errors due to small system size and heating rate have not been estimated.

Appendix 1—figure 1

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The pre-melting layer has a higher mobility than the ordered phase

In Appendix 1—figure 2 we show the mean squared displacements of lipids in the bulk ordered phase, bulk disordered phase, and the pre-melting layer induced by an orderphobic protein. As discussed in the main text, an orderphobic protein increases the mobility of the lipids in its vicinity. Note that the center of mass of the membrane fluctuates in time. These fluctuations affect the absolute positions of lipid molecules, but they are irrelevant to the issue of lipid mobility. Therefore, the mean-square displacements considered in Appendix 1—figure 2 are for tail-end particle positions relative to the instantaneous position of the membrane’s center of mass. That is to say, for ${\displaystyle ⟨|{\overline{\mathbf{r}}}_l(t)-{\overline{\mathbf{r}}}_l(0){|}^2⟩}$, where ${\displaystyle {\overline{\mathbf{r}}}_l(t)}$ is the position at time ${\displaystyle t}$ of the ${\displaystyle l}$th tail-end particle less that of the membrane’s center of mass.

Appendix 1—figure 2

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For large enough times, ${\displaystyle t}$, the mean-square displacements are asymptotic to ${\displaystyle 4Dt}$, where ${\displaystyle D}$ is the self-diffusion constant. For the disordered liquid phase, we see from Appendix 1—figure 2 that the asymptotic region is reached within 10² ns, and that ${\displaystyle D\approx 4\times {10}^{-7}}$ cm²/s. In contrast, for the ordered phase, the diffusive asymptotic limit is not reached until 10⁴ ns, and ${\displaystyle D}$${\displaystyle \approx 2\times {10}^{-9}}$ cm²/s. The mobility of lipids within the disordered layer surrounding the orderphobic protein is an order of magnitude larger than that of lipids beyond that region and in the ordered phase.

Experimental results for lipid diffusion constants in disordered and ordered bilayers are ${\displaystyle D\approx 3\times {10}^{-8}}$ cm²/s and ${\displaystyle D\approx 2\times {10}^{-10}}$ cm²/s, respectively (Korlach et al., 1999). The simulation is in harmony with experiment for the two-order of magnitude difference between the ordered- and disordered-phase values of ${\displaystyle D}$. Of course, absolute values of ${\displaystyle D}$ are beyond the scope of what can be predicted from our simulations because coarse graining omits degrees of freedom that would increase friction and decrease ${\displaystyle D}$.

A model ${\displaystyle \alpha }$-helix is orderphobic

Here, we consider the MARTINI model for KALP23—a polypeptide chain with 23 residues, consisting of alanines and leucines flanked by lysines (Schäfer et al., 2011). This molecule has a hydrophobic thickness of ${\displaystyle \ell \approx 3.0}$ nm, which means that it has essentially no hydrophobic-length mismatch with the ordered bilayer. Nevertheless, it is orderphobic because its side chains perturb the ordered lipid phase to an extent that a pre-melting layer is formed around the protein. This behavior is demonstrated with the aid of Appendix 1—figure 3. The panels render configurations from a simulation in which we have placed this model protein in the MARTINI model for the ordered DPPC bilayer system considered in the main text. The pre-melting layer that forms around the protein causes the protein to tilt so as to keep its full hydrophobic length in contact with hydrophobic tails of the lipids. The interface separating its disordered domain from the surrounding ordered phase remains stable throughout a molecular dynamics trajectory running for more than 1 µs.

Appendix 1—figure 3

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An orderphilic protein in the disordered phase nucleates an ordered domain

In a disordered bilayer, we embed a model protein with a hydrophobic thickness approximately equal to that of the ordered phase, ${\displaystyle \ell =3.3}$ nm. This embedding illustrates the orderphilic case. As a control, we also embed a model protein with a hydrophobic thickness approximately equal to that of the disordered phase, ${\displaystyle \ell =2.3}$ nm. For both cases, we then calculate the average of the bond orientational order density ${\displaystyle ⟨\varphi (r)⟩}$, and the number density ${\displaystyle ⟨\rho (r)⟩}$. We also compute the director density, ${\displaystyle ⟨u(r)⟩}$ as defined earlier, with ${\displaystyle u_l=(3/2){\cos }^2({\theta }_l)-(1/2)}$. Here, ${\displaystyle {\theta }_l}$ is the angle between the membrane normal and the ${\displaystyle l}$th chain’s orientation. The director density for ${\displaystyle \ell =3.3}$ nm as seen in Appendix 1—figure 4 changes approximately sigmoidally, connecting its values of 0.92 and 0.62 in the ordered and disordered phases, respectively. Further, we have computed the instantaneous interfaces from the coarse graining of each of these three fields ${\displaystyle \rho (\mathbf{r})}$, ${\displaystyle \varphi (\mathbf{r})}$ and ${\displaystyle u(\mathbf{r})}$. In the orderphilic case, interfaces are apparent for all three fields. The director field provides the clearest pictures, which are illustrated in Appendix 1—figures 4,5. In the control case, interfaces do not appear.

Appendix 1—figure 4

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Appendix 1—figure 5

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Distinctions between interfaces for each of the fields are microscopic effects worthy of future study. For the bulk interface between the ordered and disordered phases, the distinction disappears because there is only one order–disorder transition for the membrane model we have considered. In more complicated membrane models, those with mixtures of components exhibiting both liquid-ordered and solid-ordered phases, the different fields offer different information for both large and small length scales.

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Author details

1. Shachi Katira

   Department of Chemistry, University of California, Berkeley, Berkeley, United States

   Contribution

   SK, SK, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article, Contributed unpublished essential data or reagents

   Contributed equally with

   Kranthi K Mandadapu and Suriyanarayanan Vaikuntanathan

   Competing interests

   The author declares that no competing interests exist.

2. Kranthi K Mandadapu

   1. Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, United States
   2. Department of Chemical and Biomolecular Engineering, University of California, Berkeley, Berkeley, United States

   Contribution

   KKM, KM, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article, Contributed unpublished essential data or reagents

   Contributed equally with

   Shachi Katira and Suriyanarayanan Vaikuntanathan

   Competing interests

   The author declares that no competing interests exist.

3. Suriyanarayanan Vaikuntanathan

   Department of Chemistry, University of Chicago, Chicago, United States

   Contribution

   SV, SV, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article, Contributed unpublished essential data or reagents

   Contributed equally with

   Shachi Katira and Kranthi K Mandadapu

   Competing interests

   The author declares that no competing interests exist.

4. Berend Smit

   1. Department of Chemistry, University of California, Berkeley, Berkeley, United States
   2. Department of Chemical and Biomolecular Engineering, University of California, Berkeley, Berkeley, United States
   3. Laboratory of Molecular Simulation, Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne, Sion, Switzerland

   Contribution

   BS, BS, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article, Contributed unpublished essential data or reagents

   Competing interests

   The author declares that no competing interests exist.

5. David Chandler

   Department of Chemistry, University of California, Berkeley, Berkeley, United States

   Contribution

   DC, DC, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article, Contributed unpublished essential data or reagents

   For correspondence

   chandler@berkeley.edu

   Competing interests

   The author declares that no competing interests exist.

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