J. Phys. Chem. B 2008, 112, 2119-2127
2119
Phase Behavior of Amphiphilic Lipid Molecules at Air-Water Interfaces: An Off-Lattice Self-Consistent-Field Modeling Y. Lauw,†,‡ A. Kovalenko,†,‡ and M. Stepanova*,‡,§ Departments of Mechanical Engineering and Electrical and Computer Engineering, UniVersity of Alberta, Edmonton, Alberta, Canada, and National Institute for Nanotechnology NRC, 11421 Saskatchewan DriVe, Edmonton, Alberta, Canada ReceiVed: September 24, 2007
We introduce an extended application of the off-lattice self-consistent-field theory (SCFT) to model lipid monolayers at air-water interfaces. The off-lattice SCFT is used without a priori symmetry assumptions on equilibrium morphologies. This enables us to capture asymmetric lipid membranes at air-water interfaces which are otherwise unattainable with a conventional SCF model. Equilibrium morphologies in systems containing lipid molecules, fractions of air, and water are studied as a function of the relative amount of lipid molecules. The corresponding Langmuir isotherms are analyzed to reveal possible phase transitions. We consider both saturated and unsaturated lipid molecules with a branched structure. For saturated lipids, we find two distinct morphological phases, i.e., micellar and lamellar, showing a pronounced first-order phase transition with a well-defined region of phase coexistence. This region is sensitive to the hydrophilicity of lipid molecules and the miscibility of air with water molecules. The phase coexistence is also influenced by the size of hydrophilic and hydrophobic parts of lipid molecules. In contrast, membranes of unsaturated lipids have developed a continuous range of smooth structural transformations from a circular to an ellipsoidal micellar morphology and eventually to a lamellar structure. The shape of the lamella changes from a slightly undulated to a vigorously curved. Unlike saturated lipid membranes, there is no apparent first-order phase transition or a region of phase coexistence for unsaturated lipid membranes. We interpret this as a result of a higher flexibility of unsaturated lipid membranes which enables them to adopt a wider range of conformations in comparison with saturated lipid membranes.
I. Introduction Monolayers of lipid molecules constitute a major component of alveolar surfaces in lungs.1-2 These amphiphilic surfactants modulate the surface tension of the lung, stabilize alveoli from collapsing during expiration, and minimize the work needed to expand the alveoli during inhalation. The self-assembly of surfactant monolayers at air-liquid interfaces are fundamental for the natural biogenesis of the lung3-5 as well as for treatments of lung malfunction where surfactants are administered.6 Furthermore, the expansion and contraction of alveoli surface area during breathing are believed to be related to the spontaneous phase transition leading to reversible structural changes in self-assembled monolayers of amphiphilic surfactants.1-2,7-11 Despite intensive research efforts,1-2,7 the understanding of these functionalities lacks important details such as the number and exact mechanism of the phase transformations, as well as the molecular interaction of the relevant ingredients (saturated lipids, unsaturated lipids, proteins, and cholesterols). Extensive theoretical studies and numerical simulations of the behaviors of lipid membranes have been performed.7,12-13 Thus, a superb detailed treatment is provided by molecular dynamics (MD). Unfortunately, the performance of MD simulations is insufficient to model the entire self-assembly of diverse morphologies that lipid aggregates can adopt. It is because a †
Department of Mechanical Engineering, University of Alberta. National Institute for Nanotechnology NRC. § Department of Electrical and Computer Engineering, University of Alberta. ‡
structure close to the target (equilibrium) morphology must be predefined to reach a satisfactory convergence of the MD simulation. Furthermore, MD simulations are not capable to efficiently handle the phase transformations in the lipid membranes. A part of the reason is that MD simulations address individual molecular-level events that challenges their applications to model fundamental collective processes such as phase transformations. As a consequence, various coarse-grained approaches are very popular to study self-assembled systems such as micelles and lipid aggregates (vesicles, lamellae).13 In these approaches, molecules are considered consisting of several coarse-grained units (beads). Each bead represents group of atoms that interact through effective potentials. Understandably, such approaches are more efficient numerically than all-atom MD simulations. However, in addition to the obvious difficulties of consistently defining the effective bead interactions, these models still describe individual molecular events, thus they have similar limitations as MD in describing collective phenomena through thermodynamic characteristics such as the free energy.13 To model efficiently their phase behaviors, lipid membranes should be treated as a statistical ensemble of small systems. At the same time, molecular structures and respective interactions should be retained in sufficient detail. An approach that satisfies these requirements is known as self-consistent-field theory (SCFT).13-19 The SCF formalism iteratively minimizes the free energy functional of the system and takes into account the realistic structure of polymer chains and their connectivity in the local mean-field approximation.20-21 As a result, SCF
10.1021/jp077672x CCC: $40.75 © 2008 American Chemical Society Published on Web 01/26/2008
2120 J. Phys. Chem. B, Vol. 112, No. 7, 2008 solutions describe systems in equilibrium with a minimized free energy. The morphologies of equilibrium structures are described by the local volume fraction of the ingredients rather than particular configurations of individual molecules. The relevant thermodynamic characteristics such as the free energy and the chemical potential are then accessible directly from the SCF solutions. The SCFT has been originally introduced for modeling polymer melts.18-19,22 Further developments of the theory pursued studies on the phase separation of block copolymeric systems.23-27 The SCFT has been proven efficient for studying mechanical and thermodynamic properties of self-assembled micelles and vesicles.28-30 To date, a number of various methodologies in the SCF framework have been elaborated. In the so-called lattice SCF methodology, configurations adopted by polymer chains are confined to a predetermined discrete coordinate system.14-15 This method is relatively straightforward and numerically efficient. However, its predictive power is limited by a priori symmetry requirements of the equilibrium morphology. A more flexible yet a more numerically expensive method is the off-lattice SCF approach. Here, the self-assembled equilibrium morphology is not confined to any a priori assumptions.18-19 For the sake of numerical efficiency, off-lattice SCF modeling is often carried out in a two-dimensional space assuming a homogeneity along the third dimension.16,19,24 The vast majority of published SCF results have been obtained for linear polymer chains, whereas a general formalism for branched polymers has been introduced only recently.26-27 One of the most advantageous applications of the SCFT is the research on the phase behavior of lipid monolayers and bilayers.13,31-36 However, most SCFT studies on lipid membranes adopt the conventional lattice approach assuming planar or spherical symmetry of the equilibrium morphology and/or employing reflecting boundary conditions.30-36 Most studies consider linear amphiphilic lipid models,30-31,34-35 whereas realistic models of branched lipid molecules have not been sufficiently addressed. Only a few works consider lipid molecules with two hydrophobic tails representing the major ingredients of lung surfactants such as dipalmitoylphospatidylcholine (DPPC).32-33 Published SCF research mostly addresses qualitative mechanical properties of lipid membranes such as their bending rigidity.32-33 Other studies focus on the effects of electrostatic charges in the lipid head group to the equilibrium morphology,30 the inclusion of larger objects into lipid membranes,33 and the fusion between two membranes.34-35 Detailed studies on the thermodynamics of phase transformations in lipid membranes are rare.36 Furthermore, the majority of published works only consider lipid molecules in water disregarding the gaseous phase. Therefore, the SCFT has not yet been applied to study the whole self-assembly process of lipid membranes at the air-water interface. This is caused by two main reasons, i.e., the intensity of computational efforts required and the inability of existing SCF methodologies to capture diverse phase behaviors of such systems. In this paper, we further develop the off-lattice SCF methodology to model self-assembled lipid molecules in contact with water and air phases. A novel point in our approach is that we use the SCFT to model the phase behavior of lipid monolayers at the air-water interface where the symmetry of the equilibrium morphology is entirely unrestricted. Moreover, we explicitly introduce a significant amount of air fraction (30% of the total volume of the system). The use of an explicit amount of air in addition to the lipid and water molecules to simulate a system containing an equilibrium interface is also new and different
Lauw et al.
Figure 1. Structure of a saturated lipid AmBn (left panel) and an unsaturated lipid AmBn1CpBn2 (right panel) used in the model. The head group of both lipid molecules consists of a series of segments A with total length m. Each tail group of the saturated lipid is composed of segments B with length n. For the unsaturated lipid, one of the tail groups has the same composition as that of the saturated lipid, whereas the other one contains segments B and C in the order Bn1CpBn2, where the subindices n1, p, and n2 are the lengths of each corresponding segment chain and n1 + n2 + p ) n.
from other similar studies where only lipid and water molecules are taken into account. We use an isotropic blend of lipid, air, and water fractions as the initial condition and obtain equilibrium morphologies without any a priori assumptions on a symmetry of these morphologies. This treatment provides a sufficient flexibility to capture diverse phase behaviors of lipids at airwater interfaces. In this introductory study, we describe the SCF methodology for branched lipid molecules, each consisting of one hydrophilic head and two hydrophobic tail groups.26-27 For the sake of numerical efficiency, we consider self-assembled equilibrium morphologies in a two-dimensional space with periodic boundary conditions. As a result, our model is efficient enough to implement an extensive numerical research on self-assembled morphologies in systems containing branched lipids, water, and air in various proportions. We generate and discuss the corresponding Langmuir isotherms and present an efficient methodology to represent a phase coexistence through SCF calculations. The article is structured as follows. Section II outlines our off-lattice SCF methodology, followed by section III which describes the choice of model parameters. Results and discussions are presented in section IV, followed by the concluding remarks and highlights in section V. II. Theory We consider a canonical ensemble of small systems, each containing lipid and water molecules, as well as the air fraction. Every molecule is composed of one or several segments. The spatial distribution of a segment belonging to a molecule is described by its volume fraction φ. Segments corresponding to the air fraction are considered by analogy with units of free volume and described by their volume fractions similarly to the other components in the system. At every location in the system, the relevant volume fractions φ are determined by the corresponding local mean-field segment potential w representing the averaged interaction of segments with their neighbors. Conversely, the segment potentials depend on the volume fractions. We model lipid molecules as branched polymers, each consisting of a hydrophilic head group and two hydrophobic tails (Figure 1). The head group is composed of a chain of hydrophilic segments A, whereas the tail group consists of a chain of hydrophobic segments B (for a saturated lipid) or a combination of segments B and C (for an unsaturated lipid). Each segment located in the lipid molecule is identified by its ranking number s (the position of the segment in the chain). Segmentations are also applied for water and air fractions. They
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are represented by clusters containing two water segments W2 and two segments of free volume V2, respectively. To obtain the spatial distributions of segments, we need to account for the segment connectivity in each lipid molecule by solving the Edwards diffusion equation,20-21
∂qc(r, s) a2 2 ) ∇ qc(r, s) - wis(r)qc(r, s) ∂s 6
(1)
where the segment-weighting factor (Green’s function) qc(r, s) is the probability to find a given segment with ranking number s in the chain molecule c (c refers to lipid, water, or air) at point r. The unit a is the Kuhn length, i.e., a length scale below which a polymer chain is considered rigid, and wis(r) is the potential of segment i (where i ) {A, B, C} for lipid molecules, W for water, and V for air) with the corresponding segmentranking number s in the molecule c at point r. Equation 1 is solved numerically by applying initial condition values qc(r, s0) and periodic boundary conditions. We employ the off-lattice methodology previously used to obtain equilibrium properties of linear polymers.16-19 Here, we use the same technique to model nonlinear polymers, i.e., double tail lipid molecules. For a lipid molecule, the segment-weighting factor is calculated for each arm of the lipid branch. Each lipid molecule is composed of a single branch with three arms: one hydrophilic and two hydrophobic. An arm is considered to have a closed end at the connecting joint of the branch, whereas it is open at the opposite end (open end). The initial condition qlipid-k(r, s0) at the open-end of an arm-k (k ) {1, 2, 3}) is calculated directly from its corresponding segment potential, i.e., qlipid-k(r, s0) ) exp(-wi(r)/kBT), where i refers to the segment at the initial ranking number s0. The initial condition at the closed end of an arm-k qlipid-k(r, s′0) is calculated from the segment-weighting factors of the other arms, i.e., qlipid-k(r, s′0) ) Πp*kqlipid-p(r, sNp), where p denotes a different arm than k and qlipid-p(r, sNp) is the segment-weighting factor of the arm-p at its closed end (sNp) initially started from its open end. The initial condition of the arm-1 starting at the joint is thus qlipid-1(r, s′0) ) qlipid-2(r, sN2)qlipid-3(r, sN3) where sN2 and sN3 are the segment-ranking number of arm-2 and arm-3 located at the joint started from their respective open ends. The initial conditions at the closed end of arm-2 and arm-3 are calculated in a similar fashion. This method to account for branched structures has been employed previously by Ye et al. for nonlinear block copolymer melts.26-27 To our knowledge, this approach has never been applied before to model lipid membranes. In general, the volume fraction of segment-i φi(r) of molecule-c is calculated by convoluting the corresponding segment-weighting factors qc(r, s) with their complementary values q′c(r, s), where segments are numbered from the opposite directions.17-18,22 The φi(r) thus reads
φi(r) )
Ac
Nc
∑qc(r, si)q′c(r, si)
NcQc s)0
(2)
where Ac is the total area occupied molecules of type c, Nc is the molecular chain length, si is the segment-ranking number s of molecule c which has segment type i, and Qc ) ∫qc(r, si) q′c(r, si) dr is the partition function of the chain molecule (independent to the segment-ranking number). In our model, steric interactions between two different segments within a molecule or from different molecules are described by using the Flory-Huggins theory.37 The equilibrium segment potentials wi(r) are obtained directly from the corre-
sponding volume fractions (a saddle point approximation),17-18,22
wi(r) )
∑j χijφj(r) + λ(r)
(3)
where χij is the Flory-Huggins interaction parameter between segment-i and -j, λ(r) is the Lagrange multiplier at location r, i.e. a portion of the segment potential obtained from the incompressibility constraint, ∑iφi(r) ) 1 for all positions r. The entire SCFT modeling scheme is as follows. In every calculation, an initial random set of segment potentials is used to generate segment-weighting factors through eq 1. This corresponds to a random initial distribution of all molecules in the system. The diffusion equation is discretized using a 2D alternating-difference implicit (ADI) scheme with periodic boundary conditions. The calculated segment-weighting factors are then used to obtain the segment volume fractions via eq 2, which eventually leads to new segment potentials by eq 3. A set of segment potential values for the next iteration is provided by a linear combination of the old and new set of segment potentials (the Picard iteration scheme) with an additional use of the dynamic relaxation accelerator scheme.38 Iterations are performed until the difference between the segment volume fractions of the previous two iterations at any point r is less than 10-5. We run multiple independent calculations with random initial segment potentials to ensure the results satisfy the ergodicity condition. Once the self-consistent solution is obtained, the corresponding minimum free energy can be calculated. Here, we obtain the free energy of an equilibrium membrane Fm as a difference between the free energy of the entire system and the free energy of the bulk solution.35-36,39 For a system with a total cross sectional area A0, the free energy Fm reads
Fm
)
A0kBT
[ ( ) ( )] ∫[ ∑ ∑ fc
∑c N
A0fc
ln
c
QcRc
1
1
A0
2
-
φbc Nc
φbc
ln
Rc
-
1 A0
∫[∑wi(r)φi(r)] dr +
]
i
χij(φi(r) - φbi )(φj(r) - φbj ) dr -
i
j
1
A0
∫[λ(r)(1 - ∑φi(r))] dr
(4)
i
where the first term is the entropic contribution, the second and third terms are due to segment interactions, and the fourth is contributed from the segment-filling potentials due to the incompressibility constraint. In eq 4, fc is the total volume fraction of molecule-c in the system, Rc is the volume ratio of molecule-c and lipid molecule, and the superindex b denotes the bulk quantities. The average distance between lipid molecules at the interface (the interfacial length per lipid molecule) is denoted by S. For any given S, the interfacial pressure Π′ is calculated from the free energy of the membrane Fm as follows,
Π′ ) Π′0 -
Fm S0
(5)
where S0 is the total length of the membrane, and Π′0 is the maximum ratio of Fm and S0 (at a high S), i.e. Π′0 ) max(Fm/ S0).
2122 J. Phys. Chem. B, Vol. 112, No. 7, 2008 III. Model Parameters From the self-consistent model, we obtain thermodynamic properties of a 2D equilibrium system composed of branched lipid molecules (Figure 1). As outlined in section II, each lipid molecule consists of a hydrophilic head group and two hydrophobic tails. We denote the head group by Am, where A refers to the segment of the head group and m is the total length of the head group. Similarly, tail blocks are denoted by Bn, where B refers to the tail segment and n is the tail length. We use two models of lipid molecules representing saturated lipid molecules AmBn (Figure 1, left panel) and unsaturated lipids AmBn1CpBn2 (Figure 1, right panel). Basically, the unsaturated lipid has a similar chemical structure as the saturated lipid, with a difference in one of its tail groups. One tail group in the unsaturated lipid has the usual composition (Bn), whereas the other tail section contains two C segments representing the unsaturated alkyl groups containing the double bond CdC so that the entire tail composition is Bn1CpBn2. The subindices n1, p, and n2 are the lengths of each corresponding segment chain and n1 + n2 + p ) n. Most results presented in this work are obtained for saturated lipid molecules of the composition A10B20, and those for unsaturated lipid molecules are of the composition A10B9C2B9. In addition to the lipid molecules, our system system also includes water and air fractions. According to the conventional approach adopted in the SCFT,40 water is represented by clusters containing two segments W2. The air fraction is treated by analogy with the free volume that fills the space where the other ingredients are absent. The free space filled by air is considered similarly to the other components in the system and represented by clusters containing two “segments” of free volume V2 following the conventional approach to describe small molecules in the SCFT.40 The Flory-Huggins interaction parameters χ that appear in eqs 3 and 4 characterize the relative interactions (repulsion or attraction) between molecules in the system. In the SCFT, these empiric parameters are selected in order to reach reasonable miscibility behaviors in the system.31,33,37 The general rule is that the larger (the more positive) the χ value, the more pronounced the repulsive forces between the respective components, thus the worse their miscibility. On the contrary, moderate χ values characterize a relative affinity of the components. In this paper, unless indicated otherwise, we use the following set of χ parameters, χAB ) 0.3, χAV ) χBW ) 1, χAW ) χBV ) 0.1, and χVW ) 0.9. The large value of χAV means that the miscibility between the hydrophilic segments A and the air fraction is poor. The hydrophobicity of segments B and the immiscibility of air and water fractions also lead to relatively high values of χBW and χVW, respectively. In contrast, the hydrophillic property of segments A is represented by a relatively small χAW. Similarly, the χBV value represents the affinity of the hydrophobic segments B to air. In unsaturated lipids, the presence of the unsaturated bond is known to cause a bend in the respective tail chain.32,41 This results in an increased average interchain distance in the membranes.42 In this work, we do not explicitly account for the bending of the tail. However, we mimic the increasing average interchain distance by employing relatively high effective χ values for segments C, i.e. χAC ) χBC ) 1.5. For interactions of segments of type C with water and air, we choose χCV ) χCW ) 0, reflecting a slight permeability of water and air in the membranes of unsaturated lipids. The size of the entire 2D system is equal to L × L. This size is selected to be large enough with respect to the gyration radius Rg of one lipid molecule, i.e., L ≈ 10Rg. In our numerical
Lauw et al. scheme, the size of the grid step is chosen to be equal to 0.25 × 0.25a2 and the entire system comprises ≈ 80 × 80 grid steps. Finding equilibrium morphologies in such small systems is rather challenging due to a possible convergence to a metastable solution at some system sizes.43-44 Here, we use the conventional approach to verify the robustness of the solution by varying the number of grid steps by (10%.24 In all cases, the solution is insensitive to the system size. This indicates that our results correctly represent the equilibrium morphologies. We perform an extensive series of numerical computations by varying the initial volume fraction of the lipid molecules flipid from 1% to 30% . The volume fraction of the air fV is kept equal to 30% in all the cases presented. The volume fraction of water fW thus changes according to the condition of incompressibility from 69% to 40%, respectively. IV. Results and Discussions Typical equilibrium morphologies obtained with various relative contents of the saturated lipid molecules and water (flipid/ fW) are shown in Figure 2. The morphology in Figure 2a corresponds to a very low content of lipid, i.e., flipid/fW ) 0.03. In this case, a bubble of air containing lipid molecules is formed in water. It can be seen that the lipid molecules are oriented in such a way that their hydrophilic head groups are mostly in contact with water, whereas the hydrophobic tails are located inside the air bubble. Figure 2b shows that a similar morphology is found in the composition regimes up to flipid/fW ) 0.13. However, at the composition flipid/fW ) 0.19 the morphology changes to a planar layered (lamellar) structure. Here, the lipid head groups form two parallel monolayers separating a thin airrich layer containing the tail groups from the water-rich bulk solution, as demonstrated in Figure 2c. When the amount of lipid molecules is increased further (flipid/fW ) 0.46), the lamellar structure starts to undulate, as shown in Figure 2d. Finally, for flipid/fW ) 0.67, the lamellar structure tends to split forming multilayered structures (Figure 2e). We characterize the equilibrium morphologies by several parameters, i.e., the total perimeter P of the water-air interface, the number of lipid molecules in the interface nm lipid, and the average interlipid distance S. These parameters are related by P ) Snm lipid. Figure 3a-c shows the perimeter P, the number nm lipid, and the distance S, respectively, as a function of the ratio flipid/fW. It can be seen that the perimeter decreases sharply when the morphology changes from the micellar shape (Figure 2b) to the lamellar one (Figure 2c). However, the split-up of the lamellar morphology does not result in any significant changes of the perimeter and only a slight change of the slope is observed in this region. The number of lipid molecules in the lipid membrane nm lipid increases smoothly with the ratio flipid/fW. The interlipid distance S is also a smooth function of the ratio flipid/ fW except for the mentioned region of the morphological transition where S decreases stepwise following the similar decrease in the perimeter P. In Figure 4a, we plot the free energy of the interface Fm (see eq 4) as a function of the interlipid distance S. The figure shows that the free energy of the interface exhibits an overall decreasing trend with decreasing S (increasing number of lipid molecules at the interface). This trend is in agreement with the thermodynamic stability condition.45 Figure 4a also shows the Langmuir isotherm of our system, e.g., the interfacial pressure Π′ as given by eq 5. From the isotherm presented in Figure 4a, the interfacial pressure increases slowly when S decreases in the region S > 10a. With a further decrease of S, the increase of the interfacial pressure becomes more pronounced. In the
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Figure 3. Total interface perimeter P, the amount of lipid molecules m in the self-assembled structure nlipid , and the average inter lipid distance at the interface S (parts a-c, respectively) as functions of the ratio of lipid and water total fractions flipid/fW for the saturated lipid.
Figure 2. (a-e) Density plots of the lipid tail and head groups and the distributions of the air fraction corresponding to the relative contents of the saturated lipid molecules and water, flipid/fW ) 0.03, 0.13, 0.19, 0.46, and 0.67. Each box is of size 20 × 20a2, where a is the Kuhn length.
range of 3a < S < 4.5a, the isotherm undulates sharply with the average interfacial pressure of Π′co ≈ 2.4kBT/a. A further decrease in S results in a strong and monotonic increase of Π′. The undulating isotherm indicates the presence of a firstorder phase transition. Indeed, the change from the micellar to the lamellar morphology is observed in the undulating region as demonstrated by Figure 2b and c. Moreover, we observe a distinct coexistence of these two phases. Within the region of undulating Π′, our calculations produce either the micellar or the lamellar morphology at random (with random initial distributions of all the components) at the same S values. A detailed plot of the isotherm around the region of phase coexistence is shown in Figure 4b. The filled circles indicate the regimes where the phase coexistence is detected.
We do not observe any undulations on the isotherm or any phase coexistence regions for the split-up of the lamellar morphology shown in Figure 2d. Thus, there is no indication that the first-order phase transition occurs in this case. It should be noted that our planar morphologies found in the regimes of S < 3a differ significantly from lipid bilayers obtained in other works (see, e.g., refs 32-34). First, our morphologies are obtained from random initial distributions of the ingredients without any limiting assumptions regarding the equilibrium geometry. Here, we do not introduce any reflecting boundary conditions or other artificial symmetric constraints. Second, in contrast to other studies, our system includes a significant amount of air. This is shown in more details in Figure 5, where the cross-sectional distribution of lipid (both head and tail groups), water, and air fractions are plotted for the lamellae in Figure 2c. From the water-air interface inward, the water density drops significantly, and the air density decreases rapidly in the opposite direction. The distributions of lipid head groups peak at the interfaces, whereas the lipid tails are located within the air-rich region. The presence of air layer indicates that our system actually represents two lipid monolayers rather than a bilayer membrane, where only a minor amount of free volume is involved.32-33
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Lauw et al. TABLE 2: Comparison of the Interfacial Pressures and the Molecular Areas in the Coexistence Region (Π′coand Sco) of a Membrane Composed of Lipid Molecules with Different Head Group Lengths (Top) and Different Tail Lengths (Bottom)a lipid
Π′co (kBT/a)
Sco (a)
A7B20 A10B20 A15B20
Different Head Group Lengths 5.20 3.77 3.44
1.8 - 2.3 2.5 - 3.5 2.6 - 3.8
A10B13 A10B15 A10B20
Different Tail Group Lengths 2.95 3.62 3.77
1.4 - 2.4 1.6 - 2.4 2.5 - 3.5
a The interaction parameters are set as follows: χAB ) 0.3; χAV ) χBW ) 1; χAW ) χBV ) 0.3; and χVW ) 0.8.
Figure 4. (a) Interfacial pressure Π′ (left axis) and the membrane free energy per unit area (Fm)/(A0) (right axis) as functions of the average distance S between lipid molecules. Points labeled by a-e correspond to the morphologies shown in Figures 2a-e, respectively. (b) Zoomin of the isotherm near the region of phase coexistence. The filled circles indicate the conditions at which phase coexistence has been detected.
Figure 5. Profiles of the volume fractions φ of lipid head groups, lipid tail groups, air, and water (solid, dotted, long-dash, and shortdash lines, respectively) as functions of the coordinate z (in grid steps) for the interfacial morphology shown in Figure 2c.
TABLE 1: Comparison of the Interfacial Pressures in the Coexistence Region Π′co for Lipid A10B20 with Different Values of χAW (Top) and Different Values of χVW (Bottom)a χAW
χVW
Π′co (kBT/a)
0.2 0.3 0.5 0.6
1.2 1.2 1.2 1.2
1.80 1.51 1.39 1.19
0.1 0.1 0.1
0.9 1.0 1.1
1.94 1.82 1.61
a The other interaction parameters are the following: χ AB ) 0.3; χAV ) χBW ) 1; χBV ) 0.1.
In addition to the basic set of interaction parameters χ described in section III, we have performed an extensive modeling using tens of other values of the interaction parameters. Table 1 shows the list of the average pressures Π′co in the region of phase coexistence for an A10B20 lipid membrane with
four different values of χAW ) 0.2, 0.3, 0.5, and 0.6. The average pressure Π′co for such isotherms with three different values χVW ) 0.9, 1.0, and 1.1 are also listed in Table 1. It is apparent that the decrease in the hydrophilicity (more positive χAW) or less soluble air fraction (more positive χVW) leads to a decreasing interfacial pressure. A smaller positive value of χAW means that the surface activity of the lipid molecules is more pronounced so that the interfacial tension tends to decrease. Interestingly, these trends of the surface pressure are similar to ones obtained by reducing the temperature of first-order phase transitions.46-48 This similarity has a straightforward explanation. FloryHuggins interaction parameters consist of an enthalpic and an entropic contribution, i.e., χij ) Aij - Bij/T where Aij and Bij are the enthalpic and entropic constants and T is the temperature.31,37,49 Usually, tedious parametrization efforts are needed to simulate the exact effects of temperature variations, so that several authors have come up with a simplified approach considering a variation in only one interaction parameter from the entire set of χ parameters.29,49 By doing this, albeit the quantitative comparison of temperature variations may be lost, the corresponding qualitative trends are still retained. For Bij < 0, the increase in temperature means that the χ parameter decreases with increasing temperature, which is qualitatively mimicked in our work. Physically, it can also be explained as follows. The increasing temperature leads to the membrane hydration causing more favorable contacts between water and lipid molecules. In addition, the increasing temperature results in the increase of air pressure, which can be interpreted as smaller positive χVW causing a stronger affinity between air and water. Thus, the effect of temperature variations is similar to that resulting from changing χAW (the hydrophilicity of the lipid head group) or χVW (the miscibility of air in water), keeping the other interaction parameters constant. In short, smaller χAW or χVW values indicate that the temperature is increased. So far, we have considered the equilibrium properties of the ensemble of lipid molecules with the composition A10B20. As the next step, we consider the effect of variations in the head and tail lengths on the phase coexistence. Table 2 (top) shows the average pressure Π′co and the corresponding region of phase coexistence Sco of membranes, each composed of different lipid molecules, namely A7B20, A10B20 and A15B20. It is shown that larger head groups correspond to lower interfacial pressures at the region of phase coexistence. Furthermore, the average distance of lipid molecules at the region of phase coexistence and the width of the region increase with the size of head groups. In Table 2 (bottom), we present the Π′co and Sco values for membranes composed of lipid molecules with different tail lengths (A10B13, A10B15, and A10B20). By increasing the tail length, the region of phase coexistence moves to higher S values
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Figure 6. Density plots for the lipid tail groups of the unsaturated lipid of composition flipid/fW ) 0.06, 0.17, 0.21, 0.25, 0.40, and 0.63 (parts a-f, respectively). Each box is of size 20 × 20a2.
and the corresponding interfacial pressure increases with the size of the tail group (the interfacial tension decreases with an increasing tail length). A membrane composed of lipid molecules with a shorter tail experiences a phase transformation at a relatively low lipid density (high interlipid distance). In conclusion, both the pressure and the density regimes of the phase coexistence depend on the length of the head and tail groups. Particularly, the size of the tail group affects strongly the lipid density at which the region of phase coexistence is observed. We also consider the influence of the unsaturated groups on the phase behavior of lipid membranes. In Figure 6a-f, the equilibrium morphologies are shown for a system containing the molecules A10B9C2B9 with the compositions of flipid/fW ) 0.06, 0.17, 0.21, 0.25, 0.40, and 0.63, respectively. No welldefined phase transition is found, but the change of the morphology with the increasing content of the lipid proceeds through smooth changes from a circular to an ellipsoidal micellar structure and eventually to a lamellar/planar structure. The shape of lamellae varies from slightly undulating to vigorously curved. The respective changes of the perimeter (Figure 7a) are nonmonotonous but smooth and without any dramatic discontinuities. The intermolecular distances S (Figure 7c) are larger than those for the saturated lipids by 3-20%, which agrees qualitatively with published MD simulations and experiments.42,50 The Langmuir isotherm presented in Figure 8 is a smoothly decreasing function of S and does not show any undulations. Our results obtained for the ensemble of unsaturated lipids indicate that equilibrium morphologies in these systems do not show any dramatic structural changes that could result in firstorder phase transitions. In contrast, Figure 7 demonstrates that membranes composed of unsaturated lipids adopt a continuous range of smooth structural transformations when the composition of the system changes. This predicted flexibility of membranes of unsaturated lipids agrees very well with published MD simulations and experiments which demonstrate a high flexibility of lipid molecules containing unsaturated bonds.42,50 The explanation is that unsaturated bonds increase the fluidity of lipid membranes, and thus, fluidlike unsaturated lipid mem-
Figure 7. Total interface perimeter P, the amount of lipid molecules m in the self-assembled structure nlipid , and the average lipid distance S between lipids at the interface (parts a-c, respectively) as functions of the ratio of lipid and water total fractions flipid/fW for the unsaturated lipid.
Figure 8. Interfacial pressure Π′(S) of an equilibrium monolayer consisting of unsaturated lipid molecules A10B9C2B9 as a function of the average interlipid distance S.
branes adopt curved shapes easier than membranes composed of saturated lipids as observed experimentally.51-52 The qualitative agreement of the predicted flexibility of the membranes of unsaturated lipids with experiments prove that our model of the unsaturated lipids is valid. The predicted flexibility of membranes for unsaturated lipids also enables us to interpret available experiments on the phase separation “liquid-gel” in
2126 J. Phys. Chem. B, Vol. 112, No. 7, 2008 surfactants containing mixtures of saturated and unsaturated lipids in various proportions.7,10 This phase transformation is known to show a well-defined first-order phase transition with a region of phase coexistence of the liquid and gel phases for saturated lipids such as DPPC, whereas a presence of a significant amount of unsaturated lipids causes the region of the phase coexistence to disappear. Although here we investigate the changes of morphology in the normal direction with respect to the membrane’s plane and not in the lateral direction where the liquid-gel transition occurs, these two kinds of structural changes are closely related.50 This relation occurs due to the fact that all morphological changes of lipid membranes are driven by thermodynamic processes which tend to minimize the energy of deformation of the membrane. Therefore, morphological changes occurring in the normal direction can affect those in the plane of the membrane and vice versa.50 A flexible membrane of unsaturated lipids can therefore release some of the deformation by adopting undulating morphologies (such as shown in Figures 6e-f). This may prevent the occurrence of the distinct liquid-to-gel transformation within the membrane. V. Conclusions We contribute an extended application of the off-lattice SCFT to model self-assembled lipid membranes at air-water interfaces. Our application of the SCFT comprises the following distinguishing features: (i) No a priori assumptions regarding the symmetry of the self-assembled morphologies are made; neither are any restrictions over these morphologies imposed in our formalism. Previous comparable approaches have been employed to model linear block copolymer melts,18-19,22 but not for membranes composed of branched lipids. (ii) A significant amount of air fraction (30% of the total volume) is included in the system to form air-water interfaces. (iii) Branched lipid molecules are described through the general formalism,26-27 which has not been applied to lipid molecules before. (iv) An original approach is employed to mimic the increase of the interlipid distance in unsaturated lipid membranes by increasing the repulsion between alkyl groups containing the unsaturated bond CdC and other segments of the lipid molecules. (v) We identify the region of phase coexistence of lipid membranes by considering a canonical ensemble of small systems containing lipid, air, and water fractions. Random initial distributions of the ingredients applied in the calculations lead to different phases in the region of phase coexistence. (vi) An original methodology to derive Langmuir isotherms from SCFT results is employed to investigate phase behaviors of lipid membranes at air-water interfaces. We employ the SCFT methodology to obtain thermodynamic properties of equilibrium lipid membranes in a 2D system containing lipids, air, and water. By varying the relative amount of lipid molecules in the system, we analyze the corresponding Langmuir isotherms to reveal possible phase transformations. Our results are summarized as follows. (i) In all cases, equilibrium air-water interfaces are formed with lipid head groups embedded in water and lipid tail groups located within the air-rich region. (ii) For saturated lipids, we find two distinct morphologies depending on the relative amount of lipid and water molecules. At low lipid contents, micellar morphologies are formed, whereas at high lipid amounts lamellar morphologies are obtained. The transition from the micellar to the lamellar
Lauw et al. morphology is accompanied by a sharp change of the perimeter of the air-water interface and by a distinct undulation of the corresponding Langmuir isotherm. A region of phase coexistence of the two morphologies is detected. (iii) We study the impact of the variation of model parameters (the Flory-Huggins interaction parameters and the size of the lipid molecules) on the phase behavior of the system. The regions of phase coexistence are found to be reasonably sensitive to the change of parameters representing the hydrophilicity of lipid molecules and the miscibility of air with water. The phase coexistence is also influenced by the size of hydrophilic and hydrophobic parts of the lipid molecules. (iv) For unsaturated lipids, change in the amount of lipid molecules leads to a continuous range of smooth structural transformations from a circular to an ellipsoidal micellar morphology and eventually to a lamellar structure. The shape of the latter morphology varies from slightly undulating to vigorously curved. The respective perimeter of the interface changes smoothly and no undulating Langmuir isotherm or any regions of phase coexistence are found. The average interlipid distance is slightly higher than that for saturated lipids, in agreement with published numerical and experimental results. Our comparative analysis on the phase behavior of saturated and unsaturated lipid membranes shows that unsaturated lipid membranes are more flexible, which allows them to adopt a wider range of conformations without sharp phase transitions. This demonstrated flexibility agrees with published experimental results. The enhanced aptitude of unsaturated lipid membranes to bend allows the interpretation of available experiments on the liquid-gel phase transition in surfactant monolayers. This phase transition shows a well-defined region of phase coexistence of the liquid and gel phases in membranes composed of saturated lipid molecules such as DPPC. Significant quantities of unsaturated lipid molecules result in the disappearance of the region of phase coexistence. Our study suggests that the flexible membrane of unsaturated lipids, when exposed to pressure, can release some of the deformation by adopting undulating morphologies. This may prevent the occurrence of the distinct liquid-to-gel transition in the membrane. In conclusion, it should be noted that our comparative study of the morphologies of saturated and unsaturated lipid membranes is made possible primarily because of the use of the offlattice SCF formalism without any geometric restrictions or a priori assumptions regarding the self-assembled morphologies. This flexibility allows us to capture membrane undulations (as shown in Figure 2c) as well as to observe flexible morphologies of unsaturated lipid membranes (Figure 6a-f). All these features are not accessible from conventional methods in which geometrical restrictions are imposed. The results presented in our work demonstrate the importance of unrestricted off-lattice SCF methodologies to study amphiphilic lipid membranes, which show a rich diversity of phase behaviors that cannot be confined to any predefined symmetry. Acknowledgment. We gratefully acknowledge the financial support from the National Research Council (NRC) of Canada. References and Notes (1) Piknova, B.; Schram, V.; Hall, S. B. Curr. Opin. Struct. Biol. 2002, 12, 487-494. (2) Veldhuizen, R.; Nag, K.; Orgeig, S.; Possmayer, F. Biochim. Biophys. Acta 1998, 1408, 90-108. (3) Haagsman, H. P.; van Golde, L. M. G. Annu. ReV. Physiol. 1991, 53, 441-464. (4) Weaver, T. E.; Na, C. L.; Stahlman, M. Seminars Cell DeVelop. Biol. 2002, 13, 263-270.
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