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Letters Lipid-Diffusion-Limited Kinetics of Vesicle Growth V. P. Zhdanov*,†,‡ and B. Kasemo† Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨ teborg, Sweden, and Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received January 19, 2000. In Final Form: July 11, 2000 We derive and solve kinetic equations describing spontaneous lipid-diffusion-limited ripening of vesicles in a supersaturated solution of lipid molecules. The process is assumed to occur via the Lifshitz-Slyozov scenario; i.e., the larger vesicles are considered to grow at the expense of smaller ones via lipid desorption, diffusion, and condensation. The increase of the average vesicle radius is found to follow the power law 〈R〉 ∝ tx with x ) 1/2.
Vesicles are spherical bilayer aggregates consisting of phospholipid molecules.1 Each molecule contains a hydrophilic head and one or, more often, two hydrophobic hydrocarbon tails. To a first approximation, such molecules can be treated as a semiflexible rods with a length of about 2 nm. The thickness of a bilayer is accordingly 4-5 nm. The radius of the smallest vesicles is about 10 nm, while large vesicles can be of several micrometers. Normally, vesicles are polydisperse, i.e., there is distribution of vesicle sizes around the average size. Understanding the principles governing the evolution of the vesicle size distribution is of interest because vesicles serve as a model for the fundamental unit of biological structuressthe cell membrane. They are uniquely suited to bridge the biological and material sciences. In addition, vesicles are important for formation of so-called supported membranes, used, e.g., in biosensor technology.2 From an academic viewpoint, the formation and growth of vesicles are connected with such active current research areas as self-assembly, selforganization, and complexity.3 In the present Letter, we will discuss spontaneous lipiddiffusion-limited growth of vesicles. Experimental data on the vesicle size distribution corresponding to this case are in fact lacking, because this process is very slow. To obtain vesicles, one usually uses the techniques designed to accelerate the vesicle growth. The vesicle size distribution is then measured by employing various lightscattering techniques (see a review by van Zanten4). The results obtained in such experiments do not, however, correspond to spontaneous vesicle growth. Some experimental data relevant for the subject we treat can be obtained from studies of the exchange and/or transport of lipids into and across bilayer membranes (see numerous references in section IV of the review by Duportail and * To whom correspondence may be addressed. E-mail: zhdanov@ catalysis.nsk.su or
[email protected]. † Department of Applied Physics, Chalmers University of Technology. ‡ Boreskov Institute of Catalysis, Russian Academy of Sciences. (1) Alberts, B.; Bray, D.; Lewis, J.; Raff, M.; Roberts, K.; Watson, J. D. Molecular Biology of the Cell; Garland: New York, 1994. (2) Kasemo, B. Curr. Opin. Solid State Mater. Sci. 1998, 3, 451. (3) Rosoff, M. In Vesicles; Rosoff, M., Ed.; Dekker: New York, 1996; p III. (4) van Zanten, J. H. In Vesicles; Rosoff, M., Ed.; Dekker: New York, 1996; p 239.
Lianos5 and in section V of the review by Needham and Zhelev6). These processes, occurring between donor and acceptor vesicles on time scales from hours to weeks, are usually explored by employing radiolabeling and fluorescence labeling, light scattering, and free-flow electrophoresis. The outcome of such experiments is that the rate of uptake and transfer across the bilayer midplane depends both on the chemical characteristic of the transported molecule (such as chain length and headgroup type) and on the composition and packing of the bilayer lipids.6 The detailed data on the growth of vesicles during the exchange and transport of lipids were however not reported to our knowledge. Despite this state of the art, the understanding of the spontaneous vesicle growth is of interest from the point of view of general theory of growth processes and also for the theory of biological evolution.7,8 Theoretically, the size distribution of small vesicles was first discussed by Helfrich.9 His analysis is based on the assumption that the free energy of a spherical vesicle is given by the phenomenological expression containing the term corresponding to the planar bilayer and the bending energy
F(R) ) 4πR2(R + 2κ /R2)
(1)
where R is the vesicle radius, R the free energy (per unit area) of the planar bilayer, and κ the bending constant. Equation 1 indicates that the contribution of the bending energy to the free energy is independent of R. Taking, in addition, into account that the number of lipid molecules in a vesicle is proportional to R2 and using the grand canonical distribution, Helfrich obtained for the vesicle size distribution (5) Duportail, G.; Lianos, P. In Vesicles; Rosoff, M., Ed.; Dekker: New York, 1996; p 295. (6) Needham, D.; Zhelev, D. V. In Vesicles; Rosoff, M., Ed.; Dekker: New York, 1996; p 373. (7) Watson, J. D.; Hopkins, N. H.; Robert, J. W.; Steitz, J. A.; Weiner, A. M. Molecular Biology of the Gene; Benjamin: New York, 1989. (8) Deamer, D. W. In The Molecular Origins of Life (Assembling Pieces of the Puzzle); Brack, A., Ed.; Cambridge University Press: Cambridge, 1998; p 189. (9) Helfrich, W. J. Phys. (Paris) 1986, 47, 321.
10.1021/la000080k CCC: $19.00 © 2000 American Chemical Society Published on Web 08/18/2000
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f(R) ) (2R/〈R2〉) exp(-R2/〈R2〉)
(2)
where 〈R2〉 is the mean-square radius. Including into the model the vesicle shape fluctuations, he got
f(R) ) (8R3/〈R2〉2) exp(-2R2/〈R2〉)
(3)
Later on, similar approaches related to thermodynamics have been employed to calculate f(R) in many other works (see the paper Bergstro¨m and Eriksson10 and references therein). The pre-exponential factor in f(R) has been found to be strongly dependent on the assumptions used in calculations. Equations 2 and 3 and those discussed in ref 10 correspond to the case when the chemical potential of lipid molecules in the solution is lower than that for lipid molecules in vesicles. In other words, the concentration of lipid molecules in the solution should be so low that the formation of vesicles is thermodynamically unfavorable. In this limit, the chemical potential of lipid molecules is in fact independent of the vesicle concentration and one can use the grand canonical distribution in order to calculate f(R). If the vesicle concentration is appreciable and lipid molecules are located primarily in vesicles, the solution is supersaturated, the formation of large vesicles is favorable, and accordingly the chemical potential of lipid molecules has to be calculated self-consistently with the vesicle size distribution. The latter case should be analyzed in analogy with the late stage of the conventional firstorder phase transitions. The grand canonical distribution is here not directly applicable. One should rather solve a kinetic equation for f(R). If the vesicle concentration is still low (in the sense that vesicle-vesicle collisions are negligible), larger vesicles are expected to grow at the expense of smaller ones via lipid desorption, diffusion, and condensation. Under such circumstances, the vesicle growth is controlled either by diffusion of lipid molecules or by the interface processes including incorporation of lipid molecules from the solution into the external lipid layer, desorption from this layer, and jumps between the external and internal layers. At higher vesicle concentrations, the vesicle growth results from the vesicle-vesicle collisions. The latter growth regime was discussed by Golubovic and Golubovic.11 In our Letter, we analyze the former scenario of the vesicle growth; i.e., the growth rate is assumed to be limited by diffusion of lipids. In this case, f(R) can be calculated by using a modified Lifshitz-Slyozov (LS) approach (an excellent presentation of the LS theory12 is given in ref 13). The first LS-type treatment of the vesicle growth was published by Somoza et al.14 For the vesicle energy, they used eq 1. From this equation, one can easily obtain that the chemical potential of lipid molecules corresponding to vesicles with the radius R is independent of R. Thus, the concentration of lipid molecules near a vesicle and the lipid diffusion flux from the solution to a vesicle are independent of R as well. The latter means that the growth rate, dR/dt, is the same for all vesicles and accordingly the vesicle size distribution does not change its shape; i.e., the shape of f(R) depends only on the initial conditions. (10) Bergstro¨m, M.; Eriksson, J. C. Langmuir 1998, 14, 288. (11) Golubovic, L.; Golubovic, M. Phys. Rev. E 1997, 56, 3219. (12) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35. (13) Lifshitz, E. M.; Pitaevskii, L. P. Physical Kinetics; Pergamon: Oxford, 1981. (14) Somoza, A. M.; Marconi, U. M. B.; Tarazona, P. Phys. Rev. E 1996, 53, 5123.
Under such circumstances, the growth rate is asymptotically expected to be exponentially low. We treat below the spontaneous lipid-diffusion-limited growth of vesicles in more detail. In particular, we show that eq 1 should be complemented by a term proportional to R. With this modification, the average vesicle size is asymptotically proportional to t1/2 and the asymptotic shape of f(R) is independent of the initial conditions. This is a qualitatively different result compared to earlier treatments. To scrutinize the expression for the free energy of vesicles, we first need to define more explicitly the vesicle radius, R, to account for the different numbers of lipid molecules in the outer and inner layers. In our treatment below, R is the distance between the vesicle center and the spherical shell or spherical surface between two lipid layers. The distances between the vesicle center and the spherical shells located in the middle of the external and internal lipid layers are, respectively, R + l/2 and R - l/2, where l is the length of a lipid molecule. The areas of the latter two shells (or surfaces) are 4π(R + l/2)2 and 4π(R - l/2)2, respectively. The numbers of lipid molecules in the external and internal layers are accordingly given by
Ne ) 4π(R + l/2)2/s
(4)
and
Ni ) 4π(R - l/2)2/s where s is the shell area corresponding to one lipid molecule (in principle, s might be slightly different for the external and internal layers, but this effect is expected to be minor, because the lateral lipid-lipid interactions rapidly decreases with increasing the distance between lipids). The total number of lipid molecules
N ) Ne + Ni = 8πR2/s
(5)
and the extra number of lipid molecules in the external layer
δN ) Ne - Ni = 8πlR/s
(6)
are proportional to R2 and R, respectively. The important point is that eq 5 does not contain a term proportional to R. The free energy of a vesicle includes lipid-water and lipid-lipid interactions. The latter interactions depend on the bilayer curvature. Taking into account that the total bending energy of spherical vesicles is independent of R (eq 1), we omit below the terms corresponding to the bending energy. Then, the vesicle free energy can be represented as a sum of two terms. The first one, taking into account the lipid-water interaction and the lateral (intralayer) lipid-lipid interaction, is proportional to the total number of lipid molecules
F1 ) -1(Ne + Ni) ) -8π1R2/s
(7)
where 1 > 0 is the parameter corresponding to these interactions. The second term, F2, describes interaction between lipid molecules located in different layers. For a planar bilayer, the latter term resulting from the hydrophobic tail-tail interactions is obviously also proportional to the total number of lipid molecules. In a vesicle, the numbers of lipid molecules in the two layers are however not equal and accordingly the interaction between lipid
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Letters
molecules located in the external and internal layers is not balanced. To a first approximation, F2 is expected to be proportional to Ni because the extra molecules (δN ) Ne - Ni) located in the external layer have no partners in the internal layer, i.e.
F2 ) -22Ni ≡ -2(Ne + Ni - δN)
(8)
where 2 > 0 is the corresponding parameter (the factor 2 is introduced for convenience). Employing expressions 5 and 6, we can rewrite eq 8 as
F2 ) -8π2(R2 - lR)/s
(9)
The total free energy is accordingly given by 2
F(R) ) F1 + F2 ) -8π(0R - 2lR)/s
(10)
where 0 ) 1 + 2. The first term in this equation is physically the same as that in eq 1. The second terms in eqs 1 and 10 are however different. In particular, the latter equation contains a term proportional to R. The chemical potential of lipid molecules corresponding to a vesicle with the radius R is defined as µ ) ∂F/∂N, where F is given by eq 10. To calculate this derivative, we should take into account that R and N are interconnected by eq 5. Employing eqs 5 and 10, we get
µ(R) ) -0 + 2l/2R
(11)
This equation indicates that with increasing R the chemical potential becomes lower. Thus, the growth of vesicles is thermodynamically favorable. Assuming the vesicle growth to be limited by diffusion of lipid molecules, we can consider that the concentration of these molecules near a vesicle corresponds to the chemical potential given by eq 11, i.e.
c(R) ) c0(1 + 2l/2TR)
(12)
where T is the temperature (we use kB ) 1) and c0 the concentration of lipid molecules in the solution in the case when they are in equilibrium with a flat bilayer. Calculating the diffusion flux in the steady-state approximation with the boundary condition (12), we obtain the following equation for the vesicle radius
dR/dt ) (sD/4) [c - c(R)] ≡ (sD/4)(∆ - σ/R) (13) where c is the average concentration of monomers in the solution, δ ) c - c0 the supersaturation, and σ ) c02l/2T. According to this equation, the critical vesicle radius is Rc ) σ/∆. With increasing time, vesicles with R < Rc and R > Rc decrease and increase, respectively. The conventional kinetic equation for the vesicle size distribution is given by
∂f(R,t) ∂ + [v(R) f(R,t)] ) 0 ∂t ∂R
(14)
where v(R) ) dR/dt (eq 13). In addition, assuming the system to be closed at t > 0, we have the standard equation for conservation of the total amount of lipid molecules
∆ + (8π/s)
∫0∞R2 f(R,t) dR ) constant
(15)
where f(R,t) dR is the concentration of vesicles with the radius from R to R + dR. Initially, the solution is considered
to be supersaturated and the average vesicle size is assumed to be small. The approach to solving eqs 13-15 is described in detail in ref 13. It contains a few purely mathematical tricks related to introducing new convenient variables and careful analysis of the behavior of the obtained equations at t f ∞. Asymptotically, the growth is demonstrated to be self-similar. Omitting the cumbersome details, we present the final asymptotic results for the growth law
〈R〉 ) Rc(t) ) (σsDt/8)1/2
(16)
and the vesicle size distribution
{
(
)
8u 2u exp (2 - u) f(u) ) (2 - u)2 0
for u < 2
(17)
for u > 2
In these equations, u ) R/〈R〉 and f(u) is normalized so that
∫0∞ f(u) du ) 1
(18)
According to eq 16, the typical time of the growth up to a given value of 〈R〉 is given by
tgr ) 8〈R〉2/(σsD)
(19)
Taking into account that σ ) c02l/2T, we can rewrite this equation as
tgr ) 16T〈R〉2/(2lDc0)
(20)
On the other hand, the time characterizing the vesicle growth due to vesicle-vesicle collisions can be represented as
tcol ) 1/(16πκ〈R〉Dvcv)
(21)
where Dv is the vesicle diffusion coefficient, cv the vesicle concentration, and κ the dimensionless probability of vesicle fusion during collision. Physically, it is clear that the vesicle-vesicle collisions are negligible provided that tcol . tgr. In summary, we have shown that the spontaneous lipiddiffusion-limited growth of vesicles in supersaturated solution should follow the power law with the exponent x ) 1/2. This exponent is higher than the conventional LS exponent x ) 1/3. It is of interest that the 3D growth with x ) 1/2 has earlier been obtained by Somoza at al.14 for the case of planar membranes. Physically, the latter case can hardly be realized. Mathematically, these two cases are however in fact similar. (In ref 14, the reader can also find estimates of the orders of magnitude of various parameters used in the equations describing vesicle and membrane growth.) Finally, we should note that the assumption that the spontaneous growth of vesicles is limited by diffusion of lipid molecules is still open for discussions. We cannot exclude that the process is really controlled by the interface processes such as incorporation of lipid molecules from the solution into the external lipid layer or by jumps from this layer to the internal layer. Supporting Information Available: This work was done with financial support from the NUTEK Biomaterials Consortium (Contract 8424-96-09362) and the Engineering Science Research Council (TFR Contract 97-643). LA000080K
