How to Determine Lipid Interactions in Membranes from Experiment

Invited Feature Article Cite This: Langmuir 2019, 35, 21−40

pubs.acs.org/Langmuir

How to Determine Lipid Interactions in Membranes from Experiment Through the Ising Model Paulo F. Almeida*

Langmuir 2019.35:21-40. Downloaded from pubs.acs.org by UNIV OF FLORIDA on 01/09/19. For personal use only.

Department of Chemistry and Biochemistry, University of North Carolina Wilmington, Wilmington, North Carolina 28403, United States ABSTRACT: The determination and the meaning of interactions in lipid bilayers are discussed and interpreted through the Ising model. Originally developed to understand phase transitions in ferromagnetic systems, the Ising model applies equally well to lipid bilayers. In the case of a membrane, the essence of the Ising model is that each lipid is represented by a site on a lattice and that the interaction of each site with its nearest neighbors is represented by an energy parameter ω. To calculate the thermodynamic properties of the system, such as the enthalpy, the Gibbs energy, and the heat capacity, the partition function is derived. The calculation of the configurational entropy factor in the partition function, however, requires approximations or the use of Monte Carlo (MC) simulations. Those approximations are described. Ultimately, MC simulations are used in combination with experiment to determine the interaction parameters ω in lipid bilayers. Several experimental approaches are described, which can be used to obtain interaction parameters. They include nearest-neighbor recognition, differential scanning calorimetry, and Förster resonance energy transfer. Those approaches are most powerful when used in combination of MC simulations of Ising models. Lipid membranes of different compositions are discussed, which have been studied with these approaches. They include mixtures of cholesterol, saturated (ordered) phospholipids, and unsaturated (disordered) phospholipids. The interactions between those lipid species are examined as a function of molecular properties such as the degree of unsaturation and the acyl chain length. The general rule that emerges is that interactions between different lipids are usually unfavorable. The exception is that cholesterol interacts favorably with saturated (ordered) phospholipids. However, the interaction of cholesterol with unsaturated phospholipids becomes extremely unfavorable as the degree of unsaturation increases.

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INTRODUCTION In a short piece of fantastic literature the Argentinian writer Jorge Luis Borges tells of an empire where cartography reached such accuracy that eventually a map was produced, with the size of the whole empire and coinciding pointwise with it. Later generations, however, found this map cumbersome, and abandoned it, together with the art of cartography.1 The Ising model2 is the opposite of a detailed map. It was originally conceived to understand ferromagnetism in statistical mechanical terms, but it applies as well to the problem of lipid interactions in membranes. Because of its simplicity and generality, the Ising model has proven extremely useful for generations of physicists and chemists and, more recently, biologists. Bilayers of lipid mixtures are complex systems. One approach to understand them is to use computational descriptions of the lipid molecules that are as accurate as possible, to simulate membrane behavior using Molecular Dynamics (MD) simulations. This approach has provided very stimulating insight and opened new questions about the interactions in membranes composed of phospholipids and cholesterol.3,4 However, in complex problems, my preference is to ask questions that address basic aspects, sacrificing the accuracy of representation to the simplicity of the answer. © 2018 American Chemical Society

Thus, I have preferred to use Monte Carlo (MC) simulations of Ising models to the much more realistic MD simulations as a means to understand lipid interactions in bilayers. This is the approach described by M. E. Fisher,5 “Rather, the aim of theory of a complex phenomenon should be to elucidate which general features of the Hamiltonian of the system lead to the most characteristic and typical observed properties. Initially one should aim at a broad qualitative understanding, successively refining one’s quantitative grasp of the problem when it becomes clear that the main features have been found.” In the case of mixtures of phospholipids and cholesterol, the main features have probably not yet been found. Before discussing our work on lipid interactions using the Ising model, I want to mention a different aspect in which the Ising model has been recently used to understand the behavior of lipid membranes. This work comes mainly from the laboratories of Keller6 and Veatch.7 I especially recommend their 2009 review8 as well as some recent exciting developments in the biological sciences.9,10 Received: September 7, 2018 Revised: December 10, 2018 Published: December 27, 2018 21

DOI: 10.1021/acs.langmuir.8b03054 Langmuir 2019, 35, 21−40

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Invited Feature Article

To briefly describe that work, consider the schematic diagram of Figure 1. The figure shows a phase diagram of two

Figure 2. Compositional fluctuations in a membrane of a GUV of DPhyPC/DPPC/Chol 25:20:55 as a function of temperature.8 The light domains contain trace amounts of a fluorescent lipid for the purpose of visualization. Scale bar, 20 μm. Reprinted with permission from Honerkamp-Smith et al.8 Copyright 2009 Elsevier.

predictions about the critical point renders credibility to the model and is in itself an exciting result. However, by the same token, this universality does not allow for the derivation of any conclusions regarding lipid interactions in membranes.8 Indeed, no information regarding lipid interactions can be obtained from the critical exponents. This article, however, is about lipid interactions in membranes and how to obtain them from the Ising model in combination with experimental data. I have often been asked questions such as, How can we accurately measure interactions between lipids in a bilayer? What do they mean physically? Are these energies or free energies? If we know the values of these interactions what can we predict about the lipid behavior in complex mixtures? Can we extend those predictions to biological membranes? My answer to some of those questions is outlined here. I discussed some of these matters in a review on lipid interactions 10 years ago.11 The point of this article, however, is not to update that review. Rather, the motivation was the opportunity to convey in one place what I have understood about the measurement and meaning of interactions in lipid bilayers and what those interactions tell us about the behavior of membranes. I have been working on this problem for 20 years, but some questions have been especially difficult, and new questions have arisen. It is at the interface between different approaches that I have found research to be most interesting and scientific discussions to be most fruitful. Over the last 12 years, I have had the privilege of collaborating extensively with Steve Regen, from Lehigh University. Most of the data on the experimental measurement of lipid interactions discussed here comes from his laboratory, some of which was the result of our work together. In the case of a membrane, the essence of the Ising model is that each lipid is represented by a site on a lattice and that the interaction of each site with its nearest neighbors is represented by an energy parameter, which we call ω. If the temperature is varied so that it spans a phase transition, the melting temperature and the enthalpy change of the lipids involved are also necessary to describe the thermodynamic behavior of the membrane. But the details of the lipid molecular structures and how they are represented in the lattice do not matter. For example, phospholipids containing two identical acyl chains, such as DPPC, can be represented in various ways on a lattice. In some studies each lattice site represents an independent acyl chain.12−15 In others, each site represents a whole lipid.16−22 Yet in other studies, a phospholipid molecule is represented by two adjacent sites on the lattice, each corresponding to one acyl chain.12,23 The two chains are physically connected (they always move together) and are infinitely coupled thermodynamically (they change state in a concerted manner). In the Ising model, though, none of those details matter.12 The properties

Figure 1. Phase diagram of a mixture of two components X and Y as a function of temperature and mole percentage of component X. The shaded area is a region of coexistence of two liquid phases A and B.

components, as a function of temperature (T) and composition, given by the mole % of one of the components (X). In the shaded region below the curve, a two-phase system exists, formed by two liquids A an B that are immiscible at low temperatures. For example, a system with 50% component X at 30°C (open red symbol) consists of a mixture of liquid A, which contains 30% of component X, and liquid B, which contains 70% of this component. As the critical temperature Tc = 50 °C is approached from below, the compositions of the two liquids in equilibrium become progressively more similar, until they become identical at T = Tc. This is the critical point. Above Tc there is only one liquid. One of the predictions of the Ising model is the scaling behavior of various properties, such as the correlation length of compositional fluctuations, the line (or surface) tension, and the heat capacity. This behavior is expressed mathematically in the form of scaling laws. For example, near the critical point the line tension λ scales as λ≈

T − Tc Tc

−ν

(1)

where ν is a critical exponent. Similarly, the correlation length ξ of compositional fluctuations scales as ξ≈

T − Tc Tc

μ

(2)

where μ is another critical exponent. In lipid vesicles, it has been found that those properties scale according to the predictions of the Ising model.6−8 Namely, in the twodimensional Ising model ν = 1 and μ = 1, and the values found experimentally in membranes are consistent with these predictions. A beautiful illustration of this behavior is shown by the changes in the lipid domains in a giant unilamellar vesicle (GUV) shown in Figure 2.8 The figure shows the compositional fluctuations of liquid domains in a vesicle of a ternary mixture of diphytanoylphosphatidylcholine (DPhyPC)/dipalmitoylphosphatidylcholine (DPPC)/cholesterol (Chol) as the temperature is changed through Tc ≈ 32°C, in this mixture. The critical exponents are universal. This means that any two-dimensional system, irrespective of molecular details and interactions, has the same critical exponent in the Ising model. Thus, the agreement of the experiment with the Ising model 22


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The spin can have two values corresponding to two possible orientations, up (↑, :i = +1) or down (↓, :i = −1). Nearestneighbor spins (i,j) in the lattice interact with each other with a strength given by the interaction parameter J. In a ferromagnet J > 0. Each interaction contributes −J :: i j to the energy of the lattice. Then, the interaction is favorable if both spins are up (↑ ↑) or down (↓↓), but it is unfavorable if one spin is up and the other down (↑↓). Thus, in ferromagnetic materials, the spins tend to align with each other in a parallel orientation. However, if J < 0 the material is anti-ferromagnetic, and neighbor spins tend to adopt an antiparallel orientation. The lattice shown on the left side of Figure 3 is called a square lattice. We designate the number of nearest neighbors of each site on the lattice by z. In a square lattice z = 4. The lattice shown on the right is a triangular lattice. Each atom or molecule (A,B) has six nearest neighbors (z = 6). Now suppose that the system is under the influence of an external magnetic field Bo pointing up. (Traditionally, in the theory of the Ising model, the external field is designated by H, but H will be used for the enthalpy.) Spin up is oriented parallel to the field, and spin down is antiparallel. Each spin contributes −Bo :i to the energy of the system. The ↑ orientation is favorable and contributes −Bo; the ↓ is unfavorable and contributes + Bo. The Hamiltonian, or the energy, of the system is given by

calculated are identical in all those variants of the model, as are the conclusions and predictions regarding the experimental systems they represent. (The values of the parameters, however, may be different.) It is less clear that this approach should work in the case of phospholipids with two different acyl chains, called hybrid lipids, such as 1-palmitoyl-2-oleoylphosphatidylcholine (POPC). Indeed, it has been proposed that hybrid lipids could act as two-dimensional surfactants, or “linactants,” in a membrane, leading to the observation of small domains rather than phase separations in biological membranes.24 In POPC, for example, the palmitic chain would favor contact with domains of ordered lipids, while the oleic chain would preferentially interact with disordered lipids. According to this idea, hybrid lipids would orient themselves around a lipid domain like surfactants orient themselves at an oil/water interface to stabilize an emulsion. Yet, representing the phospholipid by a single site on the lattice is sufficient to reproduce the experimental properties of these membranes.18,20 It is also found experimentally that the different chains of POPC (palmitic and oleic acids) are not important in determining its effect of the lipid on domains in membranes. Dilauroylphosphatidylcholine (DLPC), which has two identical, saturated but short chains (12 carbons), and POPC have indistinguishable effects on the phase behavior of mixtures of distearoylphosphatidylcholine (DSPC), dioleoylphosphatidylcholine (DOPC), and Chol.25 Both POPC and DLPC have low melting temperatures (Tm = −4 °C for POPC and −2 °C for DLPC) and are therefore disordered at room temperature, but not as much as DOPC (Tm = −18 °C). They reduce the contacts and thus the unfavorable interactions between ordered (DSPC, Tm = 55 °C) and disordered (DOPC) lipids. They lead to smaller domains, but this is simply because very unfavorable interactions between DSPC and DOPC are replaced by less unfavorable ones, between DOPC and POPC and between DPPC and POPC, not because of the hybrid nature of the lipid. Thus, perhaps by a fortuitous coincidence, it appears that Nature has sided with the simplicity of the Ising model.

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/ = −Bo ∑ :i − J ∑ :: i j i

(3)

i,j

The first summation is over all lattice sites, i, and the second, over all nearest-neighbor sites, i,j. The partition function is given by Q=

∑ (all configurations of the lattice)

e−/ / kT (4)

The importance of the partition function is that all equilibrium properties of the system can be calculated from it. The partition function tells us how the energy is partitioned among the different states of the system. Of particular interest is the investigation of the order−disorder transition, from a state where the spins are largely oriented parallel to each other to a state where their orientation is random. In one dimension, an analytic solution exists, but there is no first-order phase transition.26 The solution to this problem in two dimensions at zero magnetic field (Bo = 0) was given by Onsager in a famous paper.31 No exact analytic solution exists under a finite field, although some approximations are possible.32,33

THE ISING MODEL: MAGNETIC SYSTEMS

The Ising model2,26−30 was developed to model phase transitions in ferromagnetic systems. In one dimension, the case studied by Ising,2 the model exhibits only a continuous, broad transition. However, in higher dimensions, a phase transition occurs, which can be continuous or sharp (firstorder) depending on the values of the model parameters. We now take a moment to describe the basics of the Ising model. Consider a two-dimensional lattice (Figure 3, left). Each site i on the lattice is occupied by an atom with a spin :i .

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THE ISING MODEL: LIPID MEMBRANES Following the pioneering work of Doniach,34 the Ising model has been extensively applied to lipid membranes.12−14,16−19,21,23,33,35,36 More complicated models, such as the 10-state Pink model,37,38 have also been used to interpret phase transitions in membranes.15,39 The results, however, are not significantly different from those obtained with the two-state Ising model. The simplest system to consider is a lattice representing one leaflet of a lipid bilayer. We could call it a monolayer in this sense, but the properties of lipid monolayers at the air/water interface are very different from those of bilayers. The assumption we are making, however, is that the degree of coupling between the two leaflets of the bilayers is weak. This question has been examined with various approaches,40−54 and the assumption of

Figure 3. (left) Up and down spins in a square lattice. (right) Two lipid species A and B in a triangular lattice. 23


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with their neighbors in the membrane are specified by ϵAA, ϵBB, and ϵAB. However, GA and GB do not include the configurational entropy of the lattice that arises from the distributions of the A and B in the available sites. Replacing the Hamiltonian in the magnetic system (eq 3), we now have a free energy . given by

weak coupling is essentially correct.11 The problem with explicit coupling has also been considered in the frame of the Ising model.17,55 Consider a mixture of lipids A and B. The total number of lipids is N = NA + NB, where NA and NB are the numbers of A and B molecules. Lipids A and B could represent two different chemical species or two different states, namely, gel and liquid crystalline, of the same lipid species. In the latter case, they would be connected by an equilibrium A ⇌ B corresponding to a phase transition. We will discuss first the simpler case, where A and B are distinct lipids, and extend the treatment later to the case of two states in a phase transition. The representation of the lipid layer in the Ising model is entirely analogous to that of the lattice of spins. Now, instead of spin up and spin down (Figure 3, left), we have two lipids, A and B (Figure 3, right). Typically we use a triangular lattice for lipids, because it represents better the geometry in the membrane. Each lipid interacts only with its nearest neighbors on the lattice. We define an interaction parameter ω, which replaces the magnetic parameter J, as follows.28−30 ω = ϵAB −

(ϵAA + ϵBB) 2

. = NAGA + NBG B + NAA ϵAA + NBBϵBB + NABϵAB

(6)

The first two terms of the free energy in eq 6 correspond to the term −Bo ∑ :i in the Hamiltonian of eq 3. The last three terms in eq 6 correspond to the term −J ∑ :: i j . Similarly, the partition function is now

∑

Q=

e−. / kT (7)

(all configurations of the lattice)

Before expanding this partition function, it is convenient to simplify eq 6 by expressing it as a function of only NA, NB, and NAB. To do so, we calculate the total number of contacts made by A and B. If z is the number of nearest neighbors of each site, the number of AA contacts is the total number of contacts of A (zNa/2) minus the number of contacts that each A makes with B (NAB/2), where division by 2 avoids double counting the contacts between neighbors. A similar consideration applies to B. Thus

(5)

Here the ϵAB is the actual contact interaction between unlike AB neighbors, and ϵAA and ϵBB are the contact interactions between like AA or like BB neighbors. Thus, ω, which we call the unlike nearest-neighbor interaction parameter, is the difference between the actual interactions between unlike neighbors (AB) and the average of the actual interactions between like neighbors (AA or BB). Thus, if the two lipid neighbors are identical ω = 0 (set A = B in eq 5). If the lipids are different the parameter ω can take any value. If ω > 0 we say that the interaction is unfavorable or “repulsive,” in the sense that A and B prefer to interact with like neighbors. The consequence is that the lipids A and B tend to segregate in the lattice. If ω < 0, we say that the interaction is favorable or “attractive.” The lipids A and B tend to associate with each other. If ω = 0, A and B will mix randomly on the lattice. Let us now pause briefly to note a few points. First, the interaction parameter ω and the magnetic parameter J are equivalent. According to the definition of J, ϵAB = J, ϵAA = −J, and ϵBB = −J. Therefore, from eq 5 we find that ω = 2J.29 Second, the actual interactions ϵ are in general free energies.28 Third, the actual (net) molecular interactions between lipids are of course always favorable; otherwise, they would not form a membrane. Dominant among them are the hydrophobic interactions between the lipid acyl chains, or “tails.” It is the segregation of the lipid tails from water that drives in the formation of the lipid bilayer. But other interactions are important, such as van der Waals interactions between the acyl chains (London dispersion forces) and hydrogen bonds and ionic bonds between lipid headgroups. The “likes” and “dislikes” between lipids in the membrane are determined by differences between these interactions. Let us now designate the numbers of contacts of lipids in AA, BB, and AB pairs by NAA, NBB, and NAB. In addition, we need to specify the (free) energies of the individual lipids A and B that depend only on the state or chemical nature of the lipids but not on the interactions with their neighbors. These free energies, which we will call GA and GB, include the internal energies the molecules, their interactions with the water, and the conformation entropy of the acyl chains. The interactions

NAA =

N zNA − AB 2 2

(8)

NBB =

zNB N − AB 2 2

(9)

z y z y i i . = NA jjjGA + ϵAA zzz + NBjjjG B + ϵBBzzz 2 { 2 { k k ij (ϵAA + ϵBB) yz zz + NABjjjϵAB − z 2 k {

Substituting eqs 8 and 9 into eq 6, we obtain

(10)

Now note that the last parentheses in eq 10 is just the interaction parameter ω (eq 5) ϵAB −

(ϵAA + ϵBB) =ω 2

z y z y i i . = NA jjjGA + ϵAA zzz + NBjjjG B + ϵBBzzz + ωNAB 2 2 k { k {

Therefore, we can simplify eq 10 to obtain

(11)

Interpretation of the Parameters. The description of a mixture of two different lipids or two different states of a lipid is formally identical to the Ising model for a ferromagnetic system, which is represented by a mixture of two spins in a lattice. However, we must now move beyond the formal similarity and pause to discuss the interpretation of the parameters in a lipid system. For particles without internal degrees of freedom, such as spins in a lattice, the J are pure energies. But lipid molecules have internal degrees of freedom, and in general their potential energies and interactions are free energies. In experimental work on lipid mixtures, we typically have a fixed number of molecules at constant pressure and temperature. This corresponds to an N, P, T ensemble in statistical thermodynamics,28 which means that these free energies are Gibbs energies. Furthermore, the energy is the more usual function in statistical mechanics, but the enthalpy is 24


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more convenient for work with lipid bilayers, because the pressure, not the volume, is controlled in experimental work. In general, the GA and GB in eq 11 are Gibbs energies, which comprise enthalpic and entropic components of the molecules of lipids A and B GA = HA − TSA

(12)

G B = HB − TSB

(13)

The enthalpic component HA includes the enthalpy of the acyl chains, which is lowest in the all-anti conformation (often called all-trans in the lipid literature), and interactions of the headgroups with water, for example, the formation of hydrogen bonds. The entropic component SA includes the conformational degeneracy of the acyl chains of the phospholipid, arising from the introduction of gauche conformers, and also possibly the hydrophobic effect arising from interactions of the nonpolar chains with water. If A and B represent the gel and liquid states of the same lipid, there are many more conformations, or microstates, belonging to the liquid state than to the gel. Therefore, the degeneracy associated with the liquid state is much larger. Consider now the interaction parameter ω. In general, the ϵ in eq 5 are free energies if they vary with temperature.28 In that case, ω also varies with temperature. Therefore, at constant pressure, ω is a Gibbs energy with enthalpy and entropy components.12,14,16 We call those components ΔhAB and ΔsAB. They are defined by relations analogous to those of eq 5, but they use enthalpies or entropies of interaction, respectively. The interaction enthalpy and entropy can be determined experimentally from the temperature dependence of ω.56 For example, the interaction enthalpy ΔhAB is probably positive (unfavorable) in a mixture of two lipids with different chain lengths, such as phosphatidylcholines with 14- and 18-carbon chains (dimyristoylphosphatidylcholine (DMPC) and distearoylphosphatidylcholine (DSPC), respectively). On the one hand, most likely, DSPC lipids interact better among themselves than with DMPC lipids, because of the more extensive van der Waals interactions between their long, ordered acyl chains, and this is likely to result in ΔhAB > 0. On the other hand, placing a lipid with disordered chains, for example, an unsaturated lipid such as dioleoylphosphatidylcholine (DOPC), next to an ordered lipid such as DSPC, probably results in a decrease of the conformational freedom of the disordered chains and thus in a decrease in the interaction entropy change, possibly resulting in ΔsAB < 0. If the interactions described were dominant, then in both of these mixtures we would observe ω > 0, but the origin of this unfavorable interaction would be different in the DMPC/ DSPC and DOPC/DSPC mixtures. We do not know if this is the case. One of the most useful ways of understanding these interaction parameters is that they represent the energy or free energy changes for the exchange reactions shown in Figure 4.29 In these “reactions” two different “homodimers” (AA and BB, or ↑↑ and ↓↓) exchange partners to form two “heterodimers” (AB or ↑↓). The parameters J and ω thus have a very simple interpretation: they represent the energy or the Gibbs energy change that accompanies the exchange of pairwise contact interactions in these reactions.29 Note that this free energy change relates only to the exchange of interactions; it is not the same as the Gibbs energy that would be measured in an actual chemical reaction AA +

Figure 4. Nearest-neighbor (partner) exchange reactions in a magnetic system (top) and in a membrane (bottom).

BB ⇌ 2AB, in a lattice or in a solution, as will be evident shortly. The parameter ω is also called the exchange parameter because it represents the free energy change that corresponds to exchanging an A molecule from an A-phase with a B molecule from a B-phase in a phase-separated system,30 as illustrated in Figure 5. More exactly, the exchange free energy is 2z ϵAB − z ϵAA − z ϵBB = 2zω (14)

Figure 5. Exchange reaction between an A molecule from within the A phase and a B molecule from with the B phase.

In a triangular lattice (Figure 5), z = 6, and the Gibbs energy of exchange is ΔGexch = 12ω

(15)

Note that only ωbut not the ϵAA, ϵBB, and ϵABappear in the Gibbs energy change for this exchange. Only ω is experimentally accessible. Partition Function for a Lipid Layer with Interactions. We now expand the partition function of the lattice for a mixture of lipids A and B. Consider first the case in which A and B are two independent components or chemical species. For fixed numbers of A and B (NA + NB = N), the partition function can be written by substituting eq 11 in eq 7. Q = (qA e−z ϵAA /2kT )NA (qBe−z ϵBB /2kT )NB

∑

Ω

NAB(configurations)

(N , NA , NAB)(e−ω / kT )NAB

(16)

Here, qA and qB are the molecular partition functions of A and B. They include only the properties of the molecules A or B and their interactions with the solvent (water). The molecular partition functions are related to the Gibbs energies of A and B by GA = −kT ln qA and GB = −kT ln qB. The exponential factors exp(−zϵAA/2kT) and exp(−zϵBB/2kT) correspond to nearest-neighbor interactions between A molecules only and between B molecules only. The summation in eq 16 is the configurational partition function.28,35 It represents all possible distributions of the 25


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parameters, such as ΔG and ω, are. Thus, we can simplify the partition function for the case where A and B represent gel and liquid states. With no loss of generality, we can set

molecules of A and B in the membrane and their mutual interactions. The factor e−ωNAB/ kT corresponds to the energetic contribution of the AB interactions, and the term ωNAB is called the configurational energy. The combinatorial factor Ω(N,NA,NAB) represents the numbers of ways arranging the NA molecules on the lattice with a total of N sites (NB = N − NA) under the condition that NAB contacts are formed. In general, there is no analytic solution for the sum ∑ N Ωe−ωNAB/ kT in two (or three) dimensions. We will discuss AB two common and useful approximations, the Bragg−Williams approximation and the quasi-chemical (QC) approximation. Ultimately, however, MC simulations are needed to generate a sufficiently large number of configurations of the A and B lipids in the lattice at equilibrium, so that we can obtain the thermodynamic properties of the system from those distributions instead of using approximations for the combinatorial factor. Now, let us allow for interconversion A ⇌ B. In this case, A and B represent two different states of the same lipid, such as the gel and liquid-crystalline states. The complete partition function is obtained by allowing the number NB of B molecules to vary between 0 (all A) and N (all B).

GA +

(qA e−z ϵAA /2kT )NA (qBe−z ϵBB /2kT )NB

∑ NB = 0

(N , NA , NAB)(e−ω / kT )NAB

∑

N

Ξ=

∑

Ω(N , NA , NAB)

NAB(configurations)

(23)

The configurational factor Ω × exp(−ωNAB/kT) contains all the nearest-neighbor interactions. The factor (exp( −ΔG /kT ))NB contains the Gibbs energy difference between liquid and gel, ΔG = ΔH − TΔS. Note that ΔG is analogous to the field Bo in the magnetic case.17 The “field” is zero at Tm, where the Gibbs energies of gel and liquid are identical (ΔG = 0). The question now is to evaluate the partition function and to obtain the thermodynamic properties of the lipid mixtures as functions of temperature and composition. An analytic expression for the partition function in two-dimensional lattices exists only in the absence of external field and if NA = NB.26,31 In a lipid with a phase transition, absence of field corresponds to Tm. Because of the combinatorial factor Ω in all these equations, either an approximation must be used or the calculations must be performed by Monte Carlo simulations. I will first discuss two approximations that are frequently used in the lipid literature, the Bragg−Williams approximation and the quasi-chemical approximation. Bragg−Williams Approximation. The Bragg−Williams approximation assumes that the components A and B mix randomly, even though they interact differently with each other.28 This assumption is used to calculate both the configurational entropy (Ω) and the configurational energy (ωNAB). Evidently, the approximation contains an intrinsic contradiction, because if molecules exhibit preferential interactions (ω ≠ 0) they cannot mix randomly. However, the model has been often used to describe lipid mixtures. Indeed, the same approximation is made in the theory of regular solutions.57−59 Consider the configurational (free) energy. In a random mixture, the average number of AB contacts ⟨NAB⟩ is the product of the number of contacts of all A sites (zNA) multiplied by the probability that the neighbor is B (NB/N)

Ω

NAB(configurations)

(17)

(18)

1 [kT ln(qBe−z ϵBB /2kT )N − kT ln(qA e−z ϵAA /2kT )N ] N

z y i z y i = jjjG B + ϵBBzzz − jjjGA + ϵAA zzz 2 { k 2 { k

(19)

which is the difference between the Gibbs energies of the molecules, GB − GA, and their interactions, (z/2)(ϵBB − ϵAA). Similarly, the enthalpy and entropy changes (per molecule) of the gel ⇌ liquid transition are given by z ΔH = (HB − HA ) + (hBB − hAA ) (20) 2 z ΔS = (SB − SA ) + (sBB − sAA ) 2

(e−Δ G / kT )NB

(e−ω / kT )NAB

Consider the transition of the entire lattice from the gel to the liquid state. Since no interfaces exist between A and B, the sum ∑ N ... = 1. The lattice goes from all-A (gel) to all-B AB (liquid). With eqs 17 and 18, with either N = NA or N = NB, the Gibbs energy change per molecule in the phase transition is ΔG = −

∑ NB = 0

The partition function Ξ is related to the Gibbs energy of the system by G = −kT ln Ξ

(22)

which amounts to setting the gel (A) as the reference state. The partition function acquires the simple form

N

Ξ=

z ϵAA = 0 2

⟨NAB⟩ = zNA

NB N

(24)

Thus, the configurational energy becomes ωzNANB/N. Now consider the configurational entropy. We simply replace the configurational degeneracy of the lattice Ω(N,NA,NAB) by the combinatorial factor for a random mixture.

(21)

with ΔS = ΔH/Tm, where Tm is the melting temperature. These relations show that the experimentally measured ΔH contains not only the heat of melting the acyl chains (HB − HA) but also the difference between the lipid interaction enthalpies in the liquid-crystalline phase (hBB) and in the gel (hAA) phase. Now, note that none of GA, GB, ϵAA, ϵBB, and ϵAB (and their corresponding enthalpy and entropy components) are experimentally accessible. Only the differences between those

Ω=

N! NA ! NB!

(25)

Next we substitute eqs 24 and 25 in eq 16 (or eq 17). The partition function in the Bragg−Williams approximation becomes 26


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Q = (qA e−z ϵAA /2kT )NA (qBe−z ϵBB /2kT )NB

1 K ω = − kT ln 2 4

N! NA ! NB!

(e−ω / kT )zNANB/ N

Recall that ω refers only to the differences in contact interactions between AB, AA, and BB; it does not include any statistical mixing factors. Division of K by 4 in eq 34 corrects for this statistical factor that favors AB pairs. Note also that ω corresponds to the formation of one AB contact, but in eq 28 two AB pairs are produced; hence, the factor of 1/2 in eq 34. Indeed, one of the methods to determine ω is by counting the numbers of AA, BB, and AB pairs in an actual lipid mixture.56 Monte Carlo Simulations. Instead of obtaining approximations for the combinatorial factor, the easiest solution to the configurational degeneracy problem is to use Monte Carlo simulations.14,16,36,61 Monte Carlo simulations avoid the need to actually calculate the configurational degeneracy of the lattice. Instead, the simulation generates a Boltzmanndistributed ensemble (importance sampling) from which observables can be obtained by simple averaging. The generation of a Boltzmann distribution of the states in the lattice is ensured by using the Metropolis criterion62 for the acceptance or rejection of attempted moves. Those moves are of two types: exchange between lipids in two sites on the lattice and changes of state for a lipid in a given site. Each attempted move is assigned a probability that depends exponentially on the Gibbs energy change that the move entails. The move is accepted if that probability is greater than a random number drawn between 0 and 1. The calculation is repeated for a sufficiently large number of lattice configurations generated by the simulation. The beauty of the Metropolis importance sampling method is that the average thermodynamic properties of the lattice can be obtained by simple arithmetic averages, instead of having to weigh each configuration by its Boltzmann factor, e−/ / kT .61 This is because each lattice configuration is generated with a probability that is already determined by the Boltzmann factor. The more likely configurations contribute more to the arithmetic average because they are generated more often, in proportion to their probability. For example, the enthalpy is obtained by summing over all lipids and taking an average over a sufficiently large number of lattice configurations, N H = ∑i = 1 Hi . Similarly, the configurational energy is calculated by summing over all unlike nearest-neighbor contacts between any i,j lipid states or species in the lattice, averaged over all lattice configurations generated, ⟨∑i < j ωij⟩. Most important for the present work, the excess heat capacity (ΔCp) is obtained from the enthalpy fluctuations28

(26)

Taking logarithms of eq 26, the Gibbs energy (eq 18) of mixing A and B components, per molecule, is given by ΔGmix = zωXA XB − kT (XA ln XA + XB ln XB) N

(27)

where XA and XB are the mole fractions of A and B. This is the standard expression for regular solutions, used, for example, by Anderson and McConnell in the condensed complex model of DPPC/Chol interactions.60 Quasi-Chemical Approximation. The QC is significantly better than the Bragg−Williams approximation. The assumption in the QC approximation is that the types of pairs of nearest-neighbor sites are uncorrelated, or independent.28,55 The derivation of the combinatorial factor in the QC approximation is not given here, but see Hill.28 Instead, let us focus on the gist of the idea. We begin with the equation for the pair-exchange “reaction” AA + BB F 2AB

(28)

We can write an “equilibrium constant” for this reaction in terms of the average, or most probable values, of the three possible types of pairs K=

⟨NAB⟩2 ⟨NAA ⟩⟨NBB⟩

(29)

If the sites are occupied randomly by A or B (Bragg− Williams), we can write the average numbers of AA and BB pairs in a way similar to AB pairs (eq 24). The average number of AA contacts are all contacts made with A minus those made with B (and similarly for the BB contacts) ⟨NAA ⟩ =

2 z z N z NA NA − NA B = 2 2 N 2 N

(30)

⟨NBB⟩ =

2 z z N z NB NB − NB A = 2 2 N 2 N

(31)

where the factors of 1/2 avoid double counting. To obtain the equilibrium constant for a random mixture, we substitute eqs 24, 30, and 31 in eq 29 K=

(zNANB/N )2 (z /2)(NA 2/N )(z /2)(NB 2/N )

=4 (32)

ΔCp =

Thus, in a random mixture of A and B, an AB pair is favored by a statistical factor of 2 relative to an AA or a BB pair, resulting in the factor of 4 in K. The quasi-chemical approximation can be expressed by setting ⟨NAB⟩2 = 4e−2ω / kT ⟨NAA ⟩⟨NBB⟩

(34)

⟨H2⟩ − ⟨H ⟩2 kT 2

(35)

The importance of the excess heat capacity to determine lipid−lipid interactions (ω) is that ΔCp is exquisitely sensitive to small variations of ω. We will examine several examples in well-studied lipid mixtures.

■

(33)

thus defining an equilibrium constant K = 4e−2ω/kT for the dimer exchange reaction (eq 28). Random mixing corresponds to ω = 0, in which case we recover K = 4 from eq 33. If we invert this equation, it provides a simple way to calculate ω from experimental dimer distribution data

EXPERIMENTAL DETERMINATION OF LIPID INTERACTIONS

The determination of the lipid interaction parameters relies either on the use of approximations, such as the quasi-chemical approximation, or on the comparison of the results from experiment and computer simulations. In Monte Carlo simulations, the values of ω can be varied until agreement is obtained between the calculated and experimental 27


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Table 1. Physical Properties of DPPC Vesicles diameter

Tm

ΔH

ΔCp

ω

vesicle

(μm)

(°C)

(kcal/mol)

(kcal/K/mol)

(cal/mol)

refs

MLV GUV LUV SUV

1−10 ∼10 0.1 0.02

41.4 41.7 40.8 37.2

8.7 8.7 8.7 8.7

∼10−100 5.0 3.5 2.0

∼350 310 300 280

17, 68−70 71 17, 22, 72, 73 12, 16, 17

Figure 6. Heat capacity of DPPC LUVs and SUVs measured experimentally by DSC (lines) and calculated by Monte Carlo simulations of the Ising model (points). LUV (left) The experimental data are from Ivanova and Heimburg,17 courtesy of Dr. Heimburg, renormalized slightly to our and Tm values. The simulations are from Svetlovics et al.18 SUV (right) Reprinted with permission from Jerala et al.12 Copyright average ΔCmax p 1996 The Biophysical Society. The solid symbols are from simulations where each site represents a lipid molecule. The open symbols are from simulations where each site represents an acyl chain, connected to one of its neighbors to form a whole lipid. The sharp peak at high temperature is from residual MLVs in the sample, which are extremely difficult to eliminate completely because of fusion of the highly strained SUVs.69 values of some observable quantity. This may be ΔCp(T), which displays a pronounced maximum at Tm and is very sensitive to the values of ω.12−14,16−19,21,23,36 Or it may be a spectroscopic parameter that reports on proximity between different lipid species, such as an excimer-to-monomer ratio or the Förster resonance energy transfer (FRET) efficiency between lipid fluorescent probes incorporated in the bilayer.18,20,63,64 Or it may be the experimentally determined value of the equilibrium constant K in nearest-neighbor recognition (NNR) experiments. We will examine examples of these cases. Heat Capacity of Phase Transitions. The experimental excess heat capacity (ΔCp) of lipid bilayers as a function of temperature recorded over a phase transition by differential scanning calorimetry (DSC) is one of the most informative methods to investigate lipid interactions. The reason is that the excess heat capacity, especially the value of the heat capacity maximum (ΔCmax p ) for the phase transition, which occurs at the Tm, is exquisitely sensitive to the interactions between the different states of the lipid. Those interactions are represented by the parameter ω between the gel state and the liquidcrystalline state (Ld) in the Ising model. The value of ω tells us about the nature of the phase transition. This is illustrated here with DPPC, which is the best studied case. Different types of vesicles of DPPC can be prepared in aqueous solutions (Table 1). They vary in size and in the number of lipid bilayers they contain. Dispersion of a phospholipid in water or an aqueous buffer produces multilamellar vesicles (MLVs), which typically have a diameter of 1−50 μm.65 MLVs consist of multiple bilayers, often approximately spherical and concentric. Small unilamellar vesicles (SUVs) can be prepared by sonication from dispersions of MLVs. They are typically ∼25 nm in diameter; the lipids in these vesicles are under considerable strain because of the high, unnatural curvature of the membrane. Large unilamellar vesicles (LUVs) typically have a diameter of 100 nm. This size can be varied somewhat, because LUVs can be prepared by extrusion of MLVs through filters with pores of defined size. Finally, GUVs of DPPC have a typical diameter of 5−10 μm. Their size can be substantially larger with other lipids, for example, with POPC, for which GUVs are typically much larger with diameters of 10−100 μm.66,67 The value of Tm decreases slightly with vesicle type (Table 1). The enthalpy change, ΔH = 8.7 kcal/mol,68,69 is approximately independent of vesicle size. Values in MLVs and LUVs between

∼7.5 and 9.7 kcal/mol have been reported.17,72,74,75 In LUVs we have obtained ΔH = 8.8 ± 0.2 kcal/mol on a very large number of measurements.22,73 In SUVs, ΔH ≈ 6.0 kcal/mol has been reported.69 However, this lower value is probably caused by uncertainty in the baseline of DSC curves, and the true value is probably close to 8.7 kcal/mol.12,17,21 Thus, Tm and ΔH vary little with vesicle size and type. This is not true, however, of the heat capacity at Tm. The value of ΔCmax p in MLVs has been reported between ∼10 and 100 kcal/K/mol;17,69,70,76−78 the largest value (listed) is probably the most accurate. In LUVs, ΔCmax p = ≈ 1.5−2.0;12,17 and in 3−3.5 kcal/K/mol;17,18,22 in SUVs ΔCmax p = 5.0 kcal/K/mol, assuming ΔH = 8.7 kcal/mol.71 GUVs, ΔCmax p Thus, the value of ΔCmax is significantly different in each of these p types of DPPC vesicles. It is here that the Ising model is especially useful to interpret the phase transition, because of the sensitivity of to the value of ω. ΔCmax p Figure 6 shows a comparison between the experimental ΔCp(T) obtained with DSC and Monte Carlo simulations using the Ising model for DPPC LUVs17,18 and SUVs.12 In both cases the Ising model reproduces the experimental data extremely well, with a single variable parameter, ω. The values of ω required to obtain a match are 300 cal/mol for the LUV data and 282 cal/mol for the SUV data. Those values tell us that, according to the Ising model, the transition is not first-order. Figure 7 shows a snapshot of the lattice in a Monte Carlo simulation of a DPPC LUV at Tm. Clusters of gel (white) and liquid (black) lipids coexist in approximately equal amounts, but they are not separated into two distinct phases at Tm, nor does the lattice fluctuate between all-gel and all-liquid crystalline states. The exact value of ω for the Ising model in zero field (T = Tm) is given by ω/(2RTm) = 0.274 653.5,26 In DPPC LUVs (Tm = 313.9 K) this corresponds to ω = 342.6 cal/mol. In SUVs (Tm = 310.3 K), it corresponds to ω = 338.7 cal/mol. The ω values found from comparison with experiment are significantly below those values, indicating that the transition is continuous, not first-order, in LUVs on ω in DPPC and SUVs. Figure 8 shows the dependence of ΔCmax p LUVs at Tm obtained in Monte Carlo simulations of the Ising model. To convey a more visual and intuitive perspective of what this means, the fraction of fluid lipids is shown as a function of “time” (MC simulation cycles) in Figure 9. The black lines show the 28


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Figure 7. Snapshot of a Monte Carlo simulation of DPPC LUV at Tm, with ω = 300 cal/mol, in a 100 × 100 triangular lattice. The (white) gel and (black) liquid states of DPPC molecules. Figure 9. Fluctuations of the number of lipid molecules in the liquid state at Tm = 313.9 K (Tm for DPPC LUVs) as a function of ω (given in cal/mol on the top right of each panel). The simulations were performed with the Ising model in a 100 × 100 triangular lattice. The black lines show the variation of the liquid fraction as a function of time. (insets) The histograms show the distributions of the number of liquid state lipids calculated from the black lines.

critical value. To match the shape of the curve, Ivanova and Heimburg17 used a more sophisticated approach and reached a value of ω ≈ 380 cal/mol. The problem is that an MLV does not behave as a collection of single, large bilayers. The very high cooperativity of an MLV is not due to the large size of each bilayer, because the phase transition in GUVs, which are of the same size as MLVs, is similar to that in LUVs.71 Rather, the higher cooperativity is probably a consequence of the coupling of the phase transition in the third dimension, between the bilayers of each MLV, in addition to the coupling in two dimensions through ω.

Figure 8. Dependence of ΔCmax p on ω for DPPC LUVs at Tm = 313.9 K. The points are obtained by Monte Carlo calculations using the Ising model in a 100 × 100 triangular lattice (the curve is drawn to guide the eye). The dashed vertical line marks the exact value of ω at the critical point of the Ising model in zero field (T = Tm), given by ω/(2RTm) = 0.274 653.26 Above this value, the heat capacity calculated from these simulations (1 × 107 MC cycles) is not exact, because the number of transitions of the lattice from the gel to liquid state becomes insufficient (longer simulations would be required). Indeed, above the critical value (ω > 342 cal/mol) ΔCmax p → ∞ in an infinite lattice.

■

NEAREST-NEIGHBOR RECOGNITION The nearest-neighbor recognition (NNR) method was developed in Regen’s laboratory to measure lipid interactions.79,80 In NNR, the equilibrium constant K for chemical exchange between dimers of different lipid species is obtained directly from experiment. Exchangeable analogues of different pairs of lipids are prepared so that they can establish a disulfide bond (Figure 10). For example, these could be exchangeable analogues of two phospholipids or a phospholipid and cholesterol. They are designated here with subscript ea, such as DPPCea or Cholea. These analogues are typically incorporated in low amounts (∼2.5%) in bilayers prepared from the corresponding natural lipids, such as DPPC and Chol. A thiol−disulfide exchange reaction between different pairs of dimers is then allowed to proceed at alkaline pH until equilibrium reached. (36) AA + BB F 2AB At equilibrium, the system consists of a mixture of dimers of the various types: AA, BB, and AB. The reaction is then stopped by a sudden pH drop, and the concentrations of homodimers (AA and BB) and heterodimers (AB) in the quenched mixture are determined by high-performance liquid chromatography (HPLC). The Regen constant K is the equilibrium constant for this reaction.

fluctuations in the number of lipids in the fluid state. The histograms (insets) show the distributions of fluid lipids derived from those time traces. The top left corresponds to DPPC LUVs. This is clearly a unimodal distribution with relatively small fluctuations in the number of lipids in the liquid state even at Tm. As ω increases to the critical value, the fluctuations increase considerably, and the distribution of fluid lipids becomes very broad. The critical point in these simulations is at ω ≈ 340 cal/mol. The corresponding ratio ω/(2RTm) = 0.272, just ∼1% lower than the exact value (because of finite lattice size). Above the critical value of ω, the distributions become bimodal. The membrane fluctuates between ∼10% fluid to ∼90% fluid state. Those fluctuations are frequent when ω is close to the critical value, but they become rarer when it is far above, as shown in the last panel of Figure 9. ≈ 10−100 In DPPC MLVs, the situation is less clear. The ΔCmax p kcal/K/mol, and the phase transition is apparently much more cooperative. If we treat the multiple bilayers of the vesicle as if they were a collection of large single bilayers with identical behavior, we would need ω ≈ 340 cal/mol in the Ising model to obtain a match to measured experimentally. The Tm of DPPC MLVs is the ΔCmax p slightly higher (314.5 K), but this corresponds very closely to the

K= 29

[NAB]2 [NAA][NBB]

(37) DOI: 10.1021/acs.langmuir.8b03054 Langmuir 2019, 35, 21−40

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the heat capacity, most recently in combination. Two binary mixtures of phosphatidylcholines have been examined in detail, DSPC/DMPC and DPPC/POPC. The third mixture examined in detail is DPPC/Chol. Then, we will look at several other less well-studied cases. An excellent source for the phase diagrams of these and many other lipid mixtures is Marsh’s handbook.81 Mixtures of Saturated Phosphatidylcholines. The DSPC/DMPC mixture has been studied by several groups using the Ising model to interpret heat capacity data.13,14,19,23,39,82 The heat capacity as a function of temperature was obtained by DSC for several different binary compositions in MLVs.13,14,23,68 The phase diagram was obtained from those data and from small-angle neutron scattering.83 The DSPC/DMPC phase diagram is shown in Figure 11, left. The heat capacity was accurately reproduced by Monte Carlo simulations of the Ising model as shown in Figure 11, right. Different authors have used slightly different variants of the Ising model for DSPC/DMPC, with sites representing acyl chains, isolated or connected, in a triangular lattice,13,14,23 or whole lipids in a square lattice.19,82 For consistency with the models discussed here, the calculations were repeated with each site representing a whole lipid in a triangular lattice using previously described methods.18 The interaction parameters used to match the heat capacity are given in Table 2.

Figure 10. NNR experiment. Two different lipids, A and B, are modified to allow formation of dimers through disulfide bonds. The dimer exchange reaction between the possible pairs (AA, BB, and AB) is performed until equilibrium is reached. In the example, A is a DPPC analogue, and B is a cholesterol analogue.

Table 2. Lipid Interaction Parameters, Transition Enthalpies, and Melting Temperatures Used in Monte Carlo Simulations of DSPC/DMPCa ωgl

K is the quantity directly calculated from this the NNR experiment. But note that this is exactly the same as eq 29 used in the QC approximation. Thus, we have a direct connection to the theory of lattice statistics and therefore to the interaction parameters of the Ising model. Namely, ω can be obtained from the NNR data through eq 34. 1 K ω = − kT ln 2 4

ωlg

ωgg

ωll

(cal/mol) DSPC/DMPC DSPC DMPC

450 350 320

380

210

ΔH

Tm

(kcal/mol)

(K)

12 6.3

327.9 297.1

70

The calorimetric data are from Hac et al.;23 the ω are from the present simulations. In the mixtures, the first state (g or l, for gel or liquid crystalline) in ωgl and ωlg refers to DSPC.

a

■

Table 3 summarizes NNR measurements of lipid interactions in DSPC/DMPC and other mixtures of saturated phosphatidylcholines (PC). These NNR data were not taken into account when modeling the heat capacity data by MC

INTERACTIONS IN LIPID MIXTURES We will now examine several examples of lipid mixtures that have been studied by NNR and Monte Carlo simulations of

Figure 11. Phase diagram of DSPC/DMPC (left) and the heat capacity for the DSPC/DMPC 50:50 mixture as a function of temperature (right). The experimental data were obtained by DSC and are shown in comparison with the points calculated with Monte Carlo simulations of an Ising model. The data from Hac et al.23 were courtesy of Dr. Thomas Heimburg. The open symbols are from Hac et al.;23 the solid symbols are the author’s calculations. 30


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lipid acyl chains are fairly ordered, which justifies the comparison with the gel. In the simulations ω = 210 cal/mol in the gel phase, which compares with ω ≈ 160 cal/mol by NNR in the Lo phase. At 30 °C, in the gel−Ld coexistence region of the phase diagram (Figure 11) the gel is mostly DSPC, and the fluid is mostly DMPC. Therefore, NNR measurements in this region yield an estimate of the interaction between a DSPC molecule in the gel state with a DMPC molecule in the Ld state. The NNR value of ω = 610 cal/mol compares with the simulation value of 450 cal/mol, both indicating a very strong repulsion. In DPPC/DMPC, where the acyl chains differ only by two carbons, ω = 0 by NNR in the Ld phase and even in the gel/Ld coexistence region, indicating that this mixture is close to ideal, a conclusion borne out by the phase diagram obtained by DSC.68,81 In DSPC/DLPC, the lipid chain length differs by six carbons, and the interactions are clearly much more unfavorable. In the Lo phase, the interaction is very unfavorable, as expected, but even in the Ld phase it is repulsive (ω ≈ 130 cal/mol), consistent with a very nonideal phase diagram.68 Mixtures of Saturated and Unsaturated PC. Next, we consider mixtures between saturated and unsaturated phospholipids, whose acyl chains contain one or more cis double bonds. The DPPC/POPC mixture was examined in our laboratory.18 The phase diagram of this system is shown in Figure 12, left. The heat capacity was recorded in LUVs, and MC simulations were performed in a triangular lattice, each site representing a lipid. The right panel of Figure 12 shows the heat capacity data, measured experimentally by DSC (lines) and calculated by MC simulations of the Ising model (symbols). Data are shown for POPC, DPPC/POPC 50:50, and pure DPPC. The parameters used in the MC simulations are listed in Table 4. Figure 13 shows a GUV of DPPC/POPC 40:60 at room temperature (left) and a snapshot from a Monte Carlo simulation (right) performed with the parameters given in Table 4. The gel is shown in white (mostly DPPC), and the liquid state is shown in black (mostly POPC). The similarity between the experimental result and the simulations is evident. The experimental determination of interaction parameters in mixtures involving unsaturated lipids is complicated, because double bonds do not fare very well under the conditions necessary for NNR reactions. For this reason, more recent NNR experiments have employed mimics of unsaturated lipids.

Table 3. Interactions between Saturated Phosphatidylcholinesa T phase

ω

interaction modeled

(cal/mol)

(actual mixture composition)

K

(°C)

Ld

60

4.0

0

Lo

30

2.38

156

Lo

60

4.46

−36

gel/Ld

≈30

0.53

610

Ld

53

4.0

0

gel/Ld

33

4.0

0

Ld

60

2.68

133

Lo

60

1.13

418

DSPC−DMPC (DSPCea/DMPCea 50:50) (DSPCea/DMPCea/ Cholea 30:30:40) (DSPCea/DMPCea/ Cholea 30:30:40) (DSPCea/DMPCea 50:50) DPPC−DMPC (DPPCea/DMPCea 50:50) (DPPCea/DMPCea 50:50) DSPC−DLPC (DSPCea/DLPCea 50:50) (DSPCea/DLPCea/ Cholea 30:30:40)

refs 40, 79, 84, 85 86 84 79, 86

79 79

84 84

a

Equilibrium constants K obtained by NNR for lipid dimer exchange and the corresponding lipid interaction parameters calculated with the QC approximation, ω = −1/2RT ln K/4.

simulations of the Ising model in DSPC/DMPC. As in so many fields of science, this is because the two approaches have developed separately for many years. It is only in the past 10 years that we have been trying to bring the two together. Nevertheless, some of the estimates for these interactions obtained by the two approaches are remarkably similar, and the differences do not appear difficult to reconcile. In DSPC/DMPC, where the lipid chain lengths differ only by four carbons, in the liquid-crystalline or liquid-disordered (Ld) phase, the interactions are close to neutral (ω = 0), and the mixtures are close to ideal. In the simulations ω = 70 cal/ mol, which compares to ω = 0 by NNR. NNR experiments in pure gel phase are rare and not very reliable because of the very long times required to reach equilibrium. However, we can make an approximate comparison with the simulations using NNR experiments in the liquid-ordered (Lo) phase. The Lo phase exists in PC membranes containing high levels of Chol. We will discuss the Lo phase in detail shortly, when considering the DPPC/Chol system, but for now suffice it to say that the

Figure 12. (left) Phase diagram of DPPC/POPC (center) obtained from DSC experiments.18 (right) Heat capacity of LUVs of pure POPC (blue), DPPC/POPC 50:50 (black), and pure DPPC (red) as a function of temperature. The lines are experimental DSC curves, and the points are from MC simulations of the Ising model. Data for POPC and DPPC/POPC are from Svetlovics et al.18 The DPPC experimental curve (DSC) is from Kreutzberger et al.,71 and the MC simulation data are from Almeida et al.22 31


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Table 4. Lipid Interaction Parameters, Transition Enthalpies, and Melting Temperatures Used in Monte Carlo Simulations of DPPC/POPCa ωgl DPPC/POPC DPPC POPC

450 300 240

ωlg

ωgg

(cal/mol) 240 140

ωll

ΔH

Tm

(kcal/mol)

(K)

8.7 5.8

313.9 269.15

Table 5. Interactions between Saturated and Unsaturated Phosphatidylcholinesa T

K

phase (°C)

70

Data from Svetlovics et al.18 In the mixtures, the first state (g or l, for gel or liquid crystalline) in ωgl and ωlg refers to DPPC. a

Figure 13. (left) A GUV of DPPC/POPC 40:60 at 22°C. (light) Gel. (dark) Liquid phase. (right) A snapshot of an MC simulation of the same system. Light and dark represent gel and liquid states. Reprinted with permission from Svetlovics et al.18 Copyright 2012 The Biophysical Society.

These mimics are phospholipids with one to three cyclopropane rings in place of the cis double bonds. Figure 14 shows the structure of POPC (16:0,18:1PC) and its mimic 1palmitoyl-2-dihydrosterculoyl-sn-glycero-3-phosphocholine (16:0,c1PC), which has a cyclopropane ring at position 9 of the sn-2 chain, where POPC has a cis double bond. Similar mimics have been used for polyunsaturated phospholipids. One, with cyclopropane rings at positions 9 and 12 (16:0,c2PC), is a mimic of 1-palmitoyl-2-linoleoylphosphatidylcholine (16:0,18:2PC). And another, with cyclopropane rings at positions 9, 12, and 15 (16:0,c3PC), is a mimic of 1palmitoyl-2-linolenoylphosphatidylcholine (16:0,18:3PC). The interactions between saturated and monounsaturated PC studied by NNR are unfavorable even in the Ld phase (Table 5). In DPPC/POPC the interaction is slightly repulsive, with ω = 150 cal/mol, not too different from 70 cal/mol used in the simulations. But in the presence of 40% Chol, in the Lo phase, the interaction is actually neutral (ω = 0). This suggests

ω

interaction modeled

(cal/mol)


Ld

40

3.61

30

Ld Ld

40 60

3.84 3.88

13 10

Ld

45

2.5

150

Lo

45

4.0

0

Ld

45

1.9

230

Ld

45

0.91

470

Ld

55

3.20

73

Ld

60

3.17

77

Ld

60

2.86

110

DMPC−POPC (DMPCea/POPCea 50:50) DMPC−DOPC (DMPCea/DOPCea 50:50) (DMPCea/DOPCea 50:50) DPPC−POPC (16:0,18:1PC) (DPPCea/16:0,c1PC/16:0,c1PCea 2.5:95:2.5) (DPPC/DPPCea/Chol/ 16:0,c1PCea 57.5:2.5:37.5:2.5) DPPC−16:0,18:2PC (16:0,c1PC/DPPCea/16:0,c2PCea 95:2.5:2.5) DPPC−16:0,18:3PC (16:0,c1PC/DPPCea/16:0,c3PCea 95:2.5:2.5) DPPC−DOPC (DPPCea/DOPCea 50:50) DSPC−POPC (DSPCea/POPCea 50:50) DSPC−DOPC (DSPCea/DOPCea 50:50)

refs 88 88 88 89 89

90

90

88 88 88

a


that the oleoyl chains of POPC have become so ordered that they approach the conformation of the palmitoyl chains of DPPC. This idea is borne out by MD simulations, which show that the chains of 16:0,18:2PC become as ordered as those of DSPC in the Lo phase.87 It is also evident from the NNR data that the interactions with DPPC become more repulsive as the number of unsaturations in the 18-carbon chain increases (middle block of Table 5). The exchange Gibbs energy (ΔGexch) of two lipids between two phases (Figure 5) has been recently calculated from atomistic Molecular Dynamics simulations close to 25°C.87 Some of the MD simulation results compare very well to those obtained from our Monte Carlo simulations of the heat

Figure 14. Structures of unsaturated lipids, their cyclopropane mimics, and exchangeable analogues used in NNR experiments: POPC, 16:0,c1PC (or PDSPC), and exchangeable analogues (subscript ea) of 16:0,c1PC, 16:0,c2PC, and 16:0,c3PC. 32


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capacity,18 listed in Table 4, and from NNR,88−90 listed in Table 5. In the MD simulations, ΔGexch was measured for the exchange of 16:0,18:2PC and DSPC between different phases. We can compare those calculations with ours using ΔGexch = 12ω (eq 15), which applies to a triangular lattice. Thus, we can obtain nearest-neighbor interaction parameters ω from these MD simulations. For the interaction of DSPC (gel state) with 16:0,18:2PC (Ld), application of eq 15 to the MD simulation exchange reaction yields ω = 490 cal/mol. The closest example we examined was the interaction between DPPC (gel) with POPC (Ld). In this case, ω = 450 cal/mol (Table 4), which compares well to the value from the MD simulations. In the Ld phase, the MD simulations yield ω ≈ 20 cal/mol for the interaction DSPC−16:0,18:2PC. This compares reasonably well to ω = 70 cal/mol used in MC simulations for DPPC− POPC (Table 4) and with several determinations from NNR for the interactions between saturated and unsaturated PC in the Ld phase, which yield ω ≈ 10−100 cal/mol (Table 4). However, for DPPC−16:0,18:2PC, which is the example closest to the case examined in the MD simulations, NNR yields a much larger value, ω = 230 cal/mol. In the Lo phase, the MD simulations yield ω = 360 cal/mol for DSPC− 16:0,18:2PC, whereas NNR yields ω = 0 for DPPC-POPC. We should keep in mind, however, that for all we know, interactions in the Lo phase are likely to be temperaturedependent (see these MD simulations87 and the discussion below for DPPC−Chol interactions). Finally, the MD simulations yield ω = 390 cal/mol for DSPC (Lo)− 16:0,18:2PC (Ld). We do not have a comparison for this case from NNR. However, as discussed in the last section of this article, we used MC simulations in a lattice to estimate interaction parameters by comparison with measurements of FRET. We obtained ω = 300 cal/mol between a mostly saturated sphingomyelin (SM) in the Lo state and POPC in the Ld state,20 which compares to 390 cal/mol for DSPC (Lo)− 16:0,18:2PC (Ld) from the MD simulations.87 Mixtures of Cholesterol with Saturated PC. The mixture most thoroughly investigated using a combination of NNR, DSC, and MC simulations is DPPC/Cholesterol.22 A new composition−temperature phase diagram was recently published from 2H NMR data,91 which is shown in Figure 15. The nomenclature for the phases present in DPPC/Chol was introduced by Ipsen et al. in two very influential papers.92,93 They called liquid disordered (Ld) the standard liquid-crystalline phase of DPPC (at zero or low Chol content) and coined as liquid ordered (Lo) the fluid phase that exists at high Chol concentrations. Two broad regions of phase separation were defined. One, gel−Lo, below the Tm of DPPC, and the other, Ld−Lo, above the Tm (Figure 15). The Ld phase is characterized by rapid diffusion and disordered lipid acyl chains, whereas the Lo phase is characterized by slightly slower diffusion94 but much more ordered lipid chains.74,95 The existence of a liquid−liquid phase separation region in binary mixtures of phospholipids and cholesterol, however, is controversial.91,94−100 Most experimental evidence points against the existence of a two-phase region with separation between two macroscopic phases. However, Ld−Lo phase separations have been clearly observed by fluorescence microscopy, and are indeed common, in ternary mixtures of an ordered lipid (such as a sphingomyelin, DSPC, or DPPC), a disordered, unsaturated lipid (POPC or DOPC), and Chol.96 In any case, the liquid that exists in DPPC/Chol at low Chol

Figure 15. Phase diagram of DPPC/Cholesterol based on 2H NMR.91 The data were courtesy of Dr. Jenifer Thewalt. In MLVs, DPPC-d31 has a Tm = 40.0°C, which is very close to Tm = 40.8°C for standard DPPC in LUVs. Reprinted with with permission from Almeida et al.22 Copyright 2018 American Chemical Society.

(Ld) is much more disordered that the liquid at high Chol concentration (Lo). The DPPC−Chol interactions were examined by NNR in the Ld and in the Lo phases.56,101−105 Table 6 summarizes the results from those experiments. The interaction between DPPC and Chol in the Ld phase is essentially neutral, with K ≈ 4, which means that ω ≈ 0. Further, this interaction is essentially independent of temperature (Figure 16), which indicates that the van’t Hoff interaction enthalpy is also close to zero. Thus, DPPC and cholesterol mix almost ideally in the Ld phase, with interactions that are neither favored by enthalpy nor entropy. The situation is very different in the Lo phase. Here, K is large, or ω < 0 (Table 6), which means that the interaction between DPPC and Chol is attractive. Furthermore, ω now depends on temperature, becoming less negative as T increases (Figure 16). This indicates a van’t Hoff interaction enthalpy of ΔhAB ≈ −2 kcal/mol. This experiment demonstrates that in general the interaction represented by ω is a Gibbs free energy, not a pure energy (enthalpy). The DPPC/Chol system has attracted much interest over the years.74−78,92−95,100,106−109 This is motivated in part by the biological importance of cholesterol and our still poor understanding of its role in the plasma membrane and internal cell membranes. Another reason is the peculiar shape of the excess heat capacity (ΔCp(T)) of DPPC/Chol mixtures measured by DSC. At intermediate Chol concentrations, ΔCp(T) exhibits a sharp and a broad component, which have been challenging to replicate by simulation, unlike the DSPC/ DMPC and DPPC/POPC mixtures discussed above. The Pink−Potts model with multiple DPPC states 93 and McConnell’s “condensed complex” model60,110,111 produce heat capacity profiles with sharp and broad components, but beyond this general feature the agreement with the experimental ΔCp(T) curves obtained by DSC is poor. Furthermore, those models were employed with interaction energies that are much larger than observed experimentally by NNR. 33


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Table 6. Interaction between DPPC and Cholesterol in Different Phasesa T phase

(°C)

Ld Ld

45 45 50 55 60 65 45 45 50 55 60 65 35

Lo Lo

gel

K

3.7 ± 0.2 3.60 3.58 3.81 3.73 3.72 8.9 ± 0.9 8.7 7.7 6.8 6.1 5.8 0.78

ω

interaction modeled

(cal/mol)

(actual mixture composition) DPPC−Cholesterol (DPPC/DPPCea/Cholea 95:2.5:2.5) (DPPC/DPPCea/Chol/Cholea 87.5:2.5:7.5:2.5)

101−104 56

(DPPC/DPPCea/Chol/Cholea 57.5:2.5:37.5:2.5) (DPPC/DPPCea/Chol/Cholea 57.5:2.5:37.5:2.5)

101−105 56, 105b

(DPPC/DPPCea/Cholea 95:2.5:2.5)

101

22 33 36 16 23 24 −255 −245 −210 −175 −140 −125 500

refs

a

Equilibrium constants K obtained by NNR for lipid dimer exchange and the corresponding lipid interaction parameters calculated with the QC approximation, ω = −1/2RT ln K/4. bThe K values listed are the averages of the data in the two references.

Very recently, we sought to correct this problem. We developed a different three-state Ising model,22 now based on the Pauling model or the Koshland−Nemethy−Filmer model113,114 for the cooperativity in oxygen binding by hemoglobin. In the Pauling model, the transition of a hemoglobin subunit from the T, “taut,” state to the R state only occurs concomitant with binding of oxygen. Similarly, in our new Ising model of DPPC/Chol, we postulated that the phospholipid has access to three states, gel, Ld, and Lo, but the transition of a DPPC molecule from the gel or the Ld state to the Lo state is only allowed if coupled with the (favorable) interaction with an adjacent Chol molecule. The nearestneighbor interaction parameters ω used in this work (Table 7) Table 7. Lipid Interaction Parametersa Used in the MC Simulations of DPPC/Chol22 ωoC

Figure 16. Equilibrium constant K obtained from NNR experiments as a function of temperature in DPPC/Chol mixtures in the Ld phase (10 mol % Chol) and in the Lo phase (40 mol % Chol). Data from Zhang et al., 2006.56

a

We have recently modeled the DPPC/Chol system using a three-state Ising model.21,22 First, we used a model in which a DPPC molecule was postulated to access three conformational states in the absence of Chol: gel, Ld, and Lo.21 Those states were assumed to represent the dominant conformations of the phospholipids in the three corresponding macroscopic phases defined by Ipsen et al.92,93 The Lo state was presumed to be available, although scarcely populated, even in the absence of Chol. Cholesterol prefers the Lo state and stabilizes it by nearest-neighbor interactions. This model was inspired by the Monod−Wyman−Changeux model112 for oxygen binding to hemoglobin, where the R, “relaxed,” state is available but scarcely populated in the absence of oxygen. The calculated heat capacity curves were compared with the experimental DSC curves recorded on MLVs. A qualitative agreement was obtained, but quantitatively the calculated ΔCp curves were much shallower and broader than the experimental ones, and the freezing point depression caused by Chol was too large.21 One problem of this model is that, experimentally, the existence of the Lo state is tied to the presence of Chol.

were chosen so that the value of the Regen constant K calculated by Monte Carlo simulations of the model match those obtained experimentally by the NNR method.22 We compared the heat capacity obtained from Monte Carlo calculations of this model with the experimental DSC curves measured in DPPC/Chol. This time, we used LUVs instead of MLVs because of the artificially sharp transition in DPPC MLVs. The results are shown in Figure 17. The Ising model reproduces the approximate magnitude of ΔCmax p , which is significantly lower in LUVs than in MLVs, and also the essential features of the broad component of the phase transition. However, at finite but low Chol concentrations, the sharp component of the transition observed experimentally is not replicated by the simulations. This is interesting because DPPC/Chol is the only system examined so far in which the Ising model does not quantitatively replicate the experiment. Why, is an open question, one that certainly begs further investigation.

ωgd

ωog

ωod

ωdC

ωgC

25 °C

45 °C

65 °C

300

260

300

50

200

−430

−350

−260

ω values in calories per mole. The subscripts g, o, and d refer to DPPC in the gel, Lo, and Ld states; C refers to Chol.

34


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Figure 17. 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 7. Reprinted with permission from Almeida et al.22 Copyright 2018 American Chemical Society.

and 16:0,c3PCea, for lipids with 1, 2, or 3 cyclopropane rings (unsaturations). Table 9 summarizes the NNR experiments on

Favorable interactions in the Lo phase are also observed between Chol and other saturated PC, such as DMPC or DSPC (Table 8). However, favorable interactions are clearly

Table 9. Interactions between Unsaturated PC and Cholesterola

Table 8. Interactions between Saturated PC and Cholesterola phase

T

ω

interaction modeled

(cal/mol)


K

(°C) Ld

30

4.0

0

Ld

60

4.6

−46

Lo

30

5.8

−110

Lo

60

5.0

−58

Ld

60

3.7

23

Lo

60

6.1

−114

Lo

30

8.8

−237

Lo

60

6.9

−180

gel

30

1.17

370

DMPC−Cholesterol (DSPCea/DMPCea/Cholea 42.5:42.5:15) (DSPCea/DMPCea/Cholea 42:42:16) (DSPCea/DMPCea/Cholea 30:30:40) (DSPCea/DMPCea/Cholea 30:30:40) DPPC−Cholesterol (DPPC/DPPCea/Chol/Cholea 87.5:2.5:7.5:2.5) (DPPC/DPPCea/Chol/Cholea 57.5:2.5:37.5:2.5) DSPC−Cholesterol (DSPCea/DMPCea/Cholea 30:30:40) (DSPCea/DMPCea/Cholea 30:30:40) (DSPCea/DMPCea/Cholea 42.5:42.5:15)

ω

T refs

86 115

phase

(°C)

K

(cal/mol)

Ld

45

2.4

160

Lo

45

4.0

0

Ld

45

1.8

250

Ld

45

1.0

430

86 84, 115 56

interaction modeled (actual mixture composition) 16:0,18:1PC − Chol (16:0,c1PC/16:0,c1PCea/ Cholea 95:2.5:2.5) (DPPC/DPPCea/16:0, c1PCea/Chol 57.5:2.5:2.5:37.5) 16:0,18:2PC−Chol (16:0,c1PC/16:0,c2PCea/ Cholea 95:2.5:2.5) 16:0,18:3-PC−Chol (16:0,c1PC/16:0,c3PCea/ Cholea 95:2.5:2.5)

refs

116 116

90

90

a


56, 105 86 84, 115 86

mixtures of unsaturated PC and Chol.90,116,117 The interactions determined by NNR between Chol and these mimics of unsaturated PC become increasingly “repulsive” as the degree of unsaturation increases. The values of ω increased from 160 to 250 to 430 cal/mol for 16:0,c1PCea, 16:0,c2PCea, and 16:0,c3PCea, at 45 °C. Note, however, that for the larger interactions, the QC approximation is not accurate.22,55 Indeed, in our most recent study, a match of the calculation from MC simulations with NNR experiments yielded ω = 165 cal/mol for 16:0,c1PCea−Chol and ω = 395 cal/mol for 16:0,c3PCea−Chol interactions,117 which compare with 160 and 430 cal/mol, respectively, using the QC approximation. The interactions of these unsaturated PC with Chol were also measured by NNR as a function of temperature. We found ω to be essentially independent of temperature for both

a


an exception in lipid mixtures. Interactions between all other lipids that have been measured by NNR are usually unfavorable, or close to neutral at best (Tables 3 and 5). Mixtures of Cholesterol and Unsaturated PC. Of particular interest for biological membranes are the interactions of Chol with mono- and polyunsaturated phospholipids. Figure 14 shows the structures of exchangeable mimics of unsaturated PC, here designated as 16:0,c1PCea, 16:0,c2PCea, 35


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ω = 160 cal/mol between POPC and Chol in the Ld phase determined by NNR (Table 6). The distributions of SM, POPC, and Chol in these membranes can be visualized through snapshots of the lattice in the simulations. Figure 19 shows the most revealing aspect

16:0,c1PCea and 16:0,c3PCea.117 This indicates that the interaction is determined by enthalpy. More Complex Mixtures. Once the lipid interaction parameters are known from NNR and calorimetry in combination with MC simulations, the behavior of more complex membranes can be predicted and interpreted using Ising models. But other experimental techniques, too, can be used in this approach. In the example described next, FRET was used in combination with MC simulations to investigate the phase behavior of membranes composed of sphingomyelin (SM), POPC, and Chol.20 This mixture constitutes the simplest model of the lipid in the outer leaflet of a mammalian plasma membrane. For another example of the application of the Ising model in a ternary mixture, see the study of DSPC/ DOPC/Chol.118 LUVs of SM/POPC/Chol were prepared containing trace amounts of two lipid fluorophores that constitute a FRET donor/acceptor pair. The FRET efficiency was measured experimentally and calculated in the simulations. By varying the lipid interaction parameters ω between each pair of components in the simulations, a match to the experimental FRET efficiency was obtained, as shown in Figure 18.20 The ω

Figure 19. Snapshots of Monte Carlo simulations of Ising models of mixtures of SM, POPC, and Chol at room temperature.20 SM (white), POPC (black), and Chol (red). Mixtures of SM/POPC 70:30, SM/ Chol 50:50, POPC/Chol 50:50, and SM/POPC/Chol 35:30:35 are shown. The POPC/Chol panel is original. The other panels are reprinted with permission from Frazier et al.20 Copyright 2007 The Biophysical Society.

of this study. The binary mixtures SM/POPC, SM/Chol, and POPC/Chol do not show phase separation. The lipid distribution is not far from random in POPC/Chol. Larger clusters occur in SM/POPC, and the mixture is uniform in SM/Chol. However, when all three components are present, in SM/POPC/Chol 35:30:35 (Figure 19, bottom right), phase separation occurs between POPC and SM-Chol regions. This large effect in the ternary mixture is a nontrivial consequence of the combination of a favorable interaction between SM and Chol (ω < 0) with unfavorable interactions (ω > 0) between those two lipids and POPC.20 Such combination of favorable and unfavorable interactions pointing toward the same outcome has been called the “push−pull” effect.89,116,119 The simplicity of the Ising model hides the rich behavior it is capable of displaying. As this examples shows, that behavior follows solely from simple pairwise interactions.

Figure 18. FRET efficiency between two lipid fluorescent probes incorporated in membranes of SM, POPC, and Chol, determined by experiment (●) and Monte Carlo simulations (solid line) of an Ising model of these mixtures. The best match was obtained with ω = −350 cal/mol for the interaction SM−Chol, ω = 300 cal/mol for SM− POPC, and ω = 200 cal/mol for POPC−Chol.20 Reprinted with permission from Frazier et al.20 Copyright 2007 The Biophysical Society.

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CONCLUSION AND OUTLOOK The simplicity of the Ising model is one of its greatest strengths. However, this simplicity is deceptive. As McCoy and Wu wrote in the preface of their book,120 “Of all the systems in statistical mechanics on which exact calculations have been performed, the two-dimensional Ising model is not only the most thoroughly investigated; it is also the richest and the most profound.” The same model is able to describe systems composed of very different particles or molecules with a minimum set of parameters. Here, the Ising model has been applied to describe lipid interactions in membranes. The interaction parameter ω occupies a central place in this description. A major part of the problem described here is the use or the development of experimental methods, such as DSC, NNR, or FRET, that can be used to determine ω for different pairs of lipids. Some of those methods can be used essentially on their

parameters are in reasonable agreement with the values obtained, mostly later, from a combination of DSC with MC simulations (Table 4), from NNR experiments on mixtures between DPPC and Chol (Table 6), and mixtures of DPPC and POPC (Table 5), assuming DPPC is a mimic of SM. Thus, the value of ω = 300 cal/mol used for the interaction of SM (ordered) with POPC (disordered) seems reasonable if compared with ω = 450 cal/mol between DPPC (gel) and POPC (liquid) (Table 4). The value of ω = −350 cal/mol between SM (Lo) and Chol is in good agreement with ω = −400 cal/mol between DPPC (Lo) and Chol, which is obtained by extrapolation of the temperature-dependent NNR data in Table 6 to room temperature. Finally, ω = 200 cal/mol between POPC (Ld) and Chol is in fairly good agreement with 36


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own, assuming certain approximations are valid, such as the NNR method with the QC approximation. However, in most cases, for more accurate determinations of ω, those experimental methods need to be used together with Monte Carlo simulations of Ising models. Several examples were presented in which DSC was used and one in which FRET was used together with simulations. We are now reaching a point where a wealth of information is available, much of which was described in this article. This opens new and exciting possibilities. If the interaction parameter ω is known for a pair of lipid species, Monte Carlo simulations can be used to study their mutual mixing behavior. These simulations also allow us to investigate much more complex mixtures if the mutual interactions between all their components are established. Thus, we can begin to envision the simulation of mixtures incrementally approaching the complexity found in a biological membrane, at least with regard to its major constituents. Furthermore, this approach is not restricted to lipids. Integral and peripheral membrane proteins can also be incorporated in the simulations. Indeed, we have done that in a few simple cases.63,64,121 It is true that such a goal is also approachable through MD simulations, at least with approximate lipid models such as coarse-grained MARTINI,122−128 and most recently by an innovative use of atomistic MD simulations.87 To my mind, however, a very important aspect of the approach we have taken is that it is firmly rooted on experiment. The Ising model is used to interpret experiments and to extract from them the lipid interaction parameters. But experiment is ultimately the origin of all the interaction parameters used in these MC simulations.

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and of the Biophysical Society, since 1989, where he served as chair of the Membrane Structure and Assembly Subgroup in 2011. He was editor for Biophysical Journal from 2010−2016 and joined the editorial advisory board of Langmuir in 2017. Paulo Almeida has published close to 60 peer-reviewed journal articles and is the author of a textbook on Proteins: Concepts in Biochemistry (Garland Science, 2016). His research, which has been mainly supported by grants from NIH and NSF, has concentrated on the areas of membrane-active peptides and lipid interactions in membranes, especially those involving cholesterol.

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ACKNOWLEDGMENTS This work was supported by Grant No. CHE-1464769 from the National Science Foundation. It is a pleasure to thank Steve Regen for more than 12 years of a most stimulating collaboration. I also thank Thomas Heimburg for providing experimental DSC and MC simulation data and for his critique of this manuscript.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

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

The author declares no competing financial interest. Biography

Paulo Almeida is Professor and Chair of the Department of Chemistry & Biochemistry at the University of North Carolina Wilmington (UNCW). He received his Ph.D. from the University of Virginia in 1992, where he also did his postdoc in 1994−1995. After holding faculty positions in Portugal, at the Universities of Algarve and Coimbra, he joined UNCW as assistant professor in 2001, where he has remained since. Paulo Almeida is member of ACS, since 1999, 37


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How to Determine Lipid Interactions in Membranes from Experiment

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