Langmuir 1995,11, 2696-2704
2696
Evidence for Domains in Deposited Lipid Bilayers David Pink,*>’Martin Kuhner,S Bonnie Quinn,’ Erich Sackmann,* and Hai Pham’ Physics Department, St. Francis Xavier University, P.O. Box 5000, Antigonish, Nova Scotia, Canada B2G 2W5, and Physics Department E22, Technische Universitat Munchen, James Frank Strasse, 85747 Garching, Germany Received March 23, 1994. I n Final Form: March 20, 1995@ We have modeled dimyristoylphosphatidylcholine bilayer membranes deposited on argon-sputtered glass or polyacrylamidesubstrates in order to understand the dependence of the lipid self-diffusioncoefficient upon temperature. We used a lattice model that has been successful in understanding some of the thermodynamics of lipid bilayers. We considered two modifications of this model: model I in which the bilayer was described by a distribution of noninteracting bilayer domains within and between which lipids could move and model I1 in which lipid-lipid interactions were weakened in a random way thereby permitting large diffusion coefficientsat low temperatures. We modeled the lateral exchangeoflipids as being permitted only when either an adjacent pair of lipids or a mutually adjacent trio of lipids were in their excited states. We carried out a direct computer simulation of lipid movement or related the self-diffusion coefficient t o pair or triplet correlation functions. We found that the results predicted by the unperturbed model are in general agreement with measurements of the self-diffusion coefficient on oxidized silica wafers as a prototype of a polished surface. This permitted us to apply the perturbed models to study the other cases for which we found that only model I could account for measurements on argon-sputtered glass and on polyacrylamide films. Argon-sputtered glass yielded domain distributions with a most probable size of -250 t o 600 lipids per half-bilayer, while polyacrylamide films yielded a most probable domain size of -600 lipids per half-bilayer. On the basis of these results, we have made predictions about the average number of gauche bonds per molecule as a function of temperature.
Introduction In recent years a better understanding has been obtained of how the process of diffusion of lipid molecules in lipid bilayers or biological membranes is affected by various kinds of impediments. These impediments may be due to patches of rigid bilayer1 or slowly moving proteins2 or may arise from the coupling of bilayer to associated macromolecular networks, such as the cytoskeleton beneath the plasma membrane3 or an adjacent ~ u r f a c e .Another ~ aspect of diffusion is its temperature dependence. Although phase transitions of the lipid bilayer affect the process of diffusion significantly, most interest has been focused on diffusion in the fluid phase.5 Both aspects-the effects of impediments and of phase changes-have been combined in the study of, for example, the formation ofdomains in a two-component lipid system.6 Analogously, the state of a supported lipid bilayer may reflect the underlying structure of the substrate on which it is deposited. A measurement of lipid diffusion, as a function of temperature, thus offers another way to study the physical states of lipid bilayers. Here, we report on calculations and computer simulations in order to compare and understand the results of measurements carried out on lipid bilayers deposited on argon-sputtered glass or on surfaces covered by polyacrylamide films’s8 and on oxidized silicon wafer^.^ The St. Francis Xavier University. Technische Universitat Miinchen. @Abstractpublished in Aduance A C S Abstracts, J u n e 1, 1995. (1)Saxton, M. J. Biophys. J . 1982, 39, 165-173. (2) Saxton, M. J. Biophys. J . 1987, 52, 989-997. (3) Saxton, M. J. Biophys. J . 1989, 55, 21-28. (4) Evans, E.; Sackmann, E. J . Fluid Mech. 1988, 194, 553-561. (5) Vaz, W. L. C.; Clegg, R. M.; Hallmann, D. Biochemistry 1985,24, 781-786. ( 6 )Vax, W. L. C.; Melo, E. C. C.; Thompson, T. E. Biophys. J . 1989, 56, 869-876. (7) Kuhner, M.; Tampe, R.; Sackmann, E.Biophys. J . 1994,67,217226. ( 8 ) Merkel, R.; Sackmann, E.; Evans, E. J . Phys. IPurisl 1989, 50, 1535-1555.
0743-7463/95/2411-2696$09.00/0
former measurements7,*reported differences in lipid lateral diffusion coefficients by at least 1 or 2 orders of magnitude near the “main”gel-fluid transition temperature, compared to values measured for oxidized silicon wafer^.^ In order to interpret the experimental data it is necessary to have models from which the lateral selfdiffusion can be calculated as a function of temperature. In the Theory section we shall describe such models. They will be based on one which has been used to calculate the equilibrium thermodynamics of lipid bilayers,1° and we shall show how they can be used to calculate the lateral self-diffusion coefficient of lipids as a function of temperature. The intent of these models is to identify order-ofmagnitude effects in the temperature-dependence of the lipid lateral diffusion coefficient. It will transpire that, because of the sensitivity of the lipid thermodynamic averages to the characteristics of the models, it is possible to distinguish between them. However, these calculations cannot rule out the possibility that the temperature dependences that have been observed are due to another mechanism. We can say that, given that it is believed that a bilayer is formed, there are essentially two ways to cause order-of-magnitude effects in diffusion coefficients. One is by constraining the individual lipids not to “packwell, and this is modeled by randomly weakening their interaction; the other is by constraining domains o f lipids not to “pack” well, although the individual lipids might “pack well within each domain, and this is modeled by considering a set of domains which can exchange lipids but between which the interaction energy is zero. The domains are envisaged as being of a fixed distribution of sizes and shapes with dynamical lipid molecular processes taking place within them and with lipids being exchanged (9) Tamm, L. K.; McConnell, H. M. Biophys. J . 1985,47, 105-113. (10)Pink, D. A,; Green, T. J.;Chapman, D. Biochemistry, 1980,19, 249. Pink, D. In Molecular Description of Biological Membranes by Computer Aided Conformational Analysis; Brasseur, R., Ed.; CRC Press: Boca Raton, FL, 1990.
0 1995 American Chemical Society
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Evidence for Domains in Deposited Lipid Bilayers between them. We envisage that both of these effects might be caused by the structure underlying the surface, e.g., its “roughness”, on which the bilayers are deposited. The models assume that the lipid bilayer has been formed on a substrate. We assume that the effect of the substrate can be described by one of three models. One is the unperturbed modelloin which the substrate does not perturb the bilayer. Another is a perturbed model (model I) where the structure of the substrate is such as to cause the lipid bilayer to be formed and exist as a set of essentially noninteracting, quasi-two-dimensionaldomains described by a fured distribution of sizes between which, however, lipid molecules can diffuse. Here, noninteracting means only that the interaction energies between two adjacent domains are effectively zero. The possibility that a domain containing N lipids can exist in various shapes is assumed to be accounted for in the fixed distribution of sizes. We shall return to this point, below. A third model is a perturbed model (model 11)in which the substrate causes the lipid molecules to be separated by a random distribution of distances such that the interactions between adjacent lipid molecules are randomly perturbed. Although there are many time scales associated with processes taking place in a deposited lipid bilayer, we believe that our model adequately accounts for those taking place on a time scale of about s. Transitions between states offew gauche bonds take place on -1O-lO s, and these are represented by a single effective ground state.lOJ1 Transitions between these states and excited states with many gauche bonds take place on ~ l o - ~ - l O -s.l0J1 ~ Although lipid rotations and out-ofplane motion must affect lateral movement, this is accounted for by an effective attempt t o move in one unit of s. Large deformationsof a bilayer occur on a longer time scale. Polar group motion appears to be not directly relevant since the diffusion coefficient appears to be correlated predominantly with the ‘packing” of hydrocarbon chains and the temperature of the main phase t r a n ~ i t i o n The . ~ ~model ~ calculations will be compared to measurements of lipid self-diffusion in dimyristoylphosphatidylcholine (DMPC) bilayers deposited on different substrates (argon-sputtered glass, polyacrylamide film, and oxidized silicon wafer) as a function of temperature. The model calculations may also be extended to monolayers deposited on silanated substrate^.^
Theory Equilibrium Thermodynamics of Lipid Monolayers or Bilayers: The Unperturbed Model. We shall represent the plane of one sheet of a lipid bilayer by a triangular lattice each site of which is occupied by a lipid molecule. The hydrocarbon chains of each molecule may be in one of three states: ground (g), a low-energy state in which both hydrocarbon chains are extended with few gauche bonds; excited (e), a high-energy state in which the chains contain relatively many gauche bonds; a state, i, intermediate between g and e in which one chain is extended and the other contains relatively many gauche bonds. The properties of these states and how they are calculated have been described elsewhere,lOJ1 and it suffices to say that they are characterized by sets of three parameters, {A,, E,, D,} ( n = g, i, e), where A, is the cross sectional area of the hydrocarbon chain region, E , is its internal energy, and D, is its degeneracy when the lipid chains are in the statelabeled n. A pair of nearest neighbor lipids, located at sites i and j , in states n and m, interact via a van der Waals interaction -JoJ(n,m). The bilayer (11)Pink, D. A.;MacDonald, A. L.; Quinn, B. Chem. Phys. Lipids 1988,47,83-95.
is brought into existence by the effects of the water interactions, and the consequence of this is to have, on the average, an effective lateral pressure, n, acting isotropically in the plane of the bilayer upon the hydrocarbon chains. Accordingly, the Hamiltonian operator is
E,
=E,
+ rIA,
n = g , i, e
where Li, is a projection operator for site i and is unity if the molecule at site i is in state n and zero otherwise, rIAn is the energy due to the lateral pressure and (ij) indicates that the sum is over all nearest neighbor sites i and j . H will be used to calculate equilibrium thermodynamic quantities such as the probabilities of being in the various states, @(n)>,n = g, i, e. We shall use Monte Carlo simulation techniques to model this system, and we shall make use ofthe Metropolis a1gorithm.l1J2 The appropriate energy operator for this is
F = H - k B T x Z L i nIn D, i
n
Consider the system to be in a state 1with F = f . Now choose another state 2, randomly, and let it possess F = f . Define Af = f - f . If Af 0, then the system makes a transition to state 2. If Af > 0, then choose a random number, R , where 0 5 R < 1. Then, if R Iexp(-AflkBT), the system makes a transition to state 2. Otherwise, the system remains in state 1. The model described here, and elaborated in more detail in ref 10, has been used successfully to calculate the temperature- and hydrocarbon-chain-length-dependence of the Raman “1130 cm-l” band and to predict the average number of gauche bonds excited on a lipid hydrocarbon chain as a function of temperature for T Tm,the “main” transition temperature.13 This gives us some confidence that this model essentially correctly represents the lateral packing of the lipid molecules for T < T,. The model is not concerned with studying the pretransition. We shall refer to this model, in what follows, as the unperturbed model. Models of Perturbed Lipid Bilayers. We shall consider two ways in which a lipid bilayer, which possesses lateral dimensionslarge compared to the lipid lateral scale of -1 nm, might be perturbed due to the properties of the substrate (argon-sputtered glass or polyacrylamide layers). Model I. We assume that the lipid bilayer is perturbed by the substrate on which it is deposited thereby causing it to come into existence as a set of domains. The entire lipid bilayer is described by a distribution of domains of bilayers for which the interaction energy between two such domains is zero. Lipid molecules can, however, diffuse from one domain to another. In each domain the energy is given by H,but we must consider the possibilities that if the domains are sufficiently small, then we must take account of their boundaries. The two distributions that we have considered are described by (12) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. N.;Teller,E.J. Chem.Phys. 1953,21,1087. SeealsoBinder,K.;Stauf€er, D. A Simple Introduction to Monte Carlo Simulation and Some Specialized Topics. In Application of the Monte Carlo Method in Statistical Physics; Binder, D., Ed.; Springer-Verlag: Berlin, 1984. (13)Snyder, R. G.; Cameron, D. G.; Casal, H. L.; Compton, D. A. C.; Mantsch, H. H. Biochim. Biophys. Acta 1982,684,111-116.
Pink et al.
2698 Langmuir, Vol. 11, No. 7, 1995 e-(N-No)2/2rz
p,wflo,n
(3)
= ~e-(K-No)z/2T2 K
and
wherepL andp, are the probabilities of a domain of linear dimensions N appearing in the total lipid bilayer. NOis the most probable linear dimension of the set of domains and r is the width of the distribution. Here, the distribution denoted bypL represents a Gaussian distribution of domains in linear dimension, while ps represents a Gaussian distribution of domains in area. Model II. We assume that the interaction parameter, Jo,is perturbed in a random way in a large system. There are a number of ways in which this could be realized. In addition, because a very large system could undergo thermal motion, the constraints on the distance between any pair of lipids, brought about by the underlying surface on which the bilayer is deposited, can vary in a random way with time. We shall model this as a n interaction between lipids which has, as an upper bound, the unperturbed value Jo and which changes randomly with time. We shall assume that the most probable value of the interaction parameter is its unperturbed value and that the weaker the interaction, the more unlikely is it to occur. This can be achieved, for example, by replacing JO by J(y,R),
the simplest model, in which At, = mAtl. We shall take the step magnitude to be the smallest spatial dimension of our system, which is the lattice constant, A. ’In order to identify the magnitude of A, we note that, experimentally, D decreases by about 2-3 orders of magnitude as T decreases through Tm.9This implies that lateral self-diffusion takes place, preferably, when lipids are in their e states. In order to use a s simple a model as possible with fewest parameters, we shall assume that, a t all temperatures, lateral diffusion can be modeled as a change of position of adjacent lipids such that the change takes place only when all lipids involved are in their e states. We shall return to this below. Our assumption means that A is equal to the diameter of a lipid in its e state, A = ~ ( A J Z ) ” ~At . sufficiently high temperatures, when each lipid will carry out a step of magnitude A a t every attempt, we have (r2),= ma2. We take the value of D in this case to be the reference value, DO,
(7)
We can evaluate At1 b using experimental data. If we require that for A, = 68 when T > Tm, DO= 6.8 x cm2/s which is in accord with data,15 then At1 x lo-’ s. This is also approximately the characteristic time for a lipid molecule to make transitions between low-energy extended-chain states and high-energy states.16 Thus, in each such unit of time we permit a molecule to attempt to change its state and also to move by one lattice constant. Criteria for the Lateral Movement of Lipids. We have chosen the simplest models to represent the lateral movement of a lipid, with the requirement that for any lipid to change its position it must be in its e state. The first model, the “pair-exchange”model, assumes that two adjacent lipids may exchange their positions if both of them are in their e states. It might be argued that if such J(~JE = )J,(I (5) a pair is surrounded by lipids, none of which are in their e states, then such exchange might be unlikely due to where y is a fixed number describing the perturbation packing constraints. We have, therefore, considered a and R is a random number, 0 < R < 1,which takes on a more physically realistic model, the “triplet-exchange” different value for each pair of sites a t each Monte Carlo model, in which three mutually adjacent lipids located a t the vertices of a triangle can cycle their positions by one step. This model is not trivially equivalent to that described by a quenched distribution of random bond lattice constant a t a Monte Carlo step. strengths studied before.14 In its simplest form, it involves Method A: Computer Simulation. To begin the simulaboth random bond strengths and random fields. tion, we prepared one example of a lattice (domain) of Lateral Self-Diffusion. We shall model the selflinear dimension N , with N ranging over all sizes defining diffusion of lipid molecules in the plane of a bilayer when the distributions of eqs 3 or 4. A sequence of Monte Carlo the system is in thermodynamic equilibrium. It is an steps were carried out, in each of which each site was experimental fact that, as a function of temperature, the visited in a random sequence, and an attempt was made lateral diffusion coefficient is larger when a lipid bilayer to change its state. This was carried out a t a temperature is in a fluid phase and smaller when it is in a gel p h a ~ e . ~ J ~ T > T,. A sufficiently large number of Monte Carlo steps were used so that the lattices were in states characteristic In what follows, we shall make use of this fact. We shall calculate the lateral self-diffusion coefficientin two ways: of thermal equilibrium at the end of this initialization method A which directly calculates the diffusion coefficient procedure. We then chose a lattice in accord with their probability distribution, (3) or (4),and selected a site, s, via computer simulation and method B which relates the diffusion coefficient to equilibrium correlation functions randomly, on it. For the “pair-exchange” model of lipid which are then calculated. The self-diffusion coefficient, movement we carried out the following procedure. We D,follows from the expression for the average mean-square chose a nearest neighbor site, s’, randomly. If both of distance moved in a walk of m steps, (r2),, these sites were in their excited states, we incremented a vector, r (initially, r = 0), by a vector of magnitude 2(A$ n)1/2 in the direction defined by a vector from s to s‘. We then attempted to change the states of all sites on the lattice and tested whether the state a t s’ was excited. If it was, we randomly chose a nearest neighbor site, s”. If where At, is the time elapsed in m steps. We shall adopt s” was excited, we incremented r by a vector pointing from s‘ to s”. For the “triplet-exchange” model, having (14)Aizenman, M.; Wehr, J. Phys. Rev. Lett. 1989,62,2503. Hui, chosen site s, we would then choose site s‘ as above and K.; Berker, A. N. Phys. Rev. Lett. 1989, 62, 2507. (15)Galla,H.-J.;Hartmann,W.;Theilen,U.; Sackmann,E.J.Membr. Biol. 1979, 48, 215-236.
i2,
(16) Skolnick, J.; Helfand, E. J.Chem. Phys. 1980, 72, 5489-5550.
Evidence for Domains in Deposited Lipid Bilayers
Langmuir, Vol. 11, No. 7, 1995 2699
then choose one of the sites which is adjacent to both s and s‘. If all three sites are in their excited states, then we incremented a vector r as we described above. The difference between this method and the “pair-exchange” model is that in order for r to be incremented, a third site, adjacent to both s and s’or s’ and s” and randomly chosen, must also be in its excited state. We repeated this procedure until, in choosing a nearest neighbor site, an attempt was thus made to move off the lattice. We then selected a new lattice accordingto the probabilities (3)or (4) and randomly selected a site on its edge. If both sites were excited, then we retained the new lattice and incremented r by a unit vector in the direction of the nearest neighbor. State s would now correspond to the site on the edge of the new lattice. We repeated the procedure a sufficiently large number of times, m, so as to sample all the lattices with probabilities given by (3) or (4). At the end of this run we were left with a value of r. Denote r2 by r;. We then repeated the entire procedure, using the final configuration of the previous run as the starting set for the new run, to obtain ri. We carried this out Q times to obtain
This result for (r2), was used in (6), together with At,,, = mAtl, to obtain a value of DIDO. The temperature was then reduced, and the final states of the system were used as the starting states to initialize the system before simulating r:, ri, ...,r; at the new temperature. Method B: Using Correlation Functions. An expression for (r2), has been given by Galla et al.15
(r2>,= ( ~ ~ ) v m
(9)
where vis the probability per unit of time that a movement of one such step takes place. In our model (A2) = ,I2 = 4AJn and, for high temperatures when molecules move at every step, we choose v = 1. This yields an expression for D,
DID, = v
Our assumption (above)that adjacent lipids change their positions only when they are in their e states means that, in order to make the model as simple as possible, we take v(e,e)= 1,and v(n,m)= 0 otherwise for the “pair-exchange” model, and take v(e,e,e) = 1, and v(Z,n,m) = 0 otherwise for the “triplet-exchange” model. In order to achieve this we must have v(g,g) x 10-3v(e,e)since, for T < Tm, p(g,g) 1,and we thus expect v(g,e) < lO-lv(e,e). For the “pairexchange”model, for example, the choice of v(n,m) = 0 if n, m f e is the simplest, physically-reasonable model which is in accord with experimental result^.^ Analogous statements hold for the “triplet-exchange” model. Then,
v2 = p(e,e) “pair-exchange” v3 = p(e,e,e) “triplet-exchange”
(12)
With this choice, eq 10 becomes
DID, = p(e,e) “pair-exchange” DID, = p(e,e,e) “triplet-exchange”
(13)
We now calculate the correlation functions, p(e,e) and p(e,e,e), via computer simulation using (2) and substitute the values obtained into (13). Clearly, for model I1 p(e,e), p(e,e,e), and therefore D will be functions of y. For model I, for given values of N, No, and r, we can calculate p(e,e) and p(e,e,e) to give D(N&o,r), the diffusion coefficient for a lipid in a domain of linear dimension N when the size distribution of domains is given by either (3) or (4)with N Orepresenting the most probable linear dimension and r representing the width of the distribution. We shall calculate the diffusion coefficient for lipids moving on the distribution of domains as the average of the D(N&o,r) for each of the domains, weighted by the probabilities, (3)or (4),for each domain to appear in the distribution. This cannot be exact since it does not calculate (r2), as a sequenceofmovements over many domains. However, here we are interested in orders of magnitude changes in D. Using this approximation, we obtain, for a given distribution,
(10)
The parameter v must exhibit some temperature dependence, and to a first approximation, as we stated above, we assume that v depends only upon the states of the lipids which are involved in the change of position and not upon the state of any other lipid. In this case v would have the general form
v2 = xv(n,m)p(n,m) “pair-exchange” nm
v3 = xv(Z,n,m)p(Z,n,m) “triplet-exchange” (11) Inn
Here, p(n,m) is the conditional probability for the two adjacent lipids to be in states n and m, and v(n,m) is a number giving the probability for two adjacent lipids in states n and m to attempt to exchange their positions in a time Atl. p(l,n,m)is the conditional probability for three mutually adjacent lipids, located at the vertices of a triangle, to be in states 1, n, and m, and v(l,n,m) has a meaning analogous to v(n,m) for “pair-exchange”. For T > T, both p(e,e) and p(e,e,e) are of order unity.1° The sum of the v(n,m) or the v(l,n,m)must be approximately unity since in each unit of time our models assume that each molecule can attempt to move by one lattice constant.
where a = L or S depending upon which distribution, (3) or (41, is used. We reiterate that the intention of this model is to distinguish physical processes, e.g., the formation of lipid domains brought about by the structure of the surface which supports the bilayer, which yield diffusion coefficients differing by orders of magnitude. We shall see, below, that this is indeed the case for the two models identified in the Introduction. Furthermore, in the case that a lipid bilayer is deposited on a smooth substrate, and is therefore thought to be unperturbed, this model must be able to correctly calculate the decrease in lipid diffusion coefficient, as T decreases through T,, without introducing any new parameters. Only then can we have sufficient confidence in it t o use it as a basis for the perturbed models. Note that since the lipids can change their states, there will be no nonzero temperature at which DIDO = 0. The reason for this is that at any nonzero temperature both p(e,e) and p(e,e,e) are nonzero so that a lipid attempting to move need only wait sufficiently long in order that both it and its nearest neighbor be excited. In the calculations done here we have represented all domains which contain N2 lipid molecules as triangular lattices of dimension N x N. This means that we represent
2700 Langmuir, Vol. 11, No. 7, 1995
Pink et al.
I
I .
I
I
0‘
2‘ 300
270
T (K) Figure 1. Computer simulation of lipid lateral self diffusion, method A, for model I using the “pair-exchange”model of diffusion. log(DID0)as a function of temperature for different distributionsof domains. (A)DistributionpL(eq 3). Infinitely large system, modeled using periodic boundary conditions (a, and cooling (0);NO= 8, r = 2 (b, +) and r = 4 (c, heating (0) V);No= 16,r = 2 (d, x 1andNo = 24, r = 2 (e,A). (B)Distribution p s (eq 4). Infinitely large system (a, heating (0) and cooling (0)); NO= 8, r = 28 (b, +) and r = 48 (c, v);No= 16, r = 60 (d, x ) and NO= 24, r = 100 (e, A). T,,, (295.5K)is indicated by vertical dashed lines, and other curves are to aid the eye. domain as compact and essentially isotropic. We do not address the question of what is the distribution of shapes of domains of a given size.
Results Monte Carlo simulations of model I used open boundary conditions, except for calculations of D for a single large domain when (4012lattices with periodic boundary conditions were used. In the latter case the intent is to represent a system which is sufficiently large so that the boundaries are irrelevant. We refer to this as a n “infinitely large” system. In the case of model I1 Monte Carlo simulations were performed on (4012lattices with periodic boundary conditions. When method A was employed, initialization used 5000 or 10 000 steps, and calculations used from 10 000 to 100 000 Monte Carlo steps. The large number of steps was necessary when calculating (r2),for T < T,. The results for DIDO are from averages over K such simulations, where K ranged from 100 to over 3000. For method B, 1000 to 5000 Monte Carlo steps were used for either initialization or the calculation of averages in the cases of the largest lattices. For smaller lattices 600 steps were used. Previous work concluded that the effective lateral pressure, II,in the hydrocarbon chain region of phosphotidylcholine bilayers is about 30 mN1m,l7and this value has been used in our calculations except for one case for which we used 25 mN/m. Figure 1 and 2 show results, using method A and “pairexchange” to calculate DIDO,for model I employing the distributions (3) and (4). Here, we used values of NO ranging from 8 to 24 with values of r ranging from 1 to 8 in the first case (eq 3) and from 28 to 100 in the second (eq 4). The value of @ is the number of lipid molecules in the most probable domain. (17)Georgallas, A.; Hunter, D. L.; Lookman, T.;Zuckermann, M. J.; Pink, D.A. Eur. Biophys. J. 1984,11, 79-86, and references therein.
I
270
,
ONf
1
300
T (K) Figure 2. Computer simulation of lipid lateral self diffusion, method A, for model I using the “pair-exchange”model with the distributionpL(eq 3). log(DID0)as a functionof temperature for different values of r. r = 1(O),r = 2 (+I, r = 4 (VI, and r = 8 ( x ) . (A)No=8. (B)No= 16. (C)N0=24. Theinfinitely large system is indicated by 0 (heating and cooling) with T, shown by a dashed line, and other lines are to aid the eye. Figure 1 shows DIDO for an “infinitely large” system and constant JO (a) and for four distributions of domain sizes defined by the pair [No,rl. In Figure 1A we display DIDO for the distribution of (3) for [8,21(b), [8,41(c), [16,21 (d), and [24,21 (e). In Figure 1B the distribution of (4) is used with the sets [8,281 (b), [8,481 (c), 116,601 (d), and [24,1001(e). Here, we see two things: (i)for distributions (3) and (4)which use the same value of NOand two values of r, r’, and Y‘, so that pL(N,No,r’) = p,(N,No,T”), the values of DIDO are approximately equal a t a given temperature; (ii) for a domain distribution (3) with NOas large as 24 and r = 2, DIDochanges by only 1 order of magnitude as T i s reduced by -7 “C below T,. For DIDO to decrease by 2 orders of magnitude, which is the decrease in DIDO for the “infinitely large” system a t T,, requires a decrease in T by -20 “C below T,. Since these results show that it is unnecessary to investigate both distributions (3)and (4),we shall restrict ourselves, from now on, to (3). Figure 2 shows the results of varying r for these values of NO:8,16, and 24. It can be seen that DIDO appears to be insensitive to r unless r is sufficiently large so that domains of size N < 6 have a significant probability of appearing in all cases, and then the effect is noticeable only a t a low temperature. This is seen in Figure 2A a t T = 272 K where the results for [8,11,18,21,and [8,41are separated. However, in this region the only noticeable effect is that DIDO for 18,ll lies below the other two. The results in parts B and C of Figure 2 for NO= 24 show no significant difference in the values of DIDOfor the different values of r. Figure 3 shows the results of calculating DIDO = p(e,e) for model I using method B, the “pair-exchange” model. This calculation takes much less time than does the use of method A, and accordingly, we wish to show that it gives essentially the same results as method A. Figure 3A shows DIDO =p(e,e) for the set [No,rl with NO= 8,16, and 24 and r = 1, 2, 4, and 8. Also shown is the result for an “infinitely large” lattice modeled by a (40)2lattice
Langmuir, Vol. 11, No. 7, 1995 2701
Evidence for Domains in Deposited Lipid Bilayers
B
I
C
I
0 ‘ 0
1
I I
so
L
0
270
,
I
-3
’ A
300
T (K)
0
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T (K) Figure 3. Left panel (A): model I using domain distributionpL (eq 3). log(DID0)as a function of temperature using the correlation functionp(e,e)for the “pair-exchange”model, method B, for = 1 (01,r = 2 (+I, r = 4 (V),and r = 8 (x). The results shown here are to be compared to those of Figure 2, with which they are essentially the same. (A) NO= 8. (B)NO= 16. (C)NO= 24. The infinitely large system is indicated by 0 (heating and cooling), and the vertical line indicates T,. Right panel (B): model I using domain distributionpL (eq 3). Log(DID0) as a function of temperature using the correlation function p(e,e) for the “pair-exchange” model, method B, for r = 2 (+), r = 4 (V),r = 8 (x), and r = 16 (A). (A) NO= 40. (B)NO= 48. (C)NO= 56. The infinitely large system is indicated by 0 with T, indicated by the vertical line. with periodic boundary conditions. Comparing the results of Figure 3A with those of Figure 2, we can see that both calculations are in good agreement. The results shown in Figure 3A, however, distinguish between different values of r for a fxed No. Here, for example, it can be seen that whereas [8,41 and [8,81 yield nearly identical results, only [16,81is much different from the other three values of r with No= 16. This method enables us to carry out simulations for much larger values of No than does method A, and Figure 3B shows results for No = 40,48, and 56 with r = 2,4,8,and 16. The results are insensitive to changes in for a fmed No,unless I‘ approaches about Nd3 or Nd2. Thus, the result for [40,161differs from those for smaller values of r while that for [56,161is almost the same a s for the three other cases. Figure 4 shows the results ofcalculating DIDO=p(e,e,e) for model I using method B, the “triplet-exchange” model, and the distribution of eq 3. Figure 4Ashows DIDOfor No = 8, 16, and 24 with = 1, 2 , 4 , and 8, while Figure 4B shows DIDo for No = 40,48, and 56 with r = 2 , 4 , 8 , and 16. Also shown is DIDO for a n “infinitely large” bilayer. Here, we see that the discontinuity at T, = 295.5 K is about 2l12 orders of magnitude for the unperturbed bilayer. This result will be seen to be more in‘agreement with measurements which have been carried out (below)than are those of Figure 3 using DIDO= p(e,e). The general appearance of the results of Figure 4 is similar to that of Figure 3 except that the decrease in DIDOas T decreases is more pronounced in the case of the former, and we see that the separations between DIDOare different values of r is greater in Figure 4 than in Figure 3. Figure 5 shows the results obtained from using model I1 and method B, together with results for the unperturbed “infinitely large” bilayer (a), using the “triplet-exchange”
model. Here, we have plotted the results of randomly weakening the interaction, eq 5, with y = 50,20, and 10 (b, c, and d). When y = 50, a discontinuous transition is still obsemed though the hysteresis loop is only -0.5 “C wide. Acritical point appears to occur a t T x 289 K when y 20, and it is clear that no phase transition is observed when y = 10. All these calculations used, as did those above, Il = 30 mNlm. In Figure 5 we also show the result of repeating the simulation of the unperturbed system but with ll = 25 mNlm (e). It can be seen that the discontinuity a t the discontinuous transition temperature, T x 290 K when ll = 25 mNlm, exhibits a wider hysteresis loop than that a t -296 K when l-I = 30 mNlm and that the magnitude of the discontinuity in DIDO= p(e,e,e) is -3 orders of magnitude, which is larger than the discontinuity a t -296 K. The purpose of this last calculation is that, in various experiments, the bilayers were transferred from a monolayer to the substrates a t lateral pressures of less than 30 mNlm. We wished to exhibit the effect of reducing the effective lateral pressure in the hydrocarbon chain region upon DIDO. From a comparison with the results obtained for the infinitely large case using ll = 30 mNIm, we expect that a choice of l-I = 25 mNlm will yield results for models I and I1 analogous to the results of Figures 1-4 but shifted down in temperature by -6 “ C . Finally, Figure 8 shows predictions of the average number of hydrocarbon chain gauche bonds per lipid molecule as a function of temperature for large unperturbed DMPC bilayers (curve a) and for cases NO= 16, r = 8 and NO= 24, r = 4 of Figure 4, the two cases which appear to reflect the results of the diffusion experiments to be described in the section below. We have taken the number of gauche bonds on a hydrocarbon chain in its excited state to be 4.3. This approximate number arises
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2702 Langmuir, Vol. 11, No. 7, 1995
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T (K) Figure 4. Left panel (A): “triplet-exchange”model, method B. Model I using domain distributionpL (eq 3). Log(DID0)as a function of temperature using the correlation functionp(e,e,e) for r = 1(O), r = 2 (+I, r = 4 (VI,and r = 8 (x). (A) NO= 8. (B) NO= 16. (C) NO= 24. The result for the infinitely large system is indicated by 0 (heating and cooling)with the vertical dashed line showing T m , and other lines are to aid the eye. The results here should be compared to those of Figure 3A. Right panel (B): “tripletexchange” model, method B. Model I using domain distribution PL (eq 3). Log(DID0) as a function of temperature using the (A)No= 40. (B) NO= 48. (C)No= 56. The result correlation functionp(e,e,e) for r = 2 (01,r = 4 (+I, r = 8 (V),and r = 16 (XI. for the infinitely large system is indicated by 0 (heating and cooling) with the vertical dashed line showing T,, and other lines are to aid the eye. The results here should be compared to those of Figure 3B.
-2I
,
10
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30
T (C) Figure 6. Self-diffusioncoefficient, log@), of DMPC bilayers deposited on argon-sputtered glass as functionsoftemperature. Experiment: Merkel et a1.*(+); Kiihner et ale7(x). Calculations: model 1using “triplet-exchange”model B and DO= 5.0 pm2/s.. NO= 16, r = 8 (0,Figure 4A (part B));NO= 24, r = 4 (0,Figure 4A (part C)).
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T (K) Figure 5. log(DID0)for randomly weakened interactions, model 11,as a function of temperature for different values of y (eq 5). Infinitely large system ( 0 )with T, = 295.5 K is indicated by a vertical dashed line; y = 50 ( 0 )where a vertical line at T = 293 K indicates a discontinuous transition; y = 20 ( x 1; y = 10 (+I. Also shown here is log(DID0)as a function of temperature for an infinitely large system with lateral pressure ll = 25 mN/m ( 0 )for which a vertical dashed line at T = 290 Kindicates a discontinuous transition. Other lines are to aid the eye. from the average energy of such a state, -1.94 x ergs,1° and the average energy needed to form a gauche bond, -0.45 x 10-13ergs.1° Curve b shows our predictions for case [16,81 of Figure 4, while curve c shows our predictions for case [24,41 of Figure 4. In both cases, I 7 = 30 mN/m was used.
Comparison between Simulation and Experiments Figures 6 and 7 show results of FRAP (fluorescence recovery after photobleaching) measurements reported for DMPC bilayers, supported on argon-sputtered glass, polyacrylamide films, and oxidized silica wafers. Figure 6 shows results obtained previously by Merkel et a1.8 on argon-sputtered glass together with those of Kuhner et al.7 on the same substrate. The latter experiments were performed and evaluated as described by Kuhner et al.7 with the sample preparation as described by Merkel et a1.8 In Figure 7 we compare the results of Kuhner et aL7 on polyacrylamide films together with those of Tamm and McConnellg on oxidized silicon. Although the measurements of Tamm and M ~ C o n n e l lfor , ~ the case of DMPC, were performed only for T 2 T,, those for DPPC (Figure 7) did extend to T < T,. Some results of our calculations of DIDOare included in Figures 6 and 7 where D Ois the
Evidence for Domains in Deposited Lipid Bilayers
LO
10
T (C)
Figure 7. Self-diffisioncoefficient,log(D),of DMPC and DPPC bilayers deposited on polyacrylamide or oxidized silicon substrates as functions of temperature. Experiment: DMPC on polyacrylamide (Kiihner et al.,7+); DMPC on oxidized silicon (Tamm and McConnellg,0);DPPC on oxidized silicon (Tamm and McConnellg,0).Calculations,“triplet-exchange”model B with DO= 7.94 pm2/s: infinitely large system (0,Figure 4); model I with NO= 24, r = 4 (A, Figure 4A (part C)). 9
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high-temperature limit of the diffusion coefficient which is set to 5 and 7.94pm21s,respectively. From a comparison of the experimental data with the results of our computer simulations, several points may be noted. (1) The first conclusion is that the results of the unperturbed model, making use of “triplet-exchange” for lipid movements, seem generally to be in accord with the results of Tamm and McConnell. This model predicts a sharp decrease in DIDO by -2l12 orders of magnitude as Tis reduced through T,, which is experimentally observed for DPPC and anticipated for DMPC (Figure 7). Thus, the lipid bilayer is nearly undisturbed by the hydrated, smooth, amorphous silicon oxide surface.18 (2) The experimental results of Kuhner et al.7 and Merkel et a1.8 on argon-sputtered glass and on polyacrylamide show a continuous decrease of the diffusion coefficient, D ,when cooling below the phase transition temperature. The results of Figures 1-4 for model I are in general agreement with this data. However,on physical grounds as to what mechanism actually permits the movement of a lipid molecule by one lattice constant, the “triplet-exchange” model might be more realistic than the “pair-exchange”model. This is supported by its agreement (18) Tillmann, R. W.; Radmacher, M.; Gaub, H. E. Appl. Phys. Lett. 1992,60(251,3111-3113.
Langmuir, Vol. 11, No. 7, 1995 2703 with results for, what appear to be, unperturbed bilayersg (above). In particular for the case of argon-sputteredglass, the results of Merkel et a1.8 are in best agreement with those of Figure 4A for No= 24 and r = 4 using the “tripletexchange” model, and Figure 6 shows a plot of log(D) for DO= 5.0 pm21s. This implies a distribution of bilayer domains with a most probable domain size of -600 lipid molecules per half of a bilayer. The results of Figure 4A for No= 16 are most similar to the data of Kiihner (shown in Figure 6)where log(D)for I‘= 8 is plotted. This suggests a most probable domain size of -250 molecules per halfbilayer. Thus, the bilayer seems to be perturbed by the roughness of argon-sputtered glass. Coqpared to oxidized silicon, the roughness of g l a d g and of argon-sputtered glassz0is greater and thereby induces the formation of domain-like structures in the bilayer. The differences in the average domain size of the two references might be due to different sputtering procedures or kinds of glass. The case of a polyacrylamide film, shown in Figure 7, is best represented by the results ofmodel I using the “tripletexchangen model, with No = 24 and r = 4 of Figure 4A. Thus, the most probable domain size is -600 molecules per half-bilayer. Figure 7 shows log(D) with DO= 7.94 pm2/s. (3) An inspection of the results of Figure 5 shows that the model perturbed by replacing the interaction, Jo,by the randomly weakened interaction of eq 5 does not represent the observations of Merkel and Kuhner. In this case a continuous decrease of DIDO of the kind reported appears only for a sufficiently small value of y where the onset of the decrease occurs more than 10 “C below T,. (4) In the computer simulation the temperature, T,,a t which the diffusion coefficient starts to decrease during cooling, depends on the lateral pressure of the bilayer. T, is -5.5 “C lower for Il = 25 mNlm than for ll = 30 mNlm, where T, = T , 22 “C. In the experiments the bilayers were transferred from the monolayer a t a lateral pressure of 20 mN/m (Kiihner et ale7),17-30 mNlm (Merkel et a1.9, or 36.5 mNlm (Tamm and McConnellg). Experimentally, however, T, is observed not to differ from T, (see above). This could be explained in terms of the adjustment of the effective lateral pressure in the hydrocarbon chain region during or after bilayer deposition on the supporting substrate. Thus, the internal lateral pressure of a deposited bilayer adjusts itself to a value that is independent of the lateral pressure a t which it was transferred. The prediction of the average number of hydrocarbon chain gauche bonds per lipid molecule a s a function of temperature for T < T, (Figure 8) for large unperturbed DMPC bilayers has already been confirmed by Snyder et al.13 Curve b of Figure 8, showing our predictions for model I and case NO = 16, r = 8 of Figure 4A, might represent the case of bilayers deposited on argon-sputtered glass as measured by Kuhner, while curve c, showing our predictions for model I and case NO= 24, r = 4 of Figure 3, might represent the case of bilayers deposited on polyacrylamide films7 or on argon-sputtered glass as measured by Merkel et a1.8 Ameasurement of the number of gauche bonds might be carried out via Fourier Transform infrared spectroscopy.13
Conclusion We have constructed models and carried out computer simulations in order to identify the physical states of (19) Karrasch, S.;Dolder, M.; Schabert, F.; Ramsden, J.; Engel, A. Biophys. J . 1993,65, 2437-2446. (20) Radmacher, M.Diplomarbeit, Technische Universittit Munchen, 1993.
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2704 Langmuir, Vol. 11, No. 7, 1995
DMPC bilayers deposited on argon-sputtered glass, polyor oxidized silica wafer^.^ We have acrylamide modeled lateral self-diffusion as measured using the photobleaching technique, and we considered two models. Both of them derived from one, referred to as the unperturbed model, which has already successfully predicted the number of gauche bonds per lipid molecule in large, homogeneous, lipid bilayers for T .C Tm.l0Model I assumed that the deposited bilayers were composed of distributions of noninteracting domains determined by the structure of the underlying substrate, between which, however, lipids could diffuse. Model I1 assumed that, while a single large bilayer was created, the properties of the surface on which it was deposited lead to a random weakening of the van der Waals interaction between the lipid molecules. We used two methods to calculate a lipid self-diffusion coefficient: method A, in which a computer simulation was carried out of objects diffusing through compact domains of different sizes and moving from one domain to another; method B, in which D was related to correlation functions associated with the probabilities of lipids being in excited states. We found that lateral
movement, as reflected in the diffusion coefficients, was best represented by the “triplet-exchange” of three mutually nearest neighbor lipids, occupying the vertices of a triangle on a triangular lattice. A comparison with experiment data appears to show that the unperturbed model adequately describes the results on oxidized silicon and that model I can explain the results observed on argonsputtered glass and polyacrylamide, suggesting that the domains are distributed with a most probable domain size rangingfrom about -250 to -600 lipid molecules per halfbilayer. Thus, from the measurement ofthe temperaturedependence of the diffusion coefficient the roughness of the surface may be estimated. By use of model I, predictions were made concerning the average number of gauche bonds per molecule as a function of temperature.
Acknowledgment. This work was supported by NSERC of Canada through a research grant (D.A.P.) and by the Bundesminister fur Forschung und Technologie (BMFT 0319431 A, Germany). LA9402579
