Area Compressibility Moduli of the Monolayer Leaflets of Asymmetric Bilayers from Simulations

Abstract

Extraction from simulations of the area compressibility moduli of the monolayers in a bilayer is considered theoretically. A statistical mechanical derivation shows that the bilayer modulus is the sum of the two monolayer moduli, as is often supposed but contrary to a recent study. Seemingly plausible assumptions regarding fluctuations are tested rigorously. Prospects for future research are discussed.

Significance

It is important to describe the properties of both leaflets of generally asymmetric biomembranes. One such property is the area compressibility modulus. This manuscript rigorously establishes the fundamental theory that corrects a recent Biophysical Journal article. The theory is straightforward but substantial enough that it was not readily apparent why the previous theory was incorrect. This is why this paper should be considered a new article and not just a comment. Another reason is that this paper points to an alternative method, used only once previously, for extracting the leaflet area compressibility modulus from simulations.

Introduction

Biomembranes are generally asymmetric, so increasing attention has been paid to creating asymmetric model systems, both in vitro and in silico. Then, it is appropriate to consider separately the physical properties of each of the two monolayers in asymmetric lipid bilayers. Separating some of those properties experimentally is difficult, so it is appropriate to turn to simulations. Those simulations that agree with the experiment for all the properties that experiment can measure can then be considered for extracting properties that experiments do not measure (1, 2). The property of interest in this article is the area compressibility modulus. There are two well-known methods of extracting the bilayer modulus from simulations. This article focuses on the extraction of the individual monolayer moduli.

An area compressibility modulus k is generally defined as follows:

$$k=A{\left(\partial \gamma /\partial A\right)}_T,$$ (1)

where A is the area, and γ is the surface tension. This modulus is essentially a spring constant. Assuming that there is negligible coupling between the two monolayers, j = 1 and 2, each monolayer can be thought of as analogous to a spring with modulus k_j, and the bilayer would then be two springs of equal length in parallel. The forces on the springs would be F_j = k_j x, and the force on the two springs would be F₁₂ = F₁ + F₂ = (k₁ + k₂) x, which is then identified as k₁₂ x. It would then follow by analogy from elementary mechanics of springs that the modulus k₁₂ for a bilayer is the sum of the monolayer moduli, which would be as follows:

$$k_{12}=k_1+k_2.$$ (2)

Eq. 2 for symmetric bilayers is the conventional wisdom in the lipid bilayer field that has often been used for symmetric bilayers, usually in passing without even being remarked upon (3, 4, 5, 6).

In contrast to Eq. 2, a recent article (7) derived a rather different equation as follows:

$$\frac{1}{k_{12}}=\frac{1}{2}\left(\frac{1}{k_1}+\frac{1}{k_2}\right).$$ (3)

This equation has two highly unusual features. The first comes from applying it to a symmetrical bilayer. Then, the two monolayer moduli must be equal, k₁ = k₂, so Eq. 3 requires that each monolayer modulus must equal the bilayer modulus k₁₂. This unusual feature was specifically noted, and a rationalization was provided (7). The second unusual feature comes from considering a bilayer that is highly asymmetric; for example, monolayer one might consist of gel phase 1,2-dipalmitoyl-sn-glycero-3-phosphoethanolamine at room temperature, and monolayer two might consist of highly fluid 1,2-dioleoyl-sn-glycero-3-phosphocholine. In the limit when k₁ is very much larger than k₂, Eq. 3 predicts that the bilayer modulus k₁₂ is only twice the smaller monolayer modulus k₂. This violates the definition in Eq. 1 because the tension γ₁ to change the area of monolayer 1 should be enormous compared to the tension γ₂ to effect the same change to the area of monolayer 2. A macroscopic analogy would be to construct a bilayer consisting of a sheet of rubber on a sheet of steel and claim that the area compressibility is unrelated to that of the steel. As the derivation provided for Eq. 3 has gaps and makes unproven assumptions (7), it is appropriate to return to basics.

After laying the statistical mechanical foundation in Methods, the current article provides a rigorous derivation of Eq. 2 in the Thermodynamic relations subsection. The Correlations subsection reveals exactly which assumptions employed in (7) are incorrect for the case of uncoupled monolayers considered there. It also allows for the consideration of features not considered theoretically (7) that would nevertheless affect that method of analyzing simulations. The Discussion assesses the prospects for applying the small patch method of (7), and attention is called to a different simulation method that would not be subject to the same artifacts.

Methods

The theoretical system

Consider a bilayer with fluctuating area A and average area <A> = A₀. The monolayer fluctuating areas A₁ and A₂ are necessarily constrained to be equal to the bilayer fluctuating area, A₁ = A₂ = A. The simulation method proposed in (7) analyzes the fluctuating areas of small portions of each monolayer j with fluctuating areas $a_j$. For convenience, we will set the average small areas <$a_j$> on both monolayers to be the same value $a_0$. Of course, the fluctuating areas a₁ and a₂ are not generally equal unless there is very strong coupling between the two monolayers. A schematic of this setup is shown in Fig. 1.

Figure 1.

A schematic of fluctuations of the patches in two monolayers in a bilayer. To see this figure in color, go online.

Assuming that there is also no coupling between these small fluctuating areas and between the remaining B_j = A_j − $a_j$ areas in each monolayer leads, via the equipartition theorem, to the monolayer moduli:

$$k_j=k_{B}T{a}_0/<{\left(a_j-a_0\right)}^2>,$$ (4)

where k_B is Boltzmann’s constant, and k_BT is thermal energy. This equation is well established as one of the two main ways to obtain k₁₂ when bilayer areas A and A₀ replace $a_j$ and $a_0$ (8, 9). The interesting issue is how k₁₂ is related to the k₁ and k₂ that are obtained from Eq. 4. For this, we return to the same statistical mechanics used to derive the equipartition theorem, but we now have to realize that even if the two monolayers are uncoupled locally, there is the global constraint A₁ = A₂ = A.

The formal description of the system begins by writing the basic fluctuation energy for the small patches and also for the remaining large areas B_j − B₀:

$$E_{basic}=\frac{1}{2a_0}\sum _1^2{k}_j{\left(a_j-a_0\right)}^2+\frac{1}{2B_0}\sum _1^2{k}_j{\left(B_j-B_0\right)}^2.$$ (5)

Of course, analysis of simulations would analyze many small patches to obtain better statistics, but there is no loss of generality in a derivation that considers small patches one at a time, each embedded in a reservoir that consists of the remaining small patches considered as a group. Also, the average area per molecule, suitably defined, is generally different in the two monolayers, and those areas may not coincide with the areas of the free-standing monolayers. Again, with no loss of generality, the areas of the small patches are not those of lipid molecules but have been chosen for notational convenience to contain an appropriate amount of material in each monolayer such that the average small patch area $a_0$ is the same in both monolayers.

Each of the four terms in Eq. 5 has the conventional harmonic form for the fluctuation energy with monolayer moduli k_j. This equation looks like it has four independent fluctuating variables, but there are only three because $a_j$ + B_j = A_j = A. Although this complicates the ensuing derivation, it is crucial for explaining the difference between Eqs. 2 and 3. We therefore replace (B_j − B₀) by (A − A₀) − ($a_j$ − $a_0$) in Eq. 5. It will be convenient to condense the notation in subsequent equations by writing the three independent fluctuating variables as x = A − A₀ and y_j = $a_j$ − $a_0$. Then, Eq. 5 becomes the following:

$$E_{basic}=\frac{1}{2a_0}\sum _1^2{k}_j^{\text{'}}{y}_j^2+\frac{1}{2B_0}\sum _1^2{k}_j{\left(x-y_j\right)}^2.$$ (6)

As is often the case in statistical mechanics, it is advantageous to formally distinguish nominally identical terms such as has been done for the k_j′ in the first term in Eq. 6 and to add terms to the basic energy as follows:

$$E_{add}=\frac{w}{2A_0}{x}^2+\frac{h_{12}}{2a_0}{\left(y_1-y_2\right)}^2-\frac{1}{a_0}\sum _1^2{h}_j\left(a-a_0\right)\left(B-B_0\right).$$ (7)

The first term in Eq. 7 is crucial because it will enable finding the relation between the bilayer k₁₂ and the monolayer moduli k_j by taking the derivative of the partition function with respect to w and then setting w = 0. The h₁₂ term provides for the coupling between the two monolayers. For h₁₂ > 0, the coupling energy increases when the areas of the small patches are correlated. With the usual volume conservation assumption, such correlated fluctuations correspond to total bilayer thickness fluctuations (sometimes called peristaltic modes); h₁₂ > 0 therefore suppresses thickness fluctuations, whereas h₁₂ < 0 enhances them. The h_j terms provide a kind of coupling between the small patches and the large patches in the monolayers; even more importantly, that term will enable finding the correlation functions that were previously presumed to be zero (7).

Statistical mechanical derivation

The partition function for this system is defined as follows:

$$Z=\iiint \exp \left[-\beta \left(E_{basic}+E_{add}\right)\right]dxd{y}_{2}d{y}_1,$$ (8)

where β = 1/k_BT. The result of the integrations is as follows:

$$Z^2=\frac{{\left(2\pi kT/B_0\right)}^3}{f_0\left(f_1{f}_2-f_3^2\right)}.$$ (9)

Defining R = (B₀/$a_0$) and r=(B₀/A₀), the fs are as follows:

$$f_0=\left(k_1+k_2+rw\right),$$ (10)

and for j = 1 and 2 as follows:

$$f_j=R{k}_j^{\text{'}}+k_j+2h_j+R{h}_{12}-\left({\left(k_j+h_j\right)}^2/f_0\right),$$ (11)

and

$$f_3=R{h}_{12}+\left(\left(k_1+h_1\right)\left(k_2+h_2\right)/f_0\right).$$ (12)

The evaluation of the partition function in Eq. 8 was performed by first grouping all the exponential factors involving x² and x. Completion of the square in the form (ax − c)² − c² provides a Gaussian x integral, which gives the factor 2πkT/B₀f₀ in Eq. 9. The factors involving y₁² and y₁, including those in the c² factor left over from the x integration, were then similarly treated, finally ending with a Gaussian integral over y₂. The results of the y₁ and y₂ integrations together give the remaining factor in Eq. 9.

Results

Thermodynamic relations

Derivatives of the partition function in Eq. 9 give thermodynamic quantities of interest. First, consider the average energy as follows:

$$<E>=-\partial \ln Z/\partial \beta =\left(3/2\right)k_{B}T,$$ (13)

defined by the first equality in Eq. 13. The calculation using Eq. 9 gives the second equality. This recovers the usual equipartition result for three classical harmonic degrees of freedom.

The most interesting derivative is of ln Z with respect to the parameter w. By definition of the partition function in Eq. 8 and the definition of E_add in Eq. 7, this derivative gives the first identity in the following equation:

$$-2{\left(\frac{\partial \ln Z}{\partial w}\right)}_0=\frac{<{(A-A_0)}^2>}{A_0{k}_{B}T}=\frac{1}{k_{12}}=\frac{1}{k_1+k_2}.$$ (14)

The second equality is just the identity for the bilayer modulus k₁₂ as in Eq. 4. The last equality is the result of taking the derivative in Eq. 9 and then setting w = 0 as well as h_j = h₁₂ = 0; this returns the energy to the basic terms in Eq. 6. Eq. 14 is a primary result that confirms Eq. 2 which was suggested in the Introduction by analogy to springs. This fully rigorous result proves that Eq. 3 is incorrect.

When h₁₂ is nonzero, there are corrections to Eq. 14, which, however, are of order r and therefore vanish in the small subsystem limit $a_0$≪A₀. This is consistent with the infinitely strong h₁₂ limit, which is independently calculable because then y₁ = y₂ is constrained and the tightly coupled monolayers reduce to a single layer with modulus k₁ + k₂. However, the h₁₂ coupling between the monolayers is far from innocuous for the interpretation of small patch fluctuations. These fluctuations are obtained by taking a derivative with respect to k_j′ and setting h_j = 0 = w, designated by 0′ in the first term in the following equation:

$$-2{\left(\frac{\partial \ln Z}{\partial k_1^{\text{'}}}\right)}_{0\text{'}}=\frac{<{\left(a_1-a_0\right)}^2>}{a_0{k}_{B}T}=\frac{1}{k_1^{app}}=\frac{1}{k_1+h_{12}}.$$ (15)

The first equality in Eq. 15 follows simply from Eqs. 7 and 8. The second equality in Eq. 15 defines the apparent monolayer modulus k₁^app that the small patch simulation method would report. The last equality in Eq. 15 shows the result of the calculation using Eq. 9. Importantly, k₁^app is not the true monolayer modulus k₁ but becomes k₁+ h₁₂. Encouragingly, one could determine h₁₂ = 1/2 (k₁^app + k₂^app − k₁₂) and thence obtain k₁ and k₂ using the final equality in Eq. 15. However, this assumes that the only coupling is between patches on opposite monolayers.

Although the h_j terms might appear to provide the in-plane coupling equivalent to the h₁₂ out of plane term, there is a difference that makes the h_j terms unsatisfactory for determining k₁ and k₂. For either sign of h_j, some fluctuations decrease the h_j energy term, which even leads to instability of the system for modest values of h_j. However, it may be noted that these terms decrease k₁^app and k₁₂ but only proportional to h_j² and to r = $a_0$/A₀. A better model for in-plane coupling might involve adding terms like (a−$a_0$)²(B−B₀)² to the energy, but this would introduce quartic terms, which, even if calculable, would complicate an already complicated derivation of the partition function.

Correlations

It is interesting to see exactly how plausible assumptions for correlations between the patches fail because of the A₁ = A₂ = A constraint. Let us begin with the following identity, alluded to in (7), that follows from A = 1/2 (A₁ + A₂):

$$<{\left(A-A_0\right)}^2>=<{\left(\frac{1}{2}\sum _1^2\left[\left(a_j-a_0\right)+\left(B_j-B_0\right)\right]\right)}^2>.$$ (16)

The left-hand side is just A₀k_BT/k₁₂. Expanding the right-hand side gives the following:

$$<{\left(A-A_0\right)}^2>=\frac{1}{4}{\sum }_j\left(<{\left(a_j-a_0\right)}^2>+<{\left(B_j-B_0\right)}^2>\right)+Q_1+Q_2+W,$$ (17)

where

$$Q_j=\frac{1}{2}<\left(a_j-a_0\right)\left(B_j-B_0\right)>,$$ (18)

and

$$W=\frac{1}{2}\langle \left[\left(a_1-a_0\right)+\left(B_1-B_0\right)\right]\left[\left(a_2-a_0\right)+\left(B_2-B_0\right)\right]\rangle .$$ (19)

W was previously assumed to be zero (7), but it is trivially equal to 1/2 <(A − A₀)²> by inspection. The authors of Doktorova et al. (7) have acknowledged this correction to W in a recent erratum (15) although not to the subsequent corrections in the next paragraph.

Eq. 17 can now be rewritten as follows:

$$<{\left(A-A_0\right)}^2>=\frac{1}{2}{\sum }_j\left(\langle {\left(a_j-a_0\right)}^2\rangle +\langle {\left(B_j-B_0\right)}^2\rangle \right)+2Q_1+2Q_2.$$ (20)

If there is no specific coupling between patches, application of Eq. 4 shows that <($a_j$ – $a_0$)²> + <(B_j – B₀)²> = A₀k_BT/k_j, so Eq. 20 can be further rewritten as follows:

$$\frac{A_0{k}_{B}T}{k_{12}}=\frac{1}{2}\left(\frac{A_0{k}_{B}T}{k_1}+\frac{A_0{k}_{B}T}{k_2}\right)+2Q_1+2Q_2.$$ (21)

If Q₁ + Q₂ were zero, then this would be a derivation of Eq. 3. However, the Q_j are straightforwardly determined to be nonzero by taking derivatives of the partition function with respect to h_j. The result for Q₁ is as follows:

$$Q_1=-\frac{k_2}{k_1}\frac{k_{B}T{A}_0}{4\left(k_1+k_2\right)},$$ (22)

and the result for Q₂ simply exchanges the indices 1 and 2, so Q₁ + Q₂ is not zero. The product Q₁Q₂ depends only upon the sum k₁ + k₂, but the ratio Q₁/Q₂=(k₂/k₁)² shows that the Q_j have quite different values for asymmetric bilayers with larger values of Q_j for the softer monolayer than for the stiffer one. Finally, combining 2Q_j in Eq. 22 with the k_j terms in Eq. 21 gives, for both j = 1 and 2, the result 1/2A₀k_BT/(k₁ + k₂), thereby giving the following:

$$\frac{A_0{k}_{B}T}{k_{12}}=\frac{1}{2}\left(\frac{A_0{k}_{B}T}{k_1+k_2}+\frac{A_0{k}_{B}T}{k_1+k_2}\right),$$ (23)

which again confirms that the bilayer modulus k₁₂ is the sum of the monolayer moduli as in Eq. 2. Fig. 2 plots the terms in Eq. 21 as the relative stiffness of the two monolayers varies.

Figure 2.

The terms in Eq. 21 are normalized to A₀k_BT/(k₁+ k₂). The algebraic sum of the four terms on the right-hand side of Eq. 21 equals 1/k₁₂= 1 because k₁+ k₂ is normalized to 1. To see this figure in color, go online.

It is interesting that the simple constraint that both monolayers have the same area has such a large effect on the Q_j and W correlations. Compared with a system consisting of a single monolayer, W is not defined, and the only defined Q₁ is zero as one would expect from the assumption that the small patch is uncoupled from the large patch reservoir. It is also interesting to note that W + Q₁+Q₂ = 0 for symmetric bilayers but only for symmetric bilayers.

Discussion

The difference between the derivations of Eqs. 2 and 3 is the treatment of the crucial constraint that the monolayers in a bilayer must have the same total area. Although the derivation of Eq. 3 in the previous subsection recognizes this constraint, it then had to make assumptions about various correlations. The derivation of Eq. 2 makes no such assumptions and actually calculates the correlation functions and shows that they do not agree with the assumptions in the derivation of Eq. 3. The derivation of Eq. 2 confirms what has been assumed for a long time, at least for symmetric bilayers.

Although Eq. 3 is incorrect, it is a reasonable prospect that the fluctuations in small patches will reveal differences in the monolayer moduli in asymmetric bilayers. If there is no coupling of the patches with the remainder of the bilayer, then the rigorous theory in this article shows that this analysis will give the monolayer moduli quantitatively. However, if the sum of the apparent monolayer moduli obtained from small patches k₁^app + k₂^app does not equal the well-determined bilayer modulus k₁₂, then there must be coupling. If the coupling is only between patches on opposite monolayers, then the analysis using h₁₂ allows the extraction both of the coupling and the individual k_j. One could also have equality with coupling if the in-plane coupling competes with the h₁₂ coupling, but we do not have a good theory for the effect of coupling within each monolayer.

The simulations previously reported (7) gave k₁^app + k₂^app = 2k₁₂, indicating strong coupling. However, the method employed there to convert area fluctuations to thickness fluctuations may have been flawed by the assumption that the volume was constant in a region consisting of only about half the hydrocarbon region as it was determined by the location of specific methylenes. A similar assumption has been found to be false in a current analysis of the Poisson ratio (unpublished data).

Given these problems with the small patch analysis method, it is appropriate to consider a second method that stems from the second method that has routinely been employed to obtain the bilayer modulus k₁₂. This second method simply plots the area A versus surface tension γ to obtain the area compressibility modulus directly from its definition in Eq. 1. This method was noted in (7), but it was apparently not realized that it could also be used to obtain the k_j moduli separately. For each value of γ in the simulation, one would first calculate the average lateral pressure profile Π(z) = −γ(z) of the bilayer (9, 10, 11). The integral of γ(z) along z across the whole bilayer is the value of γ. The idea is that one may also choose to integrate only over each monolayer separately to obtain γ₁ and γ₂. Then, one would calculate the separate moduli as follows:

$$k_j=-A{\left(\partial {\gamma }_j/\partial A\right)}_T,$$ (24)

using the average <A> for each value of γ_j. The only assumption in this method, as in the proposed method (7), is that it makes sense to separate the bilayer into two monolayers. One might argue that it does not make sense to apply it to bilayers with fully interdigitated hydrocarbon chains, but it does appear to be a reasonable conceptual division for most bilayers that have only mini-interdigitation (12) of the monolayers near the center of the bilayer. Furthermore, one does not have to just separate k₁₂ into two monolayer values. Indeed, k₁₂ has already been further refined into a modulus k₁₂(z) that varies with depth z for a coarse-grained simulation of a symmetric bilayer (13). When applied to an asymmetric bilayer, that method would provide an even more detailed view than just obtaining k₁ and k₂. As noted (7), this second method requires more simulations at different surface tensions, and the lateral pressure profiles would probably be subject to more noise. However, this method would not be subject to the difficulties involved in the small patch method.

Finally, it may also be noted that (7) tackled the thorny issue of the relation between the area compressibility modulus K_A (=k₁₂) and the bending modulus K_C. It focused on the appropriate thickness t to use in the following:

$$K_C=\frac{1}{24}{t}^2{K}_A,$$ (25)

rather than on the factor of 24 that comes from the polymer brush formulation that assumes independent monolayers (6). The choice of the total hydrocarbon thickness for t has worked rather well (2, 6). However, the most significant exception was found when cholesterol was added to lipid bilayers, and a large change in the definition of t was proposed (14). In agreement with (7), further analysis using reliable determinations of both K_A and K_C, both from experiment and from simulations, are indeed needed to refine what effectiveness thickness t is appropriate to relate the mechanical properties of specific lipid bilayers.

Conclusions

The statistical mechanical relation of the leaflet area compressibility moduli to that of the bilayer has been rigorously derived. Coupling between the two leaflets has been incorporated theoretically and that can be addressed using the small patch simulation method. However, an alternative method is likely to be superior for obtaining more, and more reliable, information about the area compressibility of asymmetric membranes.

Editor: D. Peter Tieleman.