Membrane-Mediated Protein-Protein Interactions and Connection to Elastic Models: A Coarse-Grained Simulation Analysis of Gramicidin A Association

Abstract

To further foster the connection between particle based and continuum mechanics models for membrane mediated biological processes, we carried out coarse-grained (CG) simulations of gramicidin A (gA) dimer association and analyzed the results based on the combination of potential of mean force (PMF) and stress field calculations. Similar to previous studies, we observe that the association of gA dimers depends critically on the degree of hydrophobic mismatch, with the estimated binding free energy of >10 kcal/mol in a distearoylphosphatidylcholine bilayer. Qualitative trends in the computed PMF can be understood based on the stress field distributions near a single gA dimer and between a pair of gA dimers. For example, the small PMF barrier, which is ∼1 kcal/mol independent of lipid type, can be captured nearly quantitatively by considering membrane deformation energy associated with the region confined by two gA dimers. However, the PMF well depth is reproduced poorly by a simple continuum model that only considers membrane deformation energy beyond the annular lipids. Analysis of lipid orientation, configuration entropy, and stress distribution suggests that the annular lipids make a significant contribution to the association of two gA dimers. These results highlight the importance of explicitly considering contributions from annular lipids when constructing approximate models to study processes that involve a significant reorganization of lipids near proteins, such as protein-protein association and protein insertion into biomembranes. Finally, large-scale CG simulations indicate that multiple gA dimers also form clusters, although the preferred topology depends on the protein concentration. Even at high protein concentrations, every gA dimer requires contact to lipid hydrocarbons to some degree, and at most three to four proteins are in contact with each gA dimer; this observation highlights another aspect of the importance of interactions between proteins and annular lipids.

Introduction

Integral membrane proteins play various essential roles in biological processes such as cell signaling and molecular transportations. To effectively carry out their functions, many membrane proteins are known to be nonuniformly distributed in the cellular membrane. Understanding the physical mechanism for such protein-sorting phenomena has been an active area of research. In some cases, the clustering of specific classes of membrane proteins is driven by direct linking of proteins (1–3) or lipid raft formation (4,5). In other cases, it is believed that protein sorting is largely driven by generic membrane-mediated interactions, such as the hydrophobic mismatch (6) between proteins and the lipid bilayer (7). Cluster formation of various proteins due to hydrophobic mismatch has been shown for different systems using various experimental techniques: for gramicidin using x-ray diffraction (8); for α-helices and rhodopsin using fluorescence resonance energy transfer (FRET) (9,10); for gramicidin A (gA) using conductance measurements (11); for α-helices using fluorescence microscopy; for gA using heat-capacity measurements; and for α-helices, gA, and bacterial mechanosensitive channels of large conductance (MscLs) using atomic force microscopy (AFM) (12–14).

Inspired by the mounting experimental evidence for membrane-mediated protein redistribution and protein-protein interactions, investigators have applied various theoretical and computational methods to elucidate the underlying physical principles. For the association of small transmembrane helices, atomistic simulations have been carried out to dissect various contributions (15). For the association of larger proteins, calculations have been limited to coarse-grained (CG) simulations (16–19), liquid-state theories (20,21), and phenomenological continuum mechanics models (7,22–26).

For example, de Meyer et al. developed a dissipative-particle-dynamics (DPD) model for studying the potential of mean force (PMF) between model membrane proteins of various sizes (16). The model proteins consist of two different types of CG beads that represent hydrophilic and hydrophobic groups. By varying the hydrophobic length and size of the model proteins, de Meyer et al. demonstrated that the effective interaction depends quite sensitively on the degree of hydrophobic mismatch, the three-dimensional structure of the protein, and the bilayer properties. Schmidt et al. (17) carried out a similar DPD simulation to study membrane-mediated cluster formation of proteins with a different model. The computed PMF showed trends similar to those in the work of de Meyer et al. (16). More recently, Parton et al. (19) reported another CG simulation study that showed spontaneous cluster formations of α-helices and β-barrels in the presence of hydrophobic mismatch using the MARTINI force field (27), which is chemically more realistic than the mesoscopic models (16,17). They also explored the effect of membrane curvature by comparing the clustering behaviors on lipid vesicles to those in bilayers. Although impressive, the results of those studies are often interpreted at a qualitative level, leaving ambiguities regarding factors that control the strength and pattern of protein association/clustering in membranes.

Another popular approach to studying membrane mediated interactions is to employ continuum elasticity theory, which has a long history in membrane biophysics (23,28,29). Most recently, Phillips and co-workers applied such a model to analyze membrane-mediated interactions between two membrane proteins that undergo transitions between conformations with different levels of hydrophobic mismatch (26). Although they are elegant and rich in qualitative physical insights, continuum models often assume that lipid membranes obey an elasticity theory of constant material properties, even for annular lipids and lipids confined between the two proteins. Since many recent studies have suggested that annular lipids have rather different properties from bulk lipids (30–32), it is likely that the approximation of constant material properties is oversimplified; to what degree this influences predictions from continuum models, however, has not been established.

The above discussion strongly suggests that it is valuable to foster connections between particle based simulations and continuum models for membrane mediated processes. Although particle based simulations are increasingly being used to describe complex processes that involve proteins and biomembranes (33,34), using them together with continuum models gives a better understanding of the underlying physical principles that govern, for example, protein association in membranes. On the other hand, once the implications of fundamental approximations in continuum models are understood by comparing them to more realistic particle simulations, sophisticated continuum models are potentially more effective for studying processes that involve very large lengthscales and long timescales (35–38).

The specific problem we have chosen to analyze in this context is the membrane-mediated interaction between two gA dimers as a function of both the distance between the two inclusions and the degree of hydrophobic mismatch. Among the membrane proteins that have been shown to be regulated by membrane properties, gA is a well established molecular-force probe in single-molecule experiments (39). This is because the dimerization energetics and kinetics of gA subunits partitioned to each monolayer are coupled to membrane mechanics, and the dimerization events can be detected by measuring the conductance of cations through the channel of a gA dimer. Moreover, a mechanism of lateral redistribution by hydrophobic coupling of a pair of gA dimers has also been experimentally well established (11); i.e., cluster formation of gA dimers is energetically favorable in the presence of hydrophobic mismatch.

Our approach is to use CG molecular dynamics simulations based on the MARTINI force field (27), which maintains the structural and chemical inhomogeneity of proteins, as well as protein/lipid interfaces. The discrete nature of the lipid molecules in our simulations also eliminates the need for fundamental assumptions made in continuum mechanics models (26). Therefore, we expect that our study will provide more realistic descriptions of the PMF between specific membrane proteins and help test the importance of approximations made in previous CG (16,17) or continuum mechanics (26) analyses. In addition to the PMF as a function of the distance between two gA dimers, we calculate the distribution of the 3-D stress field near the proteins. As demonstrated in previous studies (27,40) and in the article accompanying this one (32), the MARTINI force field leads to a detailed stress field in semiquantitative agreement with atomistic simulations. We take advantage of this feature of the MARTINI force field and demonstrate that the combination of PMF and stress field analyses gives a better understanding of the membrane-mediated interactions between two inclusions, especially as regards the importance of annular lipids. Another issue we briefly explore is the clustering behavior of multiple gA dimers. Exploration of this type of simulation has begun only very recently (19), and the results further highlight the importance of interactions between proteins and annular lipids. Moreover, the observed morphology and its dependence on gA concentration can serve as a valuable benchmark for continuum models, for which many-body effects are known to be important (24).

We first summarize our computational methods and simulation setups. Next, we present PMF results and stress field analysis for two gA molecules in lipid bilayers of different thickness and, therefore, different degrees of hydrophobic mismatch. To study the interaction between multiple gA dimers, we also report results of CG simulations with two different gA-dimer/lipid ratios. Finally, we draw a few conclusions.

Computational Methods

Coarse-grained simulations

For CG simulations, the MARTINI force field is used (27). Although we have developed an extension of the MARTINI model based on careful consideration of electrostatics (41), we use the original MARTINI model here, because the system is not featured with highly charged species at the membrane/water interface. We refer readers to our companion article for further details regarding our CG gA dimer model (32). As shown in Fig. S1 in the Supporting Material, the distribution results from CG simulations for lipids near the gA dimer resemble closely the results from atomistic simulations. Since protein-lipid and lipid-lipid interactions are expected to dictate the interaction between gA dimers, the MARTINI model is appropriate for the study presented here. All CG simulations reported in this article are carried out using the Gromacs package (42). Temperature (323 K) and surface tension (0 dyne/cm) are kept constant using the Berendsen scheme (43). The integration time step is 20 fs, and nonbonded interactions are truncated at 12 Å.

PMF calculations

Initial conformations with two gA dimers are built by duplicating a preequilibrated system of a gA dimer and 72 lipid molecules. As in the companion to this article (32), three lipid systems are studied to probe the effects of hydrophobic mismatch, with the degree of negative mismatch increasing in the sequence of DMPC, DPPC, and DSPC. The final three systems contain 1152 DMPC, DPPC, or DSPC lipid molecules, as well as two CG gA dimers and CG water beads (Fig. 1). The membrane patch is ∼200 × 200 Å². Because both membrane deformation and stress profile around a gA dimer recover the bulk value at a distance of ∼30 Å from the protein center (32), this system size is expected to be large enough for our purpose.

Figure 1.

Setup of the CG simulations for two gA dimers in the presence of different degrees of hydrophobic mismatch using the MARTINI force field. (A) A snapshot from the simulation with two gA dimers embedded in a DMPC bilayer. (B) Schematic of bilayer deformations (u) due to the inclusion of two gA dimers (effective radius $R_p$) separated by distance ξ. Throughout this report, we use a coordinate system with the origin at the center of the two gA dimers; the x and z axes are parallel to the line connecting the two protein centers of mass and the bilayer normal, respectively. For convenience, we also define $r_1$ and $r_2$ as the radial distance from the gA dimer at $x<0$ and $x>0$, respectively. u decreases monotonically over the scale of $r_{1,2}=20-30$ Å, depending on the lipid chain length (32), and the unperturbed membrane thickness, $h_0$, is recovered at large $r_{1,2}$.

We compute the PMF as a function of distance between the two gA dimers using two different and complementary methods: the weighted-histogram-analysis method (WHAM) (44) and the force-integration (FI) method (45,46). For the WHAM method, we use umbrella sampling (47) to sample configurations along the reaction coordinate, which is taken to be the distance between the centers of mass of the two gA dimers (ξ; see Fig. 1 B). The distance is varied in the range 14–74 Å with a window spacing of 1 Å using a harmonic biasing potential, $V^{\text{umb}}$,

$$V^{\text{umb}}=\frac{1}{2}{k}_{\xi }{\xi }^2,$$ (1)

where the force constant, $k_{\xi }$, is 10 kJ/mol·Å². The sampling time per window is ∼250 ns, and data from all windows are combined using WHAM to reconstruct the underlying PMF $\left(\text{Δ}G\right)$ between the two gA dimers. Errors of PMFs are estimated by standard deviations from six independent trajectory segments. For the FI method, we use umbrella sampling (Eq. 1) with $k_{\xi }=20$ kJ/mol·Å² to sample conformations at $14<\xi <60$ Å with a spacing of 1 Å. The instantaneous force along the reaction coordinate, $F_{\xi }$, is then evaluated by

$$F_{\xi }=-\frac{\partial U\left(\mathrm{x}\right)}{\partial \xi }=-\sum _{k=1}^{3N}\frac{\partial U\left(\mathrm{x}\right)}{\partial x_k}\frac{\partial x_k}{\partial \xi }=\mathrm{F}\times \frac{\partial \mathrm{x}}{\partial \xi },$$ (2)

where $U\left(x\right)$ is the potential energy. We then compute the ensemble average of $F_{\xi }$ at a fixed ξ, ${\langle F_{\xi }\rangle }_{\xi }$. Finally, we reconstruct the PMF by integrating ${\langle F_{\xi }\rangle }_{\xi }$ over the reaction coordinate:

$$\text{Δ}G=-\int {\langle F_{\xi }\rangle }_{\xi }d\xi +C,$$ (3)

where C is an arbitrary constant. An advantage of the FI method is that the PMF can be decomposed into contributions from different groups (15,48). In particular, we decompose the PMF into three components,

$$\text{Δ}{G}^{P-X}=-\int {\langle F_{\xi }^{P-X}\rangle }_{\xi }d\xi +C,$$ (4)

where $P-X$ indicates contributions from the interactions between proteins and component X (where $X=\left\{P\left(\text{protein}\right),L\left(\text{lipid}\right),W\left(\text{water}\right)\right\}$). The PMFs and the mean forces from the two different methods are in excellent agreement (see Fig. S2), and an example of the decomposed mean forces is presented in Fig. S3 for the case of DSPC. We note that the decomposition scheme according to Eq. 4 contains contributions only from interactions that have an explicit projection onto ξ; for example, changes in lipid-lipid interactions are only implicitly reflected in protein-lipid contributions. Finally, note that we correct all the PMFs by a Jacobian factor, $k_{\text{B}}T\text{ln}\left(\xi \right)$ (49).

Stress-field and membrane-deformation-energy calculations

To better understand the features of the calculated PMF curves, stress field and membrane deformation energy are calculated for three ξ values, 15, 30, and 60 Å, for each lipid type. See our companion article (32) and the Supporting Material for details of the stress field calculations and a discussion of convergence. It is worth noting that the elastic network used to maintain the structure of the gA dimers does not contribute to the stress profiles in the regions of interest here; as shown in Fig. S11, stress from the elastic network is essentially limited to the interior of the protein (r ≤ 5 Å).

For the stress analysis at short gA dimer separation, it is necessary to apply an additional restraining potential to each gA dimer, $V^{\text{rest}}$, to help maintain the orientation of the gA dimers normal to the bilayer plane (see Stress analysis at ${\xi }_{\text{min}}<\xi <{\xi }_{\text{max}}$, below):

$$V^{\text{rest}}=\frac{1}{2}{k}_r\left\{{\left({\overline{x}}_{\text{l}}-{\overline{x}}_{\text{u}}\right)}^2+{\left({\overline{y}}_{\text{l}}-{\overline{y}}_{\text{u}}\right)}^2\right\},$$ (5)

where the force constant $k_r$ is 100 kJ/mol·Å², and ${\overline{x}}_{\text{l}},{\overline{y}}_{\text{l}}$ and ${\overline{x}}_{\text{u}},{\overline{y}}_{\text{u}}$ are the $x,y$ coordinates of the lower-half and upper-half centers of mass of a gA dimer.

The bilayer deformation energy, $E^{\text{comp}}$, is computed by integrating the deformation energy density over x and y,

$$E^{\text{comp}}=\underset{r_1,r_2>10\AA }{\int }\frac{K_{\text{A}}}{2}{\left(\frac{u\left(x,y\right)}{h_0}\right)}^{2}dxdy,$$ (6)

where $r_1,r_2$ are the distances from each of the gA dimers, $u\left(x,y\right)$ is the bilayer deformation in the normal direction at $\left(x,y\right)$, $h_0$ is the unperturbed bilayer thickness, and $K_{\text{A}}$ is the bilayer area stretch modulus determined in our companion article (32). Note that the deformation energy is evaluated only for regions beyond the annular lipids, since the latter have very different mechanical properties (see discussion in our companion article (32) and below).

Configurational entropy for lipids

To better understand the difference in configurational entropy between annular and bulk lipids, we use the quasiharmonic approach (50–53) to estimate the entropy of lipid tails as a function of distance from a single gA dimer:

$$S=\frac{k_{\text{B}}T}{2}\text{ln}\text{det}\left[1+\frac{k_{\text{B}}T{e}^2}{{\hslash }^2}\underset{¯}{\text{D}}\right],$$ (7)

where $\underset{¯}{D}$ is a covariance matrix of a single tail (either sn-1 or sn-2), e is the Euler’s number, and $\hslash$ is Planck’s constant divided by 2π. To avoid complications due to lipid exchange between the annular and bulk regions, we constrain the position of the glycerol group of each lipid (type GL1 bead in the MARTINI model), which is reasonable for our purpose because we focus on the configurational entropy of the lipid tails. Without this constraint, the residence time of annular lipids (at ∼10 Å from the protein center) is at most ∼10 ns (Fig. S10). To converge the calculations, CG simulations >500 ns are carried out for each lipid type. The quantitative scale of the results is similar to those of Baron and co-workers for pure lipid bilayers using the MARTINI model (52,53). Note that previous studies have shown that, due to reduction in the number of degrees of freedom, the configurational entropy of hexadecane computed using the MARTINI model is about four times smaller than that computed using atomistic models (52,53). For relative trends among lipids at different distances from the protein, however, the MARTINI model is expected to be appropriate (also see Fig. S7).

Cluster simulations

To probe the interaction between multiple gA dimers, we carry out two cluster simulations with two different gA dimer/lipid ratios, 1:72 and 1:20. The two systems are simulated for 17 μs and 7.5 μs, respectively. See the Supporting Material for more details.

Results and Discussion

Overview of the PMF from CG simulations

The PMF as a function of the distance between the centers of mass (CM) of the two gA dimers, ξ, is shown in Fig. 2 A for three different lipid bilayers (DMPC, DPPC, and DSPC). The contributions to the PMFs by protein-protein $\left(\text{Δ}{G}^{P-P}\right)$, protein-lipid $\left(\text{Δ}{G}^{P-L}\right)$, and protein-water $\left(\text{Δ}{G}^{P-W}\right)$ interactions are also shown (Fig. 2, B–D). In general, $\text{Δ}{G}^{P-P}$ and $\text{Δ}{G}^{P-L}$ show opposite trends at ξ > 20 Å because protein-lipid contacts are gradually replaced by protein-protein contacts (Fig. 2, B–D). Interestingly, $\text{Δ}{G}^{P-L}$ significantly depends on lipid type (see discussion below).

Figure 2.

PMF (A) and its decompositions (B–D) as functions of the gA dimer-gA dimer center-of-mass separation (ξ). (A) The soft energy barriers are highlighted in inset. Only the error bars for the DSPC simulations are shown in gray shade for clarity. The error bars for other simulated systems are comparable. (B–D) Decompositions of the PMFs in DMPC (B), DPPC (C), and DSPC (D) into three interaction types: protein-protein ($\text{Δ}{G}^{P-P}$, blue), protein-lipid ($\text{Δ}{G}^{P-L}$,red), and protein-water ($\text{Δ}{G}^{P-W}$,black).

As shown in our study of stress distribution around a single gA dimer (32), compression in the normal direction of the bilayer due to hydrophobic mismatch disappears at ∼ξ > 25–30 Å from the central axis of a gA dimer. Thus, it is expected that two gA dimers start to sense each other when their CMs are separated by ∼50–60 Å. Indeed, the PMF curves start to rise at ξ ∼ 50–60 Å. The range of interaction between two gA dimers is ordered by the degree of hydrophobic mismatch: DSPC > DPPC > DMPC. From the separation of initial interactions, the PMF curve monotonically increases as ξ decreases, till it reaches a maximum, $\text{Δ}{G}_{\text{max}}$, at ξ = ξ_max; the value of ξ_max depends on lipid type (or degree of hydrophobic mismatch): ξ_max ∼ 3.6, 3.0, and 2.2 for DSPC, DPPC, and DMPC, respectively. The magnitude of $\text{Δ}{G}_{\text{max}}$, however, does not depend much on lipid type and is ∼1 kcal/mol for all three cases studied here; the physical origin of the soft repulsive potential will be discussed below using stress field calculations. At ξ < ξ_max, the PMF monotonically decreases to a minimum, $\text{Δ}{G}_{\text{min}}$, at ξ = ξ_min. In contrast to ξ_max, ξ_min does not depend sensitively on lipid type, whereas the free-energy well depth varies significantly with the degree of hydrophobic mismatch: ξ_min ∼14–16 Å for all three lipid systems, and $\text{Δ}{G}_{\text{min}}$ is ∼$-2$, $-6$, and $-11$ kcal/mol for DMPC, DPPC, and DSPC, respectively. At ξ = ξ_min, proteins are in direct contact, without any lipids between them. Therefore, the effective radius of gA, $R_{\text{p}}$, is ∼7–8 Å.

In the previous DPD simulations (16), for a model protein/bilayer system similar to the gA dimer/DMPC system we study here, the PMF exhibits a behavior qualitatively similar to that shown in this work: $\text{Δ}{G}_{\text{max}}\sim 1k_{\text{B}}T$ at ${\xi }_{\text{max}}\sim 50$ Å and $\text{Δ}{G}_{\text{min}}\sim -5k_{\text{B}}T$ at ${\xi }_{\text{min}}\sim 15$ Å. The quantitative difference (the magnitude of $\text{Δ}{G}_{\text{max}}$ and $\text{Δ}{G}_{\text{min}}$) from this work likely results from the different levels of detail in the two studies. To explain the physical origin of the soft repulsive energy barrier, Smit and co-workers proposed a concept called hydrophilic shielding (16), which suggests that lipid molecules in the vicinity of membrane proteins and in the region confined by multiple proteins reorganize so that energetic penalty is minimized. Although the concept provides valuable insights into membrane-mediated protein-protein interactions, it was not used to quantitatively explain either the soft repulsive free energy barrier or the free-energy well depth. In this work, by taking advantage of the more detailed MARTINI model, which allows realistic stress field calculations, a simple consideration of elastic deformation associated with the normal and lateral compressions of the membrane due to hydrophobic mismatch provides a useful estimate of the barrier. This is shown in the next section.

Nature of the PMF barrier and stress analysis at $\xi \sim {\xi }_{\text{max}}$

As shown in Fig. 2 A, the calculated PMF curves exhibit a soft energy barrier at $\xi ={\xi }_{\text{max}}\sim 30$ Å. Fig. 2, B–D, clearly show that the soft energy barriers mainly arise from changes in protein-lipid interactions. As mentioned above, changes in $\text{Δ}{G}^{P-L}$ imply contributions from changes in lipid-lipid interactions. To further elucidate the physical origin of this barrier, we perform stress analysis at $\xi =30$ Å for all three lipid systems. Number densities, normal stress $\left(p_{\text{N}}\right)$, and lateral stress $\left(p_{\text{L}}\right)$ at $\xi =30$ Å in DPPC averaged over $\left|y\right|<5$ Å are shown in Fig. 3, A–C. The number density map of the gA dimers in Fig. 3 A indicates that the orientations of the two gA dimers are consistently normal to the bilayer plane. The number density map of lipid hydrocarbons indicates that the confined region between the two gA dimers (①: $\left|x\right|<5$, $\left|y\right|<5$ Å) is more compressed in the normal direction than is the unconfined region (②: $25<\left|x\right|<30$, $\left|y\right|<5$ Å). Indeed, stress distributions in Fig. 3, B and C, also indicate that the confined region is more compressed in both normal and lateral directions.

Figure 3.

Distribution of lipids and stress profile around the gA dimers when they are separated by 30 Å in a DPPC bilayer. (A) Number density (nm⁻¹) of gA dimer (gold), lipid hydrocarbons (rainbow), and lipid phosphates (gray). (B and C) Normal and lateral stresses, respectively. Note that the region with $\left|p_{\text{N}}\right|,\left|p_{\text{L}}\right|>1000$ bar inside the gA dimers is not colored. Density and stress data shown here are averaged over $\left|x\right|<5$ Å. Dashed boxes depict two regions, at $\left|y\right|<5$ (Region ①) and $25<\left|y\right|<30$ Å (Region ②), which are used to represent confined and unconfined regions in Fig. 4 and Fig. S4. Note that the difference in stress profiles inside the proteins arises due to the particular choice of coordinate origin, not from poor convergence. See Supporting Material for details.

For more quantitative comparisons, hydrocarbon density, $p_{\text{N}}$, and $p_{\text{L}}$ are further averaged in regions ① and ②, and the results are plotted as functions of z for the three systems in Fig. 4, A–F, and Fig. S4, A–C. The density plots in Fig. 4, A–C, reveal that the increase in bilayer deformation, u, due to confinement depends on the lipid type; for example, u increases by ∼2 and 4 Å for DMPC and DSPC, respectively (see insets). As a result, $p_{\text{N}}$ (Fig. 4, D–F) and, to a lesser degree, $p_{\text{L}}$ (Fig. S4, A–C) are increased ($\sim 50$ bar for $p_{\text{N}}$) at $\left|z\right|<10$ Å in the confined region compared to the unconfined regions.

Figure 4.

Mean lipid hydrocarbon density and $p_{\text{N}}$ of DMPC (A and D), DPPC (B and E), and DSPC (C and F) for regions ① (blue lines) and ② (red lines), defined in Fig. 3 as representing confined and unconfined regions, from simulations at $\xi =30$ Å are shown as a function of z. In A–C, close-up views (insets) with z clearly show that hydrocarbon regions are compressed by $1-2$ Å in each leaflet, depending on lipid type.

Further, we compute bilayer deformation energy, $E_{\text{comp}}$, at $\xi =30$ and 60 Å using Eq. 6, and the results are summarized in Table 1. For DMPC, DPPC and DSPC, $E_{\text{comp}}$ is increased by 0.0, 1.0, and 1.3 kcal/mol, respectively, as ξ decreases from 60 to 30 Å. Thus, the increase in $E_{\text{comp}}$ accounts for most of the PMF barrier, suggesting that the physical origin of the soft PMF barrier is overcompression of the membrane due to two approaching gA dimers. This is consistent with our finding presented in the companion article that continuum elasticity theory holds at $r_{1,2}>10$ Å (32). Indeed, using a continuum mechanics model based on linear elasticity, Ursell et al. captured a soft energy barrier between two open MscL conformations in the presence of negative hydrophobic mismatch (26).

Table 1.

Summary of bilayer deformation energies (in kcal/mol) at different ξ

	ξ (Å)
	15	30	60
DMPC	1.1	1.2	1.2
DPPC	1.2	3.3	2.3
DSPC	2.3	5.4	4.1

gA dimer separation (ξ) values are from CG simulations of different lipid bilayers.

Origin for the PMF well—importance of annular lipids

As discussed earlier, the PMFs of two gA dimers drop significantly when the gA dimers are in direct contact: $\text{Δ}{G}_{\text{min}}$ = $-2$, $-6$, and $-11$ kcal/mol, for DMPC, DPPC, and DSPC, respectively (Fig. 2 A). Qualitatively, this is expected, since the membrane deformation energy due to hydrophobic mismatch is reduced upon dimer contact, i.e., the area of deformed membrane is smaller compared to when the two gA dimers are far apart. However, as shown in Table 1, changes in the bilayer deformation energy upon association, $\text{Δ}{E}_{\text{min}}^{\text{comp}}=E^{\text{comp}}\left(\xi =15\AA \right)-E_{\text{comp}}\left(\xi =60\AA \right),$ are only $-0.1$, $-1.1$, and $-1.8$ kcal/mol for DMPC, DPPC, and DSPC, respectively. In fact, even if all elastic membrane deformation near two distantly separated gA dimers is relieved upon dimer contact, the magnitude of the energy gain is far less compared to the calculated PMF well depth, especially for DPPC and DSPC. The significant discrepancies between computed the $\text{Δ}{G}_{\text{min}}$ and $\text{Δ}{E}_{\text{min}}^{\text{comp}}$ suggest that $\text{Δ}{G}_{\text{min}}$ is not dominated by the bilayer deformation energy in the elastic regime ($r_1,r_2>10$ Å), as is commonly assumed in analysis using elastic models.

What is missing from this comparison, clearly, is the contribution from the annular lipids; as emphasized by Eq. 6, the computation of $\text{Δ}{E}_{\text{min}}^{\text{comp}}$ pertains only to regions beyond the annular lipids. For the barrier region, which doesn’t involve any significant change in the number of lipids near each gA dimer, it is acceptable to ignore explicit contributions from the annular lipids. For the well region, however, this major approximation is no longer valid, because a few annular lipids are expected to be depleted from the region between the two gA dimers upon dimer contact, and the number of depleted lipids depends on the lipid type (see Fig. 5, G and H). Therefore, an entropic gain due to lipid depletion, which is partially compensated by the loss of protein-lipid interaction, should be considered for the dimer-formation free energy. Indeed, Fig. 2, B–D, show the critical roles of annular lipids. In Fig. 2, B–D, $\text{Δ}{G}^{P-L}$ at $\xi \sim 20\AA$, where the gA dimers are separated by at most a single layer of annular lipids, is ∼4 kcal/mol regardless of lipid type. Interestingly, the free energy drop upon the loss of this annular lipid layer (i.e., ξ changes from 20 to 15 Å) significantly depends on lipid type (e.g., $\sim 0$ for DMPC and about $-4$ kcal/mol for DSPC). To better understand the magnitude and signature of the depletion effect, we first examine the properties of annular lipids from our CG simulations, and then carry out stress analysis in the region ${\xi }_{\text{min}}<\xi <{\xi }_{\text{max}}$.

Figure 5.

Properties of annular lipids. (A–D) Orientations of CG hydrocarbon bonds in lipids depend on the distance from a gA dimer; r is measured relative to the central axis of the gA dimer. Direction cosines are calculated in a cylindrical coordinate with the gA dimer at the origin and the z axis parallel to the axis of the gA dimer. Time-averaged ${\text{cos}}^2{\alpha }_r$ and ${\text{cos}}^2{\alpha }_{\theta }$ values as functions of the bond number are shown for DMPC (A and B) and DSPC (C and D)systems. Atomistic simulation results have similar trends (Fig. S5). (E and F) Configurational entropy of CG hydrocarbon tails of lipids as a function of distance between the GL1 bead and the axis of a gA dimer. Each data point (data from sn-1 and sn-2 chains are shown as red circles and blue triangles, respectively) corresponds to the calculated entropy of a single chain from 1-μs simulations; note that the GL1 bead of each lipid is constrained in the x and y directions to avoid complications due to lipid exchange. Results are shown for DMPC (E) and DSPC (F) systems. See Fig. S6 for the DPPC result and Fig. S7 for estimates based on all-atom simulations from the accompanying article (32). (G and H) Changes in the number of CG lipid beads within 10 Å from a gA dimer ($r_1$ or $r_2<10$ Å) relative to the numbers at $\xi =60$ Å as a function of ξ: (G), head beads; (H), and tail beads. (Insets) Bead numbers.

Properties of annular lipids around a gA dimer

The differences in the motional, mechanical, and diffusional behaviors of annular and bulk lipids were clearly analyzed by Kim et al. in a recent atomistic simulation of gA dimers in several lipid bilayers (31). It was shown that the annular lipids have a notably higher area compressibility modulus, more ordered acyl chain dynamics, as reflected by the higher $S_{\text{CD}}$ order parameters, and substantially reduced diffusion constants. In our work, we observed similar trends with the CG model. For example, as shown by the directional cosine plots in Fig. 5, A–D, the orientations of the hydrocarbon bonds depend on the distance from the protein; the first-shell ($r<10$ Å) lipids have the most distinct orientations due to protein-lipid interactions. We use the directional cosines here because $S_{\text{CD}}$ characterizes the orientation only in the membrane normal direction and therefore is a suitable measure of lipid (dis)order only when there is no orientational preference in the membrane plane. For annular lipids, the nearby protein causes notable anisotropy in the lateral direction, and therefore, the directional cosines are more suitable than $S_{\text{CD}}$ for characterizing lipid-tail disorder.

As expected, the annular lipids have more limited accessible configurations compared to the bulk lipids. In the frame fixed to the glycerol group, configurations of bulk lipid tails form an umbrellalike ensemble (54). For example, ${\text{cos}}^2{\alpha }_r\sim {\text{cos}}^2{\alpha }_{\theta }$ of DSPC in Fig. 5, C and D, change from ∼0.2 to 0.3 as we move along the lipid tail from the glycerol group, consistent with the umbrellalike ensemble. On the other hand, configurations of annular lipids are more limited by the protein/lipid interface. ${\text{cos}}^2{\alpha }_r$ decreases close to 0.1 while ${\text{cos}}^2{\alpha }_{\theta }$ increases to 0.4 as the bond number increases, indicating that the bond vectors are almost tangential to the protein/lipid interface at the end of lipid tails. The orientations of lipid tails in all-atom simulations show similar behaviors (see Fig. S5).

The decrease in the configurational space available to lipid tails near the protein is illustrated more quantitatively in Fig. 5, E and F, and Fig. S6, which clearly show, despite the considerable spread of data, that the annular lipids have $TS$ values $\sim 1-2$ kcal/mol lower per tail compared to the bulk lipids. The AA entropy values (Fig. S7) point to similar trends despite the systematic difference in absolute values at the AA and CG levels (52,53). The numbers of depleted head beads upon dimerization are ∼10, 11, and 14 for DMPC, DPPC, and DSPC, respectively (Fig. 5 G). Because there are four beads/headgroup, these numbers correspond to ∼2.5, 2.5, and 3.5 lipids for DMPC, DPPC, and DSPC, respectively. Considering that the entropic gain from the depletion of a single lipid can be as high as 3–4 kcal/mol, it is likely that lipid depletion accounts for a significant portion of the $\text{Δ}{G}_{\text{min}}$, especially for DSPC. A more quantitative estimate remains difficult, however, considering the significant spread of the configurational entropy data in Fig. 5, E and F, and Fig. S6.

Stress analysis at ${\xi }_{\text{min}}<\xi <{\xi }_{\text{max}}$

The effective force between two gA dimers at ξ_min < ξ < ξ_max can be estimated using the slopes of the PMF curves in Fig. 2 A. We estimate this effective force by $-\left(\text{Δ}{G}_{\text{max}}-\text{Δ}{G}_{\text{min}}\right)/\left({\xi }_{\text{max}}-{\xi }_{\text{min}}\right)=-0.3,-0.4$, and $-0.6$ kcal/mol·Å for DMPC, DPPC, and DSPC, respectively. If we model the gA dimers as two plates of width 10 Å and height 40 Å that are separated by ∼20 Å, the difference in pressure between the confined and unconfined regions that causes an effective force of $-0.3$ to $-0.6$ kcal/mol·Å is ∼$-50$ to $-100$ bar.

To illustrate that the confinement at ${\xi }_{\text{min}}<\xi <{\xi }_{\text{max}}$ induces depletion of lipids and that the pressure difference induced by this depletion is consistent with the PMF slope, we carried out stress calculations at $\xi =20$ Å. If we carry out this calculation with only a restraint for ξ (Eq. 1), the two gA dimers form either ∨ or ∧ shape to fill the depleted volume; lipids confined between the two gA dimers appear only in one monolayer for all lipid types. For example, Fig. 6 A shows the result for DSPC; a similar asymmetric collapse occurs for other lipid types as well. To better estimate the depletion force, we carry out stress calculations at $\xi =20$ Å with additional restraints (Eq. 5) that maintain the orientations of the gA dimers normal to the bilayer plane. The resulting number densities from those simulations are shown in Fig. 6 B and Fig. S8, A and B. Lipid-tail density near $x=0$ Å is significantly lower than that at $\left|x\right|=20$ Å for all lipid types, clearly illustrating depletion of lipids due to confinement. Interestingly, phosphate positions near $x=0$ Å in Fig. 6 B and Fig. S8, A and B (gray) are rather independent of lipid type, likely because several Trp residues from the two gA dimers anchor the membrane/water interfaces strongly, as discussed in our companion article (32). Indeed, the corresponding lateral stress profiles in Fig. 6, C and D, reveal strongly attractive bands that connect Trp residues of the two gA dimers near $x=0$ and $\left|z\right|=10$ Å regardless of lipid type. This normal compaction of lipid headgroups reduces the volume of lipid hydrocarbons that can be packed, especially for lipids with longer chains, such as DSPC. Therefore, one expects that lipids with a longer chain length tend to be more depleted than those with shorter chains, as supported by Fig. 5, G and H. To illustrate the dependence of depletion force on lipid type, the lateral stress profiles at $\xi =20$ and 60 Å are averaged over $\left|x\right|,\left|y\right|<5$ Å, and the resulting stress profiles are shown in Fig. 6, E and F, and Fig. S9. Regardless of lipid type, significant drops in $p_{\text{L}}$ (i.e., attractive force) are observed at $\xi =20$ Å compared to $\xi =60$ Å, reflecting depletion of lipid hydrocarbons. The mean lateral stress measurements in $\left|x\right|,\left|y\right|<5$ Å are $-29$, $-40$, and $-121$ bar for DMPC, DPPC and DSPC, respectively. These depletion forces are consistent with the above estimates of ∼$-50$ to $-100$ bar using the slopes of the PMF curves.

Figure 6.

Distribution of lipids and stress profile around the gA dimers when they are separated by a single layer of lipids. (A and B) Number densities at $\xi =20$ Å in DSPC in the absence (A) and presence (B) of constraints on gA orientations (Eq. 5) for gA dimer (gold), lipid hydrocarbon (rainbow) and lipid phosphate (gray). (C and D) Lateral stress profiles in the simulations with upright gA dimers at $\xi =20$ Å in DMPC and DSPC, respectively. Note that the difference in stress profiles inside the proteins arises due to the particular choice of coordinate origin, not from poor convergence. See Supporting Material for details. Density and stress scale bars are identical to those in Fig. 3. (E and F) Mean lateral stress, $p_{\text{L}}$, in $\left|x\right|,\left|y\right|<5$ Å in DMPC (E) and DSPC (F) for two gA dimers with $\xi =2$ nm as a function of z in the presence of constraints on gA orientation (Eq. 5).

Implication to the application of continuum mechanics models

It is noteworthy that the depletion effect is nonelastic in nature; in other words, lipid molecules that mainly contribute to the depletion force are those in direct contact with the gA dimers at $r_1,r_2<10$ Å, which, according to our stress analysis (32), fall into the nonelastic regime. The importance of the depletion force and the differences between annular and bulk lipids suggest that commonly used continuum models should be used with caution, especially for describing processes that imply a significant change in the number of annular lipids, such as membrane-mediated protein-protein interactions and insertion of proteins into the membrane (55,56). In other words, a reliable continuum model needs to include the nonelastic free-energy components associated with the annular lipids, especially when the magnitude of hydrophobic mismatch is large.

For example, using a continuum mechanics model, Ursell et al. estimated that the change in membrane deformation energy when two open MscL channels associate in a DPPC bilayer is ∼$-15$ kcal/mol (26). Considering that the diameter of an open MscL ($\sim 70$ Å) is about seven times larger than that of a gA dimer (heights are similar) and that membrane deformation energy is proportional to the area of deformation, their estimate for MscL is qualitatively consistent with the change in the membrane deformation energy for two gA dimers upon association (∼−1 kcal/mol for DPPC in Table 1). However, the study presented here suggests that the total MscL-MscL association energy can be stronger due to contributions from the annular lipids. On the other hand, as discussed in our companion article (32), the boundary condition used in continuum models at the protein/lipid interface may also overestimate the magnitude of membrane deformation. Therefore, there can be error cancellation between the two effects.

Cluster formation of multiple gA dimers

The PMF calculations in Fig. 2 indicate formations of energetically stable gA-dimer pairs in DPPC and DSPC bilayers, yet it is uncertain whether large clusters of many gA dimers will form and, if they do form, what the preferred topologies are. To study the possibility of larger cluster formations, we carried out a large-scale simulation with 256 gA dimers and 18,432 DPPC molecules (1:72 ratio) in a fairly large simulation box (800 Å width), with gA dimers initially placed at grid points with ∼50 Å spacing. We ran this simulation for about 17 μs, and the results are shown in Fig. 7, A–C. The snapshot at the end of the simulation is shown in Fig. 7 A, in which each gA dimer is depicted with a distinct color. The mean number of gA dimers/cluster (i.e., the mean cluster size) and the mean number of gA-dimer neighbors/gA dimer (i.e., the mean contacts/gA dimer) are shown as functions of simulation time in Fig. 7 B. The histogram of the mean cluster size at the end of the simulation is shown in Fig. 7 C. The results indicate that the typical cluster size is at most three to four dimers (although larger clusters are also observed), and each gA dimer has fewer than three neighbors under this condition. Note that a basic topology unit that satisfies the observed mean cluster size and contact number is a triangle (see Fig. 7 B, inset, which is consistent with the DPD simulation study showing that a preferred topology of three membrane proteins under a negative hydrophobic mismatch is a triangle, not a single line (57). This observation is also consistent with the finding from a recent MARTINI simulation of α-helices and β-barrels (1:450 protein/lipid ratio) (19).The authors of that study also observed at most three to four neighbors per protein.

Figure 7.

Results of cluster simulations with protein (gA dimer)/lipid ratios of 1:72 (A–C) and 1:20 (D). (A) A snapshot taken at the end of the simulation. Each gA dimer is colored with a distinct color. Dashed black box depicts the simulation box. (B) Mean cluster size (blue) and mean contact number/gA dimer (red) are shown as functions of simulation time. (Inset) A close-up view of a representative cluster of seven gA dimers. (C) Histogram of the mean cluster size at the end of the simulation. (D) A snapshot taken at the end of the simulation with a protein/lipid ratio of 1:20.

To study how the cluster formation depends on protein density, we carried out another simulation with a protein/DPPC ratio of 1:20, and the snapshot at the end of the simulation (7.5 μs) is shown in Fig. 7 D. Under this condition, gA dimers form relatively larger clusters. The topology of the clusters is linear, with at most four neighbors per gA dimer, and the size of the linear dimension is about twice the gA diameter. Clusters with linear patterns have also been observed in systems of α-helices, gA, and MscL at high protein concentrations in the presence of negative hydrophobic mismatch using either fluorescence microscopy or AFM, although the size of the linear pattern was unclear (12–14,58). These results indicate that having more than four neighbors for a gA dimer is energetically unfavorable even at high gA-dimer concentrations. In other words, a gA dimer requires at least about half of the hydrophobic amino acids in direct contact with lipid hydrocarbons to be energetically stable, highlighting another aspect of the importance of protein-annular-lipid interactions.

Conclusions

In another study that aims to foster the connection between particle and continuum models for membrane-mediated processes (32,40,59,60), we carried out CG simulations for gA dimers and stress analyses to better understand how the membrane modulates protein-protein association. First, we computed the PMF of gA dimer association in three different lipid bilayers (and therefore different degrees of hydrophobic mismatch). The PMF curves exhibit a soft energy barrier (∼1 kcal/mol independent of lipid type) at $\xi =22-36$ Å and an attractive potential well (ranging from $-2$ to $-11$ kcal/mol depending on lipid type) at $\xi =14-16$ Å. We elucidate the physical origin of the soft PMF barrier and the attractive potential well using mean-force based decomposition and detailed stress fields at several ξ values; both types of analyses clearly point to the importance of lipid contributions. The stress field at $\xi =30$ Å quantitatively reveals that lipids confined between two gA dimers are compressed, resulting in an increase in membrane deformation energy comparable to the magnitude of the PMF barrier. Another stress field calculation at $\xi =20$ Å reveals that depletion of lipids is a major driving force for the lipid-dependent gA-dimer-gA-dimer association. At this separation, membrane confined between two gA dimers is severely compacted in the normal direction, resulting in a constant membrane thickness in the confined region. Due to this normal compaction of lipid, the magnitudes of depletion forces are highly dependent on lipid type.

We find that a simple continuum elasticity model that describes membrane deformation beyond the annular lipids can’t quantitatively explain the strong attraction between two gA dimers in the presence of large hydrophobic mismatch. Analysis of the lipid-tail properties as functions of distance from the gA dimer shows that the orientation of annular lipids is more tangential to the protein/lipid interface, unlike the umbrella ensemble sampled by the tails of bulk lipids. Moreover, configurational entropy of annular lipids, compared to bulk lipids, contributes about $2-4$ kcal/mol/lipid to the higher free energy. It is likely that the difference between configurational entropies of annular and bulk lipids and/or the number of depleted lipids upon dimer formation account for the lipid-dependent attraction between gA dimers.

Our analyses highlight the importance of explicitly considering contributions from annular lipids in quantitative modeling of membrane mediated processes (59). This is particularly important for processes in which lipid reorganization near the protein is significant, such as protein-protein clustering and protein insertion into membrane. To model those processes using a continuum mechanics framework requires going beyond the conventional models to include contributions from effects such as lipid depletion. This applies also to implicit membrane modeling techniques, such as various modeling techniques based on the generalized Born/Poisson-Boltzmann model for electrostatics (61,62), even when explicit membrane deformation is considered (55). In the context of implicit water models, we note that it has been well appreciated that the first layer of solvent molecules behave rather differently from bulk-solvent molecules and therefore ought to be treated explicitly, especially when describing processes that involve a major change in the solvent-accessible area (63). The effect of annular lipids is somewhat similar, with additional complications due to the larger configurational entropy potentially accessible to lipid tails.

Finally, our large-scale MARTINI simulations of spontaneous cluster formations suggest that large hexagonal clustering of the gA dimers is unlikely. Instead, it seems that every gA dimer in our simulations requires contact with lipid hydrocarbons to some degree, and at most three to four proteins can be in contact with each gA dimer; this observation highlights another aspect of the importance of protein-annular-lipid interactions. As a result, clusters of only a few gA dimers are observed at low protein concentration (protein/lipid = 1:72), and linear-pattern clusters are observed at higher protein concentration (protein/lipid = 1:20). The dimension of the linear-pattern clusters is likely to be twice the protein diameter, an observation that remains to be better characterized experimentally.

Additional details of the stress calculations and cluster simulations, as well as additional results for the decomposition of PMFs, lateral pressure profiles, and annular lipid properties, are included in the Supporting Material.