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Macroscopic Model of Phospholipid Vesicle Spreading and Rupture A. Efremov, J. C. Mauro,* and S. Raghavan Science & Technology Division, Corning Inc., Corning, New York 14831 Received November 17, 2003. In Final Form: April 20, 2004 We present a macroscopic model for the spreading and rupture of a spherical lipid vesicle on a flat, isotropic, hydrophilic surface. Formulas for the free energy of the initial and final states are derived, and the details of spreading pathways are examined. We show that the activation barrier for vesicle rupture is too large to be overcome by thermal fluctuations at room temperature and the final configuration is more likely to consist of a deflated vesicle. In order for the vesicle to rupture into a planar bilayer, it would have to be aided by increased temperature, application of an external force, or preparation of a mixed hydrophilic/ hydrophobic surface.
1. Introduction Functionalized and bifunctional surface applications such as biosensors, DNA microarrays, protein arrays, and GPCR arrays call for the deposition of lipid bilayers onto solid surfaces. Although it has been a while since such supported bilayers were discovered,1 much of the physics and chemistry concerning the interaction between membranes and surfaces needs clarification. It is generally recognized that the composition and the functionality of the surface drive the balance between the various forces at work: hydrophobic, electrostatic, and surface hydration.2 It is also quite clear that the nature of the surface determines the detailed structure of the supported lipid bilayers,3 and in particular, an ideal surface must allow the supported lipid bilayers to be laterally fluid, to be mechanically stable, and to have the right conformation.3,4 One of the first studies of the spreading of lipid membranes on surfaces in an aqueous environment was performed by Ra¨dler et al.,5 who observed the sliding and rolling behavior of bilayers unraveling and spreading over a hydrophilic surface. The problem of lipid vesicle rupture was addressed by Lingler et al.6 for the case of hydrophobic surfaces, where lipid monolayers formed from vesicles after an adsorption time of ∼6000-10 000 s. They described a possible pathway for vesicle rupture consisting of four steps: approach of the spherical vesicle to the hydrophobic surface, adhesion of hydrophobic tails, rupture of the vesicle, and lateral spreading of the monolayer. Conventional wisdom has held that a similar pathway exists for planar bilayer formation on hydrophilic surfaces1,7-9 which allows for the spontaneous self(1) Tamm, L. K.; McConnell, H. M. Biophys. J. 1985, 47, 105-113. (2) Wong, J. Y.; Park, C. K.; Seitz, M.; Israelachvili, J. Biophys. J. 1999, 77, 1458-1468. (3) Fang, Y.; Frutos, A. G.; Lahiri, J. ChemBioChem 2002, 3, 987991. (4) Fang, Y.; Lahiri, J.; Picard, L. Drug Discovery Today: HTS Suppl. 2003, 8, 755-761. (5) Ra¨dler, J.; Strey, H.; Sackmann, E. Langmuir 1995, 11, 45394548. (6) Lingler, S.; Rubinstein, I.; Knoll, W.; Offenha¨user, A. Langmuir 1997, 13, 7085-7091. (7) Cremer, P. S.; Boxer, S. G. J. Phys. Chem. B 1999, 103, 25542559. (8) Sackmann, E. Science 1996, 271, 43-48. (9) Williams, L. M.; Evans, S. D.; Flynn, T. M.; Marsh, A.; Knowles, P. F.; Bushby, R. J.; Boden, N. Langmuir 1997, 13, 751-757.
assembly of phospholipid vesicles into fluid lipid bilayers on solid supports. However, recent experiments by Jenkins et al.10 have shown that vesicle rupture is unlikely to occur on purely hydrophilic surfaces. Rather, they observed that vesicles were bound to hydrophilic surfaces in a deformed, unruptured state. The only regions where vesicles had ruptured to form planar bilayers were near the boundaries of hydrophilic and hydrophobic regions on a patterned surface, suggesting that rupture can only be induced when the vesicle comes in contact with a hydrophobic region. After rupture, a bilayer could then freely spread in the hydrophilic regions. In this report, we will be generally concerned with the spreading and rupture of a spherical bilayer vesicle on a flat, isotropic, hydrophilic surface. We derive formulas for the free energy of the initial and final states, and we examine the spreading pathway in order to calculate the activation energy that must be overcome in order for a vesicle to rupture. Our modeling supports the hypothesis of Jenkins et al.10 that vesicles cannot spontaneously rupture on a purely hydrophilic surface at room temperature. 2. Initial and Final States Let us assume that the initial state is a spherical vesicle (as shown in Figure 1) composed of N lipid molecules with outer radius Rc and inner radius Rc - t. If each lipid molecule has a “head” surface area of a0, we see that
N)
4π 2 [R + (Rc - t)2] a0 c
(1)
The free energy of the spherical vesicle is the sum of two contributions: the first is the free energy associated with the interface between water and the hyrophilic lipid heads,11 and the second is the bending energy required to form a spherical vesicle.12,13 Since each of the lipid molecules points toward the water surface with its hydrophilic head exposed, we can write the free energy of (10) Jenkins, A. T. A.; Bushby, R. J.; Evans, S. D.; Knoll, W.; Offenha¨user, A.; Ogier, S. D. Langmuir 2002, 18, 3176-3180. (11) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes, 2nd ed.; Wiley: New York, 1980. (12) Helfrich, W. Phys. Lett. 1973, 43A, 409-410. (13) Helfrich, W. Phys. Lett. 1974, 50A, 115-116.
10.1021/la036159h CCC: $27.50 © 2004 American Chemical Society Published on Web 06/05/2004
Phospholipid Vesicle Spreading and Rupture
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Figure 3. Disk with N1 molecules in the bulk and N2 molecules in the end-caps.
Figure 1. Cross section of a spherical vesicle composed of a bilayer of lipid molecules. The hydrophobic tails of the lipid molecules are shielded from the aqueous environment by the hydophilic heads.
In the first case, since the radius of the disk is rexp and there are N lipid molecules, as shown in Figure 2, we have
2πr2exp N) w rexp ) a0
x
Na0 2π
(7)
The free energy of this disk is given by the free energy due to the sum of the exposed headgroups (the same as that for a vesicle) and the exposed tails. The total surface area of the exposed tails is simply 2πrexpt. If we denote the lipid tail to water interfacial energy by γtw, then the total free energy of the rim is
Grim ) 2πrexptγtw ) γtwtx2πa0N Figure 2. Disk with N lipid molecules of radius rexp with the rim exposed.
the spherical vesicle due to the interfacial contribution as
Gves-interfacial ) γhwa0N
(2)
where γhw is the surface energy per unit area of the lipid head/water interface. The bending energy can be written as13,14
1 Gves-bend ) Kc 2
∫0π2π(R1c + R1c) R2c sin θ dθ 2
∫0π(R2c) R2c sin θ dθ 2
) πKc
∫0πsin θ dθ ) 8πKc
) 4πKc
(3)
The value of γtw varies with the size of the hydrocarbon chains that make up the lipid molecule. This variation is displayed in Figure 4. Typical values of γtw for phosphatidylcholine (PC) with chain lengths of about 12-16 carbon atoms are 50-52 mJ/m2. The free energy of the disk with the rim exposed is thus given by
Gdisk-exp ) γhwa0N + γtwtx2πa0N
where Kc is the bending modulus of the bilayer.14 Thus, the total free energy of the spherical vesicle is given by
Gves ) Gves-interfacial + Gves-bend
N1 )
2πr2 a0
1 t 2π a0 2
∫0π(r + 2t sin θ′) dθ′
N2 )
Now, let us compute the free energy of N lipid molecules forming the final state of a planar disk. We consider two cases: one in which the rim of the disk has lipid molecules with their hydrophobic tails exposed to the water surface (as shown in Figure 2) and another in which the rim of the disk has lipid molecules that have bent to form “endcaps” (as shown in Figure 3), thereby reducing the hydrophobic energy but at the cost of the bending energy. This will enable us to compute a critical stiffness of the lipid bilayer below which end-caps will form in preference to leaving the hydrophobic tails in the rim exposed to water.
)
1 πt(πr + t) a0
)
π2rt + πt2 a0
(14) Sackmann, E. FEBS Lett. 1994, 346, 3-16.
(10)
The number of end-cap molecules is equal to the area of the end-caps divided by a0. We may determine the area of the end-caps by treating them as a sphere of radius t/2 that has been expanded to accommodate a central disk of radius r. Mathematically, we write
(6)
) γhwa0N + 8πKc
(9)
We now consider the free energy of the disk with the rim made of end-caps, as shown in Figure 3. The disk consists of N1 molecules that make up the main bulk and N2 molecules that make up the end-caps. Clearly, for the bulk,
(4) (5)
(8)
(11)
To determine the radius (r) of the disk for a given N value, one has to solve the equation
N ) N 1 + N2 )
2πr2 π2rt + πt2 + a0 a0
(12)
The free energy of the disk with end-caps is quite simply
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Figure 4. Energy of the lipid tail/water interface and the tail/air interface as a function of lipid tail length.
given by
Gdisk ) Na0γhw + Gbend
(13)
where Gbend is the energy associated with bending of the lipid molecules to form end-caps. We calculate the bending energy of forming end-caps by multiplying the surface area of the end-caps by the bending energy per unit area
1 ∫0π(r +1t/2 + t/2 ) 2π 2t (r + 2t sin θ′) dθ′
1 Gbend ) Kc 2
2
(14)
1 1 1 + πt(πr + t) ) Kc 2 r + t/2 t/2 2
(
) 2πKc
)
(15)
(t + r)2(πr + t)
(16)
(r + t/2)2t
where r is determined by solving eq 12:
r)
[
πt 1 + 2 2
x( t2
)
2Na0 π2 -2 + 4 π
]
(17)
Of course, an end-cap will form only if it is energetically favorable compared to leaving the edges of the disk exposed. By comparing eqs 9, 13, and 14, we see that endcaps are likely to form if
Gdisk-exp > Gdisk γtwtx2πa0N > 2πKc
(t + r)2(πr + t) (r + t/2)2t
(18) (19)
or 2
Kc
