Lipid packing in biological membranes governs protein localization and membrane permeability

Abstract

Plasma membrane (PM) heterogeneity has long been implicated in various cellular functions. However, mechanistic principles governing functional regulations of lipid environment are not well understood due to the inherent complexities associated with the relevant length and timescales that limit both direct experimental measurements and their interpretation. In this context, computer simulations hold immense potential to investigate molecular-level interactions and mechanisms that lead to PM heterogeneity and its functions. Herein, we investigate spatial and dynamic heterogeneity in model membranes with coexisting liquid ordered and liquid disordered phases and characterize the membrane order in terms of the local topological changes in lipid environment using the nonaffine deformation framework. Furthermore, we probe the packing defects in these membranes, which can be considered as the conjugate of membrane order assessed in terms of the nonaffine parameter. In doing so, we formalize the connection between membrane packing and local membrane order and use that to explore the mechanistic principles behind their functions. Our observations suggest that heterogeneity in mixed phase membranes is a consequence of local lipid topology and its temporal evolution, which give rise to disparate lipid packing in ordered and disordered domains. This in turn governs the distinct nature of packing defects in these domains that can play a crucial role in preferential localization of proteins in mixed phase membranes. Furthermore, we observe that lipid packing also leads to contrasting distribution of free volume in the membrane core region in ordered and disordered membranes, which can lead to distinctive membrane permeability of small molecules. Our results, thus, indicate that heterogeneity in mixed phase membranes closely governs the membrane functions that may emerge from packing-related basic design principles.

Significance

Functionally important complex lateral and transverse structures in biological membranes result from the differential molecular interactions among a rich variety of lipids and other building blocks. The nature of molecular packing in a membrane is a manifestation of these interactions. In this work, using some of the ideas from the physics of amorphous materials and glasses, we quantify the correlation between heterogeneous membrane organization and the three-dimensional (3D) packing defects. Subsequently, we investigate the packing-based molecular design-level features that drive preferential localization of peptides in heterogeneous membrane and membrane permeation of small molecules.

Introduction

About 150 years ago, from their simple but ingenious experiments on movement of anesthetic and other small solutes between the surrounding and cell interior, Charles Ernest Overton and Hans Horst Meyer made the astute observation that the cell boundaries could be composed of “aliphatic” lipid molecules (1,2). The seminal work of Overton and Meyer established the idea of biological membrane as a semipermeable boundary and arguably laid the foundation for membrane biophysics. The ensuing early activities toward understanding the structure and organization of the biological membrane from the likes of Pockels and Langmuir (3), Gorter and Grendel (4), and Danielli and Davson (5) led to the now well-known information that biological membranes are made up of two layers of lipid leaflets that are stacked such that the hydrophobic fatty acid tails make the core of the bilayer and the polar headgroups face the aqueous medium on both sides. The paradigms of membrane organization has undergone a cascade of changes since then (6,7) with a large body of evidence pointing toward a complex lateral organization with existence of nonrandom localization of lipids and proteins on membrane surface resulting in existence of physiologically functional heterogeneous sub-100 nm patterns that we now call “rafts” in living membranes (8,9,10,11,12,13,14). With the advent of single-molecule tracking techniques and super-resolution microscopy and tomography (15,16,17,18) as well as lipidomics data (19,20,21,22,23,24,25,26,27), we now have access to unprecedented details on membrane composition, structure, and dynamics.

Over the last few decades, spatial and dynamic heterogeneity in membrane has been systematically and extensively investigated, in both reconstituted model membranes at carefully chosen composition and cell-derived/living cell membranes (8,10,28,29,30,31,32,33,34,35,36,37,38,39). However, our understanding of their functional implications still remains far from complete, mostly due to the lack of a comprehensive molecular level picture. We point to a few reviews that have succinctly laid down the advances made in the field and also highlighted the path forward, toward more clearly elucidating the structure, dynamics, and functional role of the membrane rafts (37,40,41,42,43,44,45). Traditionally, lipid rafts have been identified as the heterogeneous, highly dynamic, cholesterol- and sphingomyelin-rich tightly packed domains that are implicated in specific protein recruitment. In model membranes, the liquid ordered ($L_o$) domains in a phase separated membrane are considered to be the prototype for rafts (39,46,47) with tighter packing and slower diffusion compared with more fluid liquid disordered ($L_d$) regions. In general, with diffusion and packing behavior as distinguishing features of the two coexisting fluid phases in membrane systems, the relationship between the molecular packing and mobility of lipids and membrane order is seemingly obvious from the classical free volume-based theories of diffusion (48,49). However, despite the tremendous advances made in experimental resolutions in both time and space, accurate quantification and formulation of this correlation is still elusive (50,51,52,53,54). In this regard, computer simulation has proven to be a very useful tool and the success of “computational microscopy” (55,56) spans from the ability to observe phase separated nanostructures, reminiscent of the ordered raft domains, in model binary/ternary lipid systems (46,57,58) to investigating their plausible functional implications (59,60). Although our understanding of these rafts and their functions is limited to such simplified model membranes, computer simulations have been able to provide further insights into their microscopic structure and function. Only recently, relatively simple model membranes exhibiting $L_o$ and $L_d$ phases have been able to elucidate the distinct nature of lipid packing within the $L_o$ domains (58,61,62), which are observed to play a crucial role in membrane permeation of small molecules (63,64,65).

Lipid packing defects (66), the transiently exposed hydrophobic tails of the lipids, are the cooperative manifestation of both the membrane order and packing (which results from differential lipid interactions), together with the collective dynamics of the lipids in the ordered/disordered domains. It can, therefore, be loosely considered as the conjugate of local membrane orderliness. Lipid packing defects are nontrivial to identify in experiments and very recently it has been shown that lipid packing (and thus, membrane order) in membrane mimetics, such as nanodiscs that are usually used in experiments, are distinctly different from intact membranes (67). In contrast, packing defects have been extensively studied in computer simulation (68,69,70,71) and, over the last decade, the nature of packing defects in curved membranes has been well characterized, unraveling their role in the adsorption of various proteins that contain specific amphipathic helices (59,69,70,71,72). Amphipathic lipid packing sensor motifs in these proteins, can sense large preexisting packing defects and initiate binding by anchoring some of their bulky amino acid residues into such defects. Such studies on the curvature-dependent binding of amphipathic lipid packing sensor motifs have been able to elucidate the binding mechanism of various peripheral and cytosolic proteins and their physiological implications in cells (73,74,75,76,77). However, a similar understanding of the nature of lipid packing defects in coexisting fluid phase membranes is still missing, even though the lateral organization in realistic cell membranes has highly heterogeneous characteristics. One important physiological function of the raft-like domains therein is the partitioning of peripheral proteins (45,78,79,80), where large packing defects are known to provide suitable platforms for membrane adsorption (70,81). Various experiments have shown preferential domain affinities for distinct membrane binding motifs to either of the $L_o$ or $L_d$ domains or their interface. For example, while most of the peripheral proteins such as RAB proteins, RAB1, RAB5, and RAB6, preferentially bind to $L_d$ domains (82), HIV gp41 interacts predominantly at the $L_o/L_d$ domain boundary (83) and aspirin (acetylsalicylic acid) binds to $L_o$ domains (84). Intriguingly, the ordered domain affinity has been found to be further governed by the relative contrast between the ordered and disordered domains, which is known to be rather subtle in cell-derived giant plasma membrane vesicles (GPMVs) and quite distinct in giant unilamellar vesicles (GUVs) (37). Accordingly, the palmitoylated transmembrane domain of linker for activation of T cells (tLAT) is shown to bind to the $L_o$ domains in GPMV (85,86), while they are specifically excluded from $L_o$ phase in GUVs (87). While several other proteins exhibit such selective domain preference (80), the basic design principle behind this specificity of membrane-protein interactions in mixed phase membranes is so far unexplored.

Interestingly, the nature of packing defects in pure phase (homogeneous) membranes are found to be rather generic, as they exhibit a few characteristic features that depend only on the overall order (phase) of the membrane (71). In general, pure $L_o$ systems are relatively scarce in packing defects, while pure $L_d$ systems exhibit a denser distribution of defects. Furthermore, the defects in the $L_o$ systems are observed to be significantly smaller (and shallower) compared with those in the $L_d$ systems. While these observations can be trivially extrapolated to coexisting (i.e., mixed phase or heterogeneous) $L_o/L_d$ membranes, wherein the $L_d$ regions are dominated in terms of packing defects (71,88), rationalizing their role in peripheral protein partitioning is nontrivial, as proteins do not exclusively partition with the $L_d$ domains that are rich in large and deep defects. So, what factors decide the domain affinity in such membranes?

Beyond protein partitioning, lipid ordering can also play a crucial role in the membrane permeability of small molecules (89,90,91). A recent study by Pastor and co-workers (89) investigated the transport of oxygen and water molecules through ternary membrane systems exhibiting homogeneous $L_o$ and $L_d$ phases. Their observations indicate the existence of separate permeation pathways in $L_o$ and $L_d$ phase membranes, which have their origins in specific packing of of lipids. They showed a nearly free diffusion in bilayer midplane in $L_o$ membranes while the $L_d$ membranes were statistically more permeable. Their observations on relative concentration of oxygen in the bilayer midplane of both membranes were further supported by electron paramagnetic resonance spectroscopy. It is therefore evident that membrane packing and order not only dictate the peripheral membrane protein association and localization, but also the transbilayer permeation pathways. Traditionally, lateral lipid diffusion in experiments has been modeled in terms of free volume theory (92), which in its simplest form assumes that the dynamics of molecules in liquids and glasses proceed in discrete diffusive jumps and critically depend on the available free volume (48). In the case of lipid membrane, this free volume has generally been related to the fluctuation in area per lipid. Despite several rigorous attempts in the past, a complete understanding on the role of free volume in lipid diffusion still remains missing (68,93,94,95,96). In such cases, the ambiguity associated with the definition of free volume in lipid membrane is one of the major issues to be addressed (96). Unlike the case of molecular liquids, membranes with complex lipid geometry do not allow an easy or a clear estimation of free volume. The presence of multiple lipid species and the intrinsic heterogeneity of the membrane further complicates the problem. Moreover, processes such as membrane permeability, protein-membrane interaction, signaling across the membrane, and other functions, which closely depend on interleaflet coupling and involve both lateral and transverse diffusion, might demand explanations that go beyond the simple free volume theories. To understand the role of membrane free volume in small-molecule permeation and test the applicability of free volume theories, we first need to characterize the free volume unambiguously.

In this study, we investigate the functional significance of membrane orderliness in the light of packing defects. Our goal is to understand and formalize the correlation between the membrane order in heterogeneous (i.e., coexisting or mixed phase $L_o/L_d$) membranes and the packing defects therein, where homogeneous (i.e., pure phase) membranes serve as control systems for such investigations. Toward this, we characterize the local membrane orderliness in terms of the nonaffine parameter (NAP) (62,88), which captures the distinct nature of spatiotemporal evolution of lipids in their local neighborhoods without any knowledge on lipid chemistry. We identify the 3D packing defects in these membranes using our recently developed algorithm (71), which efficiently circumvents the computational bottleneck of grid-based calculations. Our results indicate a direct correlation between orderliness and defects for three different mixed phase membrane systems with distinct lipid constituents, suggesting the correlation to be rather generic. Moreover, a stronger correlation indicates the nature of mixed phase membrane to be more ordered-like, in which case, protein can also partition onto the ordered domains. Furthermore, the dynamics of these defects in the $L_o$ and $L_d$ domains are observed to be distinctly different, indicating two possible scenarios for protein partitioning onto any mixed phase membranes. Using the framework of NAP, we further investigate the membrane partitioning of tLAT and find that palmitoylation indeed increases its ordered phase affinity. By analyzing the surface defects (hydrophobic defect pockets) and core defects (free volume in the membrane midplane) in the membrane systems studied by Ghysels et al. (89), we find that, while the $L_d$ membrane clearly dominates in terms of size of surface defects, the trend is reversed for core defects, i.e., $L_o$ systems exhibit larger free volume in the membrane core. This can provide a valid explanation on the free transverse diffusion in $L_o$ domains, while $L_d$ domains remain statistically more permeable. The membrane packing and order can thus influence both its peripheral and transbilayer functions.

Materials and methods

Details of the simulation trajectories

We studied three atomistic ternary lipid membrane systems, DPPC/DOPC/CHOL, PSM/DOPC/CHOL, and PSM/POPC/CHOL, each exhibiting pure $L_o$, pure $L_d$ and mixed $L_o/L_d$ phases. We used these nine systems to understand the nature of lipid packing in pure and mixed phase membranes, i.e., the correlation between membrane order and lipid packing. The atomistic trajectories were obtained from Lyman’s group at the University of Delaware (61). The trajectories were equilibrated in NAMD (97) using the TIP3P water model (98), the CHARMM36 lipid force field (99), and Pitman et al.’s cholesterol model (100). The $\mu s$ long simulations were performed on the Anton supercomputer (101,102). The relevant details on system composition and size are summarized in Table 1, where the analysis window indicates the number of simulation snapshots used to perform the analyses reported in this work. Further technical details on system preparation and simulation methodologies can be found in the original references (58,61).

Table 1.

Details of the atomistic trajectories of pure and mixed phase systems used to analyze the correlation between membrane order and packing

Membrane system	Membrane phase	Lipid composition	Total no. of lipids	System size ($\sim$nm $\times$ nm)	Total trajectory length ($\mu \mathrm{s}$)	Analysis window (no. of snapshots)a
DPPC/DOPC/CHOL	$L_o$	0.55/0.15/0.30	512	10.4 $\times$ 10.4	5.05	500
	$L_d$	0.29/0.60/0.11	400	10.8 $\times$ 10.8	2.02	500
	$L_o/L_d$	0.37/0.36/0.27	380	9.55 $\times$ 9.55	9.43	500
PSM/DOPC/CHOL	$L_o$	0.64/0.03/0.33	560	10.62 $\times$ 10.62	0.95	195
	$L_d$	0.15/0.82/0.03	574	13.44 $\times$ 13.44	0.81	165
	$L_o/L_d$	0.43/0.38/0.19	452	10.86 $\times$ 10.86	4.54	625
PSM/POPC/CHOL	$L_o$	0.61/0.08/0.31	560	10.65 $\times$ 10.65	9.08	280
	$L_d$	0.23/0.69/0.08	560	12.4 $\times$ 12.4	1.14	500
	$L_o/L_d$	0.47/0.32/0.21	506	10.56 $\times$ 10.56	6.88	285

^aHere, the analysis windows in each trajectory are chosen so as to avoid cholesterol flipping events.

Next, we investigated the role of membrane order in governing the partitioning of membrane active peptides and proteins by considering tLAT as a model. In this case, to understand the role of palmitoylation in the ordered/disordered phase affinity of tLAT, we studied atomistic systems of a single tLAT, with and without the palmitoyl group, in mixed ($L_o/L_d$) phase DPPC/DAPC/CHOL membranes. The two atomistic trajectories were obtained from Gorfe’s lab at the University of Texas Health Science Center at Houston, who in their earlier work (60) investigated the role of palmitoylation in the partitioning of tLAT using atomistic and coarse-grained (CG) molecular dynamics (MD) simulation and potential of mean force calculations. The simulations were performed using the CHARMM36 force field (99) and the Gromacs (103) software. The relevant details of the simulation system are summarized in Table 2; as earlier the analysis window indicates the number of simulation snapshots used to perform the analyses reported in this work. Further details on simulation setup can be found in the original reference (60).

Table 2.

Details of the atomistic trajectories used to analyze partitioning of tLAT with and without palmitoylation

Membrane system	Membrane phase	Lipid composition	Total no. of lipids	System size ($\sim$nm $\times$ nm)	Total trajectory length ($\mu \mathrm{s}$)	Analysis window (no. of snapshots)
DPPC/DAPC/CHOL + tLAT	$L_o/L_d$	0.5/0.3/0.2	400	10.1 $\times$ 10.1	2.0	500
DPPC/DAPC/CHOL + palmitoylated tLAT	$L_o/L_d$	0.5/0.3/0.2	400	10.2 $\times$ 10.2	2.0	500

To understand the molecular origin behind distinct permeation pathways of small molecules in pure phase $L_o$ and $L_d$ membranes, we studied the oxygen permeation trajectories of Ghysels et al. (89). The two atomistic trajectories were obtained from Pastor’s lab at the National Heart Lung Blood Institute, NIH. The systems consisted of 50 oxygen molecules in pure phase DPPC/DOPC/CHOL membranes. The simulations were performed in CHARMM (104) using the C36 lipid force field (99). The relevant details of the systems are summarized in Table 3. Further technical details on these simulations can be found in the original reference (89) and some of the previous studies from the same group (58,64).

Table 3.

Details of the atomistic trajectories used to analyze oxygen permeation in pure phase DPPC/DOPC/CHOL membranes

Membrane system	Membrane phase	Lipid composition	Total no. of lipids	System size (nm $\times$ nm)	Total trajectory length ($\mu \mathrm{s}$)	Analysis window (no. of snapshots)
DPPC/DOPC/CHOL + 50 O${}^2$	$L_o$	0.55/0.15/0.30	512	10.88 $\times$ 10.88	1.14	1000
DPPC/DOPC/CHOL + 50 O${}^2$	$L_d$	0.30/0.62/0.08	400	11.95 $\times$ 11.95	0.95	1000

Calculation of the NAP: ${\chi }^2$

Membrane order has traditionally been characterized in terms of the area per lipid, thickness, and tail order parameter (S_CD) (58,61,62). However, the intrinsic fluidity of the membrane and its dynamic nature, wherein lipids constantly enter and exit ordered/disordered domains, necessitate the inclusion of information on the local lipid environment and its temporal evolution in the characterization. In this work, we quantify membrane orderliness in terms of the NAP. The nonaffine deformation framework, which we follow here, was originally used by Falk and Langer to identify regions of irreversible plastic deformation in 2D sheared amorphous systems (105). In our previous studies (62,88), we adopted this framework to identify locally ordered and disordered regions in lipid membranes exhibiting pure and coexisting liquid phases, and the method to calculate the NAP is thoroughly discussed in these studies. Given the importance of this analysis in the present work, the calculation of the NAP is comprehensively summarized here.

Herein, the nonaffine deformation analysis is performed on atomistic or CG trajectories from MD simulations, i.e., on a lipid membrane evolving in space and time. One leaflet of the membrane is considered at a time and a reference site is identified on every lipid in the leaflet, whose evolution is tracked over time. In our case, unless otherwise noted, the reference sites are identified as the bottom carbon atoms of the glycerol groups for lipids and the hydroxyl group oxygen atom for cholesterol. Next, a neighborhood is defined around each lipid within a cutoff radius $\mathrm{\Omega }$ (see Fig. 1 a). The N lipids surrounding the central lipid (with coordinates $r_0$) within this cutoff radius are considered as its neighboring lipids (with coordinates $r_{n\in N}$) and define its local topology. Unless specified otherwise, the radius $\mathrm{\Omega }$ is taken to be 14 Å in our calculations. The NAP is calculated for each lipid taking one simulation snapshot (at time t) as the reference configuration and the next snapshot (at time $t+\Delta t$) as the deformed configuration, where $\Delta t$ refers to the interval at which simulation snapshots are stored in the MD simulation. Mathematically, the NAP is denoted as ${\chi }^2$ and is calculated for a set of reference and deformed configurations as

$${\chi }_{\mathrm{\Omega }}^2\left(t,t+\Delta t\right)=\frac{1}{N}\sum _{n=1}^N\sum _{i=1}^d{\left(\left[r_n^i\left(t+\Delta t\right)-r_0^i\left(t+\Delta t\right)\right]-\left[\sum _{j=1}^d\left({\delta }_{ij}+{\epsilon }_{ij}\right)\times \left(r_n^j\left(t\right)-r_0^j\left(t\right)\right)\right]\right)}^2\text{,}$$ (1)

where i and j indicate the spatial indices of positions r of the lipid reference sites with dimension d, n runs through the N lipids within the neighborhood $\mathrm{\Omega }$ around the central lipid $n=0$ (Fig. 1 a), and ${\delta }_{ij}$ denotes the Kronecker delta function. The term ${\epsilon }_{ij}$ denotes the strain associated with the maximum possible affine part of the deformation and therefore minimizes ${\chi }_{\mathrm{\Omega }}^2\left(t,\Delta t\right)$, and is calculated as

$${\epsilon }_{ij}=\sum _{k=1}^d{X}_{ik}{Y}_{jk}^{-1}-{\delta }_{ij}\text{,}$$ (2)

where

$$X_{ij}=\sum _{n=1}^N\left[r_n^i\left(t+\Delta t\right)-r_0^i\left(t+\Delta t\right)\right]\times \left[r_n^j\left(t\right)-r_0^j\left(t\right)\right]\text{,}$$ (3)

and

$$Y_{ij}=\sum _{n=1}^N\left[r_n^i\left(t\right)-r_0^i\left(t\right)\right]\times \left[r_n^j\left(t\right)-r_0^j\left(t\right)\right]\text{.}$$ (4)

In Eq. 1, the first term in square brackets denotes the relative displacements of lipids within the neighborhood $\mathrm{\Omega }$ w.r.t. the central lipid $n=0$ at time $t+\Delta t$. The second term in square brackets denotes the relative displacements that would have resulted if this neighborhood was deforming under a uniform strain field ε. Consequently, the NAP, ${\chi }^2$, denotes the mean-square deviation between these two displacements, in units of Å${}^2$. For an affine deformation such as a translation, scaling, or shear, which can be associated with a uniform strain, ${\chi }^2$ is exactly zero and any deviation from an affine deformation will result in a nonzero ${\chi }^2$ (see the supporting material in (62) for a few simple examples). For an arbitrary deformation, it is reasonable to divide the same to the maximum possible affine part and a residual (minimum possible) nonaffine part. The NAP encodes this residual nonaffine content of an arbitrary deformation over time interval $\Delta t$, where a higher value of ${\chi }^2$ indicates a larger deviation from an affine deformation, i.e., a larger nonaffine content in the deformation. In a lipid membrane the NAP essentially characterizes how the local topology of lipids, i.e., the local lipid packing, evolves over a time interval. A small ${\chi }^2$ value indicates correlated evolution of the central lipid in its local environment, where all the lipids in the neighborhood have limited displacements, i.e., a tighter packing. In contrast, a large ${\chi }^2$ value indicates uncorrelated lipid evolution, where the neighboring lipids can have large-scale displacements and, thus, a less tighter packing. Such disparate dynamics leads to the formation of distinct domains in mixed phase membranes. The NAP thus characterizes the local lipid packing and, consequently, the local order.

Figure 1.

Calculating the NAP and identifying the packing defects. (a) The NAP (${\chi }^2$) is calculated using the spatial coordinates of lipid reference sites $r_n$ within a cutoff $\mathrm{\Omega }$ around the central site $r_0$ at times t and $t+\Delta t$. Schematic showing the lipid reference sites with red and green indicating different kind of lipids. (b) Schematic indicating the three-dimensional packing defects in the membrane as blue/while shaded regions, extracted from the grid-based defect analysis that specifically excludes the reference sites. Note that, while ${\chi }^2$ does not require any information on lipid identity (chemistry), the same enters in the defect analysis through the lipid topology and the van der Waals radii of lipid atoms. To see this figure in color, go online.

In contrast to traditional analyses such as the mean-square displacements or S_CD, which utilize the information on only the temporal evolution or the spatial configuration, respectively, of individual lipids, the calculation of the NAP requires the information on lipids’ neighborhoods and their evolution over a time $\Delta t$. This way, ${\chi }^2$ incorporates the information on the spatiotemporal evolution of lipids in their local environments. The NAP was shown to be able to capture the distinct nature of the spatiotemporal evolution of lipids in pure phase ($L_o$ or $L_d$) and mixed phase ($L_o/L_d$) membranes (62). Furthermore, it could also identify local molecular-scale heterogeneities in pure phase membranes indicating regions that undergo more heterogeneous topological rearrangements than their neighborhoods (62,88) and, thus, is a sensitive marker of membrane order. Finally, except for defining the reference sites, lipid chemistry has no role in the calculation of the NAP and therefore allows us to compare the results across membranes with distinct lipid chemistry.

Considering each lipid on a membrane leaflet as a center and a neighborhood $\mathrm{\Omega }$ around it, Eq. 1 results in a set of ${\chi }^2$ values for each of these lipids for a pair of reference and deformed configurations ($t,t+\Delta t$). By considering each snapshot in the trajectory as the reference and the subsequent snapshot as the deformed configuration, repeating the analysis results in a set of ${\chi }^2$ values for each lipid at every snapshot of the chosen analysis window. This set is used to calculate the distribution of ${\chi }^2$ values over the analysis window. To identify local heterogeneities in lipid ordering and their correlation to lipid packing, we also analyze ${\chi }^2$ spatial maps, as discussed later in the text.

In the NAP analysis, the values for the two free parameters, $\Delta t$ and $\mathrm{\Omega }$, are chosen based on the specific membrane system and the trajectory. Evidently, a larger value of $\Delta t$ will lead to larger relative displacements of lipid reference sites and, thus, a larger ${\chi }^2$ value. For the analysis of MD simulation trajectories, the interval between subsequent system snapshots serves as the best choice for $\Delta t$ (62). Similarly, a larger $\mathrm{\Omega }$ will result in a larger N and therefore a larger ${\chi }^2$ value. Ideally, the value of $\mathrm{\Omega }$ should be large enough to include enough neighboring lipids so as to precisely capture the local topology of the central lipid. At the same time, it should be small enough to capture the subtle differences in the local topology of individual lipids in a single membrane (62). Equation 1 differs from our previous implementations of the NAP (62,88) by the factor $1/N$. Here, we normalize ${\chi }^2$ by the total number of lipids within the neighborhood N, so as to incorporate the effect of environment in mixed phase membranes, which can be locally distinct. This also allows us to compare the results across membranes exhibiting various phases ($L_o$, $L_d$, or mixed) and sizes (see Table 1). Further details on using the NAP to characterize membrane local order can be found in our previous studies (62,88) and the source code to calculate the NAP is available on the Github repository https://github.com/codesrivastavalab/DegeneracyBiologicalMembraneNanodomains/tree/master/codes-degeneracy-membrane-nanodomains (27).

Identifying and analyzing 3D packing defects

Packing defects on the lipid membranes surface (i.e., the interface between the membrane and the aqueous medium) indicate the regions around the hydrophobic lipid tails that get transiently exposed to the aqueous medium; thus also aptly known as hydrophobic defects. We identify the lipid packing defects based on our 3D defect algorithm (71), which employs a grid-based free volume analysis framework. Defect analysis on lipid membranes have traditionally been performed in two dimensions (70,72,106) due to the computational bottleneck associated with grid-based free volume analysis, where each grid point is compared with each atom in the system to check for overlaps. To circumvent this, we employ an algorithm, which, instead of mapping the whole membrane to a global grid, constructs a small local grid around individual atoms and only compares the central atom with its neighboring local grid points. The algorithm, and its usage and merits are thoroughly discussed in the original study with ample examples (71). Below we provide a comprehensive description of the method for completeness.

We begin by translating the membrane coordinates if necessary, such that the membrane lies on the xy plane and the membrane normal is along the z direction. We define a z cutoff and only consider those atoms for the defect calculations that lie above this cutoff. The z cutoff is chosen based on the membrane system under consideration and the defects of interest. For example, to analyze the surface defects (i.e., the interfacial defects that lie at the membrane-water interface) we consider one leaflet of the membrane at a time and therefore, the z coordinate of the average membrane midplane serves as a good choice for the cutoff. Subsequently, the 3D defect algorithm is used on this selection of (heavy) atoms. For each atom in the selection, a local grid (with resolution of 1 Å) is constructed around it and grid points are compared for overlap to identify solvent-exposed regions around the lipid tails. In our calculations, the tail regions are identified as lipid atoms that lie below the z coordinates of lipid reference sites (the bottom carbon atom of the glycerol group for lipids and the hydroxyl group oxygen atom for cholesterol) (see Fig. 1 b). For overlap calculation, we use a similar approach as that of the “rolling probe method” (107) used for calculating the solvent-accessible surface area. For this, we use the van der Waals radii of the atoms to define their sizes and a probe radius of 1.4 Å corresponding to the radius of a water molecule. These two radii together define the extent of the local grid around each atom.

Using the 3D defect algorithm on the selection of atoms at every snapshot of the trajectory, we identify grid points, not occupied by the lipid atoms, that lie near the exposed hydrophobic tails of the lipids. These defect grid points, thus, indicate regions on the lipid membrane surface that are accessible to the solvent (water). Next, we group the contiguous defect grid points into a single defect (i.e., a defect pocket) by means of a distance-based clustering algorithm in which defect grid points are grouped together if they lie within a diagonal distance from each other (71). In this way we identify all surface defects at any given snapshot in the analysis window. In 2D defect analysis (106), the size of defects are characterized by their area. In our analysis, we characterize the size (or volume) of a defect in terms of the number of contiguous grid points comprising it, to avoid any assumption on the shape of the 3D defects. Given the two quantities, the exact defect volume and the number of grid points within it, are related through mathematical constants, the number of grid points serves as a reasonable quantification of the defect size. The ultimate quantities of interest are the distribution of the defect sizes, which is a histogram of defect sizes (number of grid points in a defect) and the average (not normalized) probability of their occurrence over the analysis window of the trajectory. We also analyze the defect spatial maps to investigate the persistence (life time) of defects as discussed later in the text.

The 3D defect algorithm also allows us to identify core defects, i.e., the free volume within the membrane core. In such a case, two z cutoffs are chosen to define a slab around the membrane midplane. The defect algorithm is used to identify and analyze free volumes within this membrane slab. Such core defects and their role in membrane permeation are discussed in details in later sections.

Further details on the 3D defect algorithm, the choice of free parameters and their significance, and its usage are discussed in the original reference (71). The code to identify the defect grid points in three dimensions is available on the Github repository https://github.com/codesrivastavalab/3DLipidPackingDefects along with a minimal example.

Results

Local orderliness dictates the defect profiles in mixed phase membranes

In this work, we analyze membrane order in terms of the NAP, which characterizes membrane orderliness at a local level. Membrane order is characterized by lipid packing (denser packing in $L_o$ $\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{u}\mathrm{s}$ less denser packing in $L_d$ membranes (37)) and, thus, is also expected to be intrinsically coupled to the nature of packing defects in the membrane. Characterizing local order in terms of the NAP is helpful toward corroborating this coupling. Our earlier works on pure phase membrane systems indicated that, irrespective of lipid chemistry, lipids in pure $L_o$ membranes exhibit consistently low ${\chi }^2$ values compared with those in pure $L_d$ membranes (62,88), suggesting that local lipid environments (topology) evolve very differently in these membranes, which ultimately results in distinct lipid packing. Tighter packing of lipids can lead to correlated lipid evolution, resulting in low ${\chi }^2$ values (i.e., more ordered domains) and, consequently, minimal solvent exposure of lipid tails, i.e., lesser packing defects. In contrast, less denser packing can allow uncorrelated lipid evolution, higher ${\chi }^2$ values (i.e., less ordered domains) and, therefore, larger/denser distribution of packing defects. Indeed, we observed a few generic features in these membranes in terms of the packing defects, where $L_o$ membranes exhibited a sparser spatial distribution (see Fig. S1) of defects, which were comparatively smaller and shallower compared with those in $L_d$ membranes (71,88). Moreover, in our previous work (88), we noted some preliminary visual observations on the correlation between membrane orderliness and the defect profile. This correlation can be an important factor governing the domain organization in heterogeneous (mixed phase) membranes and their functions, such as preferential peptide localization. Herein, we aim to investigate and formalize this correlation in heterogeneous (mixed phase) membranes toward which we use the observations from homogeneous (pure phase) membranes as references.

We begin by analyzing the spatial correlation between membrane local order and packing defects. Toward this, we compare the ${\chi }^2$ spatial maps against the defect spatial maps in Fig. 2 for the three mixed phase ($L_o/L_d$) membranes listed in Table 1. To generate the ${\chi }^2$ spatial maps, we consider a set of 10 consecutive system snapshots and calculate the ${\chi }^2$ values for all the lipid reference sites on one leaflet over this window. Next, we average the ${\chi }^2$ values over the 10 snapshots to get the averaged ${\chi }^2$ value for each lipid site. Finally, we plot the spatial coordinates of the lipid reference sites using the middle (sixth) snapshot of the chosen set in a scatter plot and color each lipid based on its average ${\chi }^2$ value to generate a heat map. The resulting ${\chi }^2$ spatial maps for the three $L_o/L_d$ membranes are shown in Fig. 2 a. Here, the number of data points in each subplot is equal to the number of reference sites on the corresponding membrane leaflet under consideration, and x and y indicate the membrane size in two dimensions. The same set of 10 consecutive system snapshots are used to generate the defect spatial maps shown in Fig. 2. The defect grid points are identified at each of the 10 snapshots, which are then binned in a 2D histogram along x and y coordinates with bin size 1 Å. Finally, the histogram is averaged over the 10 snapshots resulting in $P\left(x,y\right)$, which indicates the probability that a grid point $\left(x,y\right)$ belongs to a defect. A higher probability indicates both the spatial localization and temporal persistence of the defect, i.e., the defect stays in the same position over the set of snapshots. The defect spatial maps for the three membranes are shown in Fig. 2 b, whereas in Fig. 2 a, x and y indicate the membrane size. As packing defects are highly transient in nature, they are expected to disperse over a longer analysis window. Therefore, here we use an analysis window of 10 consecutive snapshots, each taken 240 ps apart, amounting to 2.4 ns simulation time. This window is small enough to distinguish between localized and dispersing defects and large enough to correlate the defect profile and membrane local order.

Figure 2.

Spatial correlation of ${\chi }^2$ values and packing defects. (a) ${\chi }^2$ spatial maps and (b) defect spatial maps for DPPC/DOPC/CHOL (left), PSM/DOPC/CHOL (middle), and PSM/POPC/CHOL (right) systems exhibiting mixed $L_o/L_d$ phases. Membrane regions exhibiting low ${\chi }^2$ values are observed to be relatively defect free. To see this figure in color, go online.

In Fig. 2, we observe that in all three membranes, regions possessing low ${\chi }^2$ value, i.e., the ordered domains, are mostly defect free, while disordered domains with high ${\chi }^2$ values exhibit denser distribution of defects. Moreover, large (extended in x and y dimensions) or persistent (high probability, $P\left(x,y\right)$) defects can almost always be mapped to reference sites with high ${\chi }^2$ values. Given that these observations are valid for all three systems, this can be considered as a general feature of all mixed phase systems where the ordered domains remain relatively defect free compared with the disordered domains. It, thus, appears that the observations on the nature of defects in pure phase membranes, where pure $L_d$ membranes exhibit a denser distribution of large defects compared with pure $L_o$ membranes, can be straightforwardly extended to mixed phase membranes.

To investigate this further, we compare the defect size distributions for the three mixed phase membranes to their pure phase counterparts. As discussed in the materials and methods, the defect size is quantified as the total number of defect grid points in a defect and the distribution indicates the average probability of occurrence (averaged over the analysis window) of such a defect. For pure phase membranes, the defect size distribution follows a generic trend, irrespective of lipid chemistry: $L_o$ membranes always exhibit smaller defects compared with their $L_d$ counterparts, as shown in Fig. 3 a. However, the distributions of defect size for the three mixed phase systems are rather distinctive compared with their pure phase membrane counterparts: in the case of DPPC/DOPC/CHOL, the distribution is very similar to its $L_d$ counterpart, for the PSM/POPC/CHOL system it is comparable with the corresponding $L_o$ one, and for the PSM/DOPC/CHOL system it is found to be intermediate to the corresponding two pure systems. Thus, the mixed phase membranes lack a distinctive trend in terms of the defect size compared with the pure phase membranes. The origin of this observation again can be related to the orderliness of the membrane, characterized in terms of the ${\chi }^2$ distribution as shown in Fig. 3 b. Pure phase $L_o$ systems exhibit sharper and narrower distributions of ${\chi }^2$ values compared with the corresponding $L_d$ ones, which is a generic feature of pure phase (homogeneous) membranes (62,71,88). Interestingly, ${\chi }^2$ distributions for the three mixed phase systems follow the exact same trend as the defect size distributions, indicating the mixed phase DPPC/DOPC/CHOL membrane to be more disordered-like and the mixed phase PSM/POPC/CHOL membrane to be more ordered-like. The mixed phase PSM/DOPC/CHOL membrane is found to be intermediate to the two pure systems. The distinct local order in the three mixed phase membranes thus leads to the characteristic defect size distributions therein, corroborating the correlation between membrane local order and defect profiles.

Figure 3.

Overall correlation between the ${\chi }^2$ values and defects. Distributions of (a) ${\chi }^2$ values and (b) defect size for DPPC/DOPC/CHOL (left), PSM/DOPC/CHOL (middle), and PSM/POPC/CHOL (right) systems exhibiting pure and mixed liquid phases. The range of axes are kept the same for better comparison. To see this figure in color, go online.

This apparent correlation between ${\chi }^2$ values and the packing defects can be formalized in terms of a 2D probability distribution P($n_d$, $\sum {\chi }^2$), which indicates the probability that a reference site (on a lipid) has “$n_d$” number of total defect grid points around it within a cutoff radius r (here taken to be 14 Å) with a cumulative ${\chi }^2$ value of $\sum {\chi }^2$. To compute P($n_d$, $\sum {\chi }^2$), we calculate the ${\chi }^2$ values for each lipid reference site on a membrane leaflet at a given system snapshot at time t (i.e., using the two snapshots at time t and $t+\Delta t$). Taking each of the lipid as the center, we add the ${\chi }^2$ values of all lipids around it within a radius r and denote it as the $\sum {\chi }^2$ value of the central lipid. Next, we identify the defect grid points at the same snapshot (at time t) and sum the defect grid points that lie within the radius r around the central lipid, which we denote as $n_d$. At any time t, we thus identify each lipid on a membrane leaflet with two quantities: $n_d$ and $\sum {\chi }^2$. These quantities are summed to address two important points. The first is the absence of a one-to-one spatial correlation between the two quantities: ${\chi }^2$ is calculated based on lipid (reference site) coordinates, while defect is a grid-based analysis that specifically excludes these sites. The second is to incorporate the effect of neighborhood, which can be a determinant factor when comparing across different mixed phase membranes. We compute $n_d$ and $\sum {\chi }^2$ values for all lipids on the leaflet for all the snapshots in the analysis window. Each of these data points (total = number of lipids on the leaflet $\times$ number of system snapshots in the analysis window) is binned along $n_d$ and $\sum {\chi }^2$ axes, with bin sizes of 1 and 2.0 Å${}^2$, respectively, to generate the 2D histogram P($n_d$, $\sum {\chi }^2$), which is averaged over the analysis window and number of lipids on the leaflet. High-probability regions in such a 2D histogram can indicate correlation between the two quantities, $n_d$ and $\sum {\chi }^2$ In Fig. 4, we present this correlation for the three mixed phase membrane systems. We already observed in Fig. 3 that packing defects are indeed correlated with membrane local order, assessed in terms of the NAP. Therefore, for a strictly one-to-one relationship between the two quantities, one can expect data points that are populated along a straight line with a positive slope. However, given that dynamics of individual lipids in a mixed phase membrane is delicately connected to its local environment, we can expect a moderately linear relationship with data points smeared around the straight line. For a pure phase $L_o$ system, with correlated lipid evolution, the data points in ($n_d$, $\sum {\chi }^2$) correlation are observed to exhibit a narrow range around the diagonal with a further narrow region exhibiting high P($n_d$, $\sum {\chi }^2$) values near the origin (high order and less defects), as shown in Fig. S2. In contrast, a pure phase $L_d$ system, wherein lipid evolution is not well coordinated, ($n_d$, $\sum {\chi }^2$) correlation is found to be much weaker with scattered data set and an extended region exhibiting high P($n_d$, $\sum {\chi }^2$) values farther away from the origin along the $\sum {\chi }^2$ axis, indicating lower order (Fig. S2). The profile of ($n_d$, $\sum {\chi }^2$) correlation in a mixed phase membrane can, thus, indicate the global order of the membrane. As observed in Fig. 4, the mixed phase DPPC/DOPC/CHOL system exhibits the characteristics of a pure phase $L_d$ system and the mixed phase PSM/POPC/CHOL system resembles a pure phase $L_o$ system. For the PSM/DOPC/CHOL system, with very different ${\chi }^2$ distributions for the two pure and mixed membranes (Fig. 3), the ($n_d$, $\sum {\chi }^2$) correlation exhibits characteristics of both pure phases: a wider range (as expected for a pure $L_d$ system), but a narrow region close to the origin exhibiting high probability (as expected for a pure $L_o$ system). Mixed phase membranes exhibiting a weak correlation, similar to the DPPC/DOPC/CHOL system here, are disordered-like and should be dominated by disordered domains. These $L_d$ domains with larger and denser packing defects can serve as binding sites for peripheral proteins. In contrast, membranes which exhibit a strong correlation, similar to the PSM/POPC/CHOL system here, are more ordered-like and should be dominated by ordered domains. In such a scenario, proteins binding in these systems should also happen in these $L_o$ domains. A quite relevant example is the case of GPMVs, which are relatively more ordered than GUVs and show protein binding onto these ordered domains (37).

Figure 4.

A one-to-one map. Defect and ${\chi }^2$ correlation map for (a) DPPC/DOPC/CHOL, (b) PSM/DOPC/CHOL, and (c) PSM/POPC/CHOL systems, each exhibiting mixed $L_o/L_d$ phase. The color bars indicate the value of probability P($n_d$, $\sum {\chi }^2$). The range of axes are kept the same for better comparison. To see this figure in color, go online.

The three mixed phase membranes studied here cover the set of all three possible cases, where a mixed phase membrane is more ordered-like, disordered-like, or intermediate. Any mixed phase membrane with distinct lipid chemistry will fall into one of these three possible cases. It should be noted that, at a given lipid composition, the ratio of ordered to disordered domains and the area fraction of domain boundary will affect the overall membrane order, which will be primarily determined by the dominant phase (see Fig. 2). Consequently, the strength of the correlation between membrane defect profile and membrane local order will be affected by this dominant phase. Despite this, the observations here indicate the molecular level connection between membrane defects and local order and can have important implications in the functionality of the membranes domain; for example, proteins binding in $L_o$ or $L_d$ domains. Finally, while finite size effects cannot be ruled out in atomistic simulation of membranes, the qualitative trends remain credible.

Defects in/around $L_o$ domains exhibit higher persistence than those in $L_d$ domains

To investigate the functional consequence of such distinct characteristics of mixed phase membranes, we investigate the time evolution of defects therein. Fig. 5 shows the defect spatial maps for the three mixed phase membranes, calculated over 2, 10, and 20 consecutive snapshots, with the snapshots taken at 240 ps time intervals. Here, the middle column, corresponding to analysis over 10 snapshots is the same as Fig. 2 b. A high probability in the map indicates both the spatial and temporal persistence of a defect, i.e., a defect that is localized on the membrane surface over the analysis window. As expected, the majority of the defects are found to be in the disordered domains (see Fig. 2). However, the ordered domains also exhibit a significant amount of defects, including large ones. Moreover, the large defects in (e.g., in the PSM/DOPC/CHOL and PSM/POPC/CHOL systems) and around (e.g., in DPPC/DOPC/CHOL) the ordered domains are found to be significantly localized and persist over 20 snapshots, i.e., 4.8 ns (highlighted in Fig. 5 with circles). This is in stark contrast to the defects in the disordered domains, which quickly disperse over time. Such nanosecond long persistent, localized defects in $L_o$ domains is a remarkable observation, since defects are quite transient owing to the fluidity of the membrane and the thermal fluctuations.

Figure 5.

Defect spatial and temporal localization. Defect spatial map of mixed phase (a) DPPC/DOPC/CHOL (top row), (b) PSM/DOPC/CHOL (middle row), and (c) PSM/POPC/CHOL (bottom row) systems calculated over 2 (left column), 10 (middle column), and 20 (right column) snapshots. Note that the middle column, corresponding to analysis over 10 snapshots is the same as Fig. 2b. The cyan circles on the right column indicate the persistent defects in and around the $L_o$ regions. To see this figure in color, go online.

These observations point us to an important inference: while the disordered domains are rich in large packing defects, the most stable defects exist in and around the ordered domains. This is a consequence of the nature of lipid packing in ordered domains, where the spatiotemporally correlated evolution of lipids leads to the localization of the packing defects in and around them that can persist over a few nanoseconds. Irrespective of the membrane composition and whether the mixed phased membrane is more ordered- or disordered-like, this is a common feature for all three mixed phase membranes studied in this work. Based on the intrinsic correlation between lipid packing and membrane defects, this might as well be a generic feature of mixed phase membranes. And likely very important for formation of early encounter sites for peripheral protein binding—especially those that are eventually stabilized due to hydrophobic insertion in the membrane.

Local membrane order governs the partitioning of membrane-active peptides

Here, we use tLAT as a paradigmatic peptide to explore the molecular origin of peptide localization in a bilayer with phase coexistence and use this example to explore the effect of posttranscriptional modification on this localization. Lin et al. (60) investigated the role of palmitoylation in the partitioning of tLAT in the DPPC/DAPC/CHOL (5:3:2) membrane exhibiting mixed $L_o/L_d$ phase using atomistic and CG MD simulation. In both cases, they observed the tLAT to partition at the domain boundary between the $L_o$ and $L_d$ domains, irrespective of the palmitoylation state. Such an observation is in contrast to those reported in experiments, wherein tLAT is shown to partition with $L_d$ domains in GUVs (87) and $L_o$ domains in GPMVs (86). While palmitoylation is implicated in the ordered phase affinity (85), the apparent discrepancy between the observations in GUV and GPMV stem from the relatively strong and weak contrast, respectively, between the raft-like and nonraft domains in the two cases (37). To further investigate the origin of this apparent discrepancy, Lin et al. (60) analyzed the energetics of the lipid-protein interaction in atomistic simulation. Considering DPPC and cholesterol as $L_o$ domain lipids and DAPC as $L_d$ domain lipids, they computed the residue-wise peptide-lipid interaction energy and observed that palmitoylated tLAT exhibited higher interactions with the $L_o$ domain lipids, while tLAT without palmitoylation interacted strongly with the $L_d$ domain lipids, in line with experiments. Finally, they confirmed the order domain preference of palmitoylated tLAT in a CG umbrella sampling simulation of a single peptide in a CG membrane model of an $L_o$ domain sandwiched between two $L_d$ domains, where the center of mass distance between the peptide and the $L_o$ domain was chosen as the reaction coordinate.

Here, we revisit the study by Lin et al. to investigate the order preference of tLAT, where membrane order is now characterized in terms of the nonaffine deformation measure, i.e., the NAP. Our previous studies on both pure phase and mixed phase membranes (62,88) have shown that membrane order is dictated by the local lipid packing and their dynamic topological evolution, rather than their specific chemistry. Therefore, there is no a priori reason to assume that all saturated lipids and cholesterol belong to the $L_o$ domain and all unsaturated lipids to the $L_d$ domain. Toward this, we analyzed the atomistic trajectories of DPPC/DAPC/CHOL membranes exhibiting mixed $L_o/L_d$ phase used by Lin et al. (60). The system consisted of 200 DPPC lipids, 120 DAPC lipids and 80 cholesterols (200 lipids per leaflet) with a single tLAT with and without palmitoylation (see Table 2). While palmitoylation, in general, is known to enhance the ordered phase affinity of tLAT (37,85), palmitoylation of a single tLAT cannot significantly affect the global membrane order, which in this case is heterogeneous or mixed phase. Nonetheless, the local environment of tLAT in the two cases should bear signature of the order preference of the palmitoyl group. To quantify this, we characterized the two tLAT-membrane systems in terms of the ${\chi }^2$ parameter, which, as we have shown earlier, is a sensitive marker of membrane order.

For the ${\chi }^2$ analysis, we considered two reference sites for the lipids (C27 and C37 tail carbon atoms for DPPC and DAPC) and one reference site for cholesterol (C9 atom). We considered 4 to 6 reference sites for tLAT with and without palmitoylation that lie close to the lipid reference sites in terms of their z coordinates. Together, we had around 365 reference sites for the C- and N-terminus side of the bilayer in both cases. We computed the ${\chi }^2$ values for each reference site using a neighborhood cutoff $\mathrm{\Omega }$ = 18 Å (see materials and methods). These values indicate the distinct spatiotemporal evolution of the lipids and tLAT in the membrane. To understand how distinct the local topological evolution is for a single tLAT with and without palmitoylation, we need to compare this metric in a local environment around the tLAT. Taking the average coordinate of the 4 to 6 reference sites of tLAT as the center, we subsequently averaged the ${\chi }^2$ values of all lipid and cholesterol reference sites within a radius r = 10 Å around it. The resulting quantity ${⟨{\chi }^2⟩}_r$ provides a quantitative measure of the average orderliness of the local membrane environment around tLAT. The choice of the cutoff radius is motivated by the fact that it should be large enough to include several neighboring lipid reference sites, while small enough to represent the local environment of tLAT. Given that we are interested in the changes in lipid packing around a single tLAT with palmitoylation, such information will be lost with a very large r. Finally, we calculated the distribution of ${⟨{\chi }^2⟩}_r$ values for the two leaflets of the bilayer and subsequently averaged them over the analysis windows of the trajectories to get the final probability distribution of ${⟨{\chi }^2⟩}_r$ for tLAT with and without palmitoylation, which is compared in Fig. 6. The distribution of ${⟨{\chi }^2⟩}_r$ clearly indicates the local environment of palmitoylated tLAT to be relatively more ordered than that of tLAT without palmitoylation. Thus, the nonaffine deformation-based analysis is sensitive enough to distinguish between the local environment of a single tLAT peptide, without any previous assumption or knowledge of lipid chemistry or membrane composition.

Figure 6.

Local order of membrane around tLAT with and without palmitoylation. Distributions of ${\chi }^2$ values averaged within a local neighborhood of radius 10 Å around tLAT and palmitoylated tLAT. The lines are guides to the eyes. To see this figure in color, go online.

Membrane defect profile governs the permeability of small molecules

The defect size distributions for pure phase DPPC/DOPC/CHOL membranes shown in Fig. 3 are computed using trajectories that Ghysels et al. (89) used to analyze water permeation through the membrane and observed a similar permeation mechanism as that of oxygen. As evident, the pure $L_d$ membrane exhibits much larger packing defects than the pure $L_o$ counterpart, which, as discussed earlier, is a generic property of pure phase membranes irrespective of the membrane composition and lipid chemistry. In the context of penetrant permeation through the membrane, these observations would indicate the abundance of permeation channels in the $L_d$ membranes, while the $L_o$ membranes remain sparse. This is in line with the observations from Ghysels et al. that $L_d$ membranes are statistically more permeable. However, the disparity between the lateral and transverse diffusion profile of water/oxygen demand a closer inspection.

To understand this, we further analyzed the oxygen permeation trajectories of Ghysels et al. (89) (pure $L_o$ and $L_d$ phase DPPC/DOPC/CHOL membranes with O${}^2$, see Table 3) in terms of the packing defects. As in the case of water permeation trajectories, defects on a leaflet in the $L_d$ system were larger in size compared with the $L_o$ one (Fig. 7 b). Subsequently, we identified two more kinds of defects following our 3D defect algorithm (71): surface and core defects. The core defects were identified for membrane slices of thickness of 14 and 8 Å around the membrane midplane, while the surface defects were identified on one leaflet of the membrane excluding this core region (Fig. 7 a). Thus, the surface defects basically are similar to the hydrophobic defect on one leaflet, but now identified on a slightly thinner section of the leaflet. The core defects indicate the free volume in the membrane midplane and are identified as the grid points that are not occupied by the lipids or cholesterol. The surface defects can be interpreted as the free volume available on the membrane surface (in lateral directions) and the core defects as the free volume in the membrane core, which has long been implicated in lipid diffusion (68,92,95,96). From the methodology point of view, it is worth mentioning that the defects or free volume in our analysis correspond to the accessible surface areas, which are calculated using a rolling probe-like algorithm with a probe radius of 1.4 Å (roughly the radius of a water molecule) (71). While this is technically justified for the surface defects, which are indeed accessible to water, the free volumes in the core should not necessarily be accessible to water unless they are connected to surface defects. In such a case, using the same probe radius allows for a fair comparison between the two kinds of free volume while providing a conservative estimate of the same. Moreover, the use of periodic boundary condition (along x and y directions) is avoided while calculating defect sizes in both cases without the loss of generality to avoid statistical implications of comparing defects in a thin slice of membrane to those on the surface. While both these points can lead to under prediction of defect sizes, unintended overpredictions (which can also be misleading) can be avoided.

Figure 7.

Role of defects/free volume in membrane permeability. (a) The definition of various kinds of defects analyzed in the DPPC/DOPC/CHOL membranes containing O${}^2$ and exhibiting pure $L_o$ and pure $L_d$ phases. One lipid from each kind is shown in van der Waals representation against all others in line representation (DPPC, green; DAPC, gray; CHOL, red). The membrane snapshot has been rendered using the visualization tool VMD (111). Distributions of defect size for: (b) defects on one leaflet, (c) surface defects, and core defects within a slab of thickness (d) 14 Å and (e) 8 Å around the membrane midplane. To see this figure in color, go online.

Fig. 7c–e show the size distributions of these defects in the two pure phase membranes. Similar to defects on a single leaflet (Fig. 7 b), the $L_d$ system was found to exhibit larger surface packing defects compared with the $L_o$ one (Fig. 7 c), implying more permeation channels and stronger transverse diffusion in the $L_d$ system. Together, this can result in significantly larger permeability compared with the $L_o$ system, as reported by the authors. Interestingly, the trend is reversed for the core defects: the $L_o$ system exhibits larger core defects than the $L_d$ system (Fig. 7 d). Furthermore, as the thickness of the membrane slice is decreased from 14 to 8 Å, the distinction between the size distributions for core defects remains apparent (Fig. 7 e), indicating that the membrane midplane of the $L_o$ system has relatively more free volume compared with the $L_d$ system. This can be attributed to the fact that lipids in the ordered subdomains in the $L_o$ phase are closely packed in the transverse direction. This circumvents interdigitation of the lipid tails between the leaflets, which can allocate substantial amounts of free volume in the midplane region. This would provide an ideal path for the penetrants to diffuse along the membrane midplane through the isolated free volumes that evolve over time: an observation that provides a possible explanation on why the authors observe stronger lateral diffusion in the $L_o$ systems. The trend reversal between surface and core defect size can lead to the anisotropy in the diffusion profile between the two phases, as reported by the authors (89).

As discussed earlier, there exists a direct correlation between membrane order and packing (here, free volume), irrespective of the membrane composition. It is, therefore, evident that the permeation profile for small molecules in other model membranes should follow the same characteristic: stronger transverse diffusion in $L_d$ membranes versus stronger lateral diffusion in $L_o$ membranes. This is indeed what Ghysels et al. (89) observed in terms of free energy profiles and nearest-neighbor analysis of water permeation in homogeneous DPPC/DOPC/CHOL, PSM/DOPC/CHOL, and PSM/POPC/CHOL membranes.

Discussion

It is evident that the relationship between membrane order and defects can be quantified as to be moderately linear. A stricter correlation indicates the mixed phase membrane to be more ordered-like with spatiotemporally correlated lipid evolution and smaller packing defects. In contrast, a moderate correlation indicates the mixed phase membrane to be more disordered-like with uncorrelated lipid evolution and larger packing defects. This correlation can govern various functional aspects of the raft-like domains including protein adsorption and membrane permeation.

While most of the peripheral proteins bind to the $L_d$ domain (108,109), the specificity of defect profiles in mixed phase membranes suggest that, in a membrane that is more ordered-like, binding should also happen in the $L_o$ domain. This is also in line with the experimental observation that some proteins (e.g., tLAT), which exclusively bind to $L_d$ domains in GUVs, can also bind to $L_o$ domains in GPMV, wherein the demarcation between the two domains is not too strong (37).

Furthermore, the distinctive time evolution of surface defect profiles, as indicated in Fig. 5, can lead to two possible routes for a binding event (see the schematics in Fig. 8). In case I, a protein can sample the membrane surface and bind to a transient, but large surface defect in the $L_d$ domain. In case II, a protein can bind to a localized, but relatively smaller defect in or around the $L_o$ domain. The two binding routes can be distinguished in terms of a number of features, the first being the timescales of binding, i.e., the time taken by the protein to bind to the membrane. This timescale should be longer in case I as the protein has to extensively sample the membrane surface to bind to large defects that are already diffusing continuously. Accordingly, the strength of the membrane-protein interactions should be stronger in case II than in case I. Finally, the mechanism of defects coalescing upon protein binding should also be different for the two cases. The binding of APLS motifs onto membrane surface proceeds via such coalescence of smaller defects so as to accommodate its hydrophobic surface (70). Here, in case II, defect coalescence can only proceed via modification of the lipid packing upon the protein binding, whereas, in case I, thermal fluctuation will suffice given that the defects are already quite transient. While substantial amount of work is needed to establish the existence of these two diverse binding routes, it can certainly elucidate the basic design principles behind membrane-protein partitioning. This in turn can unravel the physiological functions of lipid rafts and also facilitate the design and engineering of proteins that target specific membrane regions.

Figure 8.

Schematic of defect profile on a lipid membrane exhibiting mixed $L_o/L_d$ phase and how it can govern protein partitioning. The top view of the membrane is shown with different kinds of lipids in bond representation colored in silver, green, and red, and surface defect pockets analyzed over four consecutive snapshots shown in QuickSurf representation (112) with colors blue, violet, purple, and magenta. The membrane snapshot has been rendered using the visualization tool VMD (111). The overlapping defects in $L_o$ domains and near the interface indicate localized defects. A peripheral peptide with bulky side groups (shown in colored polygons) can bind to larger, dispersing defects in the $L_d$ domain (case I) or the smaller, localized defects in and around the $L_o$ domain (case II). See the main text for discussion. To see this figure in color, go online.

The effect of membrane order on protein partitioning can be further exemplified in the case of tLAT, where palmitoylation is shown to enhance the ordered phase affinity of tLAT in agreement with experiments (37,60,85). It should be noted that, unlike CG simulation, wherein there is a clear phase separation between the $L_o$ and $L_d$ domains, the distinction between domains is not so clear in atomistic simulations because of the small system size and short simulation timescales. This makes the clear characterization of the two phases, in terms of parameters such as area per lipid and bilayer thickness, quite difficult. Accordingly, the analyses on tLAT-lipid energetics by Lin et al. (60) relies on the identification of lipids as belonging to $L_o$ or $L_d$ domains based on their chemical identity as saturated lipid (and cholesterol) or unsaturated lipid, respectively. In contrast, the characterization in terms of ${\chi }^2$ relies on the information on spatial and temporal evolution of lipids without any information on membrane chemistry and can quantitatively distinguish the subtle difference in the local membrane order around tLAT with and without palmitoylation, even for such a small atomistic system.

Our inference is also in line with the recent observations by Sikdar et al. (110), which relates the impaired membrane activity of hepatitis A virus-2B protein in cholesterol-rich membranes to the scarcity and small size of defects therein. The ordering effect of cholesterol in lipid membrane is well known (57), which, in a single component lipid system, leads to cholesterol-rich interfacial regions surrounding cholesterolpoor $L_o$ and $L_d$ nanodomains (46). For a two-component lipid mixture, cholesterol preferentially partitions with unsaturated lipids, leading to ordered subdomains made of saturated lipids (58,61). The correlation between membrane order and defects thus implies significant modification in packing defects in the membrane, which for a more ordered-like membrane results in fewer, smaller, and shallower packing defects. Accordingly, the presence of cholesterol in a membrane can significantly reduce the binding events.

Beyond influencing protein partitioning, membrane packing and order can also influence the permeability of the membrane against small molecules. This can be attributed to the free volume available on the membrane surface and in its core, which provides suitable hopping paths to the penetrants. The correspondence between free volume and the permeability of the membrane is rather intuitive and was also mentioned by Ghysels et al. (89). Herein, we have been able to systematically correlate membrane order and permeability in terms of the packing defects. The membrane can be visualized as made up of ordered regions that act as platforms for strong transverse diffusion owing to the large free volume at the membrane core. The disordered and boundary (between the ordered/disordered domains) regions act as channels for penetrant permeation in/out of the membrane because of the abundant surface defects that can initialize permeation.

Conclusion

Membrane order and packing is expected to be intrinsically coupled: ordered domains in membranes are experimentally known to exhibit tighter packing compared with disordered domains. In this work, we have tried to formalize this connection based on two well-defined metrics that can precisely assess membrane order and packing in computational studies. We characterized membrane local order in terms of the NAP, ${\chi }^2$, which can capture the distinctly different local order of lipid domains in a mixed phase membrane. We characterized the lipid packing in terms of the packing defects and analyzed the 3D surface and core defects in these membranes.

The spatial correlation between the NAP and the surface defects and the averaged distribution of the NAP values and the defect sizes indicated a direct correlation between membrane local order and defects in the three mixed phase membranes under study. Moreover, based on the 2D spatial correlation between the two quantities, we found that some mixed phase membrane are more ordered-like, in which case this correlation is quite strong. In contrast, mixed phase membranes that are more disordered-like exhibit a weaker correlation. Consequently, packing defects in these membranes exhibited distinct temporal evolution based on their local order. Together, these observations indicate possible scenarios in which peripheral proteins can bind to order/disordered domains in a membrane. In addition, we also explore how the nature of surface defects and membrane core defects (free volume) in ordered/disordered membranes can lead to distinct permeation pathways of small molecules.

Our findings are based on three mixed phase membranes with distinct lipid chemistry, where one is more ordered-like, one more disordered-like, and one intermediate. Interestingly, the observations on order-defect correlation appear to be rather general across all three membranes and, therefore, should be valid for any mixed phase membrane. The local membrane order crucially governs preferential protein partitioning, while the membrane defect/free volume governs the membrane permeability of small molecules. The specificity of defects in mixed phase membranes can, thus, have important lateral and transbilayer functional implication and might also follow the same basic design principles based on membrane order and packing.

Author contributions

M.T. and A.S. designed the research. M.T. performed the research and analyzed the data with A.S. M.T. wrote the paper with help from A.S.

Editor: Ilya Levental.