Using Packing Defects in Heterogeneous Biological Membrane as a Lens to Explore Protein Localization Propensity and Small Molecule Permeability

Plasma membrane (PM) heterogeneity has long been implicated in various cellular functions. However, mechanistic principles governing functional regulations of lipid environment is not well understood due to the inherent complexities associated with the relevant length and time scales that limit both direct experimental measurements and their interpretation. In this context, computer simulation holds immense potential to investigate molecular-level interactions that lead to PM heterogeneity and the related 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 topological changes in lipid local environment using the non-affine parameter (NAP) frame-work. Furthermore, we probe the packing defects in membrane with coexisting fluid phases, which can be considered as the conjugate of membrane order assessed in terms of the NAP. In doing so, we formalize the connection between membrane packing and local membrane order and use that to explore the mechanistic principles behind preferential localization of proteins in mixed phase membranes and membrane permeability of small molecules. Our observations suggest that heterogeneity in mixed phase membranes follow some generic features, where functions may arise based on packing-related basic design principles. Significance Functionally important complex lateral and transverse structures in biological membrane result from the differential molecular interactions among a rich variety of lipids and other building blocks. The nature of molecular packing in 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 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.

Abstract: Plasma membrane (PM) heterogeneity has long been implicated in various cellular functions. However, mechanistic principles governing functional regulations of lipid environment is not well understood due to the inherent complexities associated with the relevant length and time scales that limit both direct experimental measurements and their interpretation. In this context, computer simulation holds immense potential to investigate molecular-level interactions that lead to PM heterogeneity and the related 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 topological changes in lipid local environment using the non-affine parameter (NAP) framework. Furthermore, we probe the packing defects in membrane with coexisting fluid phases, which can be considered as the conjugate of membrane order assessed in terms of the NAP.
In doing so, we formalize the connection between membrane packing and local membrane order and use that to explore the mechanistic principles behind preferential localization of proteins in mixed phase membranes and membrane permeability of small molecules. Our observations suggest that heterogeneity in mixed phase membranes follow some generic features, where functions may arise based on packing-related basic design principles.
Significance: Functionally important complex lateral and transverse structures in biological membrane result from the differential molecular interactions among a rich variety of lipids and other building blocks. The nature of molecular packing in 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 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. information that biological membrane 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 head groups face the aqueous media 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 towards a complex lateral organization with existence of non-random 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] .
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 towards more clearly elucidating the structure, dynamics and functional role of the membrane "rafts" [40][41][42][43][44][45][46] . 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 39,47,48 with tighter packing and slower diffusion as compared to a more fluid liquid disordered (L d ) regions. In general, with diffusion and packing behavior as distinguishing features of the two co-existing fluid phases in membrane systems, the relationship between the molecular packing of lipids, mobility and membrane order is seemingly obvious from the classical freevolume based theories of diffusion 49,50 . However, despite the tremendous advances made in experimental resolutions in both time and space, accurate quantification and formulation of this correlation is still elusive [51][52][53][54][55] . In this regard, computer simulation has proven to be a very useful tool and the success of 'computational microscopy' 56,57 spans from the ability to observe phase separated nano-structures, reminiscent of the ordered raft domains, in model binary/ternary lipid systems 47,58,59 to investigating their plausible functional implications 60,61 . 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 liquid ordered (L o ) and disordered (L d ) phases have been able to elucidate the distinct nature of lipid packing within the L o domains 59,62,63 , which are observed to play a crucial role in membrane permeation of small molecules [64][65][66] .
Lipid packing defects 67 , 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. While non-trivial to identify in experiments, packing defects has been extensively studied in computer simulation 60,[68][69][70][71] . Over the last decade, the nature of packing defects in curved membranes have been well characterized in various simulation studies, unraveling their role in the adsorption of various proteins that contain specific amphipathic helices 60,69,70,72 . Amphipathic lipid packing sensor (ALPS) motifs in these proteins, can sense large pre-existing 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 ALPS 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 a realistic cell membranes has highly heterogeneous characteristics. One important physiological function of the raft-like domains therein is the partitioning of peripheral proteins [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  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 coexisting (mixed phase) L o /L d membranes and the packing defects therein, where pure phase membranes serve as control systems for such investigations. Towards this, we characterize the local membrane orderliness in terms of the non-affine parameter (NAP) 63,88 , which captures the distinct nature of spatio-temporal evolution of lipids in their local neighbourhoods without any knowledge on lipid chemistry. We identify the threedimensional 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 mid-plane) 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 trans-bilayer functions. Our inferences are rather general and can be extended to any mixed phase membrane irrespective of its chemical nature.

Local orderliness dictates the defect profiles in mixed phase membranes
The nature of packing defects in mixed phase membranes is expected to be intrinsically coupled to the membrane order. The membrane order is traditionally characterized in terms of the local lipid arrangements, membrane packing (area per lipid), thickness, and lipid tailorder parameter(S CD ) 59,62,63 . However, the dynamic nature of these regions, wherein lipids constantly enter and exit them, necessitate the inclusion of temporal information in the analysis. Herein, we quantify the orderliness of three mixed phase ternary lipid systems, DPPC/DOPC/CHOL, PSM/DOPC/CHOL, and PSM/POPC/CHOL membranes, in terms of the non-affine parameter (NAP), i.e., the residual non-affine content of the deformation (χ 2 ) 96,97 , which can capture the distinct nature of the spatio-temporal evolution of lipids in their local neighborhood 63 as shown in Fig. 1a (Please see Methods sections for details on NAP calculations). Furthermore, this analysis can also identify local molecular scale heterogeneity in pure phase systems indicating regions that undergo more heterogeneous topological rearrangements than their neighborhoods 63,88 , and thus, is a sensitive marker of membrane order.
Our earlier works on pure phase systems have indicated that irrespective of lipid chem-istry, lipids in pure L o systems exhibit consistently low χ 2 values as compared to those in pure L d systems 63,88 . In mixed phase membranes, the local ordering of lipids can be distinctly different, which can subsequently result in specific packing defects profiles. To understand this, we compare the χ 2 maps, against the spatial distribution maps of the defects (see Methods) in the three membranes as shown in Fig. 1. We observe that regions in these membranes that possess low χ 2 value, i.e., the ordered domains, are mostly defect-free. Similarly, a large defect can almost always be mapped to reference sites with high χ 2 values, i,e., disordered domains. These observations are valid for all the three systems and thus, can be a general feature of all mixed phase systems. It, therefore, appears that the observations on the nature of defects in pure phase membranes can be extended to mixed phase membranes: ordered domains in mixed phase membrane remain relatively defect-free as compared to disordered domains.
However, as shown in Figure 2, the distributions of defect size for the three mixed phase systems are rather distinctive when compared to their pure phase membrane counterparts: in the case of DPPC/DOPC/CHOL, the distribution is very similar to its L d counterpart, for PSM/POPC/CHOL system it is comparable to the corresponding L o one, and for PSM/DOPC/CHOL system, it is found to be intermediate to the corresponding two pure systems. Thus, the mixed phase membranes lack a generic trend in terms of the defect size, as compared to the pure phase membranes, wherein the L o membranes always exhibit smaller defects as compared to their L d counterparts. The origin of this observation can be again related to the orderliness of the membrane, characterized in terms of the χ 2 distribution. Pure phase L o systems always exhibit a sharper and narrower distributions of χ 2 values as compared to the corresponding L d ones, which is a generic feature of pure phase (homogeneous) membranes 63,71,88 . Interestingly, χ 2 distribution 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 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.
This apparent correlation between χ 2 values and the packing defects can be formalized in terms of a two dimensional probability distribution P(n d , c), which indicates the probability that a reference site (on a lipid molecule) has 'n d ' number of total defect grid points around FIG. 1: Spatial correlation of χ 2 values and packing defects. a) The definition of neighborhood Ω in the χ 2 analysis. r i represents the coordinates of reference lipid sites, whose spatial and temporal evolution is used to calculate χ 2 . b) Schematic indicating the various defects that can be identified using the 3-dimensional defect analysis, shown as blue/while shaded regions. Note that the lipid reference sites are specifically excluded while identifying the defects c) χ 2 spatial map and d) defect spatial map for DPPC/DOPC/CHOL (left), PSM/DOPC/CHOL (middle), and PSM/POPC/CHOL (right) systems, each exhibiting mixed L o /L d phases.
it within a cutoff radius r (here taken to be 14Å) with χ 2 value 'c'. These quantities are summed in order to address two important points. The first, is the absence of a one-to-one spatial correlation between the two quantities: χ 2 is calculated based on lipid coordinates and defect is a grid based analysis that specifically excludes these coordinates. The second, is to incorporate the effect of neighborhood, which can be a determinant factor in mixed phase membrane. In Fig. 3, we present the (n d , c) correlation for the three mixed phase membrane 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. 4 shows the defect spatial maps for the three mixed phase membranes, calculated over 2, 10 and 20 consecutive snapshots, with each snapshot taken at 240 ps time interval. 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 1). However, the ordered domains also exhibit a significant amount of defects, including large ones. Moreover, the large defects in (e.g., in 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. 4 with circles). This is in stark This is a consequence of the spatio-temporally correlated evolution of lipids in the ordered domains, which 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 feature seems to be a generic one. 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 to distinguish between the local environment of a single tLAT peptide, without any prior assumption or knowledge of lipid chemistry or membrane composition.

Membrane defect profile governs the permeability of small molecules
The defect size distributions for pure phase DPPC/DOPC/CHOL membranes shown in 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 III) 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 as compared to the L o one (Fig 6b). Subsequently, we identified two kinds of defects following our 3-dimensional defect algorithm 71 : surface and core defects. The core defects were identified for membrane slices of thickness 14 and 8 A around the membrane mid-plane, while the surface defects were identified on one leaflet of the membrane excluding the core region (Fig 6a). The surface and core defects can be interpreted as the free volume available on the membrane surface (in lateral directions) and in its core, respectively, which has long been implicated in lipid diffusion 68,90,92,95 . 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 1.4Å (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 surface. While both these points can lead to under prediction of defect sizes, unintended over predictions (which can also be misleading) can be avoided. Similar to defects on a single leaflet (Fig 6b), L d system was found to exhibit larger surface packing defects as compared to the L o one (Fig 6c), implying more number of permeation channels and stronger transverse diffusion in the L d system. Together, this can result in significantly larger permeability as compared to 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 6d). Further, as the thickness of the membrane slice is decreased from 14 to 8Å, the distinction between the size distributions for core defects becomes more apparent (Fig 6e), indicating that the membrane mid-plane of the L o system has relatively more free volume as compared to the

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 spatio-temporally correlated lipid evolution and smaller packing defects. In contrast, a moderate correlation indicates the mixed phase membrane to be more 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 that 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, due to the abundant surface defects that can initialize permeation.
To summarize, we observe some generic trends in the packing defects profiles in mixed phase membranes, which correlate almost linearly with the membrane order and exhibit distinct temporal evolution. The local membrane order crucially governs the preferential partitioning of peripheral proteins, while the membrane defect/free volume governs the membrane permeability of small molecules. The specificity of defects in mixed phase membranes can, therefore, have important lateral and trans-bilayer functional implication and might also follow the same basic design principles based on membrane order and packing.

A. Details of the simulation trajectories
The pure and mixed phase atomistic trajectories of model ternary lipid systems (total 9), analyzed in the first part of the work, were obtained from Edward Lyman's group at   The χ 2 analysis is performed as described in our previous studies 63, 88 and follows the original prescription by Falk and Langer as applied for amorphous solids 96 . The lipid membrane is considered as an amorphous system evolving in space and time, where a reference site on every lipid molecule is tracked over time. In our case, unless otherwise noted, the reference sites are identified as the bottom carbon atom of the glycerol group for lipids and the hydroxyl group oxygen atom for cholesterol. A neighborhood is defined around each central lipid within a cutoff radius Ω (see Fig. 1a), which is taken to be 14Å in our calculations (unless otherwise specified). As discussed previously 63,88 , χ 2 is calculated at each reference site in the membrane as where i, j indicate spatial coordinates r of the lipid reference site with dimension d, n runs through the N lipids in the neighbourhood Ω around the central lipid n = 0 (Fig. 1a), and δ ij denotes the Kroneker delta function. ϵ ij which denotes the strain associated with the maximum possible affine part of the deformation, minimizes χ 2 Ω (t, ∆t) and is calculated as follows As opposed to the previous implementations 63, 88,96 , we now normalize the χ 2 by the total number of lipids in the neighbourhood N , so as to incorporate the effect of environment in mixed phase membranes and also to be able to compare the results across various phases (L o , L d , or mixed) and systems of various sizes (see Table I).
To generate the χ 2 spatial maps, we average the computed χ 2 values, over 10 consecutive system snapshots and map that to the spatial coordinates of the reference sites of lipids of the middle (sixth) snapshot of the chosen set (Fig. 1c).

C. Identifying and analyzing three dimensional packing defects
We identify the packing defects/free volume in lipid membranes based on our three dimensional defect algorithm 71 , which is based on a grid based free volume analysis. The defects are identified as grid points that lie around the exposed hydrophobic tails of the lipids and below the lipid reference sites (the bottom carbon atom of the glycerol group for lipids and the hydroxyl group oxygen atom for cholesterol). For the pure and mixed phase membrane systems under study, we identify the defects on one leaflet based on a suitable Z-cutoff. The defect grid points are subsequently grouped following a distance based clustering approach to identify the individual defect pockets and their size distributions are calculated.
The defect spatial maps (Fig. 1d, Fig. 4) are calculated by projecting the defects at each snapshot onto the XY grid plane, binning them (bin size: 1Å) and finally averaging over a set of snapshots. The resulting count indicates the probability that a grid point belongs to a defect and a higher probability indicates both the spatial localization and temporal persistence of the defect.
To corroborate the findings of Ghysels et al. 89 in terms of packing defects, we identify two more classes of defects. The surface defects basically are the hydrophobic defect pockets on membrane surface (same as earlier), which are identified on a membrane leaflet with Z-coordinates > 10Å. The core defects indicate the free volume in the membrane mid-plane and are identified as the grid points that are not occupied by the lipids/cholesterol. For both classes of defects, we use a probe radius of 1.4Å to identify defect grid points (as in a rolling-probe method, see the original reference for an in-depth discussion 71 ). We do not include periodic boundary condition while calculating the defect size distribution. This allows us to compare both classes of defects without any loss of generality.

AUTHOR CONTRIBUTIONS
MT and AS designed the research. MT performed the research and analyzed the data with AS. MT wrote the paper with help from AS.