Originally published as Biophys J. BioFAST on April 14, 2006.
doi:10.1529/biophysj.105.068502
Biophysical Journal 91:189-205 (2006)
© 2006 The Biophysical Society
"Entropic Traps" in the Kinetics of Phase Separation in Multicomponent Membranes Stabilize Nanodomains
V. A. J. Frolov * 
,
Y. A. Chizmadzhev * ,
F. S. Cohen 
and
J. Zimmerberg
* Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Moscow, Russia; Laboratory of Cellular and Molecular Biophysics, National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland; and Department of Molecular Biophysics and Physiology, Rush University Medical Center, Chicago, Illinois
Correspondence: Address reprint requests to Joshua Zimmerberg, E-mail: joshz{at}helix.nih.gov.
 |
ABSTRACT
|
|---|
We quantitatively describe the creation and evolution of phase-separated domains in a multicomponent lipid bilayer membrane. The early stages, termed the nucleation stage and the independent growth stage, are extremely rapid (characteristic times are submillisecond and millisecond, respectively) and the system consists of nanodomains of average radius 
5–50 nm. Next, mobility of domains becomes consequential; domain merger and fission become the dominant mechanisms of matter exchange, and line tension 
is the main determinant of the domain size distribution at any point in time. For sufficiently small , the decrease in the entropy term that results from domain merger is larger than the decrease in boundary energy, and only nanodomains are present. For large , the decrease in boundary energy dominates the unfavorable entropy of merger, and merger leads to rapid enlargement of nanodomains to radii of micrometer scale. At intermediate line tensions and within finite times, nanodomains can remain dispersed and coexist with a new global phase. The theoretical critical value of line tension needed to rapidly form large rafts is in accord with the experimental estimate from the curvatures of budding domains in giant unilamellar vesicles.
|
INTRODUCTION
|
|---|
It is agreed that cell membranes are nonuniform dynamic structures. However, there is practically no agreement whatsoever as to timescales, nature, or the forces that govern the lateral molecular assemblies that comprise membranes. Estimates of the size of these assemblies, often termed domains or rafts, are approximately tens of nanometers. Putative rafts (1
–6) are enriched in cholesterol and sphingolipid. It is difficult to measure their physical properties (3–5,7), perhaps due to their small size or their transitory nature (8,9), but there are proposals that membrane domains play important functional roles in the trafficking and sorting of proteins, cell signaling, viral-induced fusion, etc. (1–3). The mechanism of domain formation in cell membranes remains obscure.
In lipid bilayers (including those having lipid compositions matching that of cell membranes), large domains, of the order of 5–10 µm in diameter, are readily observable by fluorescence microscopy (10–14).
Rafts are thicker than surrounding membrane and represent bilayer structures (13,15,16). Lipids in such domains are in a liquid-ordered state, i.e., the cross-sectional area per lipid molecule is smaller than that of a fluid-disordered membrane (5,10,15). These domains are mobile and grow by their merger. They rapidly resume their circular shape after external perturbations (13), indicating that a significant line tension exists at the raft-bilayer interface. Line tension has been experimentally estimated for multicomponent lipid vesicles (17). Small, nanoscopic domains have also been detected in lipid bilayers (6,18,19). In vesicles, line tension leads to a three-dimensional budding of domains, and theory that accounts for budding has been extensively developed (20,21). We consider phase-separation kinetics for the case of a lipid membrane that is always flat. The results of our calculations, which ignore membrane curvature, are appropriate for and directly applicable to the multitude of experimental studies of rafts that have used planar bilayers and giant unilamellar lipid vesicles.
It is generally thought that for lipid bilayer membranes, domains form as a result of phase separation. In support of this view, domains form upon lowering the temperature of a homogeneous lipid bilayer membrane. However, it is not clear why micrometer-scale domains remain separated from each other for long times. One possibility is that repulsive forces between domains kinetically stabilize them. It is also not understood why nanodomains sometimes remain stable, rather than increasing in size up to complete phase separation. In cell membranes, nanodomains may be created by lipid wetting of proteins (22–24), rather than by phase separation. Obviously, here wetting would not be complete; instead, only a relatively thin lipid film layer would form around the protein molecule and this liquid-ordered layer could have somewhat different physical properties than that of a global liquid-ordered phase. Answering many basic questions regarding domains in model and biological membranes requires understanding kinetics of domain formation, growth, and stability. Unfortunately, even for lipid bilayer membranes, systematic experimental kinetic studies have not yet been undertaken. Theoretical understanding of kinetic phenomena is also lacking. A kinetic theory of phase separation has been quantitatively developed for three-dimensional solid solutions (25–34). However, a lipid bilayer membrane is a two-dimensional system in the liquid state and direct collisions of mobile domains should strongly affect the kinetics of matter redistribution. A rigorous theory that incorporates all pathways of redistribution of matter in liquid, multicomponent membranes, needs to be formulated.
In this study, we approach this problem by utilizing theories that faithfully describe phase separation and domain growth in fields other than membrane biophysics. We modified and/or generalized these theories so that they apply to lipid bilayers and have justified these modifications. Calculations show that domain growth is divided in time and size into two essentially different regions: at short times (approximately milliseconds) and small domain sizes (less than tens of nanometers), the system resembles a solid solution with immobile domains. Here, domains quickly absorb lipid from their supersaturated surrounding milieu. At long times, however, direct interaction of mobile domains to merge, and the budding-off of nanodomains through domain fission, are the dominant means of domain growth. The theory predicts that at low line tension, entropy and boundary energy compete to trap nanodomains in a long-lived state (approximately hours). At somewhat higher line tensions, large domains form, but nanodomains coexist with them. At even higher line tensions nanodomains cease to exist, rapidly merging into micrometer-scale domains. Large energy barriers against close domain contact may kinetically hinder domain merger for larger domains as they progress to form one global phase.
|
KINETICS OF MATTER REDISTRIBUTION
|
|---|
Setting up the problem
We consider a multicomponent liquid membrane consisting of lipids. We assume that the components are homogenously mixed, but the membrane is initially in a metastable state that can undergo a first-order phase transition. The kinetics of phase separation can be subdivided into stages. In the first stage— nucleation—fluctuations within the homogeneous medium create nuclei of a new phase. If a nucleus enlarges to above a critical size, enlargement continues spontaneously by absorption of supersaturated matter from the surrounding solution. The degree of supersaturation in the surrounding solution decreases as these nuclei enlarge, resulting in the cessation of formation of new supercritical nuclei. The second period of growth is referred to as the independent growth stage, where existing domains continue to grow by accumulating matter from the surrounding membrane.
The original theoretical analysis of the kinetics of phase separation assumed single-component, three-dimensional solid solutions (30,31). This was extended to multicomponent three-dimensional systems (25) under the assumption that the new phase has a well-defined composition that is independent of domain size. This simplifying assumption allows the domain to be assembled from elemental structural units (referred to as quasi-molecules) and domain growth consists of incorporation of additional quasi-molecules. The concept of fixed quasi-molecules imposes equations for the partial fluxes of components from the surrounding membrane to the domains. This approach can be directly applied to a two-dimensional system by using a scaling procedure. To explicitly demonstrate the validity of the approach to a two-dimensional membrane, we combine the scaling approach with direct calculations. In relaxing the condition of solid solutions, the domains are mobile.
Consequently, rafts grow after the independent growth stage continues by two processes in parallel, merger of mobile domains, and Ostwald ripening. Ostwald ripening is a process whereby the equilibrium concentration of domain material within the surrounding solution is higher near a domain of small radius than large radius; material thus diffuses from small to large domains. The small domains disappear as they dissolve their constituents into the surrounding solution; the larger domains accumulate this material to become still larger. This phenomenon is analogous to redistribution of mass from small to large water droplets, mediated by water vapor moving from high to low pressure, the dependence of pressure on radius described by the Laplace law.
These two different modes of matter redistribution superimpose. We calculate each of these rates to determine which mechanism dominates during successive stages of phase separation. To use parameters that explicitly relate to experiment, we choose a standard bilayer consisting of 1:1 DOPC/DPPC + 30% cholesterol. Both micrometer-size and nanoscopic domains have been observed for this mixture and the composition of the liquid-ordered (L0) and liquid-crystalline (L
) phases have been derived (18). The L0 phase consists of 5% DOPC, 53% DPPC, and 42% cholesterol at 20°C; the L is 59% DOPC, 14% DPPC, and 27% cholesterol. Cholesterol is only somewhat enriched in the L0 phase (i.e., the domain); the saturated lipid, DPPC, is more significantly enriched. The unsaturated DOPC is largely excluded from the L0 domain. For simplicity, we assume that the L0 domain contains a 1:1 mixture of DPPC and cholesterol and we consider this as the structural unit of the domain. The area fraction of the domain-forming phase 
, found from the lever rule, is 
0.5. Altering the value of the parameter within the range 0.1 < < 0.5, permitted for the phase diagram of our standard bilayer (18), verifies that our conclusions have general validity. Domain ensemble behavior depends strongly on the value of line tension. Unfortunately, in the literature there is only one estimate of the line tension of micrometer-sized domains for phospholipid bilayers, 0.9 pN (17). There is indirect evidence that is smaller, and it obviously depends on membrane composition. Therefore, in numerical estimates of the rates of the stages of matter redistribution we varied the value of .
Nucleation
We consider nucleation in a two-dimensional metastable multicomponent membrane. Concentrations' fluctuations lead to the formation of small nuclei of new phase. If the nucleus becomes large enough, the decrease in energy resulting from a greater number of favorable interactions within the nucleus exceeds the unfavorable energy necessary to create the one-dimensional circular interface; the nucleus, now supercritical, grows irreversibly. Because we assume that the composition of the evolving nucleus does not depend on its size, the growth of a nucleus is due to addition of structural units that are defined by stoichiometric coefficients {
i}. The area per structural unit is
 | (1) |
where ai is the cross-sectional area per molecule of ith component. Because the subcritical nuclei remain in thermodynamic equilibrium, their size distribution function is described by the equilibrium distribution function. According to classical fluctuation theory (30), the equilibrium distribution function of the nuclei f0(r) depends exponentially on the minimum work E(r) necessary to create a nucleus of radius r:
 | (2) |
This function is normalized so that f0(r)dr is the number of the nuclei with radius (r, r+dr) in an area of 1 cm2. The area and boundary terms yield
 | (3) |
where is line tension, µ is the chemical potential of the structural unit within nucleus, and µi is the chemical potentials of ith component in the surround phase. For an ideal solution, µi can be expressed as
 | (4) |
where 
is the standard chemical potential of ith component and ci is the bulk concentration of ith component measured in 1/cm2. E(r) has a maximum at a critical radius rc, given by
 | (5) |
Equations 3 and 5 yield
 | (6) |
From Eq. 6 we obtain
 | (7) |
where Q is a pre-exponential factor that cannot be expressed in terms of macroscopic properties of the system. We estimate it by supposing that Q is proportional to the number of nucleation sites (25,26). Q is determined by the number of configurations by which components can arrange into a structural unit, 
. This expression immediately follows from the intuitively appealing hypothesis that the rate of creating a minimal nucleus is proportional to the probability that all components of the structural unit meet. The normalization condition for f0(r) and the relationship 
yields 
.
In addition to the equilibrium distribution function, we require a kinetic size distribution function f(r, t) to calculate the rate of phase separation. Nuclei growth is described by the Fokker-Planck equation (30,33)
 | (8) |
where j is the flux (number of nuclei passing through the critical radius per second, per cm2) in size-space, B is a nuclear size-diffusion coefficient in cm2/s, and U is the nucleus mobility in r-space in cm/s. The relation between B and U can be found at equilibrium, j = 0, as
 | (9) |
At steady state, j = const, allowing us to rewrite Eq. 8 in the form
 | (10) |
After integration, we obtain
 | (11) |
where f0(r) is defined by Eq. 7. The constants j and p can be found from the boundary conditions that f/f0 
1 for r 0 (because equilibrium is reached in this limit), and that f/f0 = 0 as r (because f0(r) tends to infinity, whereas f(r) remains finite). Equation 11 has a solution of the form
 | (12) |
The integrand is sharply peaked at r = rc. We use Eq. 7 around this point and obtain the stationary solution of Eq. 8 for the flux of nuclei in r-space as
 | (13) |
To obtain the flux in terms of measurable quantities, we need to evaluate the diffusion coefficient, B, in r-space. We do so by a macroscopic approach. Consider a supercritical nucleus moving unidirectionally toward large radii. This allows us to ignore diffusion and to write the flux as j = Uf. The coefficient U is a velocity in size-space, dr/dt. A macroscopic nucleus grows by accumulation of structural units diffusing from the bulk to the nucleus interface. The solution of the two-dimensional equation for steady-state diffusion yields the partial flux of ith component ji,
 | (14) |
where D is the diffusion coefficient in the membrane, r* is the cutoff radius (which can be approximated by the size of whole system), and cir is the equilibrium concentration of ith component near a nucleus of radius r. In Eq. 14, we assume that all components have the same diffusion coefficients. To preserve the nucleus composition, it is necessary to impose the condition on partial fluxes of
 | (15) |
Equation 15 means that the growth of a nucleus proceeds via incorporation of structural units exclusively. Equations 14 and 15 allow us to find the velocity in size-space, dr/dt. The details of the calculations are described in Appendix, where the equation for U is obtained as
 | (16) |
Here, ci is the equilibrium concentration of ith component at a straight interface (boundary of domain with infinite radius) and 
is the effective equilibrium concentration. From Eqs. 7, 9, and 16 we obtain B,
 | (17) |
As expected, B is greater for a small nucleus than for a large one. From Eqs. 7, 13, and 17, we obtain the final expression for flux density,
 | (18) |
As supersaturation decreases during the nucleation stage, the critical radius increases and the height of the energy barrier against irreversible growth increases. Both effects cause a significant decrease in flux, because j depends exponentially on rc (Eq. 18). We assume that nucleation ceases when the flux drops 10-fold to estimate the duration of the nucleation stage 
n, the value of critical radius 
at t = n, and the total number of created nuclei Nf. The details of these calculations can be found in the Appendix. Using Eqs. A13–A18 and taking = 0.4 pN, = 0.5, and D = 3 x 10–8 cm2/s (35), we obtain 
, Nf = 6 x 109 1/cm2, and n = 2 x 10–4 s. For = 0.1, we have 
, Nf = 2 x 109 1/cm2, and n = 8 x 10–5 s. Clearly, the nucleation time is very short and is relatively insensitive to variation of .
Independent growth stage
At the conclusion of the nucleation stage, Nf supercritical nuclei have appeared in the membrane and, for all practical purposes, additional nuclei are no longer created. The nuclei that already exist continue to grow independently of each other by accumulating matter from the surrounding membrane (Fig. 1 A). This process leads to decreasing supersaturation and thus to declining growth. We estimate the time necessary for this decline (i.e., the duration of this stage, ig) and the average size of the nuclei at t = ig by utilizing the constancy of the total number of nuclei (Nf) during the independent growth stage. The nuclei that were created over time n now migrate in r-space during independent growth. That is, diffusion in r-space is not of consequence during independent growth. Because (as will be shown below) ig > n, the width of the distribution function of the nuclei is determined by diffusion during the brief nucleation phase and thus this width remains rather narrow during independent growth. In fact, the reciprocal dependence of the rate, dr/dt, on r (i.e., mobility decreases with size, Eq. 16) further assures that the distribution remains sharply peaked. Therefore, we can use the average radius 
r
in Eq. A12 to calculate the flux of matter into the nuclei. Using Eqs. A11 and A12, we obtain
 | (19) |
where 
is the total supersaturation. The condition of mass conservation has the form
 | (20) |
where in is the initial total supersaturation. Substituting Eq. 20 into Eq. 19, we obtain
 | (21) |
View larger version (13K):
[in this window]
[in a new window]
|
FIGURE 1 A schematic representation of the main stages of matter redistribution in the course of phase separation: panel A is a stage of independent growth of each domain; panel B is Ostwald ripening; panel C is domain merger; and panel D illustrates two-dimensional budding of nanodomains from a large domain.
|
|
We solved this equation numerically using the initial condition 
.
To estimate ig, we compare the rate of growth from Eq. 21 with the Ostwald ripening rate using the obvious condition (dr/dt)ig = (dr/dt)or at t = ig. The expression for dr/dt in the case of Ostwald ripening can be easily obtained from Eq. 25 (see below). Letting = 0.4 pN and = 0.5, we obtain ig = 6 x 10–3 s and r(ig) 50 nm. For = 0.1, ig = 1.5 x 10–3 s and r(ig) 40 nm. Therefore (as stated above), ig > n. However, ig is rather short and, as we showed to be the case for n, insensitive to the precise value of .
Ostwald ripening in the case of immobile domains
During Ostwald ripening, the surrounding medium is only slightly supersaturated. The subcritical domains dissolve and their material is accumulated by the remaining domains, which become larger (Fig. 1 B). Ostwald ripening in a three-dimensional dilute solid solution is quantitatively described by Lifshitz-Slezov theory (30,31), which results in the well-known asymptotic law for domain radius growth,
 | (22) |
where 
is surface tension, v is the molecular volume of the evolving phase, and c is the equilibrium concentration at a plane surface. The size-distribution function of domains is narrow and the average radius r is equal to critical radius. The number of domains, N(t), as a function of time (increase of domain size is accompanied by a decrease of the number of domains) is given by
 | (23) |
where 0 is the initial supersaturation. At the beginning of Ostwald ripening, supersaturation is small and tends to zero as t–1/3.
Numerous studies have shown that the cube-root Lifshitz-Slezov law is very general. Lifshitz-Slezov theory can be applied for arbitrary volume fractions (28). It is commonly assumed that Lifshitz-Slezov theory developed for three-dimensional systems can be applied to two-dimensional systems (29). For example, simulating a two-dimensional spin-exchange Ising model (which exhibits a second order, rather than a first-order, phase transition) yields a t1/3 law (28). This power law has been explicitly demonstrated by Marqusee (36) for a two-dimensional, one-component solid solution. In this case,
 | (24) |
where b0 is a numerically calculated factor of order one. It is easy to show that Eq. 24 follows from Eq. 22, derived for a three-dimensional system, by scaling , a.
It has also been shown that the results obtained for a one-component, three-dimensional system generalize to the case of multicomponent solid solutions (25,32). This can be readily seen by substituting Dc in Eq. 22 into 
. By assuming that all Di are the same and equal to D, we arrive at the simple substitution, 
.
We solved the Ostwald ripening problem for a two-dimensional multicomponent system (see Appendix) and proved that the substitution 
is valid for the two-dimensional case. We obtained
 | (25) |
The total number of domains depends on time as
 | (26) |
where 
0 is a numerically calculated factor of order one. Thus, during Ostwald ripening, the average radius increases slowly and the rate of increase becomes less with time. Any domains of radius less than the critical radius rc dissolve and those greater than rc slowly enlarge. In essence, domain radius remains narrowly peaked around r (which is slightly greater than rc) as domain size slowly migrates in r-space. (Below, we use Eqs. 25 and 26 to estimate the rate of Ostwald ripening.) The dependence of the kinetics of matter redistribution on radius r is readily appreciated from the characteristic time, r, for the number of domains to decrease twofold. Equations 25 and 26 yield
 | (27) |
Thus, r does not depend on . For our standard bilayer and = 0.2 pN, Eq. 27 yields r = 200 s for r = 20 nm; r = 1600 s for r = 40 nm; r = 3000 s for r = 50 nm; r = 24000 s for r = 100 nm; and r 200 h for r = 1 µm. Clearly, Ostwald ripening is not an effective means for growth of immobile domains for r >50 nm. In fluid membranes, domains are mobile and their merger can readily dominate and determine the rate of matter redistribution.
Merger of mobile domains
We first calculate the rate of domain merger by adapting Smoluchowski's theory of rapid coagulation (37,38) to a two-dimensional system. We then consider the slowing of coagulation due to repulsive forces between approaching domains.
The initial stage of rapid coagulation can be described as a second-order association between two domains (Fig. 1 C)
 | (28) |
where N0 is concentration in cm–2 of the domains of radius r0 and Kr is the rate constant. In the absence of repulsive forces between approaching domains, domains merge immediately upon contact. The rate Kr is limited by diffusion of domains toward a central domain. Placing the origin of coordinates in the center of this domain, we can write the steady-state diffusion equation as
 | (29) |
with boundary conditions
 | (30) |
where r* is the cutoff radius. The solution of Eq. 29 is
 | (31) |
The flux toward the central domain is equal to
 | (32) |
To account for the mobility of the central domain, we need to double the diffusion coefficient (or equivalently the flux) in Eq. 32. Summing up the diffusion flux over N0 domains, we obtain
 | (33) |
Therefore, we have for the coagulation rate constant
 | (34) |
We calculate Dd according to the Saffman-Delbruck equation (39),
 | (35) |
where 
is the viscosity of the lipid bilayer, w is the viscosity of aqueous solution, h is the thickness of the bilayer, and 
is Euler's constant ( 0.577). Equation 35 is valid for r0 << h/w 1 µm; taking h = 6 x 10–7 g/s and w = 10–2 g/(cm*s), we obtain Dd 10–8 cm2/s for r0 50 nm. The same value of Dd can be used for any domain of radius on the order of tens of nanometers, because Dd depends weakly on r0. For larger domains, the Saffman-Delbruck equation is
 | (36) |
In contrast to Eq. 35, the diffusion coefficient of Eq. 36 is independent of friction between the domain and water. Because this friction should be more consequential as r becomes larger, we explicitly consider it by utilizing the expression for the viscous drag, b, acting on a disk moving along the plane of a membrane (40),
 | (37) |
This leads to a diffusion-coefficient D2,
 | (38) |
The combination of the Saffman-Delbruck and the viscous drag term yields a net diffusion-coefficient Dd given by
 | (39) |
yielding Dd = 3 x 10–9 cm2/s for r0 = 1 µm.
It is worth noting that both Dd and Kr depend weakly on r0. Because the (second-order) rate constant for domain merger is relatively independent of domain size, we set Kr = const and use Eq. 28 to calculate the change of the total domain concentration 
(38),
 | (40) |
Solving Eq. 40, we come to the time-dependence of the total number of domains,
 | (41) |
Equation 41 can be easily generalized to the case of slow coagulation,
 | (42) |
where the inhibition factor W has the form (37)
 | (43) |
Here, Vmax is the height of the energy barrier hindering close contact between two circular domains of radius r0, and 
d is the effective width of the barrier.
Repulsive forces between two approaching domains could occur for several reasons. We previously showed that as a consequence of the elastic properties of a membrane, a repulsive force definitely occurs if a height (i.e., thickness) mismatch exists between the domains and the surrounding membrane (41): the height of the elastic deformations at the raft boundary oscillates as it decays and the oscillations cause repulsion between two domains. We obtain Vmax according to the Deryaguin approximation (37), yielding Vmax 
, where Emax is the height (per unit length of boundary) of the energy barrier separating two domains. The height and width of the barrier calculated according to the elastic theory of continuous membranes (41) yields d = 3 nm and Emax 0.1 pN for = 0.4 pN. This barrier depends slightly on : For = 0.2 pN, Emax 0.07 pN. Because Vmax 
, W is inconsequential for small domains, but is huge for large ones. Numerically, W 1 for r0 = 50 nm and W 104 for r0 1 µm.
The characteristic time for domain collisions can be estimated from Eq. 42 as
 | (44) |
For the number of domains to decrease twofold, we obtain for = 0.5 that m = 0.05 s for r0 = 50 nm and W 1, and that m 1 h for r0 = 1 µm and W 104. The importance of the barriers is readily seen by setting W 1 for r0 = 1 µm, yielding m 3 s. As expected, these times are larger if domains occupy a smaller fraction of the membrane area. If = 0.1, we obtain m = 0.2 s for r0 = 40 nm and W 1, m 5 h for r0 = 1 µm and W 104, and m 15 s for r0 = 1 µm and W 1.
Domain fission and two-dimensional budding
The increase in domain size that results from merger can, in principle, be reversed if the domains divide. We estimate the characteristic time for a domain of radius R to divide into two domains of radii r and 
(Fig. 1 D), assuming that the bilayer remains flat. That is, we ignore any tendency of a domain to bend out of the plane of the membrane (20). The difference in boundary energies, E, is given by
 | (45) |
If there is no activation barrier against fission, the characteristic time of division f is
 | (46) |
where 
is the characteristic frequency of the oscillation of the boundary. Boundary fluctuations, known as capillary waves, have eigenfrequencies (40) of
 | (47) |
where is the membrane density and l is the eigenvalue for wavelength = 2
R/l. We let have the size of a domain resulting from fission (i.e., 2r or l = R/r), to obtain, from Eqs. 45–47,
 | (48) |
The chance of a large domain dividing into two domains of comparable size (fission) is negligible: for 
and r = 0.1 µm, we obtain that for = 0.4 pN, f 108 s, an impossibly long time. However, a small nanodomain can split off from a large domain (two-dimensional budding). For = 0.4 pN and R = 1 µm, it would take f 0.2 s for an r = 30 nm domain to split off, but f 200 s for an r = 40 nm domain. Thus, the time-dependence is extremely sensitive to the size of the bud. The likelihood for these nanodomains to split off from a large domain also depends strongly on the line tension. A domain of r = 40 nm would take f 0.4 s to split from an R = 1 µm domain if = 0.2 pN (instead of 200 s for = 0.4 pN). Although large domains cannot, as a practical matter, split into two equally sized halves, small domains can split in half if is sufficiently low. For = 0.4 pN, f 0.1 s for a small domain to split into two r = 40 nm domains; for = 0.2 pN, f 0.01 s. Because Eq. 45 predicts E r as r 0, the maximum rate of two-dimensional budding occurs for pinching off a single structural unit.
The relative contributions of the various stages to matter redistribution
We have estimated the rates at which matter redistributes by all modes and during the various stages. As shown, the nucleation and independent growth stages are very short (10–4–10–3 s). The use of Eq. 42 shows that merger during the independent growth stage leads to a negligible (<1%) decrease in the number of domains. The membrane can thus be considered a solid solution of immobile domains during these early stages. However, for t > ig and rc > 40–50 nm, the stages of Ostwald ripening and merger/fission proceed in parallel. The rate of merger (Fig. 2 A, curve 2) is clearly much faster than the kinetics of Ostwald ripening (curve 1) for nanodomains (r > 40 nm) over the first few seconds, m/ r 10–4. Growth for micrometer-scale domains (Fig. 2 B) is also much faster by merger (curve 3, note timescale of hours) than by Ostwald ripening (curve 1), even though the domains have relatively low mobility. Even when large domains significantly repel each other (Fig. 2 B, curve 2), merger is the dominating process. Quantitatively, dNm/dt = 1010 cm–2 s–1 for r = 40 nm; dNm/dt = 102 cm–2 s–1 for r = 1 µm and W 104; and dNm/dt = 104 cm–2 s–1 for r = 1 µm and W 1.
View larger version (7K):
[in this window]
[in a new window]
|
FIGURE 2 The decrease in the number of domains caused by Ostwald ripening and domain merger. (A) The domains are initially small, r(0) = 40 nm, and the decreases in domain number are shown over a few seconds. The dotted curve (1) shows the decrease caused by Ostwald ripening (Nr(t), right-hand ordinate). The solid curve (2) illustrates the decrease caused by nanodomain merger (Nm(t), left-hand ordinate). (B) The domains are initially large, r(0) = 1 µm, and the decrease in domain number are shown over the time-course of hours. The dotted curve (1) corresponds to Ostwald ripening and solid curves (2 and 3) illustrate the consequence of microdomain merger. For curve 2, the merger inhibition factor W = 104, and for curve 3, W = 1, accounting for the much slower kinetics of curve 2. A comparison of curves 1–3 demonstrates that domain merger is much more consequential for redistribution of matter than is Ostwald ripening.
|
|
These estimates assume that = 0.5 and = 0.4 pN. However, as we have shown, the results are not strongly dependent on the value of . The mean radius at the end of independent growth is 50 nm for = 0.5 and 40 nm for = 0.1 and the Ostwald ripening characteristic time is completely independent of . The characteristic times for merger of nanodomains is also relatively insensitive to = m 0.05 s for = 0.5 and 0.2 s for = 0.1. The value of does not affect the characteristic times of merger, but it does affect characteristic times of Ostwald ripening. For instance, r = 1600 s for = 0.2 pN (for r = 40 nm), whereas r = 800 s for = 0.4 pN.
We are thus led to a general view of domain evolution. At t ig, the system consists of nanodomains with a narrow size distribution that is peaked at r rc 50 nm. The population of small nuclei (r a few nm) is in thermodynamic equilibrium with the surrounding membrane and they do not interfere with the subsequent phase separation. For t > ig, supersaturation is very small and asymptotically declines to zero. Phase transition at t > ig is not completed yet and, strictly speaking, the system is not at a state of thermodynamic equilibrium. However, Ostwald ripening is very slow and matter redistribution is overwhelmingly determined by very fast (0.1 s) merger and fission of mobile domains. When the merger and fission rates are equal, we can assume that the total area of the domain phase is virtually constant to treat the system of nanodomains as an ensemble of immiscible particles in quasi-equilibrium. Thereby standard approaches of statistical thermodynamics yield calculated domain size distributions for times ig < t < r. (See Discussion for an elaboration of the difference between equilibrium and quasi-equilibrium.)
|
STABILIZATION OF NANODOMAINS
|
|---|
We first calculated the distribution of domain size by considering the ensemble of domains after the independent growth stage (see Appendix). This approach gave the number of domains as a function of domain radius for different values of (Fig. 3). The distribution of domain size is very sharply peaked at smaller radii. The characteristic length of the decay of the distribution, 3 nm, slightly increases as increases. The area, Ar, occupied by the domains, corresponding to curves 1 and 2 in Fig. 3, is independent of . Hence, nanodomains are favorable for low line tensions (i.e., < 0.18 pN) and they are almost uniform in size, peaked at r = rmin. This peaking occurs because for small , the decrease in boundary energy is insufficient to compensate for the decreased entropy that results from domain merger.
In the coexistence region, min < < max, the approach described in the Appendix fails (see Appendix), so we formulated the following simplified model. It gives results that are self-consistent over a wide range of parameters.
We assume that the domain-forming phase exists in only two forms: one a monodisperse ensemble of n small domains of radius r = rmin, and the other a single large domain of unknown radius R. The free energy of this system can be obtained directly from Eq. A51, as
 | (49) |
Matter conservation yields
 | (50) |
If only small domains of radius r = rmin are present, 
and we can obtain the dependence F1(), which is described by the first two terms in Eq. 49. This linear function is shown in Fig. 4 A. At the other extreme, only one domain of radius 
is present and here we also have a linear function F2(), which is described by the last term in Eq. 49. However the slope of F2() is much less than that of F1(), so F2() is practically parallel to the abscissa in Fig. 4 A. The intersection point of the two curves yields a line tension *. For < *, small domains are favorable (F1 < F2); for > *, only one large domain exists (F1 > F2). We now further illustrate why the ensemble of small domains can coexist with one large domain. The total free energy (obtained from Eq. 49) of n = nmax/2 small domains in equilibrium with one large domain of radius R (its radius calculated from Eq. 50) is depicted by the dotted line in Fig. 4 A. For close to *, the free energy F3() lies below F1() and F2() (magnified in Fig. 4 B). The difference between F3() and F1() or F2() is significant, 105 kT for = *. The range of for which coexistence is favorable (i.e., F3() < F1() and F3() < F2()) depends on the number of nanodomains, n. Therefore, determining the values of n that minimize the free energy of Eq. 49 as a function of yields the limits of the coexistence region.
View larger version (7K):
[in this window]
[in a new window]
|
FIGURE 4 The dependence of the free energy of the system F, calculated from Eq. 49, on line tension . The value F1 corresponds to nmax domains of radius rmin, F2 corresponds to one domain of radius rmax, and F3 is for nmax/2 domains of radius rmin and one large domain. (A) The entire range of line tensions is used. (B) The dotted rectangle of A is expanded.
|
|
Consider the free energy F as a function of n at a line tension . Substituting R from Eq. 50 into Eq. 49 yields
 | (51) |
F(n) is depicted at different line tensions in Fig. 5 A. At small , the system disperses because F(n) decreases monotonically with n (curve 1). For large , one large domain forms because F(n) increases monotonically with n (curve 3). For intermediate , F(n) exhibits a minimum (curve 2), and so a number of nanodomains coexist with one large domain. The value of n that yields this minimum depends on (see Fig. 5 B): as becomes larger, the number of minimal domains at equilibrium becomes smaller. Clearly, F(n) exhibits a minimum only in a finite interval of (i.e., min 
max). We calculate the minimum of F(n) with
 | (52) |
where ne denotes the number of nanodomains at equilibrium. Substituting ne = nmax and ne = 1 into Eq. 52, we obtain
 | (53) |
 | (54) |
The value min depends on rmin and , but is independent of the total area A. For rmin = 40 nm and = 0.1, we have min 0.18 pN and max 0.38 pN. As readily seen from Fig. 6, min and max decrease with rmin. Physically, the smaller is rmin, the greater is the increase in entropy when the system disperses. Equivalently, an increase in rmin leads to a decrease in min. Moreover, the difference max–min decreases with rmin, as readily seen from Eqs. 53 and 54. These dependences are of consequence because rmin slowly increases during Ostwald ripening.
It is useful to consider the fractional area occupied by small and large domains (Fig. 7). We calculated, for each , the number of nanodomains at equilibrium (ne obtained from Eq. 52) and then the equilibrium radius, R, of the large domain (from matter conservation, Eq. 50) to obtain the total area of nanodomains 
(curve A1) and the area of the large domain (curve A2). For < min, the phase-separated domains maximally disperse into small domains; a large domain is not present (region A). For min < max, nanodomains of minimal size maintain quasi-equilibrium with a large domain of radius R(). The greater is the line tension, the fewer are the number of nanodomains. Equivalently, for higher line tension, the total area of nanodomains is less and the area of the large domain is greater (region B). For > max, one large domain exists and nanodomains are absent (region C).
Our simplified model is based on the assumption that the population of domains essentially divides into two distinct groups: the first is the nanodomains of radius rmin; the second is one large domain of R >> rmin, which is equivalent to a global phase. However, clearly, this separation does not occur in the vicinity of min because here the gap between small and large domains disappears, i.e., R rmin. This vicinity of min is, however, exceedingly small. Because the slope of curve A2() is extremely steep near min, even a seemingly irrelevant change in results in a great increase in the area of the large domain: Eqs. 50 and 52 yield that an increase in line tension from min = 0.18 pN to = 0.19 pN induces matter to redistribute from the class of nanodomains to one large domain of R 15 µm >> rmin (e.g., see Fig. 7). This illustrates that, except for a rather small region around min, our simplified model is valid for a wide range of .
|
DISCUSSION
|
|---|
Matter redistributes by various modes in the course of phase separation within multicomponent liquid membranes. We have quantitatively considered each of these modes and their superposition. To the best of our knowledge, ours is the first study of nucleation in a two-dimensional multicomponent system; the characteristic time n and the total number of supercritical nuclei created by this mode have now been calculated. The duration of the nucleation stage, n, and the following independent growth phase, ig, are short (approximately milliseconds), and so <1% of the nuclei merge during these stages. Also, at the end of the independent growth stage, i.e., t = ig, almost all the domains are distributed within a narrow interval of radii around r rc. After this time, domains enlarge by Ostwald ripening and the balance bet
TOP
ABSTRACT
INTRODUCTION
