Local enrichment of cardiolipin to transient membrane undulations

Abstract

Organelles such as mitochondria have characteristic shapes that are critical to their function. Recent efforts have revealed that the curvature contributions of individual lipid species can be a factor in the generation of membrane shape in these organelles. Inspired by lipidomics data from yeast mitochondrial membranes, we used Martini coarse-grained molecular dynamics simulations to investigate how lipid composition facilitates membrane shaping. We found that increasing lipid saturation increases bending rigidity while reducing the monolayer spontaneous curvature. We also found that systems containing cardiolipin exhibited decreased bending rigidity and increased spontaneous curvature when compared with bilayers containing its precursor, phosphatidylglycerol. This finding contradicts some prior experimental results that suggest that bilayers containing tetraoleoyl cardiolipin have greater rigidity than dioleoyl phosphatidylcholine bilayers. To investigate this discrepancy, we analyzed our simulations for correlations between lipid localization and local curvature. We found that there are transient correlations between curved lipids, such as cardiolipin and phosphatidylethanolamine, and curvature; these interactions enrich specific bilayer undulatory modes and cause bilayer softening. Furthermore, we show that curvature localization of some lipids such as cardiolipin can influence lipids in the opposing leaflet. These observations add to the emerging evidence that lipid geometric features give rise to local interactions, which can cause membrane compositional heterogeneities. The cross talk between composition-driven tuning of membrane properties and membrane shape has implications for membrane organization and its related functions.

Significance

The material properties of phospholipid membranes are a function of the lipid composition. The abundance of cardiolipin in the mitochondrial inner membrane implies a functional role for this special lipid. We explore the interactions of cardiolipin with other lipids with varying lipid saturation using coarse-grained molecular dynamics simulations. We find that membranes containing cardiolipin have higher spontaneous curvature and lower bending rigidity when compared with membranes without cardiolipin—in line with prior models and experiments. We also show that the low bending rigidity of cardiolipin-containing symmetric and flat bilayered systems is due to the transient partitioning of cardiolipin to undulations due to curvature sensing. This is a mechanism for forming lateral membrane heterogeneities in otherwise symmetric systems.

Introduction

Mitochondria are central to energy production in cells; their unique and characteristic ultrastructure is fundamental to their function and performance (1,2). Unlike most other organelles, mitochondria have two phospholipid bilayers called the outer mitochondrial membrane and inner mitochondrial membrane (IMM). The IMM, where the machinery for the electron transport chain and adenosine triphosphate (ATP) synthesis are localized, has a highly curved, folded, and tubulated shape (3,4). This folded geometry with a high surface area-to-volume ratio is critical for energy production (1,2,5). Recently, many studies have pointed to the important role of mitochondrial morphology in maintaining its physiological function (6). The loss of the characteristic IMM ultrastructure is associated with defects in energy production and is observed in many diseases (7,8).

Recently, we explored the consequence of changing lipid saturation on the structure and function of yeast mitochondria (9). Using a promoter strategy to control the expression of the sole lipid desaturase in yeast, we were able to tune the extent of lipid saturation in the mitochondrial membranes. We found that increasing lipid saturation led to an abrupt loss of IMM ultrastructure; that is, there is a “breaking point” of lipid saturation which, when exceeded, leads to loss of cristae folds. This loss of cristae is driven in part by the dissociation ATP synthase dimers (9). We further observed that knocking out cardiolipin synthase (CRD1), which causes the accumulation of phosphatidylglycerol ($\text{PG}$) instead of cardiolipin ($\text{CDL})$, shifts this breaking point such that a lesser increase in saturation leads to loss of shape (notably without shifting ATP synthase oligomerization). Previously, we used continuum modeling to explain this abrupt loss of cristae structure as a snap-through instability (9,10). However, such continuum models depend on the input material properties such as bending modulus and do not account for the local lipid composition or lipid-lipid interactions. Briefly, the energy of a membrane bent into a particular shape is often modeled using the Helfrich-Canham (HC) Hamiltonian (11,12,13),

$$\mathcal{H}=\int \left[2\kappa {\left(H-C_0\right)}^2\right]dA\text{,}$$ (1)

where $\kappa$ is the bending modulus, $H$ is the mean curvature, and $C_0$ is the spontaneous (mean) curvature. In the HC theory, there are two material parameters $\kappa$ and $C_0$, which dictate the energy required to bend a membrane. These parameters are also sophisticated functions of the lipid chemistry and composition. They can be estimated from molecular dynamics (MD) simulations (14,15,16,17,18,19) where advances in lipid force field parameterizations enable the study of complex lipid mixtures (20) (Fig. 1). Analyses of mean height fluctuations of the bilayer enable the characterization of the bending modulus, $\kappa$; while the first moment of the lateral pressure profile is related to the bending moment, or the product of the bending modulus and spontaneous curvature, $\kappa C_0$ (Fig. 1, B and C).

Figure 1.

Overview of the approach connecting coarse-grained molecular dynamics simulations to continuum modeling. (A) Lipid compositions from lipidomics of yeast mitochondria with the extent of lipid saturation under experimental control are input into coarse-grained molecular dynamics simulations. (B) Analysis of mean height fluctuations of the bilayer enables the characterization of the bending modulus, $\kappa$. (C) The first moment of the lateral pressure profile is related to the bending moment, or the product of the bending modulus, $\kappa$, and spontaneous mean curvature, $c_0$.

In this work, we seek to answer the following questions: How does the presence of CDL affect the mechanical properties lipid bilayers? How does the presence or absence of CDL shift the point of lipid saturation that breaks cristae curvature formation? To answer these questions, we used coarse-grained molecular dynamics (CGMD) simulations with experimentally informed lipid compositions.

The prior literature, consisting of both experimental and computational approaches, presents conflicting perspectives on the effect of $\text{CDL}$ on the bilayer. On one hand, experimental efforts investigating the partition of $\text{CDL}$ to membrane tubes pulled from giant unilamellar vesicles (GUVs) suggest that it has a conical shape, which imparts it with a large spontaneous curvature (21). Similarly, quantification of extracts of mitochondrial lipids also suggest that $\text{CDL}$-rich membranes have a very low bending rigidity (22). In contrast, measurements from scattering experiments suggest that $\text{CDL}$ has a weaker negative spontaneous curvature (cf. Fig. S4) than phosphatidylethanolamine ($\text{PE})$ (23,24,25) and an increased rigidity compared with 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) (26). Investigation using CGMD suggests a reduced bending rigidity (27,28), and a preference for $\text{CDL}$ to partition to regions of negative curvature (29,30,31,32). Reports from simulations of atomistic systems report larger bending rigidities of $\text{CDL}$ than 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) (33,34) and a weaker absolute spontaneous curvature for $\text{CDL}$ than 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) (34).

In this work, we aim to reconcile these conflicting perspectives by investigating the mechanical properties of membranes with experimentally informed lipid compositions. By modeling membranes with lipid compositions informed by lipidomics of yeast mitochondria, we predict how the compositional changes influence the material properties of the membrane (Figs. 1 and 2). We find that increasing lipid saturation leads to an increase in the bending rigidity of the membrane and a general decrease in spontaneous curvature. Our CGMD simulations also revealed that the $\text{CDL}$-containing systems have a diminished bending rigidity compared with $\text{PG}$-containing systems. While these findings are in general agreement with prior CGMD simulation studies (27), our estimated bending moduli are lower and spontaneous curvature larger than expected from experiments of $\text{CDL}$-containing systems (23,24,25,26). Examination of the height fluctuation spectra of $\text{CDL}$-containing systems suggests that these systems exhibit spectral features deviating from the predictions of the HC model (Fig. 3 C).

Figure 2.

Compositions of the modeled systems. The systems are separated into two cohorts: idealized systems (left-hand side), which are composed of either $\text{DO}$ or $\text{PO}$ forms of $\text{PC}$ and $\text{PE}$ lipids and realistic compositions (right-hand side) based on lipidomics data of yeast mitochondria (9). Systems were further categorized by the presence of ${\text{CDL}}^{1-}$, ${\text{CDL}}^{2-}$, or $\text{PG}$. The $\text{CDL}$ modeled is $\text{TOCL}$ and further differentiated by charge state. Systems containing $\text{PG}$ correspond to cardiolipin synthase knockout (crd1$\Delta$ ) yeast mutants. Systems will be referred to by the following nomenclature: ({R,I}{WT/0, +, ++}{C1, C2, PG}) (e.g., RWTC1 corresponds to the realistic ${\text{CDL}}^{1-}$ wild-type condition, and I++PG the idealized $\text{PG}$ system at elevated saturation).

Figure 3.

Quantification of bending modulus, bending moment, and spontaneous curvature for each system. For (A)–(C), the top panels are derived from system I0PG and the bottom from I0C2. (A) An example power spectrum (i.e., the first $i$ undulatory modes, $q_i$). (B) The extent of correlation within the power time series for each undulatory mode is estimated by fitting to the block average. (C) Plot of the height power spectra showing the linear fit at large wavenumber regimes. The mean values for each undulatory mode are shown by a dot and the estimated standard deviation shown by gray lines. (D) Estimated bending moduli for all systems. Error bars correspond to the standard deviation derived from parametric bootstrapping of data in (C). (E) The bending moment is calculated as a product of monolayer bending rigidity and monolayer spontaneous curvature. (F) The monolayer spontaneous curvature for each system derived by substituting the estimated bending rigidity and bending moment in (D) and (E).

Considering that the unique spectral features of $\text{CDL}$-containing systems could be caused by heterogeneities in membrane organization, we investigated the distributions of lipids around a given lipid species and found that there are preferential interactions. Investigating this further, beyond lipid chemical interactions, we found that there are transient correlations of lipids even with the curvature of the membrane. We observed that $\text{CDL}$ and $\text{PE}$ both exhibit a preference to localize with negative mean curvature, but only $\text{CDL}$ exhibits depletion across the bilayer. Altogether, the kinetics of curvature-generating or -stabilizing interactions between lipids can lead to bilayer softening, which we propose as the cause of the deviations from the HC model (35).

Materials and methods

MD simulations

Symmetric bilayers with compositions derived from systems described previously in Venkatraman et al. (9) (also shown in Figs. 2 and S5; Table S1), were modeled by CGMD using the Martini 2.2 force field parameters (36). Systems of approximately $40\times 40\times 25$ and $15\times 15\times 25$ nm (referred to as the small systems) were constructed using insane.py (37). All systems were solvated with water and charge neutralized by 150 mM $\text{NaCl}$ using the workflow provided by insane.py. The small systems were solvated with 90% water and 10% antifreeze molecules to prevent freezing of the system. Minimization and equilibration protocols were adapted from CHARMM-GUI Martini Maker (38) and summarized in the supporting material, section B.1. Production runs extend for a total of $5\mu \mathrm{s}$ of simulation time from the equilibrated structures with a 20 fs timestep. Small systems were simulated for a total of 3 $\mu$s followed by 200 ns of simulation with a 5 ps write out frequency for calculation of system stresses. Notably the simulation lengths reported reflect the actual simulation length without factoring in the conventional post hoc time scaling of Martini simulations.

The pressure was maintained in production simulations using the Parinello-Rahman semiisotropic barostat with ${\tau }_p$ of 12 ps, compressibility $3\times {10}^{-4}{\text{bar}}^{-1}$, at 1 bar pressure (39). For all simulations, electrostatics were treated using the reaction-field method with a cutoff of 1.1 nm and ${ϵ}_r$ of 15 (40), and the temperature was held at 303 K using the stochastic velocity-rescaling thermostat with ${\tau }_t$ set to 1.0 ps (41).

Analysis

From the large-box simulations, the bending modulus was estimated by analysis of the height fluctuation spectra, see supporting material, section B.1.1. Meanwhile, from the small-box simulations, the spontaneous curvatures were estimated from the moments of the lateral pressure profile and substituting the estimated bending modulus for each system, see supporting material, section B.1.3, for detailed approach. Statistical errors were propagated using conventional formulae.

Results

Model development

In this work, we focused on lipid compositions inspired by the experimental manipulations presented in (9) with the goal of understanding how mitochondrial lipid composition could affect global and local properties of the membranes. We simulated these membrane compositions using Martini 2.2 CGMD; the compositions are illustrated in Figs. 2 and S5 with details given in Table S1. To study the consequence of lipid saturation on the membrane, we designed idealized systems composed of either $\text{DO}$ or $\text{PO}$ forms of $\text{PC}$ and $\text{PE}$ lipids. These idealized systems establish a comparison for the effect of lipid saturation on bilayer properties. We also designed a “realistic” series of compositions informed by lipidomics. In these series, the populations of $\text{PC}$ and $\text{PE}$ lipids along with lipid tail identities were drawn from lipidomics of mutant yeast (9). As a further manipulation, to investigate the importance of $\text{CDL}$, we compared systems with $\text{CDL}$ against systems with $\text{PG}$ which correspond to CRD1 and crd1$\Delta$ genetic conditions, respectively. Within the CRD1 systems, we further differentiated by the charge of the $\text{CDL}$ species. This variation is designed to capture the complex chemical environment around the mitochondrial inner membrane and the fact that the protonation state of $\text{CDL}$ within the IMM may vary between the singly, ${\text{CDL}}^{1-}$, and doubly, ${\text{CDL}}^{2-}$, deprotonated charge states (42,43).

Overall, the systems are categorized by three factors: their relative realism (idealistic, realistic), the lipid saturation (WT/0, +, and ++), and the genotype (CRD1, crd1Δ). We use the following abbreviation scheme to refer to each system: {I,R}{WT/0, +, ++}{C1, C2, PG}; the system acronyms are elaborated in Table S1. In addition to these 15 systems, we also modeled symmetric bilayers mimicking the outer and inner leaflet compositions of the IMM with either ${\text{CDL}}^{1-}$ or ${\text{CDL}}^{2-}$, hence referred to as OutC1, InnC1, OutC2, and InnC2. Finally, as controls, we also modeled bilayers consisting of only ${\text{CDL}}^{1-}$ or ${\text{CDL}}^{2-}$ (referred to as C1-only and C2-only, respectively). For each condition, we constructed large ($\sim$40 nm) and small systems ($\sim$15 nm) for height-undulation and lateral pressure analysis, respectively.

Membrane properties as a function of membrane composition

To establish the relationship between the membrane composition and membrane properties, we calculated the bending modulus, bending moment, and the resulting spontaneous curvature using standard methods (14,16,44). For each system, we estimated the bending modulus (Figs. 3 D and S8) by analyzing the height fluctuation spectra (Fig. 3, A–C). The reported bending moduli for affected systems are scaled by a constant 20% to account for systematic biases discovered in the Gromacs simulation engine (45) (Fig. S6). The uncertainties of the reported bending moduli are the result of parametric bootstrapping of the data (46). Within the idealized series, we found that increasing lipid saturation leads to an increase in bending rigidity (Fig. 3 D). This increase is consistent with prior experimental measurements (47). The trends in the realistic series exhibited modest increases in bending rigidity for the ${\text{CDL}}^{1-}$-containing systems while ${\text{CDL}}^{2-}$- and $\text{PG}$-containing systems bucked the trend. Consistently across the different conditions, lipid membranes containing $\text{CDL}$ had reduced bending rigidity compared with the $\text{PG}$-containing systems; systems containing ${\text{CDL}}^{1-}$ had reduced bending rigidity compared with those with ${\text{CDL}}^{2-}$. The outer leaflet of the IMM was also found to be less stiff than the inner leaflet (Fig. S8). Consistent with prior literature (27), the bending rigidity of systems with C1-only was $\left(5.8\pm 0.2\right)k_{B}T$ while C2-only was $\left(11.5\pm 0.2\right)k_{B}T$ (Fig. S8). Notably, for all $\text{CDL}$-containing systems, the power spectrum exhibited spectral features at long wavelength modes, deviating from the constant value predicted by HC (compare top and bottom of Fig. 3, B and C).

We also considered how the changes to lipid composition could contribute to the preferred curvatures of the bilayer. We computed the monolayer bending moment, ${\kappa }_m{c}_0$, by numerically integrating the lateral pressure profile of each system (Figs. 3 E and S9). Given that the modeled systems have symmetric compositions between leaflets, we assume that the monolayer rigidity is one-half of the bilayer rigidity, ${\kappa }_m=\kappa /2$. By substituting the previously estimated bending rigidities into the bending moments, we obtained the monolayer spontaneous curvature, $c_0$, for each system (Figs. 3 F and S10). We note that the orientation of each leaflet, given by the normal, is chosen to be inward pointing, extending in the same direction as the vector pointing from lipid headgroups toward the center of the bilayer. This convention is illustrated in Fig. S4 and implies a positive mean curvature for convex shapes (e.g., a spherical micelle or outer leaflet of a vesicle would have a positive mean curvature) and a negative mean curvature for concave shapes. We found that, for the idealized compositions, with increasing saturation, there is a corresponding decrease in spontaneous curvature; for the realistic compositions, the trend is reversed. Across both idealized and realistic compositions, ${\text{CDL}}^{1-}$-containing systems had the greatest spontaneous curvature, followed by ${\text{CDL}}^{2-}$ and then $\text{PG}$. The outer leaflet of the IMM with greater $\text{CDL}$ content has greater spontaneous curvature than the inner leaflet (Fig. S10). Systems with only ${\text{CDL}}^{1-}$ exhibited a large spontaneous curvature $\left(-2.8\pm 0.1\right){\text{nm}}^{-1}$ meanwhile only ${\text{CDL}}^{2-}$ had a weaker spontaneous curvature $\left(-1.52\pm 0.03\right){\text{nm}}^{-1}$. This is in contrast with scattering experiments that report that $\text{CDL}$ has a lower spontaneous curvature than $\text{PE}$ (23,24,25).

Aberrant features in height fluctuation spectra of membranes with CDL

At this point, we were surprised by the low bending rigidities predicted for the $\text{CDL}$-containing systems. Moreover, owing to the nature of deriving the spontaneous curvature from the bending moment, the estimated bending rigidity has a strong influence on the predicted spontaneous curvature. Inspection of the approach to quantify statistical inefficiency of the power spectra revealed that the fits to theory were poor for systems with $\text{CDL}$ (Fig. 3, A and B, bottom). Further plots of the height undulation spectra of $\text{CDL}$-containing systems appear to have additional spectral features at long wavelengths. This is a substantial deviation from the HC theory, which predicts a linear dependence of the height undulation spectrum with the fourth power of the wavenumber (Fig. 3 C, bottom).

Considering the aberrant spectral features and high spontaneous curvature of $\text{CDL}$-containing systems, we considered whether interactions of lipid species could give rise to high curvature complexes (cf. (35)). To evaluate the propensity of lipid clustering, we compared the mean lipid composition within 1.5 nm of any lipid type against the composition of the bulk system (Fig. 4). We found that certain lipids are enriched or depleted against random from bulk, suggesting that there are heterogeneities driven by the interactions between lipid types. $\text{PG}$ lipids are found to be enriched against random with respect to all other lipid types (Fig. 4, E and F). $\text{CDL}$ on the other hand is found to be depleted against other $\text{CDLs}$; ${\text{CDL}}^{2-}$ around ${\text{CDL}}^{2-}$ is depleted to a larger extent than ${\text{CDL}}^{1-}$ around ${\text{CDL}}^{1-}$ lipids (compare Fig. 4, A and B with C and D). Curved lipids $\text{CDL}$ and $\text{PE}$ were found to be mutually enriched in the local neighborhood (Fig. 4, A–D). These observations suggested that lipids are heterogeneously dispersed in these complex compositions, which could lead to emergent mechanical properties.

Figure 4.

Quantification of the enrichment or depletion of y axis lipid with x axis lipid species interactions within a 1.5 nm radius compared with random. (A and B) Lipid enrichments of ${\text{CDL}}^{1-}$ containing idealized and realistic systems, respectively. (C and D) Lipid enrichments of ${\text{CDL}}^{2-}$ containing idealized and realistic systems, respectively. (E and F) Lipid enrichments of $\text{PG}$ containing idealized and realistic systems, respectively.

Coupling between membrane undulations and lipid curvature

Having established that certain lipids have preferential lipid-lipid interactions, we next asked if there was any coupling between lipid clustering and transient fluctuations of the bilayer. Given that the bilayers we modeled had compositionally identical leaflets, we note that there are no net curvatures across the bilayer—just transient thermal undulations. Overlaying a rectangular grid onto the bilayer and interpolating a height field, we computed the mean curvature of the membrane at each grid point. Evaluating the time autocorrelation of the membrane mean curvature, we found that local curvatures become uncorrelated rapidly, within a few nanoseconds. The spatial autocorrelations were found to be local in nature and on the order of the grid spacing (i.e., 2 nm). We chose a timescale of 2.5 nm wherein the time autocorrelation of mean curvature drops to about 0.5 for all systems. Using this timescale, we computed a binned lipid density by binning the positions of lipids by type over 2.5 ns (Fig. 5 A). To evaluate the extent of interactions between membrane undulations and each lipid, we computed the 2D spatial Pearson’s cross correlation coefficient of membrane mean curvature and lipid density (Fig. 5 B). We found that, in $\text{CDL}$-containing systems, $\text{CDL}$ and $\text{PE}$ are negatively correlated with mean curvature, while $\text{PC}$ has a positive correlation. $\text{PG}$, on the other hand, negatively correlates with mean curvature to a lesser extent than $\text{CDL}$.

Figure 5.

Correlations between membrane curvature and lipid localization. (A) Mean curvature of I0C1 with lipid densities (PC, PE, ${\text{CDL}}^{1-}$) obtained from averaging over 5 ns. Refer to Fig. S4 for the sign convention of the curvature. (B) 2D spatial Pearson’s cross correlation coefficient of membrane mean curvature and lipid density for I0C1- and I0PG-containing systems. (C) Histogram of local curvature around ${\text{CDL}}^{1-}$ for IWTC1 system shown as orange line compared against all membrane curvatures shown in blue (top); (bottom) difference between the distribution of mean curvatures around a lipid of interest and the general membrane mean curvature distribution. (D) Quantification of the first moment of the difference between distribution of mean curvatures around a lipid of interest and the general membrane mean curvature distribution.

With the correlations between curvature and lipid density established, we further quantified the propensity of finding a particular lipid species at a given local curvature. We assigned each lipid a “curvature value” by referencing the membrane curvature of the nearest grid point. We then compared the distribution of assigned curvature values for a given lipid type against the set of curvatures for all grid points to further quantify the curvature preference of each lipid type (Fig. 5 C). The first moment of the difference between the lipid curvatures and the membrane curvature ($\Delta$Density) is a measure of the curvature weighted extent of deviation (i.e., differences at larger curvature values are weighted more heavily) (Fig. 5 D). Comparing the first moment of the difference, we found that ${\text{CDL}}^{1-}$ associates with negative mean curvatures to a greater extent than ${\text{CDL}}^{2-}$. $\text{PE}$ was also found to be correlated with negative mean curvatures across all systems, albeit to a lesser extent than either $\text{CDL}$ species. $\text{PC}$ lipids are found to be enriched toward positive mean curvatures in all systems in a manner that appears, to an extent, relative to the shift toward negative mean curvatures of $\text{CDL}$ and $\text{PE}$ lipids. From these analyses, we conclude that $\text{CDL}$ has the capability to localize to regions of negative mean curvature in a charge-dependent manner. We find that, while $\text{PE}$ also has a preference for negative mean curvature, $\text{CDL}$ outcompetes $\text{PE}$ for negative mean curvatures. Similarly, while $\text{PC}$ has a limited preference for curvature, the presence of $\text{CDL}$ partitioning can drive $\text{PC}$ to other regions of the membrane.

CDL interleaflet cross talk through membrane undulations

Having established that transient membrane undulations can drive localization of specific lipid types, we next considered whether there are curvature-driven effects between the two leaflets of the bilayer. Since the curvature of each leaflet is opposite relative to the other as constrained by the bilayer configuration, we next asked whether the curvature-induced aggregation in one leaflet affects the composition in the opposing leaflet. We computed the 2D spatial Pearson’s cross correlation of lipid densities across leaflets (Fig. 6 A), and found that $\text{CDL}$ is negatively correlated with itself across leaflets. That is, local $\text{CDL}$ enrichment/presence in one leaflet causes a depletion of $\text{CDL}$ in the opposing region in the other leaflet. The other lipids, including $\text{PE}$ despite the curvature preference, have weak correlations across leaflets. The Pearson’s correlation of unshifted lipid densities across leaflets is shown in Fig. 6 B. The cross-leaflet depletion of $\text{CDL}$ appears to be local in nature, extending only $\sim$5 nm. Overall, we find that $\text{CDL}$ is not only enriched at negative mean curvature undulations, but also depleted in the matching section of the opposing leaflets. This behavior is unique to $\text{CDL}$ and not a detectable feature of other lipids like $\text{PE}$, despite its large spontaneous curvature.

Figure 6.

Pearson’s spatial cross correlation of lipid densities across leaflets. (A) Lipid-lipid cross correlations for systems I0C1 (top) and I0PG (bottom). Lipid densities are generated by averaging over 5 ns. The reported cross correlation is the average of 200 densities. (B) Direct Pearson’s correlation of lipid densities across leaflets for all systems. This value corresponds to the central pixel in (A), corresponding to the unshifted correlation between lipid density across leaflets.

Discussion and conclusions

In this work, we investigated the effects of changing lipid saturation on the material properties of the IMM. We found that increasing lipid saturation leads to an increase in bending rigidity and a general decrease in spontaneous curvature. Both trends point toward increasing difficulty in bending the membrane with increasing saturation. Continuum models of cristae formation suggest that the predicted parameter values lie in a range around a snap-through transition (i.e., where a small change in conditions leads to a drastic change in shape, like a jumping popper toy or slap bracelet) (9,10,48). The existence and proximity of the IMM to a snap-through instability can explain the experimental observation that there is a breaking point of saturation where cristae morphology is lost. In our analysis, we identified unexpected spectral features in the power spectrum of membrane height undulations for $\text{CDL}$-containing systems. These features were not predicted by the HC model and so we investigated whether they can be caused by heterogeneities in membrane organization.

With respect to the local lipid environment around lipids of a given species, we find that there are preferential interactions, which lead to enrichment or depletion of lipid partners. Such interactions have also been studied in the prior literature. Dahlberg and Maliniak (27) found that a decrease in charge led to an enhanced aggregation of ${\text{CDL}}^{1-}$ when compared with ${\text{CDL}}^{2-}$. They observed that ${\text{CDL}}^{1-}$ forms microdomains with negative curvature, but also that ${\text{CDL}}^{2-}$ does not. In contrast to this, we find that both ${\text{CDL}}^{1-}$ and ${\text{CDL}}^{2-}$ are transiently enriched at regions of negative mean curvature, to differing extents; ${\text{CDL}}^{1-}$ has a stronger correlation with negative mean curvature than ${\text{CDL}}^{2-}$. In apparent agreement with (27), we see that ${\text{CDL}}^{1-}$ is enriched around other ${\text{CDL}}^{1-}$, while ${\text{CDL}}^{2-}$ is depleted with respect to other ${\text{CDL}}^{2-}$. Thus, our simulations suggest that there appears to be a trade-off between specific interactions such as electrostatics and relaxation of bilayer stresses from geometric complementarity.

The correlation of other lipids with membrane curvature has also been observed in the context of asymmetric bilayers (49). Koldsø et al. (49) found that lipid species with different curvatures can partition to regions complementing their geometric preference. In our work, we modeled comparatively simpler compositions than (49), which do not undergo phase separation nor form more sophisticated heterogeneities. Nevertheless, even in our modeled symmetric bilayers, we observed transient lipid aggregation with membrane curvature. We found that lipids with strong curvature preferences such as $\text{CDL}$ species can exclude other lipids, such as $\text{PC}$, from curved regions. These steric effects are composition dependent—$\text{PC}$ shows no strong correlation with curvature in the crd1Δ systems (i.e., $\text{PG}$ containing).

Other studies have also investigated the curvature partitioning of $\text{CDL}$ and other lipids in systems with induced curvatures through buckling or other forcing mechanisms (30,31,32,50,51,52). Our results showing that $\text{CDL}$ has a greater propensity to associate with curvature than $\text{PE}$ are in agreement with simulations of their partitioning in buckled membranes (29,31,53). Unlike models with forced curvature, our work identifies localization with curvature driven by thermal undulations even in flat, symmetric bilayers without external loading. While we calculated the cross correlation between lipid density and membrane mean curvature, others have previously observed similar correlations using atomistic MD simulations of $\text{CDL}$ correlated with bilayer height (43); the mean curvature is related through Eq. S20 to the height field.

In the case of $\text{CDL}$, we also observed negative correlations of self density across leaflets of the bilayer. Our observations are consistent with prior work showing differences in number density of local $\text{CDL}$ across leaflets (27). Our work extends the observation, showing that the cross-leaflet effect is unique to $\text{CDL}$ and not observed for the other lipid species that we modeled. We suggest that this is due to the strong curvature preference of $\text{CDL}$; a negative curvature in one leaflet corresponds to a positive curvature in the opposing leaflet. This is a possible mechanism to drive membrane heterogeneity even in symmetric membranes.

With respect to the discrepancies between experimental estimates of the stiffness and spontaneous curvatures of $\text{CDL}$ containing bilayers: scattering experiments predict rigidities of $\sim$25 $k_{B}T$ and a reduced spontaneous curvature for $\text{CDL}$ compared with $\text{PE}$ (23,24,25,26); while other experiments investigating lipid partition to tubes pulled from a GUV predicted a spontaneous curvature of $-1.1{\text{nm}}^{-1}$ for $\text{CDL}$ (21), a value similar to our estimate for ${\text{CDL}}^{2-}$, $\left(-1.52\pm 0.02\right){\text{nm}}^{-1}$ (cf. Fig. S10). Beltrán-Heredia et al. (21) suggest that lipid self-associative interactions drive this curvature localization despite the fact that no raft formation was observed in their work. Our simulations suggest that curvature complementarity may be the driving force leading to transient localization/"clustering". In other words, stochastic colocalizations of $\text{CDL}$, which has a negative spontaneous curvature, will be reinforced by transient membrane undulations to minimize Eq. 1. This is a geometric mechanism which can be orthogonal to specific chemical self-interactions.

Atomistic MD simulations by others have also reported clustering to negative curvatures (43) and any curvature (34). In disagreement with our results, Wilson et al. (43) found that the membrane height fluctuations are not affected by $\text{CDL}$ content (43). Estimated bending rigidities using real space fluctuations suggest a value of $\sim$30 $k_{B}T$ (33,34), closer to scattering experiments. Perhaps the discrepancy is a result of the length scale difference between atomistic and coarse-grained simulations—systems may need to be sufficiently large to permit long wavelength undulation modes to support partitioning and thus curvature softening (35). From experiments, it has been suggested that changes to local membrane bending rigidity (from changing lipid composition due to demixing) can lead to curvature sorting (54). In line with our predicted rigidities, larger length scale experimental measurements using micropipette aspiration of mitochondrial lipid extracts estimate a bending rigidity of $5k_{B}T$ (22). Overall, we suggest that geometric frustration of $\text{CDL}$ in planar membranes facilitates the development of curvature-undulation reinforcing heterogeneities, which appear as an effective softening of the membrane (35,55). It is possible that the discrepancy between $\text{PE}$ and $\text{CDL}$ spontaneous curvature and rigidity ordering between experiments can be explained by difference in the length- and time-scale of experimental setup and fitting approach. With respect to spontaneous curvature estimates, measurements in giant vesicles via flicker spectroscopy or micropipette aspiration will capture the contributions of compositional heterogeneities driven by curvature coupling while such heterogeneities may only occur in the planar orientation of H_II systems and this may not be captured by SAXS. Regarding bending rigidity, the timescale of measurements may lead to the observed differences. Micropipette aspiration and flicker spectroscopy report equilibrium values subject to the influence of compositional heterogeneity while other approaches which infer rigidity from fast relaxing processes will not reflect the diffusional softening (35).

Beyond the membrane, the change in lipid chemistry may influence their interactions with embedded proteins. The preferences of lipids to interact with certain proteins in a structure-dependent manner has been called lipid fingerprints (56,57). The principles of how the lipid bilayer organizes, influences its properties as a solvent for these proteins. $\text{CDL}$ is asymmetrically distributed in the IMM. Curvature coupling of $\text{CDL}$ within and across leaflets may aid in the localization of transmembrane and integral proteins on the opposing leaflet. Our study suggests that lipid spontaneous curvature can drive heterogeneities in the bilayer adding to the emerging functional role that lipid composition plays in organelle physiology (58).

Acknowledgments

The authors thank Edward Lyman who provided helpful discussions. We also thank Honor Akenuwa and Nate Linden-Santangeli for critical review. This work was supported by grants from the Office of Naval Research (ONR N00014-20-1-2469 to P.R.), a Moore-Simons Project on the Origin of the Eukaryotic Cell (GBMF-9734 to I.B. and P.R.), and National Institutes of Health (R35-GM142960 to I.B.). K.V. was supported by the NIH Molecular Biophysics Training Grant (T32-GM008326). C.T.L. was supported in part by a Kavli Institute for Brain and Mind Postdoctoral Fellowship. Molecular dynamics simulations were run on hardware hosted by the Triton Shared Computing Cluster (59).

Author contributions

C.T.L., K.V., I.B., and P.R. designed research. C.T.L. performed research with feedback from K.V., I.B., and P.R. C.T.L. ran the simulations, wrote the scripts to execute analysis, and analyzed the data. C.T.L. wrote the original draft and C.T.L., K.V., I.B., and P.R. reviewed and edited the manuscript.

Declaration of interests

P.R. is an editorial board member of the Biophysical Journal. P.R. is a consultant for Simula Research Laboratories in Oslo, Norway, and receives income. The terms of this arrangement have been reviewed and approved by the University of California San Diego, in accordance with its conflict-of-interest policies.

Editor: Rainer Boeckmann.