Cholesterol Mixtures: Comparison of Single

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Heat Capacity of DPPC/Cholesterol Mixtures: Comparison of Single Bilayers with Multibilayers and Simulations Paulo F. Almeida,* Faith E. Carter, Katie M. Kilgour, Matthew H. Raymonda, and Emmanuel Tejada Department of Chemistry and Biochemistry, University of North Carolina Wilmington, Wilmington, North Carolina 28403, United States

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S Supporting Information *

ABSTRACT: The excess heat capacity (ΔCp) of mixtures of dipalmitoylphosphatidylcholine (DPPC) and cholesterol (Chol) is examined in detail in large unilamellar vesicles (LUVs), both experimentally, using differential scanning calorimetry (DSC), and theoretically, using a three-state Ising model. The model postulates that DPPC can access three conformational states: gel, liquiddisordered (Ld), and liquid-ordered (Lo). The Lo state, however, is only available if coupled with interaction with an adjacent Chol. ΔCp was calculated using Monte Carlo simulations on a lattice and compared to experiment. The DSC results in LUVs are compared with literature data on multilamellar vesicles (MLVs). The enthalpy change of the complete phase transition from gel to Ld is identical in LUVs and MLVs, and the melting temperatures (Tm) are similar. However, the DSC curves in LUVs are significantly broader, and the maxima of ΔCp are accordingly smaller. The parameters in the Ising model were chosen to match the DSC curves in LUVs and the nearestneighbor recognition (NNR) data. The model reproduces the NNR data very well. It also reproduces the phase transition in DPPC, the freezing point depression induced by Chol, and the broad component of ΔCp in DPPC/Chol LUVs. However, there is a sharp component, between 5 and 15 mol % Chol, that the model does not reproduce. The broad component of ΔCp becomes dominant as Chol concentration increases, indicating that it involves melting of the Lo phase. Because the simulations reproduce this component, the conclusions regarding the nature of the phase transition at high Chol concentrations and the structure of the Lo phase are important: there is no true phase separation in DPPC/Chol LUVs. There are large domains of gel and Lo phase coexisting below Tm of DPPC, but above Tm the three states of DPPC are mixed with Chol, although clusters persist.

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ordered (Lo)−liquid-disordered (Ld) phase separation,3,16,17 as the result of simple lipid mutual pairwise interactions,26 or as due to the formation of condensed complexes?27−32 What determines the favorable DPPC−Chol interaction in the Lo phase? van der Waals contacts (enthalpy), configurational entropy, or exposure of nonpolar groups to water, as in the umbrella model?33 It has been believed for decades that DPPC is not a biologically relevant phospholipid (PL) because it does not exist in significant amounts in eukaryotic cell membranes, with the particular exception of lung surfactant, of which it is the main component. Therefore, DPPC has been mainly studied as a model for ordered, mostly saturated sphingomyelins, which are major components of eukaryotic plasma membranes and share with DPPC the same phosphocholine headgroup and a similar main phase transition or melting temperature (Tm). However, very recent lipidomics studies34−36 have shown that

INTRODUCTION The observation of the “condensing effect” of cholesterol (Chol) in mixtures with phosphatidylcholines (PC) goes back to the 1925 paper by Leathes.1 That same year, Ising published his now famous paper on the model that bears his name,2 as if the fates of those two papers were to be linked from conception, given the importance of the Ising model to understand PC−Chol interactions in membranes. Almost 50 years later, the condensing effect of Chol was cast in terms of a phase diagram in the pioneering work of Shimshick and McConnell.3 Since then, these mixtures have been continuously investigated,4−13 in particular in the case of dipalmitoylphosphatidylcholine (DPPC) and cholesterol.14−24 Indeed, a new phase diagram of DPPC/Chol mixtures has just appeared,25 which is shown in Figure 1. Why this continued interest? First, because the nature of the interaction between DPPC and cholesterol is not understood. The problem has been exceedingly difficult, and a consensus of the best concept under which to frame the heterogeneities observed in these mixtures in the liquid state has not been reached. Should these heterogeneities be described as liquid© XXXX American Chemical Society

Received: May 28, 2018 Revised: July 21, 2018

A

DOI: 10.1021/acs.langmuir.8b01774 Langmuir XXXX, XXX, XXX−XXX

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that extends to higher temperatures (see for example the lower right panel of Figure 3). This broad component has been interpreted as a transition between Lo and Ld phases16,17 or as the melting of condensed complexes,28 whereas the sharp component has been attributed to melting of regions of almost pure DPPC. The problem is that the sharpness of the transition in pure DPPC MLVs, which has a heat capacity maximum ΔCmax ≈ 10−100 kcal/(K mol),14,15,19,21,37−40 is p largely a consequence of multilamellarity.41 The peak is much smaller in DPPC large unilamellar vesicles (LUVs), where ΔCmax ≃ 3.5 kcal/(K mol).40−42 The extreme sharpness of the p DPPC phase transition in MLVs is probably a consequence of cooperativity in the third dimension, between the multiple bilayers of the MLV. In giant unilamellar vesicles (GUVs), which are similar to MLVs in size, the DPPC phase transition is very similar to that in LUVs.41 To date, all studies of ΔCp of DPPC/Chol mixtures have been performed in MLVs. The question is, does the sharp peak persist in mixtures with Chol in LUVs? Surprisingly, this problem has never been investigated. Therefore, there is a legitimate concern that this intriguing feature of the heat capacity of DPPC/Chol mixtures may be no more than a peculiarity of MLVs. Here we have conducted a thorough examination of the heat capacity of mixtures of DPPC and cholesterol, both experimentally and theoretically. On the experimental side, we present a complete thermodynamic and statistical analysis of the phase transition in LUVs of DPPC/Chol mixtures, which has never been performed, to the best of our knowledge. We previously published limited DSC data on DPPC/Chol LUVs at a few concentrations,43 which we have now significantly improved and expanded. Further, we have performed a thorough analysis of these results and compared them with the available data on MLVs. On the theoretical side, we analyzed the heat capacity in DPPC/Chol LUVs using a new three-state Ising model, different from that previously used for MLVs.26 In the new model, DPPC also has three thermodynamic states (gel, Ld, and Lo), but the Lo state only becomes available in the presence of Chol. The interactions between different lipid states or lipid species (A, B) are specified by interaction parameters defined by44−46

Figure 1. Phase diagram of DPPC/cholesterol. The data points that indicate the phase boundaries (black symbols) are derived from 2H NMR,25 courtesy of Dr. Jenifer Thewalt. The phase diagram was obtained with DPPC-d31 multilamellar vesicles (MLVs). The melting temperature of pure DPPC-d31 MLVs is 40.0 °C, which is 0.75 °C lower than the DPPC LUVs investigated (and simulated) here. The red circles indicate the compositions and temperatures of the snapshots shown in Figure 10.

the long-held belief that DPPC is a rare component of eukaryotic membranes was in error. DPPC is not the major ordered phospholipid in the plasma membrane, but it is an abundant component, as highlighted in red in Figure 2. Hence, DPPC is not valuable merely as a model for SM, but it is a biologically important subject of research in its own right. One of the features that has intrigued physical chemists is the peculiar excess heat capacity (ΔCp) profile associated with the main phase transition of saturated PCs in mixtures with cholesterol.9,14,15,19−22 Namely, at about 15−20 mol % Chol, differential scanning calorimetry (DSC) data of multilamellar vesicles (MLVs) of DPPC/Chol exhibit a sharp ΔCp peak close to the Tm of pure DPPC, followed by a broad transition

Figure 2. Lipid distributions in two different cellular membranes showing the concentration of DPPC. Left, giant plasma membrane vesicles (GPMVs) from mammalian cells.34 Right, synaptic cell membrane.35 The full lipidomics of several cellular membranes is available.36 Data kindly provided by Dr. Ilya Levental. B


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Figure 3. Excess heat capacity associated with the phase transition of DPPC/Chol mixtures in LUVs and MLVs measured by DSC. The LUV curves for the mixtures are averages of 7−22 scans in different samples (DPPC/Chol 95:5, 7 scans; 90:10, 17 scans; 85:15, 14 scans; 80:20, 22 scans; and 70:30, 7 scans). The DPPC scan is from a single sample and is taken from our previous paper.41 The MLV data were courtesy of McElhaney and Mannock,22 except for the 5% Chol curve, which is ours. The heat capacity curves in LUVs and MLVs were normalized to the average values of ΔH and Tm observed, respectively, over all of our LUV data or all of the MLV data analyzed in this study.

ω = ϵAB −

1 (ϵAA + ϵBB) 2

pH 7.5, 0.1 mM ethylene glycol-bis(β-aminoethyl ether)-N,N,N’,N’tetraacetic acid, 0.02% NaN3, and 100 mM KCl. In this process, the lipids were first mixed in chloroform/methanol 80:20 solutions in a round-bottom flask, and the solvent was removed by rotary evaporation under vacuum over a water bath at 60−70 °C. The lipid film was then dried under high vacuum for at least 3 h using an oil pump. In some samples, to assess the possibility of improving the molecular mixing in DPPC/Chol mixtures, the lipid mixtures were lyophilized51 after the high-vacuum step. To that end, the sample was dissolved in 5 mL of cyclohexane/methanol 99:1 and thoroughly vortexed. The solution was frozen under liquid nitrogen and then attached to a solvent trap for lyophilization under high vacuum. After all of the frozen solvent had sublimed, leaving a fine lipid powder, the round-bottom flask was placed under high vacuum for 10 h. With or without lyophilization, the lipid was then hydrated with MOPS buffer at 60−70 °C, with vortexing. This process results in suspensions of large multilamellar vesicles (MLVs). The vesicle suspensions were then extruded 10−20 times (using a Lipex Biomembranes extruder, Vancouver, Canada) through two stacked 0.1 μm pore size Nuclepore polycarbonate filters (Whatman, Florham, NJ), at 60−70 °C, under 400 psi. Lipid concentrations were determined by the Bartlett phosphate assay.52 Especially in the case of pure DPPC, it is important to prepare the vesicles at lipid concentrations ≤1 mM to minimize the presence of MLVs, which may form by fusion of LUVs. Differential Scanning Calorimetry. LUV suspensions were degassed under a vacuum of 500 mmHg for 10 min prior to calorimetry. The heat capacity of the LUVs was measured by differential scanning calorimetry (DSC) using a high-sensitivity Nano DSC (TA instruments, New Castle, DE), equipped with 300 μL twin gold capillary cells, pressurized to 3 atm prior to the scan. The heating scan rate was 0.1 °C/min, which is sufficiently slow to ensure equilibrium. Repeated scans (two cycles of heating and cooling) were

(1)

which are equivalent to the parameter J in the standard Ising model with ω = 2J.45 Here, the ϵ represent actual interactions between adjacent molecules, but ω is a differential interaction. Since it is a difference, ω = 0 for like molecules (A = B). If ω > 0, the interaction is unfavorable, which is the most common occurrence in lipids.46 If ω < 0, the interaction is favorable, which is rare, but observed in DPPC/Chol in the Lo phase.47 The nearest-neighbor recognition (NNR) method48,49 provides a direct measure of the interaction parameter ω in a lipid membrane composed of two different chemical species. We tested the model in a stringent manner, requiring that it simultaneously describe the experimental DSC heat capacity data in LUVs and the NNR data available on DPPC/Chol mixtures.47 With an appropriate choice of parameters, the model is consistent with the NNR data and with most but not all features of the heat capacity data.

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MATERIALS AND METHODS

Preparation of Large Unilamellar Vesicles. Cholesterol (powder) and 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC, in chloroform solution) were purchased from Avanti Polar Lipids (Alabaster, AL). Organic solvents (HPLC, American Chemical Society grade) were from Burdick & Jackson (Muskegon, MI). Lipids and probes were tested by thin-layer chromatography and used without further purification. Large unilamellar vesicles (LUVs) were prepared by extrusion, as previously described,42,50 in a buffer containing 20 mM 3-(N-morpholino)propanesulfonic acid (MOPS), C


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However, the ω are only known from experiment as a function of temperature for DPPC/Chol in the Lo phase (ωoC) and the Ld phase (ωdC).47 Thus, we only have information on ΔhoC ≈ −2 kcal/mol and ΔhdC ≈ 0. In the absence of Δh, we assume that the lipid interactions are mainly enthalpic in origin; that is, we assume Δh ≈ ω in the calculation of the enthalpy of the lattice. This is the assumption made by all researchers in the field, including us.40,42,56,57 It is probably not a serious problem because the dominant contribution to ΔCp in eq 2 comes from the enthalpy differences between states (ΔH) and the fluctuations in the numbers of lipids in those states. Furthermore, it is the ω (not the Δh) that determines the magnitude of the fluctuations in the numbers of lipid states and in the numbers of unlike lipid contacts, thereby having the greatest influence on ΔCp. To obtain equilibrium configurations of the lattice and to calculate thermodynamic properties, Monte Carlo (MC) simulations were performed as previously described26,50 using standard methods.55−60 The lipids are exchanged by randomly selecting partners on the lattice, using a non-nearest-neighbor Kawasaki step,61 or the lipid state is changed, using a Glauber step.62 The choice between these two types of moves is aleatory. Acceptance or rejection of attempted moves is based on the Metropolis criterion63 with a probability that depends exponentially on the interaction free energy change, using a random number64 for the decision. The simulations include a preequilibration period of 105 MC cycles, followed by a period of 1−3 × 106 acquisition cycles. These periods were found to be sufficient to reach equilibrium, as judged by the heat capacity of the lattice (Figure S2) but also by the evolution of domain sizes (Figure S3). We rely mainly on ΔCp because, as a correlation (fluctuation) function, it converges much slower than the energy (or enthalpy). The simulations were performed in lattices of different sizes, from 10 × 10 to 300 × 300 sites (spanning 3 orders of magnitude in numbers of lipids), to assess the effect of system size. Size effects on the heat in the largest capacity are small (Figure S4). The value of ΔCmax p lattices is already well approximated in the 30 × 30 lattice. These validations are in agreement with our previous calculations26,50 but are included as Supporting Information for completeness and easy reference. Determination of Model Parameters. The parameters used in the model were chosen to obtain the best simultaneous match to the equilibrium constant for dimer exchange K experimentally determined by NNR and to ΔCp(T) measured by DSC. This process required extensive trial and error using both types of experimental data. The thermodynamic parameters for the transitions gel → Lo (ΔH1, ΔS1) and Lo → Ld (ΔH2, ΔS2) were chosen according to the following considerations. The Lo state should lie between the gel and the Ld states in terms of enthalpy and entropy and should result in an Lo state whose Gibbs energy is higher than both gel and Ld in the absence of Chol (Figure S1). Further, to obtain a match to the experimental heat capacity curves, we set ΔH1 < ΔH2, suggesting the Lo state is more similar to gel than to Ld. The values of ΔH1, ΔH2, ΔS1, and ΔS2 affect to some extent the ω parameters that provide the best match to experiment. However, small variations have little influence on the results. Interactions between different states of the same chemical species (DPPC) cannot be determined by NNR. Thus, the interaction ωgd = 300 cal/mol between gel and Ld states of DPPC was obtained by matching the maximum of ΔCmax p of pure DPPC in the simulations. In addition, the interaction between DPPC and Chol in the gel state (ωgC) is very difficult to obtain by NNR because of slow equilibration of the spatial lipid distributions. The value of ωgC chosen represents a compromise between reproducing the freezing point depression of DPPC induced by Chol and matching the ΔCmax p in LUVs, in addition to being consistent with the NNR data. Matching the experimental K in DPPC/Chol 90:10, in the Ld phase, imposes constraints mainly (but not only) on the values of ωdC and ωgC. To obtain the ωoC that provided the best match for K, we concentrated primarily on DPPC/ Chol 60:40, in the Lo phase. In the end, we settled on the parameter values listed in Table 1.

routinely performed to verify reversibility and reproducibility. The DSC curves were corrected by baseline subtraction as previously described.53 Statistical analyses of the calorimetry data were performed with R software.54 Model and Monte Carlo (MC) Simulations. The equilibrium properties of the membrane were calculated using a new three-state Ising model. The relation between this and the model previously used26 will be addressed in the discussion. The lipid membrane is represented by a triangular lattice, typically of 100 × 100 sites, with skew-periodic boundary conditions.55 Each site on the lattice is occupied by a phospholipid (DPPC) or a Chol molecule. The model postulates that a DPPC molecule has access to three conformational states: gel (used as reference), Lo, and Ld. These are internal molecular states of the phospholipid in a statistical−mechanical sense. But we establish a correspondence between these conformational states and the phases originally defined in the phase diagram.16,17 Since Chol does not undergo a melting transition, its contribution to the enthalpy can be ignored. There are three enthalpy and entropy changes (assumed independent of temperature) associated with transitions between these states. The enthalpy change associated with the main phase transition of pure DPPC (ΔH0) and the melting temperature (Tm) are obtained experimentally by DSC. The entropy change is ΔS0 = ΔH0/Tm. In DPPC/Chol mixtures, the model postulates additional enthalpy and entropy changes between gel and Lo (ΔH1, ΔS1) and between Lo and Ld (ΔH2, ΔS2). The values of these parameters are listed in Table 1. They were chosen in such a

Table 1. Thermodynamic Parameters Used in the MC Simulationsa gel → Ld

gel → Lo

Lo → Ld

Tm

ΔH0

ΔS0

ΔH1

ΔS1

ΔH2

ΔS2

313.9 ω gd

8.7

27.72

3.0

8.05

5.7

ωog

ωod

ωdC

ωgC

ωoC

19.67 ΔhoC

300

260

300

50

200

4.3T − 1717

−1.72

ΔH in kcal/mol, ΔS in cal/(K mol), ω in cal/mol, T in K.

a

way that the Gibbs energy of the Lo state is above those of the gel and Ld states in the absence of interaction with Chol. Figure S1 shows a schematic of the transitions and the approximate values of these parameters. Thus, in the simulations, when a DPPC molecule transitions from gel → Ld, ΔH = 8.7 kcal/mol; for the transition gel → Lo, ΔH = 3.0 kcal/mol; and for Lo → Ld, ΔH = 5.7 kcal/mol. The distinct feature of the new model is that the transition of DPPC from the gel or the Ld state to the Lo state is only allowed if coupled with the interaction with an adjacent Chol molecule. If separated from Chol, the Lo state converts to gel or Ld. The interactions between unlike nearest neighbors on the lattice are specified by six interaction parameters ω defined by eq 1 and listed in Table 1: ωgd (gel−Ld), ωog (Lo−gel), ωod (Lo−Ld), ωdC (Ld−Chol), ωgC (gel−Chol), and ωoC (Lo−Chol). The excess heat capacity at constant pressure is obtained from the enthalpy fluctuations44,56,57 ΔCp =

⟨H2⟩ − ⟨H ⟩2 RT 2

(2)

where R is the gas constant and T is temperature. In a membrane with several states of the various lipids, the enthalpy includes the enthalpies (ΔH) of the various lipid states relative to a common reference, which we chose as the gel state. The enthalpy of the lattice also includes the mutual interaction enthalpies (Δh) between unlike lipids or lipid states, which can be obtained from the derivative of the corresponding ω with respect to reciprocal temperature Δh =

d(ω /T ) d(1/T )

(3) D


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RESULTS AND DISCUSSION Heat Capacity of DPPC/Cholesterol in LUVs: Comparison with MLVs. Here we present a thorough investigation of the phase transition of DPPC/Chol mixtures in LUVs. This transition has been extensively studied in MLVs but not in LUVs. The only published DSC data on DPPC/Chol LUVs that we know of are a limited set from our own laboratory,43 which we have now significantly improved and expanded. The goal of that work was to assess the difference in the interaction of DPPC with Chol and 18,19-dinorcholesterol, a “smooth” version of cholesterol, lacking the methyl groups on the steroid rings. We found those interactions to be very similar.43 It is important to carefully examine the transition in LUVs because they are the most common and the best model for the lipid bilayer that is accessible to thermodynamic measurements, using calorimetry or fluorescence binding experiments, for example. LUVs are largely unstrained, as opposed to small unilamellar vesicles (SUVs) prepared by sonication. GUVs are perhaps better models for biological membranes, but they are not easily amenable to those experiments. To our knowledge, the only DSC study of GUVs came from our laboratory, where we investigated the DPPC main phase transition and compared it to the transition in MLVs and LUVs.41 We found that the DPPC phase transition in GUVs, which have a single bilayer, is similar to that in LUVs. The transition widths are similar and the maxima of the heat capacity are similar, with ΔCmax ≈ 3.5 p kcal/(K mol) in LUVs and 5 kcal/(K mol) in GUVs. But the transition is much sharper in MLVs, with ΔCmax about 10 p times larger. This is probably a consequence of coupling of the phase transition between the multiple bilayers of the MLV.41 Thus, major features of the transition in MLVs may not be representative of single lipid bilayers. Therefore, we have performed extensive DSC experiments in mixtures of DPPC/Chol in LUVs and now present the results in comparison with MLVs. There is more variability in the DSC scans in LUVs. Hence, it was important to record a large data set to obtain representative results. Figure 3 compares ΔCp(T) in LUVs and MLVs.22 A few aspects are immediately evident. First, in pure DPPC and in DPPC/Chol mixtures with small amounts of cholesterol (≤10%, top panels; notice the different scales of Cp on the left and right), ΔCmax is much p larger in MLVs than in LUVs, but the drop in the value of ΔCmax as Chol increases is much smaller in LUVs. Second, the p Tm shift to lower temperatures (freezing point depression) is much more pronounced in MLVs. This means that the interaction of DPPC and Chol in the gel phase is not as unfavorable in LUVs as it is in MLVs. Moreover, the Tm in DPPC LUVs (40.7 °C) is about 0.6 °C lower than in MLVs (41.3 °C). These observations suggest that the gel phase is slightly less ordered in LUVs: it is more tolerant of Chol and suffers less additional disordering with increasing amounts of Chol. Now we examine the three most important thermodynamic parameters of the phase transition: the heat capacity maximum (ΔCmax p ), the melting temperature (Tm, understood as the temperature of ΔCmax p ), and the enthalpy change (ΔH) of entire phase transition. Table 2 summarizes the thermodynamic data in LUVs. Figure 4 compares the Tm in LUVs and MLVs. Broadly, the behavior is very similar, but the freezing point depression is less pronounced in LUVs. Figure 5 compares the enthalpy change per lipid (DPPC + Chol) in LUVs and MLVs. Here, the behavior is even more strikingly

Table 2. Thermodynamic Parameters in DPPC/Chol LUVs (Average ± Standard Error of the Mean)a lipid

Tm (°C)

ΔH (kcal/mol)

ΔCmax p (cal/(K mol))

N samples

DPPC DPPC/Chol 95:5 90:10 85:15 80:20 70:30

40.71 ± 0.04

8.8 ± 0.2

3.53 ± 0.04

27

± ± ± ± ±

7 21 15 25 6

a

40.6 40.3 40.2 40.9 42.8

± ± ± ± ±

0.1 0.1 0.2 0.2 1.1

7.6 6.8 5.1 3.5 1.6

± ± ± ± ±

0.7 0.3 0.4 0.2 0.2

2.6 2.3 1.3 0.41 0.09

0.4 0.2 0.14 0.03 0.01

Enthalpy and heat capacity are reported per lipid (DPPC + Chol).

similar. The regression lines are identical in both panels of Figure 5, indicating that the enthalpy changes are the same in LUVs and MLVs. Finally, Figure 6 compares the dependence of ΔCmax p on Chol concentration. Here the differences between LUVs and MLVs are large. The heat capacity is much larger in MLVs and drops more steeply with Chol concentration. Simulations of the Heat Capacity. The experimentally observed heat capacity curves of DPPC/Chol LUVs over the transition from the gel to the Ld phase were interpreted with a new three-state Ising model, where Lo is a state of DPPC associated with Chol. The aim is to reproduce the experimental heat capacity of DPPC/Chol LUVs. Our original three-state model26 was able to reproduce several features of the dependence of ΔCp on Chol concentration in MLVs, but could not quantitatively yield the values of ΔCmax p , except in pure DPPC. In the mixtures, the curves were significantly broader than the experimental ones.26 The question is whether this was due to the very sharp peak in MLVs, which is probably caused by cooperativity between bilayers. Here we examine the issue in LUVs, where the peaks are not nearly as sharp (Figure 3). Our original three-state model26 was conceptually based on the allosteric model of Monod, Wyman, and Changeux (MWC) for the binding of oxygen by hemoglobin.65 In the MWC model, the hemoglobin conformation that binds oxygen, the R state, is available but scarcely populated in the absence of oxygen; it becomes the dominant state when hemoglobin binds oxygen. The same assumption was made then for the Lo state of DPPC, which was available in the absence of cholesterol but only became likely if a cholesterol molecule was adjacent to DPPC.26 The new model is conceptually similar to the classical alternative for binding of oxygen to hemoglobin, which is the Pauling model,66 better known today as the sequential or KNF model, since its expansion by Koshland, Némethy, and Filmer, 30 years later.67 The essence of this model is that the R state does not exist if hemoglobin is free (has no oxygen bound). The change from the T to the R conformation occurs concomitant with oxygen binding (induced fit). Similarly now, the transition of a DPPC molecule from the gel or the Ld state to the Lo state is only allowed if coupled with the interaction with a Chol molecule that is a nearest neighbor of this DPPC in the membrane. Further, the new model differs from the original one26 in the way the lipid−lipid interaction parameters are chosen. Fundamental to the Ising model are the interactions between nearest neighbors in the lattice. Those interactions occur between different lipids (DPPC, Chol) or different states (gel, Ld, Lo) of the phospholipid in the membrane, represented as a lattice. These differential interactions are represented by E


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Figure 4. Melting temperature (Tm) in LUVs and MLVs. The LUV data is from the present study. The red points correspond to individual samples, and the black points are the means at each composition. The error bars represent the 95% confidence interval of the mean. The MLV data are from the complete data sets available, colored according to the laboratory of origin.14,15,19,20 The lines in both panels are the best smooth lines through the data.

Figure 5. Enthalpy change (ΔH) per lipid molecule (DPPC + Chol) in LUVs and MLVs. The LUV data is from this study. The red points correspond to individual samples, and the black points are the means at each composition. The error bars represent the 95% confidence interval of the mean. The regression line is ΔH = 8.7 − (0.243 ± 0.008) × (Chol mol %) kcal/mol. The MLV data are from the complete data sets available, colored according to the laboratory of origin.14,15,19,20 The regression line is ΔH = 8.7 − (0.242 ± 0.016) × (Chol mol %) kcal/mol, identical to that in LUVs (the intercept was fixed at the standard 8.7 kcal/mol).

The ω between different lipid species can be obtained from experiment through the NNR method.48,49 In this method, analogs of the different lipids (DPPC, Chol) are prepared so that they can establish a disulfide bond. Thiol−disulfide exchange in mixed membranes containing these analogs is allowed to occur until the system reaches equilibrium. Then, the reaction is quenched (by a sudden pH drop), and the concentrations of homodimers (AA and BB) and heterodimers (AB) are determined by HPLC. The quantity directly calculated from experiment is the equilibrium constant

parameters ω44,46 defined by eq 1, also known as Flory parameters.68 In eq 1, the interactions ϵ between adjacent molecules are free energies if they depend on temperature.44 Therefore, in general, ω is also a free energy. Under the experimental conditions used (thermodynamic variables T and P), ω is a Gibbs energy. Hence, ω = Δh − TΔs, where Δh and Δs are defined by relations analogous to eq 1 for enthalpy or entropy differences. The contact enthalpy change Δh can be detected experimentally as the temperature dependence of ω (eq 3). Indeed, the interaction parameter ωoC between DPPC and Chol in the Lo phase depends on temperature, yielding ΔhoC ≈ −2.0 kcal/mol.47 Because of the coupling of the DPPC transition from Ld (or gel) to Lo with the interaction with a Chol nearest neighbor, the enthalpy of the phospholipid changes concomitant with the change in nearest-neighbor interaction. That is, ΔhoC < 0 effectively lowers the enthalpy of the Lo state.

K=

[AB]2 [AA][BB]

(4)

Previously, we calculated ω from K through 1 K ω = − RT ln 2 4 F

(5) DOI: 10.1021/acs.langmuir.8b01774 Langmuir XXXX, XXX, XXX−XXX

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Figure 6. Heat capacity maxima (ΔCmax p ) per lipid molecule (DPPC + Chol) in LUVs and MLVs. The LUV data is from this study. The red points correspond to individual samples, and the black points are the means at each composition. The error bars represent the 95% confidence interval of the mean. The best smooth line through the means is shown. The MLV data are from the complete data sets available, colored according to the laboratory of origin.14,15,19,21 The lines are simply to guide the eye.

This equation is obtained from the quasi-chemical (QC) approximation in lattice statistics. The QC approximation assumes that pairs of nearest neighbors are independent, thus ignoring their correlations in the lattice.44,69 The factor of 1/2 arises because ω is defined for one AB dimer, whereas K is defined for the formation of a pair of dimers (eq 4). The factor of 4 in eq 5 arises because AB pairs are statistically favored 2:1 compared to AA or BB pairs. This results in K = 4 in eq 4 for a random mixture. Thus, in a random mixture ω = 0 (the interaction ϵ between A and B is identical to the average of the interactions between A and B). Values of K > 4 indicate a favorable interaction between A and B and therefore a negative free energy change (ω < 0) for the formation of an AB heterodimer. We say that interactions with ω < 0 are “attractive.” Values of K < 4 indicate an unfavorable interaction between A and B, compared to the A−A and B−B interactions. Therefore, a positive free energy change (ω > 0) occurs for the formation of an AB heterodimer. We say that interactions with ω > 0 are “repulsive.” (Of course, the true interactions ϵ in eq 1 are always attractive.) As recently pointed out, the QC approximation only works well if the deviation from ideal mixing is not too large.69 Therefore, we assessed the deviations of the QC approximation from the value obtained by simulation. Figure 7 shows a comparison of ω calculated from MC simulations and from the QC approximation in lipid mixtures at 45 °C. PL designates any phospholipid. If each species is entirely in one state and at least one of them is dilute (white circles), the QC approximation is excellent, and essentially exact for ω < 200 cal/mol in absolute value. (We use a triangular lattice instead of a square lattice, so the parameter values are somewhat different from May’s.69) However, in the mixtures of DPPC/Chol investigated here, where the phospholipid can access different states (gel, Lo, Ld) in the temperature interval of interest, the deviations of the QC approximation from the simulation results can be considerable (Figure 7). Therefore, to determine ω in DPPC/Chol mixtures, the equilibrium constant K was directly calculated in the MC simulations through

Figure 7. Comparison of dependence ω on K obtained from MC simulations (circles) and the QC approximation (line). The simulations were performed on a mixture of an entirely fluid phospholipid (PL) and cholesterol (PL/Chol 97.5:2.5, open circles) and in the DPPC/Chol model studied here (red circles), at 45 °C, in a 100 × 100 lattice.

K=

⟨AB⟩2 ⟨AA⟩⟨BB⟩

(6)

where ⟨AA⟩, ⟨BB⟩, and ⟨AB⟩ are the simulation averages of the numbers of AA, BB, and AB types of nearest neighbors in the lattice, respectively. The values of ω were varied in the simulations until a match between the calculated and the experimental K was obtained. Determining the parameters that provide the best simultaneous match to K (NNR) and ΔCp (DSC) required an extensive process of trial and error using both types of experimental data. We finally settled on the set of parameter values listed in Table 1. We obtained ωgC = 200 cal/mol (gel phase) and ωdC = 50 cal/mol (Ld phase). For reference, the value of ωdC = 20 cal/mol in the Ld phase is calculated directly with the QC approximation (eq 5) from K = 3.74, which is the G


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freezing point depression than observed experimentally. We are surprised by the inability of the model to quantitatively reproduce the sharp component of ΔCp in DPPC/Chol mixtures. The Ising model works very well for pure DPPC MLVs, LUVs, and SUVs,40,42,56 in mixtures of DPPC with 1palmitoyl-2-oleoylphosphatidylcholine,42 and also in mixtures of distearoylphosphatidylcholine and dimyristoylphosphatidylcholine.58,74 In DPPC/Chol MLVs, only a qualitative agreement was obtained.26 In LUVs, the model performs better, but the agreement with the DSC experiments is not complete. We took enormous care to ensure that the sharp heat capacity peak observed in LUVs between ≈5 and 15% Chol is not an artifact from macroscopic demixing that could result in the presence of a vesicle population much richer in DPPC. To that end, we varied the method of LUV preparation, including sometimes a second mixing and lyophilization from cyclohexane/methanol. But the sharp peak remains, though less prominent than in MLVs. We are quite certain that this sharp component is not artifactual. If so, it suggests we are missing an element, in the simple Ising model used, that affects the heat capacity mainly at low Chol concentrations. Nevertheless, the model captures some of the most important features of the experiment. The agreement is best with the broader component of the DSC curves, which becomes dominant as the cholesterol concentration increases and the system shifts toward the Lo phase. Therefore, results of the simulations should be especially relevant for this broad component of the phase transition and for the structure of the Lo phase. A useful aspect of MC simulations is the visual insight they provide into the organization of the membrane at the molecular level. Figure 10 shows a few snapshots of the lattice at different temperatures and membrane compositions (gel is white, Ld is black, Lo is blue, Chol is red). These temperature−composition points are indicated by the red circles in the phase diagram shown in Figure 1. In DPPC/Chol 95:5 at 30 °C, which is in the gel phase but close to the line that separates the gel from the gel−Lo coexistence region, the membrane is mainly uniform, with a few Lo clusters (red-blue) in the middle of the gel phase. In DPPC/Chol 80:20 at 30 °C, corresponding to gel−Lo coexistence regions below the Tm of DPPC in the phase diagram, the simulations yield a membrane with large gel and Lo domains. In this region, two-component 2 H NMR spectra have been observed,25,75 suggesting phase separation. The domains in the simulations are large but do not constitute macroscopic phase separation (Figure 10, top row, middle snapshot), contrary to the case in MLVs.26 The difference arises because of the much smaller unfavorable interaction between Chol and gel in LUVs (ωgC = 200 cal/ mol) than in MLVs (350 cal/mol). In DPPC/Chol 80:20 at about 45 °C, which is in the middle of the Ld−Lo coexistence region shown in the phase diagram (Figure 1), the simulations do not show a two-phase region, but a single phase with Chol and the various states of DPPC well mixed (Figure 10). This is in agreement with a variety of physical measurements that do not show liquid phase separation in this region76,77 but in disagreement with the classical papers16−18 and with recent X-ray diffraction and NMR works.4,78 However, the most recent 2H NMR spectra in the Ld−Lo coexistence region exhibit broadening but not two distinct components.25 This indicates compositional heterogeneities rather than true phase separation, in agreement with our simulations.

average value for DPPC/Chol mixtures containing only 2.5% Chol at 45 °C in all available NNR data together.70−73 Thus, ωdC ≈ 0 (K ≈ 4), indicating ideal mixing of Chol and DPPC in the Ld phase. Furthermore, ωdC is essentially independent of temperature, which means that ΔhdC ≈ 0 in the Ld phase.47 The parameter ωoC for the DPPC−Chol interaction in the Lo phase is temperature dependent. We obtained ωoC = −1717 + 4.3T (cal/mol), which yields ΔhoC = −1.7 kcal/mol and ωoC ≈ −410 cal/mol at 30 °C, −350 cal/mol at 45 °C, and −280 cal/ mol at 60 °C. The parameters listed in Table 1 were used to calculate K in the simulations for all mixtures. Figure 8 shows the comparison of the calculated K with those derived from NNR experiments47 in DPPC/Chol mixtures containing 10, 25, 35, and 40 mol % Chol, as a function of temperature.

Figure 8. Equilibrium constant K obtained experimentally by NNR and calculated in the MC simulations as a function of temperature in DPPC/Chol mixtures. The experimental data is from ref 47. The match was obtained in DPPC/Chol 60:40 between 45 and 60 °C (Lo phase) with ωoC = −1717 + 4.3T (cal/mol). The values of K were calculated using ωdC = 50 cal/mol (Ld phase) and ωgC = 200 cal/mol (gel phase) for DPPC/Chol 90:10, 75:25, 65:35, and 60:40.

The results of the MC simulations of ΔCp(T) are compared to the experimental data in Figure 9. The simulations match the experiment in several aspects. First, the Tm is well matched as a function of Chol. Second, the simulations correctly capture the broadening of the experimental DSC curves with increasing Chol concentration. In fact, they match very well the shape and the maximum of the broad components of the experimental heat capacity. Note that the uncertainty in the baseline correction of the experimental curves increases with Chol concentration. In fact, what the simulations suggest is that for Chol concentrations >15 mol %, there has been perhaps an excessive baseline subtraction in the experimental DSC curves, both in this work and that published on MLVs. Experimentally, ΔH of the phase transition decreases linearly with Chol concentration, identically in LUVs and MLVs, reaching zero at about 36 mol % Chol (Figure 5). In the simulations, ΔH does not approach zero as in the experiment (Figure 9). However, we must keep in mind that, in mixtures, ΔH also includes contributions from interactions between unlike lipid neighbors, which are model dependent. What the simulations do not match is the sharp component, except in pure DPPC. It is possible to increase the heat capacity maximum by increasing ωgC, but this results in a larger H


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Figure 9. Experimental (solid lines) and simulated (symbols) heat capacity curves of DPPC and DPPC/Chol mixtures. The MC simulations were performed with the parameters listed in Table 1. See legend of Figure 3 for additional details on the experimental curves.

Figure 10. Snapshots of the MC simulations for DPPC/Chol 95:5 (gel phase); DPPC/Chol 80:20 at 30 °C (gel−Lo coexistence) and at 45 °C (Ld−Lo coexistence); and DPPC/Chol 70:30 (Lo phase) at 30, 45, and 60 °C. The lattices shown have 100 × 100 sites. White represents DPPC in the gel state; black, in the Ld state; and blue, in the Lo state; red represents Chol.

Finally, consider the structure of the Lo phase. The bottom panels of Figure 10 show snapshots of DPPC/Chol 70:30, in the Lo phase, just outside the gel−Lo region of the phase diagram (Figure 1) as a function of temperature. At 30 °C, the Lo phase contains a few gel clusters (white). At 45 °C, the system is essentially homogeneous, with lipids of all species and states present but mixed. At 60 °C, which is close to the putative critical point of the Ld−Lo two-phase region in the phase diagram, the membrane consists essentially of small Ld

and Lo clusters. In recent atomistic molecular dynamics (MD) simulations of ternary mixtures of DPPC, Chol, and an unsaturated phospholipid, the Lo phase appears to consist essentially of “broken-up” gel, where Chol partitions to interfaces between gel and Ld regions.79,80 The Lo phase we observe is a more intimate mixture of DPPC and Chol than seen in the atomistic simulations, and it shares similarities with that observed in coarse-grained MD simulations in DPPC/ Chol mixtures.7 It will be interesting to find out what our I


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model yields for ternary mixtures similar to those examined by the atomistic simulations.79,80 This work is under way in our laboratory.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by grant CHE-1464769 from the National Science Foundation. We are grateful to Carolyn and David McLemore for a scholarship award to Katie Kilgour. It is a pleasure to thank Ron McElhaney and David Mannock for their DSC data in MLVs, Jenifer Thewalt for the phase diagram data, and Ilya Levental for the lipidomics data.

CONCLUSIONS The phase transition of DPPC/Chol mixtures is similar in LUVs and MLVs with regard to Tm, ΔH, and the general shape of ΔCp(T) measured by DSC. There is, however, one important difference: the ΔCp curves are much broader in LUVs, with a concomitant large decrease in their maximum value (ΔCmax p ). The experimental results are interpreted using a modified Ising model, in which a DPPC molecule has access to three conformational states: gel, Lo, and Ld. Those molecular states are defined to correspond to the phospholipid conformations that dominate the three phases traditionally identified in the original phase diagram of DPPC/Chol mixtures. Monte Carlo simulations of this model in a triangular lattice were used to calculate the thermodynamic properties and visualize the distribution of lipids in the membrane using the thermodynamic parameters of DPPC from experiment (Tm, ΔH) and lipid interaction parameters (ω) derived from comparison of the simulations with ΔCp and nearest-neighbor recognition (NNR) experiments. The modified Ising model reproduces well the NNR data, the freezing point depression induced by Chol (Tm), and the broad component of the DSC curves in DPPC/Chol mixtures, although the sharper component of ΔCp at low Chol concentrations is not matched. The broad component of ΔCp(T) corresponds to the region of the phase transition that involves the Lo state, which dominates the system at high Chol concentrations. Therefore, the results of the simulations are especially interesting in this region. Between about 10 and 30 mol % Chol and below the Tm of DPPC, large gel and Lo domains coexist, but true macroscopic phase separation is not observed. Above the Tm, there is certainly no liquid phase separation between Ld and Lo phases. Rather, an intricate mixture of all states of DPPC and Chol exists, albeit with some clustering. The Lo phase revealed by the MC simulations at 30 mol % Chol is a well-mixed membrane of Chol and DPPC in the Lo state, containing clusters of gel at low temperatures or Ld at high temperatures.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b01774. Schematic of the enthalpy, entropy, and Gibbs energy changes for the transitions between the three conformational states of DPPC (Figure S1); convergence of ΔCp with its final value in the MC simulations for DPPC, DPPC/Chol 80:20, and DPPC/Chol 60:40 (Figure S2); size distribution of DPPC clusters in the Ld state at Tm (Figure S3); dependence of ΔCp on the lattice size for DPPC, DPPC/Chol 80:20, and DPPC/Chol 60:40 (Figure S4) (PDF)

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Paulo F. Almeida: 0000-0003-4591-938X J


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